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This is to certify that the
dissertation entitled

DATA MINING FOR A WEB-BASED EDUCATIONAL
SYSTEM

presented by
Behrooz Minaei-Bidgoli

has been accepted towards fulfillment
of the requirements for the

 

 

 

 

Doctoral degree in Computer Sciences and
ErLcLineering
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Major P'rofessor’s Signature
7/ / / o 3
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MSU is an Affirmative Action/Equal Opportunity Institution

LIBRARIES
MICHIGAN STATE UNIVERSITY
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DATA l

 

DATA MlNlNG FOR A WEB-BASED EDUCATlONAL SYSTEM

By

Behrooz Minaei—Bidgoli

A DlSSERTATlON
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHlLOSOPHY

Department of Computer Science and Engineering

2004

     

 

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ABSTRACT

DATA MINING FOR A WEB-BASED EDUCATIONAL SYSTEM
By

Behrooz Minaei-Bidgoli

Web-based educational technologies allow educators to study how students learn
(descriptive studies) and which learning strategies are most effective (causal/predictive
studies). Since web-based educational systems are capable of collecting vast amounts of
student profile data, data mining and knowledge discovery techniques can be applied to
find interesting relationships between attributes of students, assessments, and the solution
strategies adopted by students. The focus of this dissertation is three-fold: 1) to introduce
an approach for predicting student performance; 2) to use clustering ensembles to build
an optimal framework for clustering web-based assessment resources; and 3) to propose a
framework for the discovery of interesting association rules within a web-based
educational system. Taken together and used within the online educational setting, the
value of these tasks lies in improving student performance and the effective design of the
online courses.

First, this research presents an approach to classifying student characteristics in order
to Predict performance on assessments based on features extracted from logged data in a
Web-based educational system. We show that a significant improvement in classification
Performance is achieved by using a combination of multiple classifiers. Furthermore, by

“learning” an appropriate weighting of the features via a genetic algorithm (GA), we have

successfully impr
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successfully improved the accuracy of the combined classifier performance by another
10-12%. Such classification is the first step towards a “recommendation system” that will
provide valuable, individualized feedback to students.

Second, this project extends previous theoretical work regarding clustering
ensembles with the goal of creating an optimal framework for categorizing web-based
educational resources. We propose both non-adaptive and adaptive resampling schemes
for the integration of multiple clusterings (independent and dependent). Experimental
results show improved stability and accuracy for clustering structures obtained via
bootstrapping, subsampling, and adaptive techniques. These improvements offer insights
into specific associations within the data sets.

Finally, this study turns toward developing a technique for discovering interesting
associations between student attributes, problem attributes, and solution strategies. We
propose an algorithm for the discovery of “interesting” association rules within a web-
based educational system. The main focus is on mining interesting contrast rules, which
are sets of conjunctive rules describing interesting characteristics of different segments
within a population. In the context of web-based educational systems, contrast rules help
to identify attributes characterizing patterns of performance disparity between various
groups of students. We propose a general formulation of contrast rules as well as a
framework for finding such patterns. Examining these contrasts can improve the online
educational systems for both teachers and students —— allowing for more accurate

assessment and more effective evaluation of the learning process.

This dissertation is dedicated to:
my parents
my wife,
my son Mohsen,
my daughters Maryam and Zahra,

and to whoever serve the Truth for the Truth itself.

iv

 

his Mill much .1
titdicaztd themstl \ ts
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completion of this
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O’htr teachers

Acknowledgements

It is with much appreciation and gratitude that I thank the following individuals who
dedicated themselves to the successful completion of this dissertation. Dr. Bill Punch, my
major professor and advisor, maintained his post by my side from the inception to the
completion of this research project. He gave me guidance throughout my research.
Without his knowledge, patience, and support this dissertation would have not been
possible. The knowledge and fiiendship I gained from him will definitely influence the
rest of my life.

Other teachers have influenced my thinking and guided my work. I would also thank
professor Anil Jain. It was my honor to be a student in his pattern recognition classes,
which defined my interest in the field for many years to come. His professional guidance
has been a source of inspiration and vital in the completion of this work. I owe many
inspirational ideas to Dr. Pang-Ning Tan whose keen insights and valuable discussions
often gave me stimulating ideas in research. Though kept busy by his work, he is always
willing to share his time and knowledge with me. For these reasons, I wish he would have
been at Michigan State University when I began my research.

I also owe many thanks to Dr. Gerd Kortemeyer for his extraordinary support and
patience. His productive suggestions and our discussions contributed enormously to this
work. From the first time I stepped through the door of Lite Lab in the Division of
Science and Mathematics Education, until now, Gerd has allowed me freedom to explore
my own directions in research. He supported me materially and morally for many years,

and for that I am very grateful. I am grateful to Dr. Esfahanian for taking time to serve on

the gttdancc commit
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the guidance committee and overseeing this work. I deeply appreciate his support and
insightful suggestions. His great organization of the graduate office in the Computer
Science Department helps advance graduate research. I have learned much from him,
both as a teacher and as a friend.

I am also grateful to my colleagues in the GARAGe Laboratory especially
Alexander Topchy, and all my friends in the LON-CAPA developers group: Helen Keefe,
Felicia Berryman, Stuart Raeburn, and Alexander Sakharuk. Discussion with our
colleagues Guy Albertelli (LON-CAPA head developer), Matthew Brian Hall, Dr. Edwin
Kashy, and Dr. Deborah Kashy were particularly useful. The discussions with them
substantially contributed to my work and broadened my knowledge.

I offer my special thanks to Steven Forbes Tuckey in the Writing Center for editing
this dissertation and his proof-reading. I am grateful to him for the countless hours of
constructive discussion and his informative revisions of my work. Last but not the least,
many thanks go to my wife, without her love, care, encouragement and continuous
support, I would not be who I am today. Above all, I thank God the Almighty, for
blessing me with the strength, health and will to prevail and finish this work.

I and my colleagues in LON—CAPA are thankful to the National Science Foundation
for grants supporting this work through the Information Technology Research (ITR
0085921) and the Assessment of Student Achievement (ASA 0243126) programs.
Support in earlier years of this project was also received from the Alfred P. Sloan and the
Andrew W. Mellon Foundations. We are grateful to our own institution, Michigan State

University and to its administrators for over a decade of encouragement and support.

vi

 

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Table of Content

 

 

List of Tables ................................................................................................................ xiii
List of Figures .............................................................................................................. xvi
CHAPTER 1 INTRODUCTION 1
1.1 STATEMENT OF THE PROBLEM ................................................................................... l
1.2 DATA MINING ............................................................................................................ 5
1.2.1 What is Data Mining? ....................................................................................... 6
1.2.2 Data Mining Methods ........................................................................................ 8
1.2.3 Predictive tasks .................................................................................................. 9
1.2.4 Descriptive tasks ................................................................................................ 9
1.2. 5 Mixed tasks ...................................................................................................... 10
1.3 ONLINE EDUCATION SYSTEMS ................................................................................. 11
1.3.1 LON—CAPA, System Overview ......................................................................... 12
1.3.2 LON-CAPA T opologt ...................................................................................... 12
1.3.3 Data Distribution in LON-CAPA .................................................................... 15
1.3.4 Resource Variation in LON- CAPA .................................................................. I 6
1.3.5 LON-CAPA Strategy for Data Storage ............................................................ 1 7
1.3.6 Resource Evaluation in LON-CAPA ............................................................... 19
1.4 INTELLIGENT TUTORING SYSTEMS (ITSS) ............................................................... 20
1.4.1 Learning Enhancement in ITSs ....................................................................... 22
1.4.2 Basic Architecture of an ITS ............................................................................ 24
1.4.3 Learning and Cognition Issuesfor ITS Development and Use ....................... 27
1.5 SUMMARY ................................................................................................................ 29

vii

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2.1 CLASSIFICATION AND PREDICTIVE MODELING ........................................................ 30
2.1.1 Bayesian Classifier .......................................................................................... 31
2.1.2 Decision tree-based method ............................................................................ 33

2.1.2.1 What is the best feature for splitting? ................................................................................... 34
2.1.2.1.] Entropy impurity ........................................................................................................... 35
2.1.2.1.2 Gini impurity ................................................................................................................. 36
2.1.2.1.3 Twoing impurity ............................................................................................................ 36

2.1.2.2 How to Avoid Overfitting ..................................................................................................... 37
2.1 .2.2.1 Cross-Validation ............................................................................................................ 37
2.1.2.2.2 Setting a threshold ......................................................................................................... 37
2.1.2.2.3 Pruning .......................................................................................................................... 38

2.1.2.3 Drawbacks of decision tree ................................................................................................... 39

2.1.3 Neural Network Approach ............................................................................... 3 9
2.1.4 k—Nearest Neighbor (kNN) Decision Rule ....................................................... 42
2.1.5 Parzen Window classifier ................................................................................ 43

2.2 CLUSTERING ............................................................................................................. 44

2. 2. 1 Partitional Methods ......................................................................................... 45
2.2.1.1 k-mean Algorithm ................................................................................................................. 45
2.2.1.2 Graph Connectivity ............................................................................................................... 48
2.2.1.3 Nearest Neighbor Method ..................................................................................................... 49
2.2.1.4 Mixture density clustering .................................................................................................... 49
2.2.1.5 Mode Seeking ....................................................................................................................... 50
2.2.1.6 k-medoids .............................................................................................................................. 51

2.2.1.6.] Partitioning Around Medoids (PAM) ............................................................................ 51
2.2. 1.6.2 CLARA ......................................................................................................................... 52
2.2.1.6.3 CLARANS .................................................................................................................... 52

2.2.1.7 Other partitional methods for large data sets ......................................................................... 53

2.2.2 Hierarchical Methods ...................................................................................... 54

viii

2.2.2.1 Traditional Linkage Algorithms ............................................................................................ 55

2.2.2.2 BIRCH .................................................................................................................................. 56
2.2.2.3 CURE .................................................................................................................................... 57

2.3 FEATURE SELECTION AND EXTRACTION .................................................................. 58
2.3.1 Minimizing the cost .......................................................................................... 59
2.3.2 Data Visualization ........................................................................................... 59

2. 3.3 Dimensionality Reduction ............................................................................... 59
2.3.4 Feature Selection ............................................................................................. 60
2.3.5 Feature Extraction ........................................................................................... 61
2.4 SUMMARY ................................................................................................................ 63

CHAPTER 3 DATA REPRESENTATION & ASSESSMENT TOOLS IN LON-CAPA ..64

 

3.1 DATA ACQUISITION AND EXTRACTING THE FEATURES ........................................... 64
3.1.1 Preprocessing student database ...................................................................... 64
3.1.2 Preprocessing Activity Log .............................................................................. 6 7
3.1.3 Extractable Features ....................................................................................... 68

3.2 FEEDBACK TO THE INSTRUCTOR FROM ONLINE HOMEWORK ................................... 69
3.2.1 Feedback tools ................................................................................................. 69
3.2.2 Student Evaluation ........................................................................................... 71
3. 2. 3 Conceptual Problems ...................................................................................... 7 7
3.2.4 Homework and Examination Problem Evaluation .......................................... 83

3.3 SUMMARY ................................................................................................................ 89

CHAPTER 4 PREDICTING STUDENT PERFORMANCE 91

4.1 DATA SET AND CLASS LABELS ................................................................................. 92

4.2 CLASSIFIERS ............................................................................................................. 95
4.2.1 Non-tree based classifiers ............................................................................... 95

4.2.1.1 Combination of Multiple Classifiers (CMC) ......................................................................... 96

ix

4.2. 1.2 Normalization ....................................................................................................................... 97

 

4.2.1.3 Comparing 2-fold and lO-fold Cross-Validation .................................................................. 98
4.2.1.4 Results, Error Estimation .................................................................................................... 100

4. 2. 2 Decision T ree-based software ....................................................................... 105
4.2.2.1 C5.0 .................................................................................................................................... 106
4.2.2.2 CART .................................................................................................................................. 109
4.2.2.3 QUEST, CRUISE ............................................................................................................... 1 13

4. 2.3 F inaI Results without optimization ................................................................ 118
4.3 OPTIMIZING THE PREDICTION ACCURACY .............................................................. I 19
4. 3 . 1 Genetic Algorithms (GAs) ............................................................................. I 19
4.3.1.1 What is a Simple GA (SGA)? ............................................................................................. 120
4.3.1.2 Specific use of GAs in pattern classification ....................................................................... 120

4.3.2 Implementation of a GA to optimize the prediction accuracy ....................... 122
4.3.2.1 GA Operators ...................................................................................................................... 123
4.3.2.1.1 Recombination ............................................................................................................ 123

4.3.2.1.2 Crossover ..................................................................................................................... 124

4.3.2.1.3 Mutation ...................................................................................................................... 124

4.3.2.2 Fitness Function .................................................................................................................. 125

4.3.3 Experimental Results of GA Optimization ..................................................... 126
4.4 EXTENDING THE WORK TOWARD MORE LON-CAPA DATA SETS .......................... 131
4.4.1 Experimental Results ..................................................................................... 1 3 4
4.5 SUMMARY .............................................................................................................. 138
CHAPTER 5 ENSEMBLES OF MULTIPLE CLUSTERINGS 140
5.1 INTRODUCTION ....................................................................................................... 141
5.2 TAXONOMY OF DIFFERENT APPROACHES ............................................................... 144
5.3 NON-ADAPTIVE ALGORITHMS ................................................................................ 148
5.3.1 Similarity-based algorithm ............................................................................ 1 4 9
5.3.2 Algorithms based on categorical clustering .................................................. 150

5.4 CONSENSUS FUNCTIONS ......................................................................................... 153

 

5.4.1 Co—association based functions ..................................................................... 1 5 3
5.4.2 Quadratic Mutual Information Algorithm (QM) .......................................... I53

5. 4.3 Hypergraph partitioning ............................................................................... l 5 4

5. 4. 4 Voting approach ............................................................................................ 1 54
5.5 CRITICAL ISSUES IN RESAMPLING ........................................................................... 155
5.5.1 Variable number of samples .......................................................................... 1 5 5
5.5.2 Repetitive data points (objects) ..................................................................... 1 5 6
5.5.3 Similarity estimation ...................................................................................... 1 5 6

5. 5. 4 Missing labels ................................................................................................ 1 5 7

5. 5. 5 Re-labeling .................................................................................................... 15 7
5.5.6 Adaptation of the k-means algorithm ............................................................ 158
5.6 ADAPTIVE SAMPLING SCHEME ............................................................................... 158
5.7 EXPERIMENTAL STUDY ON NON-ADAPTIVE APPROACHES ..................................... 163
5. 7.1 Data sets ........................................................................................................ 164

5. 7.2 The role of algorithm 's parameters ............................................................... 165

5. 7.3 The Role of Consensus Functions (Bootstrap algorithm) .............................. 166

5. 7.4 Eflect of the Resampling method (Bootstrap vs. Subsampling) ..................... 169
5.8 EMPIRICAL STUDY AND DISCUSSION OF ADAPTIVE APPROACH ............................. 174
5.9 CONCLUDING REMARKS ......................................................................................... 176
CHAPTER 6 ASSOCIATION ANALYSIS IN LON-CAPA 178
6.1 INTRODUCTION ....................................................................................................... 178
6.2 BACKGROUND ........................................................................................................ 1 81
6.2.1 Association analysis ...................................................................................... 181
6.2.2 Data mining for online education systems .................................................... [83

xi

6.2.3 Related work .................................................................................................. 184

 

 

6.3 CONTRAST RULES .................................................................................................. 185

6.4 ALGORITHM ............................................................................................................ 190

6.5 EXPERIMENTS ......................................................................................................... 192

6. 5. I Data model .................................................................................................... 193

6. 5.2 Data sets ........................................................................................................ I 96

6. 5. 3 Results ........................................................................................................... I 97

6.5.3.1 Difference of confidences ................................................................................................... 199

6.5.3.2 Difference of Proportions ................................................................................................... 200

6.5.3.3 Chi-square ........................................................................................................................... 200

6.6 CONCLUSION .......................................................................................................... 201
CHAPTER 7 SUMMARY 203
7.1 SUMMARY OF THE WORK ...................................................................................................... 203

7.1.1 Predicting Student Performance ................................................................................ 204

7.1.2 Clustering ensembles ................................................................................................. 205

7.1.3 Interesting association rules ...................................................................................... 206

7.2 FUTURE WORK ...................................................................................................................... 207
APPENDIX A: TREE CLASSIFIERS OUTPUT 208
gig ........................................................................................................................................... 208

CARI ......................................................................................................................................... 213

QU_E_T ...................................................................................................................................... 221

w ..................................................................................................................................... 227
BIBLIOGRAPHY 230

 

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Table 1.] Different Specific I TSs and their affects on learning rate ............................... 23

Table 3.1 A sample of a student homework results and submissions ............................... 73

Table 3.2 Statistics table includes general statistics of every problem of the course
(Homework Set 1) ............................................................................................................. 85

Table 3.3 Analysis of Examination Problems (N =3 93) DoDiff = Difficulty Index
DoDisc = Discrimination Index ....................................................................................... 87

Table 4.1. 9-Class labels regarding students’ grades in course PH Y 1 83__ 5502 ......... 93
Table 4. 2. 3-Class labels regarding students ’ grades in course PH Y] 83 5502 ............ 93
Table 4. 3. 2-class labels regarding students ’ grades in course PH Y1 83 5502 ............ 93

Table 4.4 Comparing Error Rate of classifiers with and without normalization in the
case of 3 classes ................................................................................................................ 98

Table 4.5 Comparing Error Rate of classifiers 2-fold and 10-fold Cross- Validation in the
case of 3 classes .............................................................................................................. 100

Table 4.6: Comparing the Performance of classifiers, in all cases: 2-Classes, 3 -Classess,
and 9-Classes, Using 10-fold Cross- Validation in all cases. ......................................... 105

Table 4. 7 Variable (feature) Importance in 2-Classes Using Gini Criterion ............... 111
Table 4.8 Variable (feature) Importance in 2-Classes, Using Entropy Criterion ....... I 11

Table 4. 9: Comparing the Error Rate in CART, using 10-fold Cross- Validation in
learning and testing set. .................................................................................................. I 12

Table 4.10: Comparing the Error Rate in CART, using Leave-One-Out method in
learning and testing test. ................................................................................................. 112

Table 4.11: Comparing the Error Rate of all classifiers on PH Y I 83 data set in the cases
of 2-Classes, 3-Classes, and 9-Classes, using IO-fold cross-validation, Without
Optimization .................................................................................................................... I 18

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without GA in the cases of 2-Classes, 3-Classess, and 9-Classes, 95% confidence

interval. ........................................................................................................................... 128
Table 4.13. Feature Importance in 3-Classes Using Entropy Criterion ........................ 130
Table 4.14. 14 of LON-CAPA courses at MS U ............................................................. 131
Table 4.15 Characteristics of 14 of MS U courses, which held by LON-CAPA ............. 132

Table 4.16 Comparing the average performance% often runs of classifiers on the given
datasets using I 0-fold cross validation, without GA ...................................................... 134

Table 4.1 7 Comparing the classification fusion performance on given datasets, without-
GA, using-GA (Mean individual) and improvement, 95% confidence interval .............. 135

Table 4.18 Relative Feature Importance%, Using GA weighting for BS] 1 I 2003 course
......................................................................................................................................... 13 7

Table 4.19 Feature Importance for BS 1 I 1 2003, using decision-tree software CART,
applying Gini Criterion ................................................................................................... 1 3 8

Table 5.1 (a) Data points and feature values, N rows and d columns. Every row of this
table shows a feature vector corresponding to N points. (b) Partition labels for resampled
data, n rows and B columns. ........................................................................................... 151

Table 5.2 An illustrative example of re-labeling difliculty involving five data points and
four different clusterings of four bootstrap samples. The numbers represent the labels
assigned to the objects and the “.7 ” shows the missing labels of data points in the

bootstrapped samples ...................................................................................................... 15 7
Table 5. 3. Consistent re-labeling of 4 partitions of 12 objects ...................................... 160
Table 5. 4. A summary of data sets characteristics ......................................................... 164

Table 5.5 “Star/Galaxy ” data experiments. Average error rate (% over 10 runs) of
clustering combination using resampling algorithms with diflerent number of components
in combination B, resolutions of components, k, and types of consensus fimctions. ...... 169

Table 5. 6 The average error rate (%) of classical clustering algorithms. An average over
100 independent runs is reported for the k-means algorithms ....................................... I 72

xiv

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Table 5.8 Subsampling methods: trade-off among the values of k, the number of
partitions B, and the sample size, S. Last column denote the percentage of sample size
regarding the entire data set. (Bold represents most optimal) ...................................... I 73

Table 6. I A contingency table of student success vs. study habits for an online course

......................................................................................................................................... 1 79
Table 6.2 A contingency table proportional to table 6.1 ................................................ 185
Table 6.3 A contingency table for the binary case ........................................................ 185

Table 6.4 Characteristics of three MS U courses which used LON-CAPA in fall semester

2003 ................................................................................................................................. 1 9 7
Table 6.5 LBS_2 71 data set, difference of confidences measure .................................. 199
Table 6. 6 CEM_I41 data set, difference of confidences measure ................................ 199
Table 6. 7 BS_111 data set, difference of proportion measure ..................................... 200
Table 6.8 CEM_I 41 data set, chi-square measure ....................................................... 200
Table 6.9 LBS_2 71 data set, difference of confidences measure ................................... 201

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Figure 1.1 Steps of the KDD Process (F ayyad et al., 1996) ............................................ 6
Figure 1.2 A schema of distributed data in LON-CAPA ................................................ 14
Figure 1.3 Directory listing of user ’s home directory .................................................... 1 7
Figure 1.4 Directory listing of course ’s home directory ................................................ 18
Figure 1.5 Distributions for different learning conditions (Adapted fiom Bloom, 1984)22
Figure 1.6 Components of an Intelligent Tutoring System (ITS) .................................... 24
Figure 2.1 The Bayesian Classification Process (Adapted from Wu et al., 1991) ....... 31
Figure 2.2 A Three Layer F eedforward Neural Network (Lu et al., I 995) .................... 40
Figure 3.1 A sample of stored data in escape sequence code ......................................... 65
Figure 3 .2 Perl scrip code to retrieve stored data ........................................................... 65
Figure 3.3 A sample of retrieved data from activity log ................................................. 65
Figure 3.4 Structure of stored data in activity log and student data base ..................... 66
Figure 3.5 A sample of extracted Activity. log data ......................................................... 67
Figure 3.6 A small excerpt of the performance overview for a small introductory physics

class ................................................................................................................................... 72
Figure 3. 7 Single-student view of a problem ................................................................... 74
Figure 3.8 Compiled student responses to a problem .................................................... 75
Figure 3.9 One Early Measure of a Degree of Difliculty ............................................... 76

Figure 3.10 Success (%) in Initial Submission for Selecting the Correct Answer to Each
of Six ‘Concept’ Statements .............................................................................................. 78

Figure 3.11 Success Rate on Second and Third Submissions for Answers to Each of Six
‘Concept ’ Statements ....................................................................................................... 79

xvi

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Figure 3.12 Randomly Labeled Conceptual Physics Problem ........................................ 79
Figure 3.13 Vector Addition Concept Problem ............................................................... 81
Figure 3. 14 Upper Section: Success Rate for Each Possible Statement. Lower Section:

Relative distribution of Incorrect Choices, with Dark Gray as “greater than Light Gray
as “Less Than ” and Clear as “Equal to ” ........................................................................ 83

Figure 3.15 Grades on the first seven homework assignments and on the first two

midterm examinations ....................................................................................................... 88
Figure 3.16 Homework vs. Exam Scores. The highest bin has 18 students .................... 89
Figure 4.1 Graph of distribution of grades in course PH Y 1 83 SS02 ............................ 92

Figure 4.2: Comparing Error Rate of classifiers with 10-fold Cross- Validation in the
case of 2-Classes ............................................................................................................. 101

Figure 4.3: Table and graph to compare Classifiers ’ Error Rate, 10-fold CV in the case
of 2-Classes ..................................................................................................................... 102

Figure 4. 4: Comparing Error Rate of classifiers with 10-fold Cross- Validation in the
case of 3-Classes ............................................................................................................. 103

Figure 4.5 Comparing Classifiers ’ Error Rate, 10-fold CV in the case of3-Classes 104
Figure 4.6. Graph of GA Optimized CMC performance in the case of 2-Classes ........ 127
Figure 4. 7. Graph of GA Optimized CMC performance in the case of 3-Classes ......... 127
Figure 4. 8. Graph of GA Optimized CMC performance in the case of 9—Classes ......... 128
Figure 4.9. Chart of comparing CMC performance, using GA and without GA. .......... 129
Figure 4.10. LON-CAPA: BS111 SS03, Grades distribution ......................................... 133

Figure 4.11 GA-Optimized Combination of Multiple Classifiers ' (CMC) performance in
the case of 2-Class labels (Passed and Failed) for BS 1 1 I 2003, 200 weight vectors

individuals, 500 Generations .......................................................................................... I3 6
Figure 5.1 Diflerent approaches to clustering combination ......................................... 14 6
Figure 5 .2 Taxonomy of different approaches to clustering combination .................... 14 7

Figure 5.3 First algorithms for clustering ensemble, based on co-association matrix and
using diflerent similarity-based consensus functions ..................................................... 150

xvii

  
  

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Figure 5.5 Two possible decision boundaries for a 2-cluster data set. Sampling
probabilities of data points are indicated by gray level intensity at different iterations (to
< t, < 1;) of the adaptive sampling. True components in the 2-class mixture are shown as
circles and triangles ........................................................................................................ I 6 0

Figure 5.6 Algorithms for adaptive clustering ensembles .............................................. 162

Figure 5. 7 “Halfrings ” data set with 400 patterns (100-300 per class) , “2-Spirals "
dataset with 200 patterns (100-100 per class) ................................................................ 163

Figure 5.8 “Iris ” data set. Bootstrapping for fixed consensus function MCLA, different
B, and diflerent values of k ............................................................................................ 1 6 7

Figure 5.9 “Halfrings " data set. Experiments using subsampling with k=10 and B=100,
different consensus function, and sample sizes S ............................................................ I 70

Figure 5.10 “Star/Galaxy " data set. Experiments using subsampling, with k = 4 and B
= 50 and different consensus fimction and sample sizes S. ............................................ 1 71

Figure 5.11 Clustering accuracy for ensembles with adaptive and non-adaptive
sampling mechanisms as a junction of ensemble size for some data sets and selected

consensus fimctions ......................................................................................................... I 75
Figure 6.1 A contrast rule extracted from Table 6.1 ..................................................... 179
Figure 6.2 A contrast rule extracted from Table 6.1 ..................................................... 180
Figure 6.3 A contrast rule extracted fiom Table 6.1 ..................................................... 180
Figure 6.4 A contrast rule extracted from Table 6.1 ..................................................... 180
Figure 6.5 Set of all possible association rules for Table 6.3. ..................................... 186
Figure 6.6 Formal definition of a contrast rule ........................................................... 186

Figure 6. 7 Mining Contrast Rules (MCR) algorithm for discovering interesting
candidate rules ................................................................................................................ 1 91

Figure 6.8 Attribute mining model, Fixed students ’ attributes, Problem attributes, and
Linking attributes between students and problem .......................................................... 193

Figure 6.9 Entity Relationship Diagram for a LON-CAPA course .............................. 194

xviii

Chapter 1

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Chapter 1 Introduction

The ever-increasing progress of network-distributed computing and particularly the
rapid expansion of the web have had a broad impact on society in a relatively short period
of time. Education is on the brink of a new era based on these changes. Online delivery of
educational instruction provides the opportunity to bring colleges and universities new
energy, students, and revenues. Many leading educational institutions are working to
establish an online teaching and learning presence. Several different approaches have
been developed to deliver online education in an academic setting. In particular,
Michigan State University (MSU) has pioneered some of these systems which provide an
infrastructure for online instruction (Multi-Media Physics; CAPA; LectureOnline;
PhysNet; Kortemeyer and Bauer, 1999; Kashy et al., 1997, LON-CAPA). This study
focuses on the data mining aspects of the latest online educational system developed at
MSU, the Learning Online Network with Computer-Assisted Personalized Approach

(LON-CAPA).
1.1 Statement of the problem

In LON-CAPA, we are involved with two kinds of large data sets: 1) educational
resources such as web pages, demonstrations, simulations, and individualized problems

designed for use on homework assignments, quizzes, and examinations; and 2)

 

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information about users who create, modify, assess, or use these resources. In other
words, we have two ever-growing pools of data. As the resource pool grows, the
information from students who have multiple transactions with these resources also
increases. The LON-CAPA system logs any access to these resources as well as the
sequence and frequency of access in relation to the successful completion of any
assignment.

The web browser represents a remarkable enabling tool to get information to and
from students. That information can be textual and illustrated, not unlike that presented in
a textbook, but also include various simulations representing a modeling of phenomena,
essentially experiments on the computer. Its greatest use however is in transmitting
information as to the correct or incorrect solutions of various assigned exercises and
problems. It also transmits guidance or hints related to the material, sometimes also to the
particular submission by a student, and provides the means of communication with fellow
students and teaching staff.

This study investigates data mining methods for extracting useful and interesting
knowledge from the large database of students who are using LON-CAPA educational
resources. This study aims to answer the following research questions:

0 How can students be classified based on features extracted from logged data? Do
groups of students exist who use these online resources in a similar way? Can we
predict for any individual student which group they belong to? Can we use this
information to help a student use the resources better, based on the usage of the

resource by other students in their groups?

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How can the online problems that students engage in be classified? How do different
types of problems impact students’ achievements? Can the classifications of
problems be employed to find patterns of questions that help student success?

How can data mining help instructors, problem authors, and course coordinators
better design online materials? Can we find sequences of online problems that
students use to solve homework problems? Can we help instructors to develop their
homework more effectively and efficiently? How can data mining help to detect
anomalies in homework problems designed by instructors?

How can data mining help find patterns of student behavior that groups of students
take to solve their problems? Can we find some associative rules between students'
educational activities? Can we help instructors predict the approaches that students
will take for some types of problems?

How can data mining be used to identify those students who are at risk, especially in
very large classes? Can data mining help the instructor provide appropriate advising
in a timely manner?

The goal of this research is to find similar patterns of use in the data gathered from

LON-CAPA, and eventually be able to make predictions as to the most beneficial course

of studies for each student based on a minimum number of variables for each student.

Based on the current state of the student in their learning sequence, the system could then

make suggestions as to how to proceed. Furthermore, through clustering of homework

problems as well as the sequences that students take to solve those problems, we hope to

help instructors design their curricula more effectively. As more and more students enter

the online leamin
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the online learning environment, databases concerning student access and study patterns
will grow. We are going to develop such techniques in order to provide information that
can be usefully applied by instructors to increase student learning.

This dissertation is organized as follows: The rest of this first chapter provides basic
concepts of data mining and then presents a brief system overview of LON-CAPA that
shows how the homework and student data are growing exponentially, while the current
statistical measures for analyzing these data are insufficient. Chapter 2 introduces the
research background: the important algorithms for data classification and some common
clustering methods. Chapter 3 provides information about structure of LON-CAPA data,
data retrieval process, representing the statistical information about students, problem and
solution strategies, and providing assessment tools in LON-CAPA to detect, to
understand, and to address student difficulties. Chapter 4 explains the LON-CAPA
experiment to classify students and predict their final grades based on features of their
logged data. We design, implement, and evaluate a series of pattern classifiers with
various parameters in order to compare their performance in a real dataset from the LON-
CAPA system. Results of individual classifiers, and their combination as well as error
estimates are presented. Since LON-CAPA data are distributed among several servers
and distributed data mining requires efficient algorithms form multiple sources and
features, chapter 5 represents a framework for clustering ensembles in order to provide an
optimal framework for categorizing distributed web-based educational resources. Chapter
6 discusses the methods to find interesting association rules within the students’

databases. We propose a framework for the discovery of interesting association rules

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within a web-based educational system. Taken together and used within the online
educational setting, the value of these tasks lies in improving student performance and the
effective design of the online courses. Chapter 7 presents the conclusion of the proposal

and discusses the importance of future work.

1.2 Data Mining

Presently, the amount of data stored in databases is increasing at a tremendous speed.
This gives rise to a need for new techniques and tools to aid humans in automatically and
intelligently analyzing huge data sets to gather useful information. This growing need
gives birth to a new research field called Knowledge Discovery in Databases (KDD) or
Data Mining, which has attracted attention from researchers in many different fields
including database design, statistics, pattern recognition, machine learning, and data
visualization. In this chapter we give a definition of KDD and Data Mining, describing its
tasks, methods, and applications. Our motivation in this study is gaining the best
technique for extracting useful information from large amounts of data in an online
educational system, in general, and from the LON-CAPA system, in particular. The goals
for this study are: to obtain an optimal predictive model for students within such systems,
help students use the learning resources better, based on the usage of the resource by
other students in their groups, help instructors design their curricula more effectively, and
provide the information that can be usefully applied by instructors to increase student

learning.

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1.2.1 What is Data Mining?

Data Mining is the process of analyzing data from different perspectives and
summarizing the results as useful information. It has been defined as "the nontrivial
process of identifying valid, novel, potentially useful, and ultimately understandable

patterns in data" (Frawley et al., 1992; Fayyad et al., 1996).

 

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Figure 1.] Steps of the KDD Process (Fayyad et al., 1996)

The process of data mining uses machine learning, statistics, and visualization
techniques to discover and present knowledge in a form that is easily comprehensible.
The word “Knowledge” in KDD refers to the discovery of patterns which are extracted
from the processed data. A pattern is an expression describing facts in a subset of the

data. Thus, the difference between KDD and data mining is that

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data while data mining refers to application of algorithms for extracting
patterns fiom data without the additional steps of the KDD process.”

(Fayyad et al., 1996)

However, since Data Mining is a crucial and important part of the KDD process,

most researchers use both terms interchangeably. Figure 1.] presents the iterative nature

of the KDD process. Here we outline some of its basic steps as are mentioned in

Brachman & Anad (1996):

Providing an understanding of the application domain, the goals of the system
and its users, and the relevant prior background and prior knowledge (This step
in not specified in this figure.)

Selecting a data set, or focusing on a subset of variables or data samples, on
which discovery is to be performed

Preprocessing and data cleansing, removing the noise, collecting the necessary
information for modeling, selecting methods for handling missing data fields,
accounting for time sequence information and changes

Data reduction and projection, finding appropriate features to represent data,
using dimensionality reduction or transformation methods to reduce the number
of variables to find invariant representations for data

Choosing the data mining task depending on the goal of KDD: clustering,

classification, regression, and so forth

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data

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0 Evaluating or interpreting the mined patterns, with a possible return to any
previous steps

0 Using this knowledge for promoting the performance of the system and
resolving any potential conflicts with previously held beliefs or extracted
knowledge

These are the steps that all KDD and data mining tasks progress through.

1.2.2 Data Mining Methods

The objective of data mining is both prediction and description. That is, to predict
unknown or future values of the attributes of interest using other attributes in the
databases, while describing the data in a manner understandable and interpretable to
humans. Predicting the sale amounts of a new product based on advertising expenditure,
or predicting wind velocities as a function of temperature, humidity, air pressure, etc., are
examples of tasks with a predictive goal in data mining. Describing the different terrain
groupings that emerge in a sampling of satellite imagery is an example of a descriptive
goal for a data mining task. The relative importance of description and prediction can
vary between different applications. These two goals can be fulfilled by any of a number
data mining tasks including: classification, regression, clustering, summarization,
dependency modeling, and deviation detection. (Harrell and Frank, 2001; Montgomery et

al., 2001)

1.2.3 Predictive tasks

The following are general tasks that serve predictive data mining goals:

Classification — to segregate items into several predefined classes. Given a
collection of training samples, this type of task can be designed to find a
model for class attributes as a function of the values of other attributes (Duda
et al., 2001).

Regression — to predict a value of a given continuously valued variable based
on the values of other variables, assuming either a linear or nonlinear model of
dependency. These tasks are studied in statistics and neural network fields
(Montgomery et al., 200]).

Deviation Detection —— to discover the most significant changes in data from
previously measured or normative values (Aming et al., 1996; Fayyad et al.,
1996). Explicit information outside the data, like integrity constraints or
predefined patterns, is used for deviation detection. Arning et al., (1996)
approached the problem from the inside of the data, using the implicit

redundancy.

1 .2.4 Descriptive tasks

Clustering -— to identify a set of categories, or clusters, that describe the data
(Jain & Dubes, 1988).
Summarization — to find a concise description for a subset of data. Tabulating

the mean and standard deviations for all fields is a simple example of

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summarization. There are more sophisticated techniques for summarization
and they are usually applied to facilitate automated report generation and
interactive data analysis (Fayyad et al., 1996).

Dependency modeling — to find a model that describes significant
dependencies between variables. For example, probabilistic dependency
networks use conditional independence to specify the structural level of the
model and probabilities or correlation to specify the strengths (quantitative

level) of dependencies (Heckerman, 1996).

1.2.5 Mixed tasks

There are some tasks in data mining that have both descriptive and predictive

aspects. Using these tasks, we can move from basic descriptive tasks toward higher-order

predictive tasks. Here, we indicate two of them:

Association Rule Discovery — Given a set of records each of which contain
some number of items from a given collection, produce dependency rules
which will predict the occurrence of an item based on patterns found in the
data.

Sequential Pattern Discovery — Given a set of objects, where each object is
associated with its own timeline of events, find rules that predict strong
sequential dependencies among different events. Rules are formed by first
discovering patterns followed by event occurrences which are governed by

timing constraints found within those patterns.

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So far we briefly described the main concepts of data mining. Chapter two focuses
on methods and algorithms of data mining in the context of descriptive and predictive
tasks. The research background of both the association rule and sequential pattern
mining —- newer techniques in data mining, that deserve a separate discussion - will be
discussed in chapter five.

Data mining does not take place in a vacuum. In other words, any application of this
method of analysis is dependent upon the context in which it takes place. Therefore, it is
necessary to know the environment in which we are going to use data mining methods.

The next section provides a brief overview of the LON-CAPA system.

1.3 Online Education systems

Several Online Education systems1 such as Blackboard, WebCT, Virtual University
(VU), and some other similar systems have been developed to focus on course
management issues. The objectives of these systems are to present courses and
instructional programs through the web and other technologically enhanced media. These
new technologies make it possible to offer instruction without the limitations of time and
place found in traditional university programs. However, these systems tend to use
existing materials and present them as a static package via the Internet. There is another
approach, pursued in LON-CAPA, to construct more-or-less new courses using newer
network technology. In this model of content creation, college faculty, K-12 teachers, and

students interested in collaboration can access a database of hypermedia sofiware

 

1 See http://wwwedutools.info for an overview of current web-based educational systems.

11

modules that car.

CAPA system is I‘

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modules that can be linked and combined (Kortemeyer and Bauer, 1999). The LON-

CAPA system is the primary focus of this chapter.

1.3.1 LON-CAPA, System Overview

LON-CAPA is a distributed instructional management system, which provides
students with personalized problem sets, quizzes, and exams. Personalized (or
individualized) homework means that each student sees a slightly different computer-
generated problem. LON-CAPA provides students and instructors with immediate
feedback on conceptual understanding and correctness of solutions. It also provides
faculty the ability to augment their courses with individualized, relevant exercises, and
develop and share modular online resources. LON-CAPA aims to put this functionality
on a homogeneously distributed platform for creating, sharing, and delivering course
content with emphasis on cross—institutional collaboration and intellectual property rights

management.

1.3.2 LON-CAPA Topology

LON-CAPA is physically built as a geographically distributed network of constantly
connected servers. Figure 1.2 shows an overview of this network. All machines in the
network are connected with each other through two-way persistent TCP/IP connections.
The network has two classes of servers: library servers and access servers. A library
server can act as a home server that stores all personal records of users, and is responsible
for the initial authentication of users when a session is opened on any server in the

network. For authors, it also hosts their construction area and the authoritative copy of

12

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every resource that has been published by that author. An Access Server is a machine that
hosts student sessions. Library servers can be used as backups to host sessions when all
access servers in the network are overloaded.

Every user in LON-CAPA is a member of one domain. Domains could be defined by
departmental or institutional boundaries like MSU, FSU, OHIOU, or the name of a
publishing company. These domains can be used to limit the flow of personal user
information across the network, set access privileges, and enforce royalty schemes. Thus,
the student and course data are distributed amongst several repositories. Each user in the
system has one library server, which is his/her home server. It stores the authoritative

copy of all of their records.

13

The Learr

The LearningOnline Network

Library Server

at

 

1:1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Access
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Figure 1.2 A schema of distributed data in LON-CAPA

 

 

 

 

 

 

 

l4

LON-CAPA
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accessed by the st

uses mod_perl ins

1.3.3 Data Di.

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LON-CAPA currently runs on Redhat-Linux Intel-compatible hardware. The current
MSU production setup consists of several access servers and some library servers. All
access servers are set up on a round-robin IP scheme as frontline machines, and are
accessed by the students for “user session.” The current implementation of LON-CAPA

uses mod _perl inside of the Apache web server sofiware.

1.3.3 Data Distribution in LON-CAPA

Educational objects in LON-CAPA range from simple paragraphs of text, movies,
and applets, to individualized homework problems. Online educational projects at MSU
have produced extensive libraries of resources across disciplines. By combining these
resources, LON-CAPA produces a national distributed digital library with mechanisms
to store and retrieve these objects. Participants in LON-CAPA can publish their own
objects in the common pool. LON-CAPA will allow groups of organizations
(departments, universities, schools, commercial businesses) to link their online
instructional resources in a common marketplace, thus creating an online economy for
instructional resources (lon-capa.org). Internally, all resources are identified primarily by
their URL.

LON-CAPA does enable faculty to combine and sequence these learning objects at
several levels. For example, an instructor from Community College A in Texas can
compose a page by combining a text paragraph from University B in Detroit with a
movie from College C in California and an online homework problem from Publisher D

in New York. Another instructor from High School E in Canada might take that page

15

from Communtt}

section. Those in

1.3.4 Resour

LON-CAPA :
refers to these res
i130 “ES Still“ ’Z

or SITJ‘CILHC. for UT.

0 A Cont

 

 

 

from Community College A and combine it with other pages into a module, unit or

section. Those in turn can be combined into whole course packs.

1.3.4 Resource Variation in LON-CAPA

LON-CAPA provides three types of resources for organizing a course. LON-CAPA
refers to these resources as Content Pages, Problems, and Maps. Maps may be either of
two types: Sequences or Pages. LON-CAPA resources may be used to build the outline,
or structure, for the presentation of the course to the students.

0 A Content Page displays course content. It is essentially a conventional html
page. These resources use the extension “.html”.

0 A Problem resource represents problems for the students to solve, with
answers stored in the system. These resources are stored in files that must use
the extension “.problem”.

0 A Page is a type of Map which is used to join other resources together into
one HTML page. For example, a page of problems will appear as a problem
set. These resources are stored in files that must use the extension “.page”.

0 A Sequence is a type of Map, which is used to link other resources together.
Sequences are stored in files that must use the extension “.sequence”.
Sequences can contain other sequences and pages.

Authors create these resources and publish them in library servers. Then, instructors
use these resources in online courses. The LON-CAPA system logs any access to these
resources as well as the sequence and frequency of access in relation to the successful

completion of any assignment. All these accesses are logged.

l6

1.3.5 LON-C

Internally. th
home httpdl
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1.3.5 LON-CAPA Strategy for Data Storage

Internally, the student data is stored in a directory:

/home/httpd/lonUsers/domain/ 1 st.char/2nd.char/ 3 rd.char/usemame/

For example /home/httpd/lonUsers/msu/m/i/n/minaeibi/

Figure 1.3 shows a list of a student’s data. Files ending with .db are GDBM files

(Berkeley database), while those with a course-ID as name, for example

msu_12679c3ed54 3a25msul 1 .db, store performance data for that student in the course.

 

 

ls —a1F /home/httpd/lonUsers/msu/m/i/n/minaeibi

-rw-r--r-- 1 www users 13006 May 15 12:21 activity.log

-rw—r ----- 1 www users 12413 Oct 26 2000 coursedescriptions.db
-rw-r--r-- 1 www users 11361 Oct 26 2000 coursedescriptions.hist
-rw-r ----- 1 www users 13576 Apr 19 17:45 critical.db

-rw—r--r-- 1 www users 1302 Apr 19 17:45 critical.hist

~rw-r ----- 1 www users 13512 Apr 19 17:45 email_status.db

-rw-r-—r-- 1 www users 1496 Apr 19 17:45 email_status.hist
-rw-r--r-- 1 www users 12373 Apr 19 17:45 environment.db

-rw-r--r-- 1 www users 169 Apr 19 17:45 environment.hist

-rw-r ----- 1 www users 12315 Oct 25 2000 junk.db

-rw—r—-r-- 1 www users 1590 Nov 4 1999 junk.hist

-rw-r ----- 1 www users 23626 Apr 19 17:45 msu_12679c3ed543a25msu11.db
—rw—r—-r—- 1 www users 3363 Apr 19 17:45 msu_12679c3ed543a25msu11.hist
-rw-r ----- 1 www users 18497 Dec 21 11:25 msu_1827338c7d339b4msu11.db
—rw—r-—r-- 1 www users 3801 Dec 21 11:25 msu_1827338c7d339b4msu11.hist
-rw—r ————— 1 www users 12470 Apr 19 17:45 nohist_annotations.db

-rw-r ----- 1 www users 765954 Apr 19 17:45 nohist_email.db

-rw-r--r-- 1 www users 710631 Apr 19 17:45 nohist_email.hist
-rw-r--r-- 1 www users 13 Apr 19 17:45 passwd

-rw-r--r-- 1 www users 12802 May 3 13:08 roles.db

—rw-r--r-- 1 www users 1316 Apr 12 16:05 roles.hist

 

Figure 1.3 Directory listing of user’s home directory

Courses are assigned to users, not vice versa. Internally, courses are handled like
users without login privileges. The username is a unique ID, for example
msu_12679c3ed543a25msull — every course in every semester has a unique ID, and
there is no semester transition. The user-data of the course includes the full name of the
course, a pointer to its top-level resource map (“course map”), and any associated

deadlines, spreadsheets, etc., as well as a course enrollment list. The latter is somewhat

l7

 

retardant since

users. and lookin-

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redundant, since in principle, this list could be produced by going through the roles of all

users, and looking for the valid role for a student in that course.

 

 

ls -alF /home/httpd/lonUsers/msu/1/2/6/12679c3ed543a25msu11/

-rw-r ----- 1 www users 17155 Apr 25 16:20 classlist.db

—rw—r--r-- 1 www users 60912 Apr 25 16:20 classlist.hist

—rw-r ----- 1 www users 12354 Jan 4 16:40 environment.db

-rw—r-—r-- 1 www users 82 Jan 4 16:40 environment.hist

—rw-r ----- 1 www users 103030 May 15 14:47 nohist_calculatedsheets.db
—rw-r ----- 1 www users 13050 May 9 21:04 nohist_expirationdates.db
-rw-r--r~- 1 www users 6 Jan 4 16:40 passwd

-rw-r ----- 1 www users 17457 May 9 21:04 resourcedata.db

-rw-r--r-- 1 www users 8888 May 9 21:04 resourcedata.hist

 

Figure 1.4 Directory listing of course’s home directory

An example of course data is shown in Figure 1.4. classlist is the list of
students in the course, environment includes the course’s full name, etc, and
resourceda ta are deadlines, etc. The parameters for homework problems are stored
in these files.

To identify a specific instance of a resource, LON-CAPA uses symbols or “symbs.”
These identifiers are built from the URL of the map, the resource number of the resource
in the map, and the URL of the resource itself. The latter is somewhat redundant, but

might help if maps change. An example is
msu/korte/parts/partl.sequence 19 msu/korte/tests/partlz.problem

The respective map entry is

<resource id="19" src="/res/msu/korte/tests/partlZ.problem"

title="Problem 2">

</resource>

l8

 

SymbS 3’ e U'
specific to a cert:
|

ofthe system.

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Symbs are used by the random number generator, as well as to store and restore data
specific to a certain instance of a problem. More details of the stored data and their exact
structures will be explained in chapter three, when we will describe the data acquisition

of the system.

1.3.6 Resource Evaluation in LON-CAPA

One of the most challenging aspects of the system is to provide instructors with
information concerning the quality and effectiveness of the various materials in the
resource pool on student understanding of concepts. These materials can include web
pages, demonstrations, simulations, and individualized problems designed for use on
homework assignments, quizzes, and examinations. The system generates a range of
statistics that can be useful in evaluating the degree to which individual problems are
effective in promoting formative learning for students. For example, each exam problem
contains attached metadata that catalog its degree of difficulty and discrimination for
students at different phases in their education (i.e., introductory college courses,
advanced college courses, and so on). To evaluate resource pool materials, a standardized
format is required so that materials from different sources can be compared. This helps
resource users to select the most effective materials available.

LON-CAPA has also provided questionnaires which are completed by faculty and
students who use the educational materials to assess the quality and efficacy of resources.
In addition to providing the questionnaires and using the statistical reports, we
investigate here methods to find criteria for classifying students and grouping problems

by examining logged data such as: time spent on a particular resource, resources visited

19

(other web page
sutzsticallyl and
gathered and sort
and choose hem:
ethire demonstrs

resoLHCC p001

git

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CAPA cats anal} \t

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(other web pages), due date for each homework, the difficulty of problems (observed
statistically) and others. Thus, floods of data on individual usage patterns need to be
gathered and sorted — especially as students go through multiple steps to solve problems,
and choose between multiple representations of the same educational objects like video
lecture demonstrations, a derivation, a worked example, case-studies, and etc. As the
resource pool grows, multiple content representations will be available for use by
students.

There has been an increasing demand for automated methods of resource evaluation.
One such method is data mining, which is the focus of this research. Since the LON-
CAPA data analyses are specific to the field of education, it is important to recognize the
general context of using artificial intelligence in education.

The following section presents a brief review of intelligent tutoring systems — one
typical application of artificial intelligence in the field of education. Note that herein the
purpose is not to develop an intelligent tutoring system; instead we apply the main ideas
of intelligent tutoring systems in an online environment, and implement data mining

methods to improve the performance of the educational web-based system, LON-CAPA.

1.4 Intelligent Tutoring Systems (lTSs)

Intelligent tutoring systems are computer-based instructional systems that attempt to
determine information about a student’s learning status, and use that information to
dynamically adapt the instruction to fit the student’s needs. Examples of educational
researchers who have investigated this area of inquiry are numerous: Urban—Lurain,
1996; Petrushin, 1995; Benyon and Murray, 1993; Winkkels, 1992; Farr and Psotka,

20

1992'. Yemeni." 3‘
Gauthier. 1953." l
breed rotors. bi"
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For many years.

1992; Venezky and Osin, 1991; Larkin and Cabay, 1991; Goodyear, 1990; Frasson and
Gauthier, 1988; Wenger, 1987; Yazdani, 1987. ITSs are often known as knowledge-
based tutors, because they have separate knowledge bases for different domain
knowledge. The knowledge bases specify what to teach and different instructional
strategies specify how to teach (Murray, 1996).

One of the fundamental assumptions in ITS design is from an important experiment
(Bloom, 1956) in learning theory and cognitive psychology, which states that
individualized instruction is far superior to class-room style learning. Both the content
and style of instruction can be continuously adapted to best meet the needs of a student
(Bloom, 1984). Educational psychologists report that students learn best “by doing”,
learn through their mistakes, and learn by constructing knowledge in a very
individualized way (Kafaei and Resnik, 1996; Ginsburg and Opper, 1979; Bruner, 1966).
For many years, researchers have argued that individualized learning offers the most
effective and cognitively efficient learning for most students (Juel, 1996; Woolf, 1987).

Intelligent tutoring systems epitomize the principle of individualized instruction.
Previous studies have found that intelligent tutoring systems can be highly effective
Ieaming aids (Shute and Regine, 1990). Shute (1991) evaluates several intelligent
tutoring systems to judge how they live up to the main promise of providing more
effective and efficient learning in relation to traditional instructional techniques. Results

of such studies show that ITSs do accelerate learning.

21

1.4.1 Learni r

In one study
least effectiye
ndnidsali/ed. l"

\ .

using three for?

 

 

 

 

 

alti‘u idualized tutu.

hum:
81
8: ~-

1.4.1 Learning Enhancement in ITSs

In one study, Bloom (1984) states that conventional teaching methods provide the
least effective method of Ieaming. As instruction becomes more focused and
individualized, learning is enhanced. He compares students’ scores on achievement tests
using three forms of instruction: conventional teaching, mastery teaching, and
individualized tutoring. Mastery teaching is an instructional technique whereby a teacher
supplements a lecture with diagnostic tests to determine where students are having
problems, and adjusts the instruction accordingly. The results of this comparison are
shown in Figure 1.5. Students receiving conventional teaching scored in the 50th
percentile, students receiving mastery teaching scored in the 84‘h percentile, while

students receiving individualized tutoring scored in the 98th percentile.

 

r I I I I I

Individualized Tutoring (1 : l)

Mastery Learning (1 : 30) \

Number ~ -

of
Students _ Conventional (l : 30) \

 

 

 

l l l l L

 

50% 84% 98%

Percentiles: Summative Achievement Scores

Figure 1.5 Distributions for different learning conditions (Adapted from Bloom,
1984)

22

Bloom repli.
ditierent domains
etiectn'e educstw

Since llSs .
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Bloom replicates these results four times with three different age groups for two
different domains, and thus, provides concrete evidence that tutoring is one of the most
effective educational delivery methods available.

Since ITSs attempt to provide more effective Ieaming through individualized
instruction, many computer-assisted instruction techniques exist that can present
instruction and interact with students in a tutor-like fashion, individually or in small
groups. The incorporation of artificial intelligence techniques and expert systems
technology to computer-assisted instruction systems gave rise to intelligent tutoring
systems — i.e., systems that model the learner’s understanding of a topic and adapt
instruction accordingly. A few examples of systematically controlled evaluations of ITSs

reported in the literature are shown in Table 1.1.

Table 1.1 Different Specific ITSs and their affects on learning rate

 

 

 

ITS Literature Objective progress
LISP tutor (Anderson, 1990) “5mm“ 1:18P 1/3-2/3 less time
programming
. (Shute and Glaser, Teach scientific inquiry 1/2 time, same
S‘m‘hmw“ 1990) skills knowledge

 

Sherlock (Lesgold etal.,1990) Avionics troubleshooting “5 time, same

 

 

 

 

knowledge
Pascal ITS (Shute, 1991) Teach Pascal 1/3 time, same
programmrng knowledge
Stat Lady (Shute et al., 1993) Instruct statlstrcal More performance
procedures
Geometry (Anderson et al., Teach geometry theorems Better solving

 

 

 

Tutor 1985)

 

23

 

Shute and Po
the rotors do aeeel
snuld be evaluate
cases. indlylduals
leaning from t:
:ndnldsalized tut.

mouse the mean ~

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81

Figure ].

 

Shute and Poksta (1996) examine the results of these evaluations, which show that
the tutors do accelerate learning with no degradation in outcome performance. The tutors
should be evaluated with respect to the promises of ITSs — speed and effectiveness. In all
cases, individuals using ITSs learned faster, and performed at least as well as those
learning from traditional teaching environments. The results show that these
individualized tutors could not only reduce the variance of outcome scores, but also

increase the mean outcome dramatically.

1.4.2 Basic Architecture of an ITS

There is no standard architecture for an ITS. Nevertheless, four components emerge
from the literature as part of an ITS (Wasson, I997; Costa, 1992; Polson and Richardson,
1988; Yazdani, 1987; Wenger, 1987; Sleeman and Brown, 1982). These are the student
model, the pedagogical module, the expert model, and the communication module or

interface. These four components and their interactions are illustrated in Figure 1.6.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘ Expert Model
Student < > Pedagogical
\ J,
Communication

 

 

 

A

l

Figure 1.6 Components of an Intelligent Tutoring System (ITS)

 

24

The Student v
model tracks ho‘»
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model is to pros
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The student model stores information of each individual learner. For example, such a
model tracks how well a student is performing on the material being taught or records
incorrect responses that may indicate misconceptions. Since the purpose of the student
model is to provide data for the pedagogical module of the system, all of the information
gathered should be usable by the tutor.

The pedagogical module provides a model of the teaching process. For example,
information about when to review, when to present a new topic, and which topic to
present is controlled by this module. As mentioned earlier, the student model is used as
input to this component, so the pedagogical decisions reflect the differing needs of each
student.

The expert model contains the domain knowledge, which is the information being
taught to the learner. However, it is more than just a representation of the data; it is a
model of how someone skilled in a particular domain represents the knowledge. By using
an expert model, the tutor can compare the learner's solution to the expert's solution,
pinpointing the places where the learner has difficulties. This component contains
information the tutor is teaching, and is the most important since without it, there would
be nothing to teach the student. Generally, this aspect of ITS requires significant
knowledge engineering to represent a domain so that other parts of the tutor can access it.

The communication module controls interactions with a student, including the
dialogue and the screen layouts. For example, it determines how the material should be

presented to the student in the most effective way.

25

Thege four C
motel. and the C
mditldualized 6&3
structure of each t
ITS.

Current resea:
lfilflilal sense oft
for areas of intell:
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These four components — the student model, the pedagogical module, the expert
model, and the communication module — shared by all ITSs, interact to provide the
individualized educational experience promised by technology. The orientation or
structure of each of these modules, however, varies in form depending on the particular
ITS.

Current research trends focus on making tutoring systems truly “intelligent,” in the
artificial sense of the word. The evolution of ITSs demands more controlled research in
four areas of intelligence: the domain expert, student model, tutor, and interface.

0 The domain knowledge must be understood by the computer well enough for the
expert model to draw inferences or solve problems in the domain.

0 The system must be able to deduce a student’s approximation of that knowledge.

0 The tutor must be intelligent to the point where it can reduce differences between the
expert and student performance.

0 The interface must possess intelligence in order to determine the most effective way
to present information to the student.

For ITSs to have a great impact on education, these and other issues must be
resolved. To take advantage of newer, more effective instructional techniques, ITSs of
the future will have to allow for increased student initiative and inter-student
collaboration (Shute and Psotka, 1996). ITSs must also assess Ieaming as it transfers to
authentic tasks, not standardized tests, and establish connections across fields so that
topics are not learned in isolation. A more fruitful approach for ITS development may be

to develop specific cognitive tools, for a given domain or applicable across domains.

26

Such a transition

explain. cntique. ;

 

1.4.3 Learning‘
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Such a transition would allow future ITSs to be everywhere, as embedded assistants that

explain, critique, provide online support, coach, and perform other ITS activities.

1.4.3 Learning and Cognition Issues for ITS Development and Use

There are some findings in the areas of cognition and Ieaming processes that impact
the development and use of intelligent tutoring systems. Many recent findings are paving
the way towards improving our understanding of learners and learning (Bransford, Brown
et al., 2000). Learners have preconceptions about how the world works. If their initial
understanding is not referenced or activated during the Ieaming process, they may fail to
understand any new concepts or information.

One key finding regarding competence in a domain is the need to have a more than a
deep knowledge base of information related to that domain. One must also be able to
understand that knowledge within the context of a conceptual framework — the ability to
organize that knowledge in a manner that facilitates its use. A key finding in the Ieaming
and transfer literature is that organizing information into a conceptual framework allows
for greater transfer of knowledge. By developing a conceptual framework, students are
able to apply their knowledge in new situations and to learn related information more
quickly. For example, a student who has learned problem solving for one topic in the
context of a conceptual framework will use that ability to guide the acquisition, of new
information for a different topic within the same framework. This fact is explained by
Hume (1999): "When we have lived any time, and have been accustomed to the
uniformity of nature, we acquire a general habit, by which we always transfer the known

to the unknown, and conceive the latter to resemble the former.”

27

A relations}.
stations. Trans:
and what is tests.
must eoneeise of

Recent TCSL‘s‘.‘
lemeen tasks is
tSmgley and An.

see i text editor

 

 

 

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A relationship exists between the learning and transfer of knowledge to new
situations. Transferring is usually a function of the relationships between what is learned
and what is tested. For students to transfer knowledge successfully across domains, they
must conceive of their knowledge base in terms of continuous, rather than discrete steps.

Recent research by Singley and Anderson indicates that the transfer of knowledge
between tasks is a firnction of the degree to which the tasks share cognitive elements
(Singley and Anderson, 1989). In their study, Singley and Anderson taught students
several text editors, one after the other. They found that students learned subsequent text
editors more rapidly and that the number of procedural elements shared by the two text
editors predicted the amount of transfer. Their results showed that there was large transfer
across editors that were very different in surface structures but had common abstract
structures. Singley and Anderson were able to generate similar results for the transfer of
mathematical competence across multiple domains.

Emerging computer-based technologies hold great promise as a means of supporting
and promoting Ieaming. There are several ways that such technology can be used to help
meet the challenges of developing effective learning environments (El-Sheikh, 2001):

o Bringing real-world problems to the Ieaming environment.

0 Providing “scaffolding” support to students during the Ieaming process.
Scaffolding allows students to participate in complex cognitive experiences,
such as model-based learning, that is more difficult without technical support.

0 Increasing opportunities for learners to receive feedback and guidance from

sofiware tutors and learning environments.

28

0 But.

and-

0 Exp.
Learning e
understanding '0
tiese new learn.
can facilitate le..

hing these new '

1.5 Summa

This researef

mol'lled'gT llt‘tm j

The purpose is to

’- flied by

ll‘htr‘g

 

0 Building local and global communities of teachers, administrators, students,
and other interested learners.
0 Expanding opportunities for teachers’ learning.

Learning environments need to be developed and implemented with a full
understanding of the principles of learning and deveIOpmental psychology. In addition,
these new learning environments need to be assessed carefully, including how their use
can facilitate Ieaming, as well as the cognitive, social, and learning consequences of

using these new tools.

1.5 Summary

This research addresses data mining methods for extracting useful and interesting
knowledge from the large data sets of students using LON-CAPA educational resources.
The purpose is to develop techniques that will provide information that can be usefully
applied by instructors to increase student learning, detect anomalies in homework
problems, design the curricula more effectively, predict the approaches that students will
take for some types of problems, and provide appropriate advising for students in a
timely manner, etc. This introductory chapter provided an overview of the LON-CAPA
system, the context in which we are going to use data mining methods. In addition, a
brief introduction to Intelligent Tutoring Systems provided examples of expert systems
and artificial intelligence in educational software. Following this, it is necessary to

analyze data mining methods that can be applied within this context in greater detail.

29

Chapter 1

 

In the prexi

toerses on metl

tau mining \V;

Chapter 2 Background on Data Mining Methods

In the previous chapter we described the basic concepts of data mining. This chapter
focuses on methods and algorithms in the context of descriptive and predictive tasks of
data mining. We describe clustering methods in data mining, and follow this with a study
of the classification methods developed in related research while extending them for
predictive purposes. The research background for both association rule and sequential

pattern mining will be presented in chapter five.

2.1 Classification and Predictive Modeling

Classification is used to find a model that segregates data into predefined classes.
Thus classification is based on the features present in the data. The result is a description
of the present data and a better understanding of each class in the database. Thus
classification provides a model for describing firture data (Duda et al., 2001; McLachlan,
1992; Weiss and Kulikowski, 1991; Hand, 1987). Prediction helps users make a decision.
Predictive modeling for knowledge discovery in databases predicts unknown or future
values of some attributes of interest based on the values of other attributes in a database
(Masand and Shapiro, 1996). Different methodologies have been used for classification
and developing predictive modeling including Bayesian inference (Kontkanen et al.,
1996), neural net approaches (Lange, 1996), decision tree-based methods (Quinlan, 1986)

and genetic algorithms-based approaches (Punch et al., 1995).

30

2.1.1 Baye

One of the

Bayesian class.

    

assumes that :-
thou-2h this {lbs

classifier where

 

(lam et al,‘ jinn:

fzgare ll can be

:eIern is sense.

pat-em is tons;

..t-.;‘, .,
\lbbtlltalltln 5L

eaten a, For t

aid each part-en

p"!.. l
"“tnl l
\
u. l

i ‘.

2.1.1 Bayesian Classifier

One of the major statistical methods in data mining is Bayesian inference. The naive
Bayesian classifier provides a simple and effective approach to classifier learning. It
assumes that all class-conditional probability densities are completely specified. Even
though this assumption is often violated in real world data sets, a naive Bayesian
classifier (where a small number of parameters are not completely specified) is employed
(Jain et al., 2000; Duda et al., 2001; Wu et al., 1991). The Bayes classifier shown in
Figure 2.1 can be explained as follows: A set of patterns aj, j = 1, ...,n, is given, and every
pattern is sensed by a sensing device which is capable of capturing the features. Each
pattern is considered in terms of a measurement vector x,. A pattern a; belongs to a

classification set a)” which includes all the possible classes that can be assigned to

pattern a}. For the sake of simplicity, all feature measurements are considered identical

and each pattern belongs only to one of the m-possible classes ‘0.» i = 1, ...,.m

 

Bayesian Classification

 

 

 

 

 

 

 

 

 

g 1

Evaluation ’———"
Likelihood g2 . . .-

Pattem xj Functions . Maxrmum 0961510" ’

Feature - Selector
$ Extraction v
a' n
I
.— . gr

{1—1, ...n} g...

{1:1 .m}

 

 

 

 

 

 

 

 

 

 

Figure 2.1 The Bayesian Classification Process (Adapted from Wu et al., 1991)

To classify a pattern into one of the m classes, a feature space is constructed

according to the measurement vector x, which is considered to be a measurement of true

31

taluc’S damiiii ‘

functions esttnt

Ba) 65 d3“:

calculated aet‘tt.’

Etentuall): i

CiLRSll‘

leation. Th

 

prooabilzt} or likel

The quadrant
art-rm '
ui.On mtlhtll i

S 53 GaUSstan m

values damaged by random noisy data. The class-conditional probability density

firnctions estimated from training data represent the uncertainty in discovered knowledge.

p(x|o),.), i = l,...,m. (2.1)

Bayes decision theory states that the a-posteriori probability that an event may be

calculated according to the following equation:

p(w. Ix): p(X|w,-)p(w.) , l. : l,...,m. (2.2)
pm

 

Eventually, the decision criteria can be applied for classification. To gain the optimal
solution, the maximum likelihood classification or the Bayesian minimum error decision
rule is applied. It is obtained by minimizing the misclassification and errors in

classification. Thus, a pattern is classified into class a), with the highest posteriori
probability or likelihood:

g, = max]. {gt},j =1,...,m. (2,3)

The quadratic discriminant function using the Bayesian approach is the most
common method in supervised parametric classifiers. If the feature vectors are assumed
to be Gaussian in distribution, the parameters of the Gaussians are estimated using
maximum likelihood estimations. The discriminant function decision rule and the a-
posteriori probabilities for each classification are calculated for each sample test, x, using
the following equation (Duda et al., 2001):

l 7- -. l
gi(x):_§(x_lui) 2: (x—p,)—§ln|zil+lnp(a),) (2-4)

9

32

 

where r is
MmaXm“

die sample to.

 

 

SOl'JllOll. the til
role is applied

p’i_i.'572(';rl P’L‘l’l

 

assitication

2.1.2 Decisii

Decision in

The oeeisioii ire

or
\\i »:

X In the if:
torrespond [Q a
$6 Set ofdata 1

Lu '

:hé HAQ ‘
~~ speed} e

b\ th A, ‘

~ 6 PM}? ireir

fifth-1-2.. ~
" ““6 nllL‘S i

‘36: [in

he attrzh

l)“
.u;

where x is a dxl vector representing any point in the d—dimensional feature space, ,ui

(also a d x 1 vector) is the sample mean of the ith class training sample, and Z, (d x d) is

the sample covariance matrix of the ith class training sample. To obtain the optimal
solution, the maximum likelihood classification or the Bayesian minimum error decision
rule is applied. The sample is then assigned to the class that produces the highest a-
posteriori probability. It is obtained by minimizing the misclassification and errors in

classification.

2.1.2 Decision tree-based method

Decision tree-based methods are popular methods for use in a data mining context.
The decision tree classifier uses a hierarchical or layered approach to classification. Each
vertex in the tree represents a single test or decision. The outgoing edges of a vertex
correspond to all possible outcomes of the test at that vertex. These outcomes partition
the set of data into several subsets, which are identified by every leaf in the tree. A leaf of
the tree specifies the expected value of the categorical attribute for the records described
by the path from the root to that leaf. Learned trees can also be represented as sets of if-
then-else rules. (Mitchell, 1997)

An instance is classified by starting at the root node of the tree. At each level of the
tree the attributes of an instance are matched to a number of mutually exclusive nodes.
The leaf nodes assign an instance to a class. The classification of an instance therefore
involves a sequence of tests where each successive test narrows the interpretation. The
sequence of tests for the classifier is determined during a training period. Given some

new data, the ideal solution would test all possible sequences of actions on the attributes

33

of the new it
misclassifieatzi

Tree-base;
they are partn
methods are It ‘
and errors in th.
item the data e.

Most alga:
flgofilhm that L.
tension trees. I”:

siecessor C45 (1

 

each
"3‘ .s‘
&.}“]A i -‘
I' “C “it
i‘Z‘fi-ii
its“
Untteis Ent
Fir-.4.
QsC- this n”,

of the new data in order to find the sequence resulting in the minimum number of
misclassifications.

Tree—based classifiers have an important role in pattern recognition research because
they are particularly useful with non-metric data (Duda et al., 2001). Decision tree
methods are robust to errors, including both errors in classifying the training examples
and errors in the attribute values that describe these examples. Decision tree can be used
when the data contain missing attribute values. (Mitchell, 1997)

Most algorithms that have been developed for decision trees are based on a core
algorithm that uses a top-down, recursive, greedy search on the space of all possible
decision trees. This approach is implemented by ID3 algorithmZ (Quinlan, 1986) and its
successor C4.5 (Quinlan, 1993). C45 is an extension of ID3 that accounts for unavailable
values, continuous attribute value ranges, pruning of decision trees, and rule derivation.

The rest of this section discusses some important issues in decision trees classifiers.

2.1.2.1 What is the best feature for splitting?

The first question that arises in all tree-based algorithms concerns which properties
are tested in each node? In other words, which attribute is the “most informative” for the
classifier? We would like to select the attribute that is the most informative of the
attributes not yet considered in the path from the root. This establishes what a "Good"
decision tree is. Entropy is used to measure a node’s information. Claude Shannon (1984)

introduced this notion in “Information Theory”. Based on entropy, a statistical property

 

21D3 got this name because it was the third version of “interactive dichotomizer” procedure.

34

called informs

 

examples in re.

2.12.1.1 Entr
Entropy c?

Emil-1C node ‘\

l’\/

 

called information gain measures how well a given attribute separates the training

examples in relation to their target classes.

2.1.2.1.1 Entropy impurity

Entropy characterizes the impurity of an arbitrary collection of examples S at a

specific node N. Sometimes (Duda et al., 2001) the impurity of a node N is denoted by

W).

Entroy(S) :- i(N) z —ZP(a)j)log2 P(a)j) (2.5)

where P((oj) is the fraction of examples at node N that go to category to]. .

If all the patterns are from the same category the impurity is 0, otherwise it is
positive; if all categories are equally distributed at node N then the impurity has its
greatest value 1.

The key question then is, on the decision path from the root to node N, what features
and their values should we select for the test at node N when property query T is used? A

heuristic is suggested to select a query that decreases the impurity as much as possible

(Duda et al., 2001).

MN)=itNi—PiitNii—(l—PiiitN.) (2.6)

where N L and N R are the left and right descendent nodes, and the i(NL) and i(NR) are
their impurities respectively, and PL is fraction of patterns at node N that will go to N 1.

when the property query T is used. The goal of the heuristic is to maximize Ai , thus

35

minimizing [in

query.

2.1.2.1.2 Gini

One can i
enrentin. Gini :
rile. Essentiall}

the parent node

I1

 

‘11

minimizing the impurities corresponds to an information gain which is provided by the

query.

2.1.2.1.2 Gini impurity

One can rank and order each splitting rule on the basis of the quality-of-split
criterion. Gini is the default rule in CART because it is often the most efficient splitting
rule. Essentially, it measures how well the splitting rule separates the classes contained in
the parent node (Duda et al., 2001 ).

i(N) = ZP(wj)P(w,-) =1‘ 21’2“”) (2.7)

It” I

As shown in the equation, it is strongly peaked when probabilities are equal. So what
is Gini trying to do? Gini attempts to separate classes by focusing on one class at a time.
It will always favor working on the largest or, if you use costs or weights, the most

"important" class in a node.

2.1.2.1.3 Twoing impurity

An alternative criterion also available in CART is Twoing impurity. The philosophy
of Twoing is different from that of Gini. Rather than initially pulling out a single class,
Twoing first segments the classes into two groups, attempting to find groups that together
add up to 50 percent of the data. Twoing then searches for a split to separate the two
subgroups (Duda et al., 2001). This is an ideal split. It is unlikely that any real-world
database would allow you to cleanly separate four important classes into two subgroups
in this way. However, splits that approach this ideal might be possible, and these are the

splits that Twoing seeks to find.

36

2.1.22 Hov

If we eo'
impmt}' then l
single training
noisy problems

01'. ll‘it‘ other 11.:

 

 

not be achieied

tree-based class:

2.12.2.1 Cros

Crossq 111 d

“133 lDJda et a

1‘1“. ‘

”5‘1 l C 111 ‘
ill: _
‘ 1‘ gel ll):
1419-
“*L and {h .n
\
ski. :h 21 ifyi

2.1.2.2 How to Avoid Overfitting

If we continue to grow the tree until each leaf node corresponds to the lowest
impurity then the data is typically overfit. In some cases every node corresponds to a
single training input. In such cases we cannot expect an appropriate generalization in
noisy problems having high Bayes error (Duda et al. 2001; Russell and Norvig, 1997).
On the other hand if we stop splitting early, then a high performance tree classifier will
not be achieved. There are several approaches to avoid overfitting in the training phase of

tree-based classification:

2.1.2.2.1 Cross-Validation

Cross-validation is a technique to eliminate the occurrence of overfitting. The main
idea of cross-validation is to estimate how well the current hypothesis will predict unseen
data (Duda et al. 2001; Russell and Norvig, 1997). This is done by randomly dividing the
data into two subsets, training and test. Usually, the test subset is a fraction of all of the
data, i.e., 10%. The hypothesis induced from the training phase is tested on the rest of
data to get the prediction performance. This should be repeated on different subsets of
data, and then the result averaged. Cross-validation should be used in order to select a

tree with good prediction performance.

2.1.2.2.2 Setting a threshold

Another method for overfitting avoidance is to consider a small threshold value in
minimizing the impurity. We stop splitting when the impurity at a node is reduced by less

than the considered threshold. The benefit of this method over the cross-validation is that

37

the tree is in“

2.1.22.3 Pru

The print“
approach. call:
tree to be cant
sub-tree rooted .
classification or
it"h—e resuiting ;

I. .‘ .. a
duttnell. l99‘i

 

(’3

‘4-

the tree is trained using all the training data. Another benefit of this method is that leaf

nodes can lie at different levels of the tree.

2.1.2.2.3 Pruning

The principal alternative of stop-splitting is pruning (Duda et al., 2001). One
approach, called reduced-error pruning (Quinlan, 1987), sets each node in the decision
tree to be candidate for pruning. “Pruning a decision node consists of removing the
subtree rooted at that node, making it a leaf node, and assigning it the most common
classification of the training examples affiliated with that node. Nodes are removed only
if the resulting pruned tree performs no worse than the original over the validation set.”
(Mitchell, 1997)

In C45, Quinlan (1993) applied a successful technique for finding high accuracy
hypotheses during the pruning process, which is called rule post pruning. It involves the
following steps:

1. Induce the decision tree from the training set, growing the tree until the training data
is fully fitted as well as possible, allowing overfitting to occur.

2. Convert the learned tree into an equivalent set of rules by creating a rule
corresponding to a path from the root to a leaf node.

3. Prune each rule by deleting any preconditions that lead to promoting the estimated
accuracy.

4. Sort the pruned rules by their estimated accuracy, and set them in a sequence for

classifying the instances.

38

llhy 5h“

 

that records C‘

"Prtjer tlte’ si‘I’

2.1.2.3 DraV

 

The draiih
Single data poin
tie decision. bo.

anode is split

decision tree cl

“ ‘
l

U JCCISIOII ll"'
\\

‘5'“1’ l- ’
~ 3.11. Kiltkhcl‘

2.1.3 Neur:

\ lie

llru':

.996). The e3

:13“.- w '
unskilo.

Illsl

l. 43th In SK

\'
.‘Uf‘t‘n ’
1%. l99<J

Why should we prefer a short hypothesis? We wish to create small decision trees so
that records can be identified after only a few questions. According to Occam's Razor,

“Prefer the simplest hypothesis that fits the data” (Duda et al., 200]).

2.1.2.3 Drawbacks of decision tree

The drawback of decision trees is that they are implicitly limited to talking about a
single data point. One cannot consider two or more objects to make a decision about, thus
the decision boundary in this classifier is linear and often too coarse. In other words, once
a node is split, the elements in a child stay in that sub-tree forever. Therefore, the
decision tree classifier often yields a sub-optimal decision boundary. Another drawback
of decision trees is that they are very instable in the presence of noisy data (Duda et al.,

2001; Mitchell, 1997).

2.1.3 Neural Network Approach

A neuron is a special biological cell with information processing ability (Jain et al.,
1996). The classification approach based on an Artificial Neural Network (ANN) (a
connectionist model) generates a lower classification error rate than the decision tree
approach in several cases, yet it requires more training time (Quinlan, 1994; Russle and
Norvig, 1995). A neural network is usually a layered graph with the output of one node
feeding into one or more nodes in the next layer.

The Multi-layer Perceptron (MLP) is a basic feedforward artificial neural network
using a back-propagation algorithm for training. That is, during training, information is
propagated back through the network and used to update connection weights. According

to Ruck et al., (1990), multi—Iayer perceptron training uses the back-propagation Ieaming

39

algorithm
The outp.
being trai.

VCCIOT S Ill

—— s.

<-
f

algorithm to approximate the optimal discriminant function defined by Bayesian theory.
The output of the MP approximates the posteriori probability functions of the classes
being trained. The Sigmoidial activation function is used for Ieaming the input weight

vectors in the training phase as follows:

1
f (X) = 1+ e‘”) (2.8)

 

Tuning each of the Ieaming rate, the number of epochs, the number of hidden layers,
and the number of neurons (nodes) in every hidden layer is a very difficult task and all
must be set appropriately to reach a good performance for MLP. In each epoch the input

data are used with the present weights to determine the errors, then back-propagated

errors are computed and weights updated. A bias is provided for the hidden layers and

output.

  
  

 

g .
/' ‘\

 

   
 
  

Hidden

’\

  

Input

Figure 2.2 A Three Layer Feedforward Neural Network (Lu et al., 1995)

40

Adopting data mining techniques to MLP is not possible without representing the
data in an explicit way. Lu et al.(1995) made an effort to overcome this obstacle by using
a three-layer neural network to perform classification, which is the technique employed
in this study. ANNs are made up of many simple computational elements (neurons),
which are densely interconnected. Figure 2.2 shows a three-layer feedforward network,
which has an input layer, a hidden layer, and an output layer. A node (neuron) in the

network has a number of inputs and a single output. Every link in the network is

associated with a weight. For example, node N,- has x: , ..., x; as its inputs and a" as its

output. The input links of N,- have weights w: , ..., w; . A node generates its output (the

activation value) by summing up its input weights, subtracting a threshold and passing

the result to a non-linear function f (activation function). Outputs from neurons in a layer

are fed as inputs to next layer. Thus, when an input tuple (xl , x") is applied to the

input layer of a network, an output tuple (cl , cm) is obtained, where c,~ has value 1 if

the input tuple belongs to class c,- and 0 otherwise.

Lu et al.'s approach uses an ANN to mine classification rules through three steps

explained as follows:

I. In the first step, a three-layer network is trained to find the best set of weights to
classify the input data at a satisfactory level of accuracy. The initial weights are
selected randomly from the interval [-1, 1]. These weights are then updated
according to the gradient of the error function. This training phase is terminated

when the norm of the gradient falls below a preset threshold.

41

2. Redur.
any er":
obtain L‘

:. Compr.
I

1161“ Of.-

Where a

 

2. Redundant links (paths) and nodes (neurons) that is, those nodes that don’t have
any effects on performance are removed and therefore, and a pruned network is
obtained.

3. Comprehensible and concise classification rules are extracted from the pruned
network in the form of: “if (a, 6 vi) & (a2 6 vz) & & (an 6 V“) then Q
where an a,- is an input attribute value, v, is a constant, 6 is a relational operator

(=, S, 2, <,>), and C,- is one of the class labels.

2.1.4 k-Nearest Neighbor (kNN) Decision Rule

The k-nearest neighbor algorithm makes a classification for a given sample without
making any assumptions about the distribution of the training and testing data. Each
testing sample must be compared to all the samples in the training set in order to classify
the sample. In order to make a decision using this algorithm, the distances between the
testing sample and all the samples in the training set must first be calculated. In this
proposal, the Euclidean distance is calculated, but, in general, any distance measurement
may be used. The euclidean distance metric requires normalization of all features into the
same range. At this point, the k closest neighbors of the given sample are determined
where k represents an integer number between 1 and the total number of samples. The
testing sample is then assigned to the label most frequently represented among the k
nearest samples (Duda et al., 2001). The value of k that is chosen for this decision rule
has an affect on the accuracy of the decision rule. The k-nearest neighbor classifier is a

nonparametric classifier that is said to yield an efficient performance for optimal values

of k.

42

 

ZLSPWZ

InUhSap
apd then. bd‘
estimates of t':

lflfll

Lg, ,
u....e ie is a

 

probability distr

Jr‘?%' -
deitiatit‘in for t.

l.__ ‘ ~
kL‘DAn‘ but [b.lri

tktL

prob} >
mi under

Hélilmjyi
Jig the 1

en
T- In our ex

:fl‘"
.1471“ '.'.
“he matri
\

2.1.5 Parzen Window classifier

In this approach a d-dimensional window is formed around all the training samples
and then, based on the number of patterns that fit in those windows, the probability

estimates of the different classes are made. This can be stated as follows (Duda et al.,

2001)

 

1 " 1 (x—xi)
x =— , 2.9
p..() "Ev w h ( )

n I?

where v” is a d—dimensional hypercube in the feature space, and (0 is a general
probability distribution function. p” (x) is the probability that the pattern fits in the given
class. It is necessary to choose the form of (a. One can assume a multivariate normal
distribution fore). The windows are centered on the training points, hence, the mean is

known, but there is no predefined method to determine the variance. Depending upon the
problem under study, the variance is estimated by minimizing the error rate and
maximizing the classifier performance. Therefore, it needs to be determined by trial and
error. In our experiment we assume that the classes are independent and thus the

covariance matrices for the Gaussian distribution are diagonal.

43

2.2 Clust

Data ClLls
for finding sin
data and lllldll
darn it Ddi‘t‘r

members ll'lSldi

1h ' ' '
he oo'ieets Witt

et al.. 3111:1131. T

1h» ~ . -» ‘
with (113 (“[3

T‘C‘fir

"id hi 3 ii

dd 5
d“ el al. 3111];

filth--

”Mia:
The gillupjn

LEme,
”Ll/T"
“13 Mod: 1“

l

‘n

emf/£3 a

 

Clugk...

 

2.2 Clustering

Data clustering is a sub-field of data mining dedicated to incorporating techniques
for finding similar groups within a large database. Data clustering is a tool for exploring
data and finding a valid and appropriate structure for grouping and classifying the data
(Jain & Dubes, 1988). A cluster indicates a number of similar objects, such that the
members inside a cluster are as similar as possible (homogeneity), while at the same time
the objects within different clusters are as dissimilar as possible (heterogeneity) (Hoppner
et al., 2000). The property of homogeneity is similar to the cohesion attribute between
objects of a class in software engineering, while heterogeneity is similar to the coupling
attribute between the objects of different classes.

Unlike data classification, data clustering does not require category labels or
predefined group information. Thus, clustering has been studied in the field of machine
Ieaming as a type of unsupervised learning, because it relies on “learning from
observation” instead of “learning from examples.” The pattern proximity matrix could be
measured by a distance function defined on any pairs of patterns (Jain & Dubes, 1988;
Duda et al., 2001). ). A simple distance measure i.e., Euclidean distance can be used to
express dissimilarity between every two patterns.

The grouping step can be performed in a number of ways. “Hierarchical clustering
algorithms produce a nest series of partitions based on a criterion for merging or splitting
clusters based on similarity. Partitional clustering algorithms identify the partition that

optimizes a clustering criterion” (Jain et al. 1999). Two general categories of clustering

44

methods are

in analysis 01

2.2.1 Part

A partitil
input paramei
each object it
depentt on ih
. . - I
nastingutsh the
m multidimer.

|
stmeture for
partitioning mel

|

connectiyity. ar

 

methods are partitioning method, and hierarchical method — both of which are employed

in analysis of the LON-CAPA data sets.

2.2.1 Partitional Methods

A partitional algorithm assuming a set of n objects in d-dimensional space and an
input parameter, k, organizes the objects into k clusters such that the total deviation of
each object from its cluster center is minimized. The deviation of an object in a cluster
depends on the similarity function, which in turn depends on the criterion employed to
distinguish the objects of different clusters. Clusters can be of arbitrary shapes and sizes
in multidimensional space. Every particular clustering criterion implies a specified
structure for the data. These criteria are employed in some of the most popular
partitioning methods: square error approach, mixture model, mode estimation, graph

connectivity, and nearest neighbor relationship.

The most common approach in these methods is to optimize the criterion function
using an iterative, hill-climbing technique. Starting from an initial partition, objects are
moved from one cluster to another in an effort to improve the value of the criterion
function (Jain & Dubes, 1988). Each algorithm has a different way for representing its

clusters.

2.2.1 .1 k-mean Algorithm

The k-means algorithm is the simplest and most commonly used clustering algorithm
employing a square error criterion (McQueen 1967). It is computationally fast, and
iteratively partitions a data set into k disjoint clusters, where the value of k is an
algorithmic input (Jain & Dubes, 1988; Duda et al. 2001). The goal is to obtain the

45

 

partition (USU-
clusters :0 (

(I

 

where n. is nuri
r is the point it:
The total s.

‘shieh is comp '

l- Cit-0.056 ;

 

311d n is

3
~- Generate

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partition (usually of hyper-spherical shape) with the smallest square-error. Suppose k
clusters {C1, C2, Ck} such that Ck has nk patterns. The mean vector or center of cluster

Ck

1 ""
#(k) 2 _Z xltk) (2.10)

where n,- is number of patterns in cluster C,, (among exactly k clusters: Cl, C 2, C.) and

x is the point in space representing the given object.
T

K "k
The total squared-error: E: :23: where e: : 205k) __ #k) (xfk) _ #1:)
k=1 i=1
which is computed in this way:

The steps of the iterative algorithm for partitional clustering are as follows:

1. Choose an initial partition with k < n clusters (.11', ,uz , , M) are cluster centers
and n is the number of patterns).

2. Generate a new partition by assigning a pattern to its nearest cluster center pi.

3. Recompute new cluster centers ,ui.

4. Go to step 2 unless there is no change in ,u'.

5. Return ,u', #2 , , ,uk as the mean values ofC,, C2, Ck.

The idea behind this iterative process is to start from an initial partition assigning the
patterns to clusters and to find a final partition containing k clusters that minimizes E for
fixed k. In step 3 of this algorithm, k-means assigns each object to its nearest center
forming a set of clusters. In step 4, all the centers of these new clusters are recomputed

with function E by taking the mean value of all objects in every cluster. This iteration is

46

repeated Lint;
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objects. d is Z
iterations sud

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repeated until the criterion function E no longer changes. The k-means algorithm is an
efficient algorithm with the time complexity of 0(ndkr), where n is the total number of
objects, d is the number of features, k is the number of clusters, and r is the number of
iterations such that r<k<n.

The weaknesses of this algorithm include a requirement to specify the parameter k,
the inability to find arbitrarily shaped clusters, and a high sensitivity to noise and outlier
data. Because of this, Jain & Dubes, (1988) have added a step before step 5: “Adjust the
number of clusters by merging and splitting the existing clusters or by removing small, or
outlier clusters”.

Fuzzy k-means clustering (soft clustering). In the k-means algorithm, each data point
is allowed to be in exactly one cluster. in the fuzzy clustering algorithm we relax this
condition and assume that each pattern has some “fuzzy” membership in a cluster. That
is, each object is permitted to belong to more than one cluster with a graded membership.
Fuzzy clustering has three main advantages: 1) it maps numeric values into more abstract
measures (fuzzification); 2) student features (in LON-CAPA system) may overlap
multiple abstract measures, and there may be a need to find a way to cluster under such
circumstances; and 3) most real-world classes are fuzzy rather than crisp. Therefore, it is
natural to consider the fuzzy set theory as a useful tool to deal with the classification
problem (Dumitrescu et al., 2000).

Some of the fuzzy algorithms are modifications of the algorithms of the square error
type such as k-means algorithm. The definition of the membership function is the most
challenging point in a fuzzy algorithm. Baker (1978) has presented a membership

function based on similarity decomposition. The similarity or affinity function can be

47

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based on the different concept such as Euclidean distance or probability. Baker and Jain
(1981) define a membership function based on mean cluster vectors. Fuzzy partitional
clustering has the same steps as the squared error algorithm, which is explained in the k-

means algorithm section.

2.2.1.2 Graph Connectivity

A graph can represent the relationship between patterns. Every vertex represents a
pattern and every edge represents the relation between two patterns. The edge weights are
distances between two adjacent vertices. The criterion function here is that the pairs of
patterns, which belong to the same cluster, should be closer than any pair belonging to
different clusters. Several graph structures, such as minimum spanning trees (MST)
(Zahn, 1971), relative neighborhood graphs (Toussaint, 1980), and Gabriel Graphs
(Urquhart, 1982), have been applied to present a set of patterns in order to detect the
clusters. For example, Zahn’s algorithm consists of the following steps: 1) Create the
MST for the entire set of N patterns. 2) Determine the inconsistent edges. 3) Delete the

inconsistent edges from MST. 4) The remaining components are our clusters.

This algorithm can be applied recursively on the resulting components to determine
new clusters. The heart of this algorithm lies in the definition of inconsistency. Zahn
(1971) presents several criteria for inconsistency. An edge is inconsistent if its weight is
much larger than the average of all other nearby edge weights. Other definitions of
inconsistency are dependent on either the ratio between, or the standard deviation in

which an edge weight differs from the average of nearby edge weights.

48

 

 

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2.2.1.3 Nearest Neighbor Method

Since the distance matrix gives us an intuition of a cluster, nearest-neighbor
distances can be used as the core of a clustering method. In this method every pattern is
assigned to the same cluster as its nearest neighbor. An iterative procedure was proposed
by Lu and Fu (1978) where each unlabeled pattern is assigned to the cluster of its nearest
labeled neighbor pattern, provided the distance to that labeled neighbor is below a
threshold. The process continues until all patterns are labeled or no additional labeling
can occur. This algorithm partitions a set of {x1, x2, xn} patterns into a set of k clusters
{C,, C 2, Ck}. The user should specify a threshold r, which determines the maximum
acceptable nearest neighbor distance. The steps of this algorithm are described in Jain &
Dubes (1988). The number of clusters k which are generated is a function of the

parameter r. As the value of r is increased the number of clusters k is decreased.

2.2.1.4 Mixture density clustering

The mixture resolving method assumes that the patterns are drawn from a particular
distribution, and the goal is to identify the parameters of that distribution. Usually it is
assumed that the individual components of the target partition are the mixture of a
Gaussian distribution, and thus the parameters of the individual Gaussians are to be
estimated (Jain et al., 1999). There are traditional approaches to use a maximum
likelihood estimate of the parameter vectors for the component densities (Jain & Dubes,
1988). Dempster et al. (1977) proposed a general framework for maximum likelihood
using the Expectation Maximization (EM) algorithm to estimate the parameters for

missing data problems. The EM algorithm widely employed to estimate maximum

49

likelihood estimation in in-complete data problems where there are missing data
(McLachlan & Krishnan, 1997).

The EM algorithm is an iterative method for Ieaming a probabilistic categorization
model from unlabeled data. In other words, the parameters of the component densities are
unknown and EM algorithm aims to estimate them from the patterns. The EM algorithm
initially assumes random assignment of examples to categories. Then an initial
probabilistic model is learned by estimating model parameters from this randomly
labeled data. We then iterate over the following two steps until convergence:

I Expectation (E-step): Rescore every pattern given the current model, and
probabilistically re-label the patterns based on these posterior probability estimates.

- Maximization (M-step): Re-estimate the model parameters from the probabilistically
re-labeled data.

A practical description of this algorithm has been provided in Mitchell (1997).

2.2.1.5 Mode Seeking

One of the simplest ways to determine the modes in a dataset is to construct a
histogram by portioning the pattern space into k non-overlapping regions. Regions with
relatively high pattern frequency counts are the modes or cluster centers. In non-
parametric density estimation, the clustering procedure searches for bins with large
counts in a multidimensional histogram of the input patterns (Jain & Dubes, 1988). The
regions of pattern space in which the patterns are the densest would represent the
partition components. The regions that include fewer numbers of patterns separate the

clusters.

50

 

 

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As with other clustering methods, there are benefits and drawbacks. The advantage
of the density estimation method is that it does not require knowing the number of
clusters and their prior probabilities. The disadvantage of this approach is that the process
of looking for the peaks and valleys in the histogram is difficult in more than a few
dimensions and requires the user to identify the valleys in histograms for splitting

interactively.

2.2.1 .6 k-medoids

Instead of taking the mean value of the data points in a cluster, the k-medoids
method represents a cluster with an actual data point that is the closest to the center of
gravity of that cluster. Thus, the k-medoids method is less sensitive to noise and outliers
than the k-means and the EM algorithms. This, however, requires a longer computational
time. To determine which objects are good representatives of clusters, the k-medoids
algorithm follows a cost function that dynamically evaluates any non-center data point

against other existing data points.

2.2.1.6.1 Partitioning Around Medoids (PAM)

PAM (Kaufman & Rousseeuw, 1990) is one of the first k-medoids clustering
algorithms which first selects the initial k cluster centers randomly within a data set of N
objects. For every k cluster centers, PAM examines all non-center (N — k) objects and
tries to replace each of the centers with one of the (N — k) objects that would reduce the
square error the most. PAM works well when the number of data points is small.

However, PAM is very costly, because for every k X (N — k) pairs PAM examines the (N

51

— k) data points to compute the cost function. Therefore, the total complexity is 0(k X (N

— Ioz ).

2.2.1 .6.2 CLARA

Because of the complexity inherent in PAM, an enhanced version of PAM was
developed for use in large data sets. CLARA (Clustering LARge Applications) is a
sampling-based method that was developed by Kaufman & Rousseeuw (1990). CLARA
selects a small portion of data to represent all data points therein. It selects the medoids
from these samples using PAM. The cost function is computed using the whole data set.
The efficiency of CLARA depends on the sample size, S. CLARA searches for the best k-
medoids among the sample of the data set. For more efficient results, CLARA draws
multiple samples from the data set, runs PAM on each sample and returns the best
clustering. The complexity of each iteration becomes 0(k x S2 + k X (N — k)) where S is
the sample size, k is the. number of clusters, and N is the number of data objects. It is
important to note, when comparing this complexity with that of the standard PAM, the

major distinction lies in the power of N — k, the largest value within the expression.

2.2.1 .6.3 CLARANS

CLARANS (Clustering Large Applications based upon RANdomized Search) was
proposed by Ng and Han (1994) to improve the quality and scalability of CLARA. This
algorithm tries to find a better solution by randomly picking one of the k centers and
replacing it with another randomly chosen object from the remaining objects. The goal of
CLARANS is not to find the best set of data points that represent the cluster centers.

Instead, it trades accuracy for efficiency and tries to find the local optimum. The

52

 

 

l0 ll'i‘il
it no
hat q.

lit-i ti

randomized procedure applied by CLARANS is the strong point of this algorithm. Ng
and Han (1994) have shown that CLARANS outperforms the PAM and CLARA
algorithms, however, the computational complexity has been reported to be
approximately 0(N2) (Han et al., 2001). The clustering quality is highly dependent upon
the sampling method employed. Ester et al. (1995) improved the performance of

CLARANS using special clustering methods, such as R‘-trees.

2.2.1 .7 Other partitional methods for large data sets

The main drawback of the CLARA and CLARAN S algorithms is their requirement
to hold the entire data set in the main memory. In general, large data mining applications
do not allow the entire data set to be stored in the main memory, so clustering algorithms
that can handle this situation are required. Some approaches were proposed to cluster the
data from such sets that we explain two of them as follows.

1. Divide and conquer approach. The entire data set is stored in a secondary
memory. The stored data is divided into p subsets. Each subset is clustered
separately into k clusters. One or more representative samples from each of these
clusters are stored individually. In the final step, all these p subset clusters are
merged into k clusters to obtain a clustering of the entire set. This method can be
extended to any number of levels (subsets); more subsets are necessary if the
main memory size is small and the data set is very large (Murty and Krishna,
1980). The drawback of this algorithm is that it works well only when the points
in each subset are realistically homogenous, e.g. in image data sets (Jain et al.,

1999)

53

 

2.2

den;

J

2. Incremental clustering. In this method the entire pattern matrix is stored in a

secondary location from which data objects are loaded into main memory one at a
time and assigned to existing clusters. Only the cluster representatives are stored
in the main memory. Each new data is allocated to an existing cluster or assigned
to a new cluster depending on criteria like the distance between the loaded data
point and the cluster’s representative. This method is naturally non-iterative, thus

the time complexity requirement is as small as the memory requirement (Jain et

al., 1999).

2.2.2 Hierarchical Methods

Hierarchical methods decompose the given set of data items forming a tree, which is

called dendrogram. A dendrogram splits the dataset recursively into smaller subsets. A

dendrogram can be formed in two ways:

1.

The Bottom-up approach, also referred to as the agglomerative approach, starts
with each object forming a distinct group. It successively merges the groups
according to some measure, such as the distance between the centers of the
groups, which continues until all of the groups are merged into one — the top most
level of hierarchy.

The Top-down approach, also referred to as the divisive approach, starts with all
the objects in the same cluster. In every successive iteration, a cluster is split into
smaller groups according to some measure until each object is eventually in one
cluster, or until a termination condition is met.

Hierarchical methods are popular in biological, social and behavioral systems,

which often need to construct taxonomies. Due to rapidly increasing data

54

densities, dendrograms are impractical when the number of patterns exceeds a few
hundred (Jain & Dubes, 1988). As a result, partitional techniques are more
appropriate in the case of large data sets. The dendrogram can be broken at

different levels to obtain different clusterings of the data (Jain et al., 1999).

2.2.2.1 Traditional Linkage Algorithms

The main steps of hierarchical agglomerative algorithms are as follows: We compute

the proximity matrix including the distances between each pair of patterns. Each pattern

is treated as a cluster in the first run. Using the proximity matrix we find the most similar

pair of clusters and merge these two clusters into one. At this point, the proximity matrix

is updated to imitate the merge operation. We continue this process until all patterns are

in one cluster. Based on how the proximity matrix is updated a range of agglomerative

algorithms can be designed (Jain et al., 1999).

1.

single-link clustering: The similarity between one cluster and another cluster is
equal to the greatest similarity between any member of one cluster and any
member of another cluster. It is important to note that, by “greatest similarity,” a
smallest distance is implied.

complete-link clustering: The distance between one cluster and another cluster is
equal to the greatest distance from any member of one cluster to any member of
another cluster. Unlike the single-link algorithm, this is focused on the lowest
similarity between clusters.

average-link clustering: The distance between one cluster and another cluster is
equal to the average distance from any member of one cluster to any member of

the another cluster.

55

2.2.2.2 BIRCH

The BIRCH (Balanced Iterative Reducing and Clustering) algorithm (Zhang et al.,
1996) uses a hierarchical data structure, which is referred to as a CF-tree (Clustering-
Feature-Tree) for incremental and dynamic clustering of data objects. The BIRICH
algorithm represents data points as many small CF-trees and then performs clustering
with these CF-trees as the objects. A CF is a triplet summarizing information about the
sub-cluster in the CF-tree; CF = (N, LS, SS) where N denotes the number of objects in the
sub-cluster, LS is the linear sum of squares of the data points, and SS is the sum of
squares of the data points. Taken together, these three statistical measurements become
the object for further pair-wise computation between any two sub-clusters (CF-trees). CF-
trees are height-balanced trees that can be treated as sub.clusters. The BIRCH algorithm
calls for two input factors to construct the CF-tree: the branching input factor B and
threshold T. The branching parameter, B, determines the maximum number of child
nodes for each CF node. The threshold, T, verifies the maximum diameter of the sub-
cluster kept in the node (Han et al, 2001).

A CF tree is constructed as the data is scanned. Each point is inserted into a CF node
that is most similar to it. If a node has more than B data points or its diameter exceeds the
threshold T, BIRCH splits the CF nodes into two. After doing this split, if the parent node
contains more than the branching factor B, then the parent node is rebuilt as well. The
step of generating sub-clusters stored in the CF-trees can be viewed as a pre-clustering
stage that reduces the total number of data to a size that fits in the main memory. The
BIRCH algorithm performs a known clustering algorithm on the sub-cluster stored in the

CF-tree. If N is the number of data points, then the computational complexity of the

56

BIRCH algorithm would be 0(N) because it only requires one scan of the data set —
making it a computationally less expensive clustering method than hierarchical methods.
Experiments have shown good clustering results for the BIRCH algorithm (Han et al,
2001). However, similar to many partitional algorithms it does not perform well when the
clusters are not spherical in shape and also when the clusters have different sizes. This is
due the fact that this algorithm employs the notion of diameter as a control parameter
(Han et al, 2001). Clearly, one needs to consider both computational cost and geometrical
constraints when selecting a clustering algorithm, even though real data sets are often

difficult to visualize when first encountered.

2.2.2.3 CURE

The CURE (Clustering Using REpresentatives) algorithm (Guha et al., 1998)
integrates different partitional and hierarchical clusters to construct an approach which
can handle large data sets and overcome the problem of clusters with non-spherical shape
and non-uniform size. The CURE algorithm is similar to the BIRCH algorithm and
summarizes the data points into sub-clusters, then merges the sub-clusters that are most
similar in a bottom-up (agglomerative) style. Instead of using one centroid to represent
each cluster, the CURE algorithm selects a fixed number of well-scattered data points to
represent each cluster (Han et al., 2001).

Once the representative points are selected, they are shrunk towards the gravity
centers by a shrinking factor a which ranges between 0 and 1. This helps eliminate the
effects of outliers, which are often far away from the centers and thus usually shrink
more. After the shrinking step, this algorithm uses an agglomerative hierarchical method

to perform the actual clustering. The distance between two clusters is the minimum

57

 

 

distance between any representative points. Therefore, if a = 1, then this algorithm will
be a single link algorithm, and if a = 0, then it would be equivalent to a centroid-based
hierarchical algorithm (Guha & Rastogi, 1998). The algorithm can be summarized as
follows:
1. Draw a random sample 5 from the data set.
2. Partition the sample, 3, into p partitions (each of size |s| / p).
3. Using the hierarchical clustering method, cluster the objects in each sub-cluster
(group) into |s| /pq clusters, where q is a positive input parameter.
4. Eliminate outliers; if a cluster grows too slowly, then eliminate it.
5. Shrink multiple cluster representatives toward the gravity center by a fraction of
the shrinking factor a.
6. Assign each point to its nearest cluster to find a final clustering.
This algorithm requires one scan of the entire data set. The complexity of the
algorithm would be 0(N) where N is the number of data points. However, the clustering
result depends on the input parameters |s|, p, and a. Tuning these parameters can be

difficult and requires some expertise, making this algorithm difficult to recommend (Han

et al. 2001).

2.3 Feature Selection and Extraction

Feature extraction and selection is a very important task in the classification or
clustering process. Feature selection is the procedure of discovering the most effective
subset of the original features to use in classification/clustering. Feature extraction is the
process of transforming the input features to produce new relevant features. Either or
both of these techniques can be used to obtain an appropriate set of features to use in

58

classification or clustering. Why do we need to use feature selection or extraction? We

can denote the benefits (Jain, 2000) of feature selection and extraction as follows:

2.3.1 Minimizing the cost

In many real world applications, feature measurement is very costly, especially with
a large sample size. Pei, et al. (1998) presented in the context of a biological pattern
classification that the most important 8 out of 96 features gives 90% classification
accuracy. They showed that feature selection has a great potential effect in minimizing

the cost of extracting features and maintaining good classification results.

2.3.2 Data Visualization

For explanatory purposes it is useful to project high dimensional data down to two or
three dimensions. The main concern is protecting the distance information and
deployment of the original data in two or three dimensions. The traditional approach in
data visualization is linear projection. A more convenient choice is projecting the data
onto a graph (Mori 1998). "Chemoff F aces" project the feature space onto cartoon faces,

by which one can visualize more than three features (Chernoff, 1973).

2.3.3 Dimensionality Reduction

Based on an ideal situation where we have an infinite number of training samples,
classification would be more accurate when we have more features because generally,
more features gives more information. Nevertheless, in real applications with finite
sample sizes, the maximum performance is inversely proportional to the number of

features (Duda & Heart, 1973).

59

The demand for a large number of samples grows exponentially with the
dimensionality of the feature space. This is due to the fact that as the dimensionality
grows, the data objects becomes increasingly sparse in the space it occupies. Therefore,
for classification, this means that there are not enough sample data to allow for reliable
assignment of a class to all possible values; and for clustering, the definition of density
and distance among data objects, which is critical in clustering, becomes less meaningful
(Duda & Heart, 1973).

This limitation is referred to as the “curse of dimensionality” (Duda et al., 2001).
Trunk (1979) has represented the curse of dimensionality problem through an exciting
and simple example. He considered a 2-class classification problem with equal prior
probabilities, and a d-dimensional multivariate Gaussian distribution with the identity
covariance matrix for each class. Trunk showed that the probability of error approaches
the maximum possible value of 0.5 for this 2-class problem. This study demonstrates that
one cannot increase the number of features when the parameters for the class-conditional
density are estimated from a finite number of training samples. Therefore, when the
training sets are limited, one should try to select a small number of salient features.

This puts a limitation on non-parametric decision rules such as k-nearest neighbor.
Therefore it is often desirable to reduce the dimensionality of the space by finding a new

set of bases for the feature space.

2.3.4 Feature Selection

A thorough review of feature selection has been presented in Jain (2000). Jain et al.

(1997) presented a taxonomy of available feature selection algorithms based on pattern

60

recognition or ANN, sub-optimal or optimal, single solution or multi-solution, and
deterministic or stochastic.

What are the criteria for choosing a feature subset? A subset of features might be
chosen in regards to the following points:

1. The relevance to the classification result: that is, we remove the irrelevant
features based on prior knowledge of the classification task. Langley (1994)
has a usefirl review on relevance and feature selection.

2. The correlation with the other features: High correlation among features will
add no more efficiency to classification (Harrell & Frank, 2001). That is, if
two features are highly correlated, one of the features is redundant, even
though it is relevant.

John et al. (1994) has presented the definitions of Strong Relevance and Weak
Relevance by considering the correlations among feature samples. Hall and Smith (1998)
formulated a measure of “Goodness of feature” as follows:

“Good feature subsets contain features highly correlated (predictive of) with the

.7

class, yet uncorrelated with (not predictive of) each other. ’

2.3.5 Feature Extraction

Feature extraction can be either a linear or non-linear transformation of the original
feature space. Principal Component Analysis (PCA) and Linear Discriminant Analysis
(LDA) are the most commonly used techniques for feature extraction. PCA is an

unsupervised technique which is intended for feature extraction, while LDA is

supervised.

61

The idea of the PCA is to preserve the maximum variance after transformation of the
original features into new features. The new features are also referred to as “principal
components” or “factors.” Some factors carry more variance than other, but if we limit
the total variance preserved afier such a transformation to some portion of the original
variance, we can generally keep a smaller number of features. PCA performs this
reduction of dimensionality by determining the covariance matrix. Afier the PCA
transformation in a d—dimensional feature space the m (m < d) largest eigenvalues of the
d x d covariance matrix are preserved. That is, the uncorrelated m projections in the
original feature space with the largest variances are selected as the new features, thus the
dimensionality reduces from d to m (Duda al et., 2001).

LDA uses the same idea but in a supervised learning environment. That is, it selects
the m projections using the criterion that maximizes the inter-class variance while
minimizing the intra-class variance (Duda al et., 2001). Due to supervision, LDA is more
efficient than PCA for feature extraction.

As explained in PCA, m uncorrelated linear projections are selected as the extracted
features. Nevertheless, from the statistical point of View, for two random variables that do
not hold normal distribution, uncorrelated between each other does not necessarily lead to
independent between each other. Therefore, a novel feature extraction technique,
Independent Component Analysis (ICA) has been proposed to handle the non-Gaussian
distribution data sets (Comon, 1994). Karhunen (1997) gave an simple example when the
axes found by ICA is different than those found by PCA for two features uniformly

distributed inside a parallelogram.

62

2.4 Summary

A body of literature was briefly explained which deals with the different problems
involved in data mining for performing classification and clustering upon a web-based
educational data. The major clustering and classification methods are briefly explained,
along with the concepts, benefits, and methods for feature selection and extraction. In the
next chapter, we design and implement a series of pattern classifiers in order to compare
their performance for a data set from the LON-CAPA system. This experiment provides
an opportunity to study how classification methods could be put into practice for future

web-based educational systems.

63

Chapter 3 Data Representation and Assessment

Tools in LON-CAPA

This chapter provides information about the structure of LON-CAPA data namely:
its data retrieval process, how we provide assessment tools in LON-CAPA on many
aspects of teaching and learning process. Our ability to detect, to understand, and to
address student difficulties is highly dependent on the capabilities of the tool. Feedback
from numerous sources has considerably improved the educational materials, which is a

continuing task.

3.1 Data Acquisition and Extracting the Features

3.1.1 Preprocessing student database

Preprocessing and finding the useful student data and segmenting may be a difficult
task. As mentioned earlier, LON-CAPA has two kinds of large data sets: 1) Educational
resources such as web pages, demonstrations, simulations, and individualized problems
designed for use on homework assignments, quizzes, and examinations; 2) Information
about users, who create, modify, assess, or use these resources.

The original data are stored with escape sequence codes as shown in Figure 3.1:

64

 

 

 

1007070627:msullz1007070573%3a%2fres%2fadm%2fpages$2fgrdst2egif%3aminaeibi$3amsu$26100707
0573%3a%2fres%2fadm%2fpages%2fstat%2eqif%3aminaeibii3amsu$261007070574%3amsu%2fmmp%2flabq
uiz%2flabquiz%2esequence___1___msu%2fmmp%2flabqui2%2fnewclass%2ehtmlt3aminaeibi%3amsu%261
007070589$3amsu%2fmmp%2flabqui2%2flabquiz%2esequence___5___msu%2fmmp%2flabquiz%2fproblems
%2fqui22part2%2eproblem%3aminaeibi$3amsu%26l007070606%3a%2fadm%2fflip%3aminaeibi%3amsu%26
1007070620%3a%2fadm%2fflip%3aminaeibi%3amsu%26l007070627%3a%2fres%2fadm%2fpages%2fs%2egif
%3aminaeibi$3amsu%261007070627%3a%2fadthflogout%3aminaeibi%3amsu

 

 

Figure 3.1 A sample of stored data in escape sequence code

To sense the data we use the following Perl script function as shown in Figure 3.2

 

my $str; my Sline;

open (LOG ,Sfile);

while (Sline =<LOG>l l
my ($dumptime,$host,$entry)=split(/\:/,$line);
my $str = unescape($entry);
my ($time,$ur1,$usr,$domain,$store,$dummy)=split(/\:/,$str);
my $string = escape($store);
foreach(split(/\&/,$string)){

print ”Stime $url Susr domain \n";

l

}

sub unescape {
my $str=shift;
$str =~ s/%([a-fA—FO—9][a-fA-F0-9])/pack("C",hex($1) )/eg;
return $str;

 

 

. Figure 3.2 Perl scrip code to retrieve stored data

After passing the data from this filter we have the following results as shown in

Figure 3.3:

 

 

1007070573 /res/adm/pages/grds.gif minaeibi /res/adm/pages/stat.gif

1007670091 /res/adm/pages/grds.gif minaeibi /adm/flip

1007676278
msu/mmp/labquiz/labquiz.sequence___2___msu/mmp/labquiz/problems/quizlpartl.problem
1007743917 /adm/logout minaeibi

1008203043 msu/mmp/labquiz/labquiz.sequence___1___msu/mmp/labquiz/newclass.html minaeibi
1008202939 /adm/evaluate minaeibi /adm/evaluate

1008203046 /res/adm/pages/g.gif minaeibi /adm/evaluate

1008202926 /adm/eva1uate minaeibi

 

Figure 3.3 A sample of retrieved data from activity log

The student data restored from .db files from a student directory and fetched into a
hash table. The special hash keys “keys ", “version ” and “timestamp ” were obtained from
the hash. The version will be equal to the total number of versions of the data that have
been stored. The timestamp attribute is the UNIX time the data was stored. keys is

available in every historical section to list which keys were added or changed at a specific

65

 

 

 

 

 

 

“b.
1.0%
'I

 

historical revision of a hash. We extract some of the features from a structured homework

data, which is stored as particular URL’s. The structure is shown in figure 3.4.

 

 

resource.

resource

resource

resource.

resource

resource.

resource

resource.

resource.

resource.

resource

resource

resource

partid

.partid.

.partid.

.partid.

.partid.
.partid.

.partid.

partid.
.partid.
partid.

partid.

partid.

partid.

opendate #unix time of when the local machine should let the
#student in

duedate #unix time of when the local machine should stop
#accepting answers

answerdate #unix time of when the local machine should

#provide the correct answer to the student
weight Q points the problem is worth
maxtries # maximum number of attempts the student can have

tol # lots of possibilities here
# percentage, range (inclusive and exclusive),
8 variable name, etc
i 3%
# 0.5
# .05+
ll 395+
# 0.5+,.005
sig # one or two comma sepearted integers, specifying the
# number of significatn figures a student must use
feedback # at least a single bit (yes/no) may go with a
3 bitmask in the future, controls whether or not
# a problem should say "correct" or not
solved # if not set, problem yet to be viewed

# incorrect_attempted == incorrect and attempted

# correct_by_student == correct by student work

# correct_by_override == correct, instructor override

# incorrect_by_override == incorrect, instructor override
# excused == excused, problem no longer counts for student
# " (empty) == not attempted

# ungraded_attempted == an ungraded answer has been

sumbitted and stored
tries # positive integer of number of unsuccessful attempts
# made, malformed answers don't count if feedback is
# on
awarded 1 float between 0 and 1, percentage of
# resource.weight that the stundent earned.
responseid.submissons
# the student submitted string for the part.response
responseid.awarddetail
# list of all of the results of grading the submissions
# in detailed form of the specific failure
1 Possible values:
# EXACT_ANS, APPROX_ANS : student is correct
# NO_RESPONSE : student submitted no response
8 MISSING_ANSWER : student submitted some but not

# all parts of a response
9 WANTED_NUMERIC : expected a numeric answer and
# didn't get one

# SIG_FAIL : incorrect number of Significant Figures
# UNIT_FAIL : incorrect unit
# UNIT_NOTNEEDED : Submitted a unit when one shouldn't
# NO_UNIT : needed a unit but none was submitted
# BAD_FORMULA : syntax error in submitted formula
# INCORRECT : answer was wrong
# SUBMITTED : submission wasn't graded

 

Figure 3.4 Structure of stored data in activity log and student data base

For example, the result of solving homework problem by students could be extracted

from resource. partid.solved, the total number of the students for solving the

66

 

 

problem could be extracted from resource.partid. tries, and so forth. One of the
difficult phases to data mining in the LON—CAPA system is gathering the student and
course data, which are distributed in several locations. Finding the relevant data and

segmentation phase is complicated as well.

3.1.2 Preprocessing Activity Log

LON-CAPA records and dynamically organizes a vast amount of information on
students' interaction with and understanding of these materials. Since LON-CAPA logs
every activity of every student who has used online educational resources and their
recorded paths, the activity.log usually grows faster when students have more access to
the educational resources. A sample of different types of data, which are logged in

activity.log after a preprocessing phase, is shown in figure 3.5.

studentX --> /adm/navmaps eb. This information is stored in an

144) 1010955846:

“activity.log" which is located in a course’s directory. The data stored in the

activity.log includes user name,

time and resource URL.

 

145) 1010955205: studentx --> /res/msu/mmp/kap14/picts/beta_eqn.gif

147) 1010955988: studentx —-> /adm/navmaps

148) 1010955998: student)! --> msu/mp/kapll/kapld.eequence_5_meu/uup/kap14/cd396.htrn
149) 1010955999: studentx -—> /res/msu/mmp/kap14/picts/velocity_eqn3.gif

150) 1010956000: studentx --> /res/msu/mmp/kap14/picts/time_eqn.gif

151) 1010954609: studentx --> /res/adm/pages/grds.gif

.152) 1010954611: studentx -—> /res/msu/mmp/wordproc.gif

153) 1010954626: studentx —-> /res/adm/pages/i.gif

1554) 1010955717: student}! -—> neu/nnp/kapll/kapll.sequence_1_neu/nnp/kap14/cd392.htn
15.5) 1010955717: studentx -—> /res/msu/mmp/kap14/picts/backsoun.gif
156) 1010955920: student)! --> mulmp/kap14/kap14.eequence_3_meu/mp/kep14/cd394.htm
157) 1010955921: studentx --> /res/msu/mmp/gifs/demo.gif
163) 1010955754: studentx --> neu/rrnp/kapld/kaplit.sequence_2_rneu/uup/kep14/cd393.htrn
1¢;4) 1010955756: studentx —-> /res/msu/mmp/kap14/picts/asound.jpg
166) 1010955999: studentx --> /res/msu/mmp/gifSZ/example.gif
1(73) 1010955687: studentX -—> /res/adm/pages/u.gif
174) 1010955688: studentx --> /res/adm/pages/e.gif
175) 1010956528: student)! -->

mu/nnp/keplllkepltl . sequence 33 men/nap] kep14/prob1ene/ cd418e . problem
176) 1010956536: student}! -->

 

 

 

uu/Mapld/kepld . sequence 33 men/unp/kap14/prob1ene/cd418a .problen
178) 1010956536: student}! -->
neu/gékepld/kepld . sequence 33 mu/mrp/kep14/problems/cd418a . problem

 

Figure 3.5 A sample of extracted Activity.log data

67

3.1.3 Extractable Features

An essential step to perform classification is selecting the features used for
classification. The following features are examples of those stored by the LON-CAPA
system:

0 Total number of correct answers.

0 Getting the problem right on the first try.

0 Number of attempts before correct answer is derived.

0 Total time that passed from the first attempt, until the correct solution was
demonstrated, regardless of the time spent logged in to the system. Also, the time
at which the student got the problem correct relative to the due date.

0 Total time spent on the problem regardless of whether they got the correct answer
or not. Total time that passed from the first attempt through subsequent attempts
until the last submission was demonstrated.

0 Participating in the communication mechanisms, versus those working alone.
LON-CAPA provides online interaction both with other students and with the
instructor.

0 Reading the supporting material before attempting homework vs. attempting the
homework first and then reading up on it.

0 Submitting a lot of attempts in a short amount of time without looking up material
in between, versus those giving it one try, reading explanatory/supportive
material, submitting another one, and so forth.

0 Giving up on a problem versus students who continued trying up to the deadline.

68

0 Time of the first log on (beginning of assignment, middle of the week, last
minute) correlated with the number of submissions or number of solved problems.
These features enable LON-CAPA to provide many assessments tools for instructors as it

will be explained in the next section.

3.2 Feedback to the instructor from online homework

LON-CAPA has enabled instructors to efficiently create and distribute a wide variety
of educational materials, assignments, assessments, etc. These include numerous types of
formative conceptual and algorithmic exercises for which prompt feedback and assistance
can be provided to students as they work on assigned tasks. This section presents recent
developments that allow rapid interpretation of such data in identifying students'
misconceptions and other areas of difficulty, so that concurrent or timely corrective
action can be taken. This information also facilitates detailed studies of the educational

resources used and can lead to redesign of both the materials and the course.

3.2.1 Feedback tools

While several meta-analyses of the effects of assessment with immediate feedback to
the student on their learning are positive (Mason & Bruning, 2003; Azevedo & Bernard
1995), the range of effect size is considerable (Bonham et al., 2001), and can even be
negative (Mason & Bruning, 2003; Bransford et al., 2000; Kluger & DeNisi, 1996;
Kluger & DeNisi, 1998). Even within our own model systems CAPA, LectureOnline, and
LON-CAPA, when used just for homework, a range of partly contradictory observations
were made (Kotas, 2000; Pascarella, 2004). The timely feedback is crucial for ensuring

effective use.

69

 

 

1*:

As it has been mentioned earlier LON-CAPA do record all information transmitted
to and from the student. That large amount of data, especially in large courses, is much
too voluminous for the faculty to interpret and use without considerable pre-processing.
We discuss functions that make that vast amount of data useful in a timely fashion. The
instructor can then give students useful feedback, either promptly enough that student can
benefit while still working on current task, or at a later date to clarify misconceptions and
address lack of understanding. A preliminary report on some of this work was presented
in Albertelli et al., (2002).

LON-CAPA outperforms many web-based education systems in three important
aspects relevant to the current discussion.

1. The first is its ability to individualize problems, both algorithmic numerical
exercises as well as problems that are qualitative and conceptual so that
numbers, options, images, etc. differ from student to student. (Kashy et al.,
1995)

2. The second is in the tools provided that allow instructor to collaborate in the
creation and sharing of content in a fast and efficient manner, both within and
across institutions, thus implementing the initial goals of the W3.

3. And the third is its one-source multiple target capabilities: that is, its ability
to automatically transform one educational resource, for example a numerical

or conceptual homework question, into a format suitable for multiple uses.

 

3 See http://www.w3.org/History/19921l03-hypertext/hypertext/WWW/Proposal.html and also

http://www.w3 .org/History/ 19921 l03-hypertext/hypertext/WWW/Designlssues/Multiuserhtml
70

 

 

 

 

501

has

0);

3.2.2 Student Evaluation

An important task of the feedback tools for the instructor is to help identify the
source of difficulties and the misconceptions students have about a topic. There are
basically three ways to look at such homework data: by student, by problem, or cross-
cutting (per student, per problem).

The amount of data gathered from large enrollment courses (ZOO-400 students) with
over 200 randomizing homework problems, each of them allowing multiple attempts, can
be overwhelming. Figure 3.6 shows just a small excerpt of the homework performance in
an introductory physics course, students in the rows, problems in the columns, each
character representing one online homework problem for one student. A number shown is
the number of attempts it took that particular student to get that particular problem correct
(“*” means more than nine attempts), “.” denotes an unsolved problem, blank an un-
attempted problem. This view is particularly useful ahead of the problem deadline,
where columns with a large number of dots or blank spaces indicate problems that the

students have difficulties with.

71

LBS 272 Spring 2004 Thu Apr 1 20:14:39 2004

Number of Tries before success on each Problem Part

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

username Electrostatics Electric Field Capacitors
m2 142822271744 12/ 12 25121615‘2 10/ 10 11213211222 11/ 11
m 191111531111 12/ 12 1121112113 10/ 10 11111111111 11/ 11
3:31-14. 211111121111 12/ 12 2113124159 10/ 10 11211111111 11/ 11
3M 12321*2412*2 12/ 12 23.*l98158 9/ 10 13321141125 11/ ll
3;:-.33.: 212.12143.*2 10/ 12 ._1 .41 3/ 10 14121141138 11/ 11
m 111112111111 12/ 12 1111111121 10/ 10 12111111111 11/ 11
m 3326221211*2 12/ 12 2323241313 10/ 10 21311121122 11/ 11
.2.-£75- 112116121113 12/ 12 1511111111 10/ 10 11311111112 11/ 11
3:33;} 121.151*11.6 10/ 12 2211422237 10/ 10 11941112111 11/ 11
m 111111151211 12/ 12 2221113111 10/ 10 2111111111; 11/ 11
1.9: correct by student in 1.9 submissions *: correct by student in more than 9 submissions

+: correct by override -: incorrect by override

.: incorrect attempted #: ungraded attempted

‘ ‘: not attempted x: excused

 

 

Figure 3.6 A small excerpt of the performance overview for a small
introductory physics class

We extract from student data some reports of the current educational situation of
every student as shown in table 3.1. A ‘Y’ shows that the student has solved the problem
and an ‘N’ shows a failure. A ‘-‘ denotes an un-attempted problem. The numbers in the
right column show the total number of submissions of the student in solving the

corresponding problems.

72

Table 3.1 A sample of a student homework results and submissions

 

 

 

 

 

 

|| Homework —Set Ti 19 fiestas Tries
ltnmplphyl 83 .se quence

Hmsdmmplkapllcaldcqalsequence WYYYYYYYYYYY 1,1,l,2,1,1,1,l,1,7,9,3,2
llmmmptkapztcaraapzseqmme [V'rmmmmm 10,1,o,1,1,2,1,4,5,1,1,1,3,2,1,1,1

 

ansdmmp/kapfiilcalckafisecpeme uwwvmmvwmw 4,3,5,1,1,2,2,3,20,1,1,1,1,1,2,3,2,2,3

 

“mm/mmp/kaptllcalckap4sequence J mmmmw 20,111s,3,2,3,4,2,3,2,1,1,2,5

 

 

 

 

 

 

 

 

 

 

lfnwmmpncapstcarmpiseqmme wmmmw p,2,1,9,12,1,3,12,1,2,1,,1,3
ansulmmplkapfilcalckapésequeme wmmmm B,2,4,2,1,1,2,1,1,9,2,3,2,2
"r'nswmmplkap'llcalckqi'lsequence wmmmmmv 4,1,3,1,10,4,1,1,2,1,1,2,1,2,1,3,1,3
lrnaimmp/kapSl cal ckap8.sequence WYYYYYYWYYYYY 4,3, l,2,3,3, 4,3 ,3,1 ,1,4,l,l ,7
Ibsflmmpl‘kapfllcalckapSJsequeme Wm.mm 1,1,1,2,l,,2,2,l,6,4
lE'naImmp/kaplmcdckapwsequence wmmmx p,1,1,1,1,1,1,1,1,1,1,1

lfn sulmmp/kapl 1 lcalckapl 1 sequence r----------..---- P,0,0,0,0,0, 0,0 ,0,0,0,0 ,0,0 ,0,0 ,0

 

For a per-student view, each of the items in the table in Figure 3.6 is clickable and
shows both the students’ version of the problem (since each is different), and their
previous attempts. Figure 3.7 is an example of this view, and indicates that in the
presence of a medium between the charges, the student was convinced that the force
would increase, but also that this statement was the one he was most unsure about: His
first answer was that the force would double; no additional feedback except “incorrect”
was provided by the system. In his next attempt, he would change his answer on only this
one statement (indicating that he was convinced of his other answers) to “four times the
force” — however, only ten seconds passed between the attempts, showing that he was

merely guessing by which factor the force increased.

73

Resource: Two Charges
View ofthe problem - gen-.m.!-

Two opposite charges are placed some distance apart in a vacuum.
What will happen if ...?

One forth the force: The distance between the charges is doubled.

Double the force: The magnitude of one of the two charges is doubled.

Four times the force: The magnitude of both charges is doubled.

Four times the force: The distance between the two charges is cut in half.

Half the force: The charges are placed in a medium with a factor two higher permittivity.
You are correct. - .

Your receipt is 498-1666 3

Correct answer: .
[Answer for Part0 brie forth the force Double the force {Four times the force [Four times the force [Half the forceJ

 

 

 

F
Fullname: 32:11: rare: 2.1m

 

 

 

 

 

 

Darefirme Subm'ssion

Mon Jan l9 20:15:]9 P811011“ lliTriall

2004
{Answer One forththe fDoublethe lFourtimesthe TI-‘ourtimesthe
! force _ _ 7f°!7°{_-- ___> _ Home 7‘ ,. force _ 7 __
it"riii'z'.‘ l _n_ l _4_: [lg—(LU; Pug};- 1761-1 __‘
i it

 

 

 

Mon Jan I9 20: I 5:29
2004

Parl0ill) llzTriaIZ

 

 

 

 

{Kniuva‘fiofeime’T-lpbufii the '5 jig... m’ Firié“"‘{_roifnm‘“miiie ‘

Figure 3.7 Single-student view of a problem

The per-problem view Figure 3.8 shows which statements were answered

correctly course-wide on the first and on the second attempt, respectively, the graphs on

the right which other options the students chose if the statement was answered

incorrectly. Clearly, students have the most difficulty with the concept of how a medium

acts between charges, with the absolute majority believing the force would increase, and

about 20% of the students believing that the medium has no influence — this should be

dealt with again in class.

74

 

Foil Number ‘ Foil Name Foil Text Correct Value

 

 

 

 

 

l l_6_ l_l_2 The distance between the two charges is cut in half. Four times the force
2 l_6_l_2_2 The magnitude of both charges is doubled. Four times the fbrce
3 l_6_l_3_2 The magnitude of one of the two charges is doubled. Double the force

4 l‘6_l_4_2 The distance between the charges is doubled. One forth the force
5 _ l_6_l_5_2 FI'he charges are placed in a medium with a factor two higher permittivity. Half the force

 

 

 

 

 

 

Rtternpt 1, 186 submissi attempt 1, 126 submissi
1" a I"

§ ” E 3‘ .Onc forth the force
g .. f .. .Half the force
.5. n g u I Same force
0 u I Double the force
5’ 2' :5: " .Four times the force
. I 2 3 9 5 8 ' I 2 3 1 5
Fort umber N Foil timber

 

Attempt 2, 125 submissi fittempt 2, 83 submissit
1.0 fi 100

El
§ w E " .Oneforththeforce
g .. 5 .. .Halftheforce
3 n if: ” .Same force
E 5 .Doublethcforce
3 2' g 2' .Four timestheforce
. 1 2 3 4 3 g . 1 2 J 4 5
Foil Number Foil Mme:-

 

 

 

 

Figure 3.8 Compiled student responses to a problem

The simplest function of the statistics tools in the system is to quickly identify areas
of student difficulties. This is done by looking at the number of submissions students
require in reaching a correct answer, and is especially useful early after an assignment is
given. A high degree of failure indicates the need for more discussion of the topic before
the due date, especially since early responders are often the more dedicated and capable
students in a course. Sometimes a high degree of failure has been the result of some
ambiguity in wording or, mostly in newly authored problem resources, the result of errors
in their code. Difficulty is then ‘sky high’. Quick detection allows correction of the
resource, often before most students have begun the assignment. Figure 3.9 shows a plot

of the ratio of number of submissions to number of correct responses for 17 problems,

75

from a weekly assignment before it was due. About 15% of the 400 students in an

introductory physics course had submitted part or most of their assignment.

 

 

Homework 8, PHY183

Submissions per Correct Answer
Vt

 

 

4.
3,
21
1.
0- . .
12 3 4 s 6 7 8 9|0lllZl3l4lSl6l7
Problem Number

 

 

 

Figure 3.9 One Early Measure of a Degree of Difficulty

The data of Figure 3.9 is also available as a table which also lists the number of
students who have submissions on each problem. Figure 3.9 shows that five of the
questions are rather challenging, each requiring more than 4 submissions per success on
average. Problem 1 requires a double integral in polar coordinates to calculate a center of
mass. Problem 14 is a qualitative conceptual question with six parts and where it is more
likely that one part or another will be missed. Note that incorrect use of a calculator or
incorrect order of operation in a formula would not be detected in Figure 3.9 because of
their relatively low occurrence. Note also that an error in the unit of the answer or in the
formatting of an answer is not counted as a submission. In those instances, students re-
enter their data with proper format and units, an important skill that students soon acquire

without penalty,

76

3.2.3 Conceptual Problems

An important task of the feedback tools for the instructor is to help identify the
source of difficulty in numerical algorithmic questions, but it also allows for the
identification of misconceptions students may have on qualitative questions. Student
responses to two qualitative exercises, one from physics and the second from vector
math, illustrate the way that the analysis tool detects difficulties and their source, specific
misconceptions. The physics question is Problem 14 from assignment 8 (Figure 3.12),
which as indicated above, had five days before it was due. As shown in Figure 3.9 that
problem averaged at that time slightly more than 4 submissions per successful solution.
There were 50 correct solutions as a result of 208 submissions by 74 students. The order
in which the six statements are presented varies among students. Each statement is
selected randomly from one of the six concept groups. Each concept group focuses on a
particular aspect in the question. Success rate on each concept for the initial submission

is shown in Figure 3.10.

77

 

 

 

 

 

 

 

 

 

 

 

 

Submission 1, N = 74
100 — , fl
*5 80
O)
8 60 —
U
5 40
8
9.. 20
O , ,
1 2 3 4 5 6
Concept Number

 

 

 

Figure 3.10 Success (%) in Initial Submission for Selecting the Correct Answer
to Each of Six ‘Concept’ Statements

While concept ‘3’ is quite clearly the most misunderstood, there is also a large error
rate for concepts ‘2’, 4’ and ‘6’. About one third of the students succeeded on their first
submission for all six concepts groups and thus earned credit on their first submission.
This can be seen by looking at the decreasing number of submissions from Figure 3.10 to
Figure 3.11. Note the pattern in the initial submissions persists in subsequent submissions

with only minor changes.

78

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

- - = l “ W‘s—{155365314 = 25 _
100 r Submrssron 2, N 47 . . 100 _

i 7‘7 __ ”—l i
3 soil—l ————l g 80»: l - — >———~'

D I _ { —.
b I "—7 l b ‘ F... .
- . _ . . o 60 . —.—————_- - i— .-
8 60'L4 i {-1 l U , l
g 40:1 _ . - t l l g 404 3 ~— ~

8 l l i g s
8 20 3 +— 4 1 a. 20“: _. _— 7
l l J 0 l - .
l 2 3 4 5 6 _ 1 2 3 4 5 6
Concept Number 1 Concept Number

 

 

 

 

Figure 3.11 Success Rate on Second and Third Submissions for Answers to
Each of Six ‘Concept’ Statements

The text of the problem corresponding to the data in Figures 3.10 and 3.11 is shown

in Figure 3.12.

 

 

 

i AT:

3 Mb .. ,‘ '

i 0 M. «
l

l
t

 

A frictionless pulley with mass Ma is attached to the ceiling,
in a gravity field g. Mass MC is smaller than mass Mb. The
tensions T1,, Ty, T1, and the constant g are magnitudes. For
each statement, select a response.
Choices: Greater than, Less than, Equal to, True, False.
1. T; is Ty.
2. (A{b+Mc+Ma)g is T3.
3. Tu is (Mb)g.
4. The magnitude of the acceleration of M), is that of
MC.
The center-of-mass of A45, ll/IC, and Ma accelerates.
6. T3 is Tz+Ty.

9‘

 

 

 

Figure 3.12 Randomly Labeled Conceptual Physics Problem

79

 

The labels in the problem are randomly permuted. In the version of the problem
shown in Figure 3.12 the first question is to compare tension T2 to Ty. It is the most
commonly missed statement, corresponding to concept ‘3’ of Figures 3.10 and 3.11. The
incorrect answer given by over 90% of the students is that the two tensions are equal,
which would be the answer for a pulley with negligible mass. That had been the case in
an assignment two weeks earlier. This error was addressed by discussion in lecture and
by a demonstration showing the motion for a massive pulley with unequal masses. This
quickly impacted the subsequent response pattern. Note that solutions to the versions of
the problems use as illustrations are given at the end of this section. (Solution to Figure
3.12: l-less, 2-greater, 3-less, 4-equal, 5-true, 6-greater)

The next example is shown in Figure 3.13. It deals with the addition of two vectors.
The vectors represent the possible orientations and rowing speed of a boat and the
velocity of water. Here also the labeling is randomized so both the image and the text
vary for different students. Students are encouraged to discuss and collaborate, but cannot
simply copy from each other (Solution to Figure 3.13: l-less, 2-greater, 3-less, 4-equal, 5-

greater).

80

 

 

 

A river is to be crossed by a boy using a row boat.

RowiniSfieed
' . ' n f
amt}; /

 

 

 

 

\i

 

 

 

 

 

// /
s ZCM/””
> r—
I §:\N\. 5
River Velocfly \\ N

 

\‘F

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Assume that the water has uniform velocity, represented
above by the vector labeled B. The rowing speed of the boy
and a set of possible orientation of his boat are also shown.

Select an answer for each below.

Choices: Greater than, Less than, Equal to.
Time to row across for A is for H

Time to row across for K is for C

The distance traveled in crossing for H is for K
Time to row across for K is for H

F‘PPNE“

For an observer on shore, the speed of the boat for K is
for C

 

 

 

Figure 3.13 Vector Addition Concept Problem

The upper graphic of Figure 3.14 shows once again the success rate of 350 students
on their initial submission, but this time in more detail showing all the possible
statements. There are two variations for the first three concepts and four for the last two.

The lower graph in Figure 3.14 illustrates the distribution of incorrect choices for the
282 students who did not get cam credit for the problem on their first submission. The
stacked bars show the way each statement was answered incorrectly. This data gives
support to the ‘concept group’ method, not only in the degree of difficulty within a group
as reflected by the Percent Correct in Figure 3.14, but also by the consistency of the
misconception as seen from the Incorrect Choice distribution. Statements 3 and 4 in

81

Figure 3.14 present ‘Concept 2’, that greater transverse velocities result in a shorter
crossing time, with the vectors in reverse order. Statement 3 reads ‘Time to row across
for K is for C’, and statement 4 is ‘Time to row across for C is for K’. Inspection of
the graph indicates the students made the same error, assuming the time to row across for
K is less than the time to row across for C, regardless of the manner in which the question
was asked. Few students believed the quantities to be equal. In concept group 3,
statements 7, 8, 9 and 10, “equal to” is predominantly selected instead of ‘greater than’ or
‘less than’ as appropriate. This detailed feedback makes it easier for the instructor to
provide help so that students discover their misconceptions. Finally, as in the previously
discussed numerical example, particular hints can be displayed, triggered by the response
selected for a statement or by triggered by a combination of responses for several

statements.

82

 

 

H
0
CD

80 ———~ r—err——% ‘

60 r, , 7 V.»

40 i ,
20L # _ H _
0 , ,

,. l
12 34 5 6 7891011121314
Submission 1, N = 282

12 34 5 6 7891011121314
Statement Number 7 i

Percent Correct

 
  

 

 

   

 

 

Iv

 

Incorrect Choice
D /\

UH

 

 

 

 

 

 

Figure 3.14 Upper Section: Success Rate for Each Possible Statement. Lower
Section: Relative distribution of Incorrect Choices, with Dark Gray as “greater
than”, Light Gray as “Less Than” and Clear as “Equal to”

3.2.4 Homework and Examination Problem Evaluation

The same source code which is used to present problems for on-line homework can
also generate them for an on-line examination or for a printed version suitable for a
proctored bubble sheet examination which is later machine scored (Albertelli et al.,
2003)

LON-CAPA can provide statistical information about every problem in a table (see
Table 3.2), which is called “Stats Table”. Every part of a multi-part problem is
distinguished as a separate problem. The mum-instance problem is also considered
separately, because a particular problem or one part of it might be used in different

homework sets. Finally, a table is created which includes all computed information from

83

all students, sorted according to the problem order. In this step, LON-CAPA has provided

the following statistical information:

1.

5.

6.

#Stdnts: Total number of students who take a look at the problem.(Let

#Stdnts is equal to n)

Tries: Total number of submissions to solve the problem (2x, where 3‘: denote

i=1
a student try).

Mod: Mode, maximum number of submissions for solving the problem.

. . _ l ”
Mean: Average number of the submissmns. x =—Zx,
n

i=1
#YES: Number of students solved the problem correctly.

#yes: Number of students solved the problem by override.

Sometimes, a student gets a correct answer after talking with the instructor. This type

of correct answer is called “corrected by override".

7.

%Wrng: Percentage of students tried to solve the problem but still incorrect.

 

( n—(#YES+#yes) )
n

100*

8. S.D.: Standard Deviation of the students’ submissions.

 

 

J 1 :(xi—f)2

"-114

84

Table 3.2 Statistics table includes general statistics of every problem of the

course (Homework Set 1)

 

131361666611 Seti

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I 33333 I 3:3 D..F II
Order .. I#Stdntst'r193 Mod IMean IfYES %ngIDoDiffI I.Skew I 15! I2" d II
CalculatorSkIlls‘I _256 I 267 I 3 I104 I32_56 1| 60 I3304 I231: 573 I0031000!
INumbers 3256 34143 I317 I162 I' 255 _ 64 3038 I 16 1 35.7 0113 0021
ISpeed __ _”- -256 I 6938 I313 I273 3I 3255 I333 64 'I 063 I 2332 3'. 19006 0021
66616616 3, 256 388 7_ I 1._52 i 255 64 I 034 1' 63.93 i 2.4 5000 6362,
Reduc“ I 256 I 315 I 4 I 1.23 I 256 I 0.0 I019 I05 3 2.3 3.2001 I000?
IF’aCI.‘.‘1‘1..-.____‘ _—____.__. T -__. '. ..._ .1 . . .I f .- ,__,_ -__ -__. ._-____ ,.__.—.-..
Ica‘cu‘amlgw‘m 256 I 393 I 7 I 1.541 255 I 0.4 I 0.35 I09 I 2.0 015% I002I
IFractionsw _ _; -__. I._ "I _ I _ ‘—‘3 a: 3’” .__. 33-..- I
Breaorasanoon 3254 I601 . 12 2.37 I 247 I 2.83 ' 059 I3318 I 1.8 1-005I—002.
_.__._-._..-._.--.. ...m“; _-_---:._.__ ..m-: ; . -3-3 .. “77% "“‘I
game“ 5 252 I 565 'I11 I224I 243 I 3.6 I057 19 I 2.0 .'-006.-003
alloon___ __ _ _ ---.3—_ ___—_I_- 1 __ _, _: 3...._..__I._-- -..'.-. -
I068; __ 33__235_63 111_16I 2OI_4.36I 246 I 33.9 I078142i 139 1018 0031
‘Numencalva'ueI 256 I 268 II 4 I105 I256 I 00 I004 I02 I 3.4 I001I 00 I
’of Fraction -- , -- j: _ . .
. _-1. -______-._.____. -__... ._ _.____-_
Igftflg'vmus 254 I 749 I 11 I 2.95 I 251 I 1.2 I 066 I 22 1.1 1-0305 -005‘
[4661663966166 .253 31362620314063, 250 I 1.2 I 03.76 . 33.36 3333 1.38 3 _33014 I000.
IProximity 249 I 663 3; 19 I266 I 323339 3.63 I 0.64 I233 , 328 011 -0101
9. Skew.: Skewness of the students’ submissions.
1 n _ 3 1 n _ ,
—Z(x.—x> —Z(x.-—x)
’7 i=1 _ n i=1
(5.0.)3 1 ,. _ 2
—Z(xi 3 X)
‘1 i=1
lO. DoDiff: Degree of Difficulty of the problem.

1- (

# Y__E__S+#yes )

2-x..-

Clearly, the Degree of Difficulty is always between 0 and 1. This is a useful factor
for an instructor to determine whether a problem is difficult, and the degree of this

difficulty. Thus, DoDiff of each problem is saved in its meta data.

85

 

 

11. DoDisc: Degree of Discrimination4 (or Discrimination Factor) is a standard
for evaluating how much a problem discriminates between the upper and the
lower students. First, all of the students are sorted according to a criterion. Then,
27% of upper students and 27% lower students are selected from the sorted
students applying the mentioned criterion. Finally we obtain the Discrimination
Factor from the following difference:

Applied a criterion in 27% upper students - Applied the same Criterion in 27% lower
students.

Discrimination Factor is a number in interval [-1,l]. If this number is close to 1, it
shows that only upper students have solved this problem. If it is close to 0 it shows that
the upper students and the lower students are approximately the same in solving the
problem. If this number is negative, it shows that the lower students have more success in
solving the problem, and thus this problem is very poor in discriminating the upper and
lower students.

We compute the Discrimination Factor from two criteria:

Partial Credit Awarded
1st Criterion for Sorting the Students: 2 Z33
x.
i=l '

Zoe; YES+# yes)

I!

x.

i=l '

 

 

2nd Criterion for Sorting the Students:

These measures can also be employed for evaluating resources used in examinations.
Examinations as assessment tools are most useful when the content includes a range of

difficulty from fairly basic to rather challenging problems. An individual problem within

 

4 This name has been given by administration office of Michigan State University for evaluating the
exams’ problem. Here we expanded this expression to homework problems as well.

86

an examination can be given a difficulty index (DoDiff) simply by examining the class
performance on that problem. Table 3.3 shows an analysis for the first two mid-term

examinations in Spring 2004.

Table 3.3 Analysis of Examination Problems (N=393) DoDiff = Difficulty Index
DoDisc = Discrimination Index

 

 

 

 

 

 

 

 

 

 

 

 

Problem Number DoDiff DoDisc DoDiff DoDisc
Exam 1 Exam 1 Exam 2 Exam 2
1 0.2 0.4 0.7 0.24
2 0.16 0.31 0.13 0.2
3 0.4 0.4 0.19 0.31
4 0.44 0.57 0.41 0.57
5 0.32 0.38 0.52 0.11
6 0 0 0.18 0.26
7 0.23 0.33 0.7 0.36
8 0.21 0.24 0.57 0.35
9 0.36 0.63 0.55 0.58
10 0.4 0.59 0.87 0.14

 

 

 

 

 

 

 

We can see that Exam 1 was on the average somewhat less difficult than Exam 2.
Problem 10 in Exam 2 has DoDisc=0.l4 and DoDiff=0.87, indicating it was difficult for
all students. The students did not understand the concepts involved well enough to
differentiate this problem from a similar problem they had seen earlier. In Exam 1,
problems 3, 4, 9, and 10 are not too difficult and nicely discriminating. One striking entry
in Table 3.3 is for problem 6 in Exam 1. There both DoDiff and DoDisc are 0. No
difficulty and no discrimination together imply a faulty problem. As a result of this
situation, a request was submitted to modify LON-CAPA so that in the future an
instructor will be warned of such a circumstance.

The distribution of scores on homework assignments differs considerably from that

on examinations. This is clearly seen in Figure 3.15.

87

 

 

 

 

 

 

     

 

 

 

 

‘1 60
g l 5
'O ‘ '0
a ‘ B 40‘
m 80 i 2
“a g 2
8 ‘1’ 20
-° 40 r E
g l E I]
Z : 0--_v--ll.. 7,74,",

. 0 20 40 60 80 100
0 2_0 40 60 80 100 Cummulative Exam Scores (percent)
Cummulative Homework Scores (percent)

 

Figure 3.15 Grades on the first seven homework assignments and on the first
two midterm examinations

The correlation of homework and examinations is moderate (1:043). Students with a
good exam score tend to score high on homework but the reverse is not as true. This can
be seen in the 3-D plot of the Figure 3.15 data in Figure 3.16. Homework grades peak
near 100% as motivated students tend to repeat problems until a correct solution is
obtained.

Students also often interpret a high homework grade as indication that they are doing
well in the course. To counter that misconception, a readily accessible on-line grade
extrapolator provides students a review of their performance to date in the various
components of the class, quizzes, mid term exams, and homework. They enter their own
estimate of their future performance for the remainder of the semester, as well as for the
final examination. This tool then projects a final grade, thus keeping students aware of

their progress. As a result of feedback on students’ work, those doing very poorly can be

identified quite early (Minaei et al., 2003; Thoennessen et al., 1999).

88

 

Figure 3.16 Homework vs. Exam Scores. The highest bin has 18 students.

3.3 Summary

LON-CAPA provides instructors or course coordinators full access to the students’
educational records. With this access, they are able to evaluate the problems presented in
the course afier the students have used the educational materials, through some statistical
reports. LON-CAPA also provides a quick review of students’ submissions for every
problem in a course. The instructor may monitor the number of submissions of every
student in any homework set and its problems. The total numbers of solved problems in a
homework set as compared with the total number of solved problems in a course are
represented for every individual student.

LON-CAPA reports a large volume of statistical information for every problem e. g.,
“total number of students who open the problem,” “total number of submissions for the

problem,” “maximum number of submissions for the problem,” “average number of

89

9, u

submissions per problem, number of students solving the problem correctly,” etc. This
information can be used to evaluate course problems as well as the students. More details
can be found in Albertelli et al. (2002) and Hall et al. (2004). Aside from these

evaluations, another valuable use of data will be discussed in the next chapter.

90

Chapter 4 Predicting Student Performance

The objective in this chapter is to predict the students’ final grades based on the
features which are extracted from their (and others’) homework data. We design,
implement, and evaluate a series of pattern classifiers with various parameters in order to
compare their performance in a real data set from the LON-CAPA system. This
experiment provides an opportunity to study how pattern recognition and classification
theory could be put into practice based on the logged data in LON-CAPA. The error rate
of the decision rules is tested on one of the LON-CAPA data sets in order to compare the
performance accuracy of each experiment. Results of individual classifiers, and their
combination, as well as error estimates, are presented.

The problem is whether we can find the good features for classifying students! If so,
we would be able to identify a predictor for any individual student after doing a couple of
homework sets. With this information, we would be able to help a student use the
resources better.

The difficult phase of the experiment is properly pre-processing and preparing the
data for classification. Some Perl modules were developed to extract and segment the
data from the logged database and represent the useful data in some statistical tables and

graphical charts. More details of these tasks have been explained in a part of previous

chapter which is dedicated for data acquisition and data representation.

91

4.1 Data set and Class Labels

As the first step in our study, in order to have an experiment in student classification,
we selected the student and course data of a LON-CAPA course, PHY183 (Physics for
Scientists and Engineers 1), which was held at MSU in spring semester 2002. Then we
extend this study to more courses. This course integrated 12 homework sets including
184 problems. About 261 students used LON-CAPA for this course. Some of the students
dropped the course after doing a couple of homework sets, so they do not have any final
grades. After removing those students, 227 valid samples remained. You can see the

grade distribution of the students in the following chart (Figure 4.1)

 

Grade Distribution

 

 

O 10 20 30 40 so 60
# of students

 

Figure 4.1 Graph of distribution of grades in course PHY183 $802

92

are:

We can group the students regarding their final grades in several ways, 3 of which

1. The 9 possible class labels can be the same as students’ grades, as shown in Table
4.1.

2. We can group them into three classes, “high” representing grades from 3.5 to 4.0,
“middle” representing grades from 2.5 to 3, and “low” representing grades less
than 2.5, as shown in table 4.2.

3. We can also categorize students with one of two class labels: “Passed” for grades

above 2.0, and ”Failed” for grades less than or equal to 2.0, as shown in table 4.3.

Table 4.1. 9—Class labels regarding students’ grades in course PHY183_ 8802

 

 

Class Grade # of Student Percentage
1 0.0 2 0.9%
2 0.5 0 0.0%
3 1.0 10 4.4%
4 1.5 28 12.4%
5 2.0 23 10.1%
6 2.5 43 18.9%
7 3.0 52 22.9%
8 3.5 41 18.0%
9 4.0 28 12.4%

 

Table 4.2. 3-Class labels regarding students’ grades in course PHY183 SSOZ

 

 

Class Grade Student # Percentage

High Grade >= 3.5 69 30.40%
Middle 2.0 < Grade < 3.5 95 41.80%

Low Grade <= 2.0 63 27.80%

 

Table 4.3. 2-class labels regarding students’ grades in course PHY 183 8802

 

 

Class Grade Student # Percentage
Passed Grade > 2.0 164 72.2%
Failed Grade <= 2.0 63 27.80%

 

93

We can predict that the error rate in the first class grouping should be higher than the

others, because the sample size among the 9-Classes differs considerably.

The present classification experiment focuses on the first six extracted students’

features based on the PHY183 Spring 2002 class data.

1.

2.

Total number of correct answers. (Success rate)

Getting the problem right on the first try, vs. those with high number of
submissions. (Success at the first try)

Total number of attempts before final answer is derived

Total time that passed from the first attempt, until the correct solution was
demonstrated, regardless of the time spent logged in to the system. Also, the
time at which the student got the problem correct relative to the due date.
Usually better students get the homework completed earlier.

Total time spent on the problem regardless of whether they got the correct
answer or not. Total time that passed from the first attempt through
subsequent attempts until the last submission was demonstrated.

Participating in the communication mechanisms, vs. those working alone.
LON-CAPA provides online interaction both with other students and with the

instructor.

94

4.2 Classifiers

Pattern recognition has a wide variety of applications in many different fields;
therefore it is not possible to come up with a single classifier that can give optimal results
in each case. The optimal classifier in every case is highly dependent on the problem
domain. In practice, one might come across a case where no single classifier can perform
at an acceptable level of accuracy. In such cases it would be better to pool the results of
different classifiers to achieve the optimal accuracy. Every classifier operates well on
different aspects of the training or test feature vector. As a result, assuming appropriate
conditions, combining multiple classifiers may improve classification performance when

compared with any single classifier.

4.2.1 Non-tree based classifiers

We compare some popular non-parametric pattern classifiers and a single parametric
pattern classifier according to their error estimates. Six different classifiers over one of
the LON-CAPA data sets are compared. The classifiers used include Quadratic Bayesian
classifier, I—nearest neighbor (I-NN), k-nearest neighbor (k-NN), Parzen-window, multi-
layer perceptron (MP), and Decision T ree.5 These classifiers are some of the most
common classifiers used in practical classification problems. After some preprocessing
operations were made on the data set, the error rate of each classifier is reported. Finally,

to improve performance, a combination of classifiers is presented.

 

5 The first five classifiers are coded in MATLABTM 6.0, and for the decision tree classifiers we have use
some available software packages such as C5.0, CART, QUEST, CRUISE. We will discuss the Decision
Tree-based software in the next section. In this section we deal with non-tree classifiers.

95

4.2.1.1 Combination of Multiple Classifiers (CMC)

In combining multiple classifiers we seek to improve classification accuracy. There

are different ways one can think of combining classifiers:

The simplest way is to find the overall error rate of the classifiers and choose
the one which has the lowest error rate for the given data set. This is called an
oflline CMC. This may not really seem to be a CMC; however, in general, it
has a better performance than individual classifiers. The output of this
combination will simply be the best performance in each column in Figures
4.3 and 4.5.

The second method, which is called online CMC, uses all the classifiers
followed by a vote. The class getting maximum votes from the individual
classifiers will be assigned to the test sample. This method seems, intuitively,
to be better than the previous one. However, when we actually tried this on
some cases of our data set, the results were not more accurate than the best
result from the previous method. Therefore, we changed the rule of majority
vote from “getting more than 5 0% of the votes” to “getting more than 75% of
the votes”. We then noticed a significant improvement over offline CMC.
Table 4.6 shows the actual performance of the individual classifier and online
CMC over our data set.

Woods et a1. (1995) suggest a third method, which is called DSC-LA
(Dynamic Selection of Classifiers based on the Local Accuracy estimates).

This method takes a particular test sample, investigates the local

96

neighborhood of that sample using all the individual classifiers and the one

which performs best is chosen for the decision-making?
Besides CMC, we also show the outcomes for an “Oracle” which chooses the correct
results if any of the classifiers classified correctly, as Woods et a1. (1995) has presented

in their article.

4.2.1 .2 Normalization

Having assumed in Bayesian and Parzen-window classifiers that the features are
normally distributed, it is necessary that the data for each feature be normalized. This
ensures that each feature has the same weight in the decision process. Assuming that the
given data is conforms to a Gaussian distribution; this normalization is performed using
the mean and standard deviation of the training data. In order to normalize the training

data, it is necessary first to calculate the sample mean, p , and the standard deviation, 0' ,

of each feature (column) in the data set, and then normalize the data using the following

equation:

 

x. = ‘ (4.1)

This ensures that each feature of the training data set has a normal distribution with a
mean of zero and a standard deviation of one. In addition, the kNN method requires
normalization of all features into the same range. However, we should be cautious in

using the normalization before considering its effect on classifiers’ performances. Table

 

6 We have not implemented this method in this proposal yet.

97

4.4 shows a comparison of Error Rate and Standard Deviation, using the classifiers in

both normalized and un-normalized data in the case of 3 classes.

Table 4.4 Comparing Error Rate of classifiers with and without normalization

in the case of 3 classes

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3-Classes With Normalization Without Normalization
Classifier Error rate S.D Error rate S. D.
Bayes 0.4924 0.0747 0.5528 0.0374
INN 0.5220 0.0344 0.5864 0.041
KNN 0.5144 0.0436 0.5856 0.0491
Parzen 0.5096 0.0408 0.728 0
MLP 0.4524 0.0285 0.624 0
CMC 0.2976 0.0399 0.3872 0.0346
Oracle 0.1088 0.0323 0.1648 0.0224

 

Thus, we tried the classifiers with and without normalization. Table 4.4 clearly

shows a significant improvement in most classification results after normalization. Here

we have two findings:

1. The Parzen-Window classifier and MLP do not work properly without normalizing

the data. Therefore, we have to normalize data when using these two classifiers.

2. Decision tree classifiers do not show any improvement on their classification

performance after normalization, so we ignore it in using tree classifiers. We will

study the decision tree classifier later, though the Decision Tree classifiers’ results

are not introduced in Table 4.4.

4.2.1.3 Comparing 2-fold and 10-fold Cross-Validation

In k-fold cross-validation, we divide the data into k subsets of approximately equal

size. We train the data k times, each time leaving out one of the subsets from training, but

98

 

using only the omitted subset to compute the error threshold of interest. If k equals the
sample size, this is called "Leave-One-Out" cross-validation. (Duda et al. 2001; Kohavi,
1995). Leave-One-Out cross-validation provides an almost unbiased estimate of true
accuracy, though at a significant computational cost. In this proposal both 2-fold and 10-
fold cross validation are used.

In Z-fold cross-validation, the order of the observations, both training and test, are
randomized before every trial of every classifier. Next, every sample is divided amongst
the test and training data, with 50% going to training, and the other 50% going to test.
This means that testing is completely independent, as no data or information is shared
between the two sets. At this point, we classify7 the test sets after the training phase of all
the classifiers. We repeat this random cross validation ten times for all classifiers.

In I0-fold cross-validation, the available data (here, the 227 students’ data) are
divided into 10 blocks containing roughly equal numbers of cases and class-value
distributions. For each block (10% of data) in turn, a model is developed using the data in
the remaining blocks (90% of data, the training set), and then it is evaluated on the cases
in the hold-out block (the test set). When all the tests (10 tests) are completed, each
sample in the data will have been used to test the model exactly once. The average
performance on the tests is then used to predict the true accuracy of the model developed
from all the data. For k-values of 10 or more, this estimate is more reliable and is much

more accurate than a re-substitution estimate.

 

7 All the code was written in MATLABTM 6.5

99

Table 4.5 Comparing Error Rate of classifiers 2-fold and 10—fold Cross-
Validation in the case of 3 classes

 

 

 

 

 

 

 

 

 

 

3-Classes 10-fold Cross-Validation 2-fold Cross-Validation
Classifier Error Rate S.D. Error Rate S.D.
Bayes 0.5 0.0899 0.5536 0.0219
INN 0.4957 0.0686 0.5832 0.0555
KNN 0.5174 0.0806 0.576 0.0377
Parzen 0.5391 0.085 0.4992 0.036
MLP 0.4304 0.0806 0.4512 0.0346
CMC 0.313 0.084 0.3224 0.0354
Oracle 0.1957 0.0552 0.1456 0.0462

 

 

 

 

 

Table 4.5 shows comparison of Error Rate and Standard Deviation using the
classifiers in both 2-fold and 10-fold cross-validation in the case of the 3-Classes. You
can see that the 10-fold cross-validation in relation to individual classifier has slightly
more accurate than 2-fold cross validation, but in relation to combination of classifiers

(CMC) there is no a significant difference. Nonetheless, we selected 10-fold cross

validation for error estimation in this proposal.

4.2.1.4 Results, Error Estimation

The experimental results were averaged and are presented in the charts below. They
show the effect of selecting the data randomly on the average error rate. The average
error rate and its standard deviation, which is associated with each classifier, is shown in

the tables as well as the chart. Each table summarizes the results of all the classifiers on

our data set.

100

 

 

 

 

 

 

Figure 4.2: Comparing Error Rate of classifiers with lo-fold Cross-Validation
in the case of 2-Classes

The standard deviation of error rate shows the variance of the error rate during cross
validation. The error rate is measured in each round of cross validation by:

Total missclassfied of test examples
Total numberof test examples

 

Error Rate in each round =

Afier 10 rounds, the average error rate and its standard deviation are computed and
then plotted. This metric was chosen due to its ease of computation and intuitive nature.
Figure 4.2 and 4.3 show the comparison of classifiers’ error rate when we classify the
students into two categories, “Passea” and “Failed’. The best performance is for kNN
with 82% accuracy, and the worst classifier is Parzen-window with 75% accuracy. CMC

in the case of 2-Classes classification has 87% accuracy.

10]

 

.__..._.. -__ '._.w .__—E

LON-CAPA, Classifiers Camparison on PHY183 $802,
10-fold Cross-Validation, 2 Classes

0.3

 

0.25

 

 

 

 

0.2 ~

0.15 e -- ~ -- ——_ __- , - _. _____1

 

 

 

 

0.05

      

 

o .

 

 

 

        

Bayes 1 1NN MLP CMC

1 lAverage Error Rate 0.2364 0.2318 0.1773 0.25 0.2045 0.1318 1 0.0818
I DSlandard Deviation 0.0469 1 0.0895 0.0725 1 0.089 0.0719 0.0693 1 0.0559

Parzen Oracle

 

 

 

 

 

 

 

 

1
1
1
1
1
1
1
1
1
|
1
1
1
|
1
1
1 0.1 a
1
1
1
1
1
1
1
1

 

 

 

 

 

 

 

Figure 4.3: Table and graph to compare Classifiers’ Error Rate, 10-fold CV in
the case of 2—Classes

It is noticeable that these processes were done after we had found the optimal k in the
kNN algorithm and after we had tuned the parameters in MLP and after we had found the
optimal h in the Parzen-window algorithm. Finding the best k for kNN is not difficult,
and its performance is the best in the case of 2-Classes, though is not as good as the other

classifiers in the case of 3-Classes, as is shown in Figure 4.4.

102

 

Oracle

Figure 4.4: Comparing Error Rate of classifiers with lO—fold Cross-Validation in the
case of 3-Classes

Working with Parzen-window classifier is not as easy because finding the best width
for its window is not straitforward. The MLP classifier is the most difficult classifier to
work with. Many parameters have to be set properly to make it work optimally. For
example, after many trials and errors we found that the structure of the network in the
case of 3-c1asses, the 4-3-3 (one hidden layer with 3 neurons in hidden layer) works
better, and in the case of 2-classes, if we have 2 hidden layer with 2 or 4 neurons in each
hidden layer, would lead to a better performance. There is no algorithm to set the number
of epochs and learning rates in the MLP. However, sometimes MLP has the best

performance in our data set. As shown in the Table 4.6, MLP is slightly better than the

other individual classifier. In the case of 9-Classes we could not set the MLP to work

103

properly, so we have not brought the result of MLP classifier into the final result in table

4.6.

 

LON-CAPA, Classifiers Comparison on PHY183 data set,
10-fold Cross-Validation, 3 classes

0.6

 

0.5 ~

 

0.4 4 7

 

0.3 -.

0.2 «

 

 

 

0.1 -

 

0 a

 

1

1 Bayes 1 WM 1 KNN CMC Oracle 1
1 IAverageErrorRate 0.5143 1 0.4952 1 0.4952 0.519 , 0.4905 0.2905 0.1619
1

1

 

 

 

 

 

1 1

, DStandard Deviation 0.1266 L00751 1 0.0602 0.1064 1 0.1078 0.0853 0.0875 J

 

 

 

 

 

 

 

 

Figure 4.5 Comparing Classifiers’ Error Rate, 10—fold CV in the case of 3-Classes

As predicted before, the error rate in the case of 9-Classes is much higher than in
other cases. The final results of the five classifiers and their combination in the case of 2-
Classes, 3-Classes, and 9-Classes are shown in Table 4.6.

In the case of 9-Classes, l-NN works better than the other classifiers. Final results in
Table 4.6 show that CMC is the most accurate classifier compared to individual
classifiers. In the case of 2-Classes it improved by 5%, in the case of 3-Classes it
improved by 20%, and in the case of 9-Classes it improved by 22%, all in relation to the

best individual classifiers in the corresponding cases.

104

Table 4.6: Comparing the Performance of classifiers, in all cases: 2-Classes, 3-
Classess, and 9-Classes, Using 10-fold Cross-Validation in all cases.

 

 

 

 

 

 

 

 

 

 

 

Error Rate
Classifier Z—Classes 3-Classes 9-Classes
Bayes 0.2364 0.5143 0.77
INN 0.2318 0.4952 0.71
KNN 0.1773 0.4952 0.725
Parzen 0.25 0.519 0.795
MLP 0.2045 0.4905 —
CMC 0.1318 0.2905 0.49
Oracle 0.0818 0.1619 -

 

 

 

 

One important finding is that when our individual classifiers are working well and
each has a high level of accuracy; the benefit of combining classifiers is small. Thus,
CMC has little improvement in classification performance while it has a significant
improvement in accuracy when we have weak learner8 classifiers.

We tried to improve the classification efficiency by stratifying the problems in
relation to their degree of difliculty. By choosing some specific conceptual subsets of the
students’ data, we did not achieve a significant increase in accuracy with this parameter.
In the next section, we explain the results of decision tree classifiers on our data set,

while also discussing the relative importance of student-features and the correlation of

these features with category labels.

4.2.2 Decision Tree-based software

Decision trees have proved to be valuable tools for the description, classification and
generalization of data. Many users find decision trees easy to use and understand. As a

result, users more easily trust decision tree models than they do "black box" models, such

 

8 “ Weak learner” means that the classifier has accuracy only slightly better than chance (Duda et al., 2001)

105

as models produced by neural networks. Many tools and sofiware have been developed to
implement decision tree classification. Lim et al. (2000) has an insightful study about
comparison of prediction accuracy, complexity, and training time of thirty-three
classification algorithms; twenty—two decision trees, nine statistical and two neural
network algorithms are compared on thirty-two data sets in terms of classification
accuracy, training time, and (in the case of trees) number of leaves. In this proposal we
used C5.0, CART, QUEST, and CRUISE software to test tree-based classification. Some
statistical sofiware is employed for multiple linear regression on our data set. First we
have a brief view of the capabilities, features and requirements of these software
packages. Then we gather some of the results and compare their accuracy to non-tree

based classifiers.

4.2.2.1 05.0

Decision tree learning algorithms, for example, ID3, C50 and ASSISTANT (Cestnik
et al., 1987), search a completely expressive hypothesis space and are used to
approximate discrete valued target functions represented by a decision tree. In our
experiments the C50 inductive Ieaming decision tree algorithm was used. This is a
revised version9 of C45 and ID3 (Quinlan 1986, 1993) and includes a number of
additional features. For example, the Boosting option causes a number of classifiers to be
constructed - when a case is classified, all of these classifiers are consulted before making

a decision. Boosting will ofien give a higher predictive accuracy at the expense of

 

9 It is the commercial version of the C45 decision tree algorithm developed by Ross Quinlan. See5/C5.0
classifiers are expressed as decision trees or sets of if-then rules. RuleQuest provides C source code so that
classifiers constructed by See5/C5.0 can be embedded in your own systems.

106

increased classifier construction time. For our experiments, however, data set boosting

was not found to improve prediction accuracy.

When a continuous feature is tested in a decision tree, there are branches
corresponding to the conditions: “Feature Value 5 Threshold " and “Feature Value >
Threshold” for some threshold chosen by C50. As a result, small movements in the
feature value near the threshold can change the branch taken from the test. There have
been many methods proposed to deal with continuous features (Quinlan, 1988; Chan et
al., 1992; Ching et al., 1995). An option available in C50 uses fuzzy thresholds to soften
this knife-edge behavior for decision trees by constructing an interval close to the
threshold. This interval plays the role of margin in neural network algorithms. Within this
interval, both branches of the tree are explored and the results combined to give a

predicted class.

Decision trees constructed by C50 are post pruned before being presented to the
user. The “Pruning Certainty Factor " governs the extent of this simplification. A higher
value produces more elaborate decision trees and rule sets, while a lower value causes
more extensive simplification. In our experiment a certainty factor of 25% was used. If
we change the certainty factor, we may obtain different results.

C5.0 needs four types of files for generating the decision tree for a given data set, out
of which two files are Optional:

The first file is the .names file. It describes the attributes and classes. The first line of
the .names file gives the classes, either by naming a discrete attribute (the target attribute)

that contains the class value, or by listing them explicitly. The attributes are then defined

107

in the order that they will be given for each case. The attributes can be either explicitly or
implicitly defined. The value of an explicitly defined attribute is given directly in the
data. The value of an implicitly-defined attribute is specified by a formula. In our case,
data attributes are explicitly defined.

The second file is the .data file. It provides information on the training cases from
which C5.0 will extract patterns. The entry for each case consists of one or more lines
that give the values for all explicitly defined attributes. The '?' is used to denote a value
that is missing or unknown. Our data set had no missing features. Also, 'N/A' denotes a
value that is not applicable for a particular case.

The third file used by C50 consists of new test cases on which the classifier can be
evaluated and is the .test file. This file is optional and, if used, has exactly the same
format as the .data file. We gave a .test file for our data set.

The last file is the .costs file. In applications with differential misclassification costs,
it is sometimes desirable to see what affect costs have on the construction of the
classifier. In our case all misclassification costs were the same so this option was not
implemented.

Afier the program was executed on the PHY183 8802 data set we obtained results
for both the training and testing data. A confusion matrix was generated in order to show
the misclassifications. The confusion matrices for three types of classification in our data
set that are, 2-Classes, 3-Classes and 9-Classes are in Appendix A. You can also find
some rule set samples resulted from the rule-set option in C50, as well as a part sample

of the tree produced by C50 in Appendix A.

108

Using lO-fold cross validation we got 79.3% accuracy in 2-Classes, 56.8%
accuracy in 3-Classes, 25.6% accuracy in 9-Classes.

One of the important and exciting experiments of C50 is to use a training, and test
set; and thus at least a 10—fold cross-validation to get a desirable result. For example, in
the case of 3-Classes, we might get approximately 75% accuracy in the training set, while
boosting might improve accuracy up to 90% or 95%. Unfortunately, this is overfltting, or
overtraning, and so we therefore would not be able to generalize these results or this

complex training model to test the unseen data because of overfitting.
4.2.2.2 CART

CART10 uses an exhaustive search method to identify useful tree structures of data. It
can be applied to any data set and can proceed without parameter setting. Comparing
CART analyses with stepwise logistic regressions or discriminant analysis, CART
typically performs better on the Ieaming sample. Listed below are some technical aspects
of CART:

CART is a nonparametric procedure and does not require specification of a
functional form. CART uses a stepwise method to determine splitting rules, and thus no
advance selection of variables is necessary, although certain variables such as ID
numbers and reformulations of the dependent variable should be excluded fi'om the

analysis. Also, CART’s performance can be enhanced by proper feature selection and

 

10 CARD“) (Classification And Regression Trees) is a data mining tool exclusively licensed to Salford
Systems (http://www.salford-systems.com). CART is the implementation of the original program by
Breiman, Friedman, Olshen, and Stone. We used CART version 5.02 under windows for our classification.
Using CART we are able to get many interesting textual and graphical reports, some of which are presented
in Appendix A. It is noticeable that CART does not use any description files to work with. Data could be
read as a text file or any popular database or spreadsheet.

109

creation of predictor variables. There is no need to experiment with monotone
transformations of the independent variables, such as logarithms, square roots or squares.
In CART, creating such variables will not affect the resulting trees unless linear
combination splits are used. Outliers among the independent variables generally do not
affect CART because splits usually occur at non-outlier values. Outliers in the dependent
variable are often separated into nodes where they no longer affect the rest of the tree.
CART does not require any preprocessing of the data. In particular, continuous variables
do not have to be recoded into discrete variable versions prior to analysis. While the
CART default is to split nodes on single variables, it will optionally use linear
combinations of non-categorical variables. For each split in the tree, CART develops
alternative splits (surrogates), which can be used to classify an object when the primary
splitting variable is missing. Thus, CART can be used effectively with data that has a
large fraction of missing values.

One of the advantages of CART is presenting the importance of independent
variables in predicting both classification mode and regression mode. Each variable in the
CART tree has an importance score based on how often and with what significance it
served as primary or surrogate splitter throughout the tree. The scores reflect the
contribution each variable makes in classifying or predicting the target variable, with the
contribution stemming from both the variable’s role in primary splits and its role as a
surrogate splitter (Dan and Colla, 1998). In Table 4.7 and 4.8, the importance of the six
features (independent variables) are scored in the case of 2-classes with Gini splitting

criterion and 3-classes with Entropy splitting criterion respectively.

110

Table 4.7 Variable (feature) Importance in 2—Classes Using Gini Criterion

 

 

 

 

 

 

 

 

Variable
TOTCORR 100.00 111111111111111111111111111111111111111111
TRIES 56.32 11111111111111111111111
FIRSTCRR 4.58 1
TOTTIME 0.91
SLVDTIME 0.83
DISCUSS 0.00

 

 

 

Table 4.8 Variable (feature) Importance in 2-Classes, Using Entropy Criterion

 

 

 

 

 

 

 

 

 

Variable
TOTCORR 100-00 “1111111111111111111111111111111111111111
TRIES 58.61 111111111111111111111111
FIRSTCRR 27.70 11111111111
SLVDTIME 24.60 1111111111
TOTTIME 24.47 1111111111
DISCUSS 9.21 111

 

 

 

 

The results in Tables 4.7 and 4.8 show that the most important feature (which has the
highest correlation with the predicted variables) is the “Total number of Correct
answers,” and the least useful variable is the “Number of Discussions. ” If we consider the

economical aspect of computation cost, we can remove the less important features.

In our experiment, we used both lO-fold Cross-Validation and Leave-One-Out
method. We found that the error rates in training sets are not improved in the case of 2-
Classes and 3-Classes, but the misclassifications in the test sets are improved when we
switch from 10-fold Cross-Validation to Leave-One-Out. Yet, in the case of 9-Classes
both training and testing sets are improved significantly when switching occurs, as is
shown in Table 4.9 and 4.10. How can we interpret improvement variation between

classes?

lll

Table 4.9: Comparing the Error Rate in CART, using 10-fold Cross-Validation
in learning and testing set.

 

 

 

 

 

 

 

 

 

 

Splitting Criterion 2-Classes 3-Classes 9-Classes
Training Testing Training Testing Training Testing
Gini 17.2% 19.4% 35.2% 48.0% 66.0% 74.5%
Symmetric Gini 17.2% 19.4% 35.2% 48.0% 66.0% 74.5%
Entropy 18.9% 19.8% 37.9% 52.0% 68.7% 76.2%
Twoing 17.2% 19.4% 31.3% 47.6% 54.6% 75.3%
Ordered Twoing 17.2% 20.7% 31.7% 48.0% 68.3% 74.9%

 

 

 

 

 

Table 4.10: Comparing the Error Rate in CART, using Leave-One-Out method
in learning and testing test.

 

 

 

 

 

 

 

 

Splitting Criterion 2-Classes 3-Classes 9-Classes
Training Testing Training Testing Training Testing
Gini 17.2% 18.5% 36.6% 41.0% 46.7% 66.9%
Symmetric Gini 17.2% 18.5% 36.6% 41.0% 46.7% 66.9%
Entropy 17.2% 18.9% 35.2% 41.4% 48.0% 69.6%
Twoing 17.2% 18.5% 38.3% 40.1% 47.1% 68.7%
Ordered Twoing 18.9% 19.8% 35.2% 40.4% 33.9% 70.9%

 

 

 

 

 

 

 

Discussion of improvement variation:

In the case of 2-Classes, there is no improvement in the training phase and a slight

(1%) improvement in the test phase. It shows that we can have a more reliable model

with the Leave-One-Out method for student classification. It shows that the model we

obtained in training phase is approximately complete.

In the case of 3-Classes, when we switch from 10-fold to Leave-One-Out the results

in the training phase become slightly worse, but we achieve approximately 7.5%

improvement. It shows that our model was not complete in 10-fold for predicting the

unseen data. Therefore, it is better to use Leave-One-Out to get a more complete model

for classifying the students into three categories.

112

 

 

In the case of 9-Classes when we switch from 10—fold to Leave-One-Out, the results
in both training and test sets improve significantly. However, we cannot conclude that
our new model is complete. This is because the big difference between the results in
training and testing phase shows that our model is suffering from over/itting. It means
that our training samples are not enough to construct a complete model for predicting the
category labels correctly; more data is required to come to an adequate solution.

After discussing the CART results it is worth noting that by using CART we are able
to produce many useful textual and graphical reports, some of which are presented in
Appendix A. In next section we discuss the other decision tree software results. One of
the advantages of CART is that it does not require any description files; therefore data

can be read as a text file or any popular database or spreadsheet file format.

4.2.2.3 QUEST, CRUISE11

QUEST is a statistical decision tree algorithm for classification and data mining. The
objective of QUEST is similar to that of the algorithm used in CART and described in
Breiman, et al. (1984). The advantages of QUEST are its unbiased variable selection
technique by default, its use of imputation instead of surrogate splits to deal with missing
values, and its ability to handle categorical predictor variables with many categories. If
there are no missing values in the data, QUEST can use the CART greedy search

algorithm to produce a tree with univariate splits.

 

1 l QUEST (Quick, Unbiased and Efficient Statistical Tree) A classification tree restricted to binary splits
CRUISE (Classification Rule with Unbiased Interaction Selection and Estimation) A classification tree
that splits each node into two or more sub-nodes. These new software were developed by Wei-Yin Loh at
the University of Wisconsin-Madison, Shih at University of Taiwan, and Hyunjoong Kim at University of
Tennessee,. vastly-improved descendant of an older algorithm called FACT.

113

QUEST needs two text input files: I) Data file: This file contains the training
samples. Each sample consists of observations on the class (or response or dependent)
variable and the predictor (or independent) variables. The entries in each sample record
should be comma or space delimited. Each record can occupy one or more lines in the
file, but each record must begin on a new line. Record values can be numerical or
character strings. Categorical variables can be given numerical or character values. 2)
Description file: This file is used to provide information to the program about the name
of the data file, the names and the column locations of the variables, and their roles in the
analysis.

The following is the description file:

phy183.dat

n?"

column, var, type
1 lstGotCrr n
2 TotCorr n
3 Angries n
4 TimeCorr n
5 TimeSpent n
6 Discuss n
7 ClassZ
8 Class3
9 Class9
10 Grade

><><Cl><

In the first line, we put the name of data file (ph y183 . da t), and in the second line we
put the character used to denote missing data (2). In our data set we have no missing data.
The position (column), name (var) and role (type) of each variable follow, with one line
for each variable. The following roles for the variables are permitted: “c” stands for
categorical variable; “(1” for class (dependent) variable; only one variable can have the d
indicator; “n” for a numerical variable; and “x” which indicates that the variable is

excluded from the analysis.

114

QUEST allows both interactive and batch mode. By default it uses “discriminant
analysis” as a method for split point selection. It is an unbiased variable selection method
described in Loh and Shih (1997). However, in advanced mode, the user can select
“exhaustive search” (Breiman et al., 1984) which is used in CART. The former is the
default option if the number of classes is more than 2, otherwise the latter is the default
option. If the latter option is selected, the program will ask for the user to choose the
splitting criterion including one of the following five methods which are studied in Shih
(1999):

1 Likelihood Ratio 6‘2

2 Pearson cn1‘2

3 Gini

4 MPI (Mean Posterior Improvement)

5 Other members of the divergence family

The likelihood criterion is the default option. If instead the CART-style split is used,
the Gini criterion is the default option. In our case, we selected the fifth method of
exhaustive search which was optimal regarding the misclassification ratio.

QUEST asks for the prior for each class. If the priors are to given, the program will
then ask the user to input the priors. If unequal costs are present (like in this example),
the priors are altered using the formula in Breiman et al. (1984, pp. 114-115). In our cases
the prior for each class is estimated based on the class distribution. This asks for the
misclassification costs. If the costs are to be given, the program will ask the user to input
the costs. In our cases the misclassification costs are equal. The user can choose either
split on a single variable or linear combination of variables. We used split on a single

variable.

115

QUEST also asks for the number of SEs which controls the size of the pruned tree.
O-SE gives the tree with the smallest cross-validation estimate of misclassification cost or
error. QUEST enables user to select the value of V in V-fold cross-validation. The larger
the value of V, the longer running time the program takes to run. 10-fold and 226-fold
(Leave-One-Out) are used in our cases.

The classification matrices based on the learning sample and CV procedure are
reported. Some samples of these reports are shown in Appendix A. You can see a table
gives the sequence of pruned subtrees. The 3rd column shows the cost complexity value
for each subtree by using the definition in Breiman et al. (1984, Definition 3.5 p. 66). The
4th column gives the current or re-substitution cost (error) for each subtree. Another table
gives the size, estimate of misclassification cost and its standard error for each pruned
sub-tree. The 2nd column shows the number of terminal nodes. The 3rd column shows
the mean cross-validation estimate of misclassification cost and the 4th column gives its
estimated standard error using the approximate formula in Breiman et a1. (1984, pp. 306-
309). The tree marked with an “’1‘” is the one with the minimum mean cross-validation
estimate of misclassification cost (also called the O—SE tree). The tree based on the mean
cross-validation estimate of misclassification cost and the number of SEs is marked with
“**” (See Appendix A).

QUEST trees are given in outline form suitable for importing into flowchart
packages like alICLEAR (CLEAR Software, 1996). Alternatively, the trees may be
outputted in LaTeX code. The public domain macro package pstricks (Goossens, Rahtz
and Mittelbach, 1997) or TreeTEX (Bruggemann-Klein and Wood, 1988) is needed to

render the LaTeX trees.

116

CRUISE is also a new statistical decision tree algorithm for classification and data
mining. It has negligible bias in variable selection. It splits each node into as many sub-
nodes as the number of classes in the response variable It has several ways to deal with
missing values. It can detect local interactions between pairs of predictor variables.

CRUISE has most of QUEST capabilities and reports (See Appendix A). We have
brought the results of tree-based classification with QUEST and the CRUISE into the
final reporting table (Table 4.11), which includes all tree-based and non-tree based

classifiers on our data set in the cases of 2-Classes, 3-Classes, and 9-Classes.

117

4.2.3 Final Results without optimization

The overall results of classifiers’ performance on our data set are shown in the Table
4.11. Regarding individual classifier, for the case of 2-classes, kNN has the best
performance with 82.3% accuracy. In the case of 3-classes and 9-classes, CART has the
best accuracy of about 60% in 3-classes and 43% in 9-Classes. However, considering the
combination of non-tree-based classifiers, the CMC has the best performance in all three
cases. That is we got the 86.8% accuracy in the case of 2-Classes, 71% in the case of 3-

Classes, and 51% in the case of 9-Classes.

Table 4.11: Comparing the Error Rate of all classifiers on PHY183 data set in
the cases of 2-Classes, 3-Classes, and 9-Classes, using IO—fold cross-validation,

 

 

 

 

 

 

 

 

Without Optimization
Error Rate
Classifier 2-Classes 3-Classes 9-Classes

C5.0 20.7% 43.2% 74.4%
Tree CART 1 8.5% 40.1% 66.9%
Classifier QUEST 19.5% 42.9% 80.0%
CRUISE 19.0% 45.1% 77.1%
Bayes 23.6% 51.4% 77.0%
INN 23.2% 49.5% 71.0%
Non- tree kNN 17.7% 49.6% 72.5%
Classifier Parzen 25.0% 51.9% 79.5%

MLP 20.5% 49.1% -
CMC 13.2% 29.1% 49.0%

 

 

 

 

 

 

 

 

 

 

 

 

 

So far, we have grouped students using multiple classifiers, comparing their
prediction accuracy with CMC. In the next section we will study a way to optimize these

results in order to find more efficient and more accurate classifiers.

118

4.3 Optimizing the prediction accuracy

We found that a combination of multiple classifiers leads to a significant
improvement in classification performance. Through weighting the feature vectors using
a Genetic Algorithm we can optimize the prediction accuracy and get a marked
improvement over raw classification. We further show that when the number of features

is few; feature-weighting works better than just feature selection.

4.3.1 Genetic Algorithms (GAs)

This Ieaming procedure can be considered a search through a space of data points. A
genetic algorithm presents a powerful alternative to traditional search techniques. It has
been inspired by a similar predictive model in nature. Evolution in nature is controlled
through the following principles:

Natural Selection: the strongest specimens have the highest chance to survive and
reproduce, while the weak ones are likely to die before the reproduction stage.

Reproduction: the fittest specimens recombine their genetic information, thus
creating new specimens with somewhat new characteristics.

Mutation leads to random changes in genetic information.

The success of this “search technique” in nature inspired some researchers to
propose methods and develop algorithms that could be encoded in computer programs. A
clear and simple introduction to the discipline of genetic algorithm has been made by
Goldberg (1989). To start casting a real problem in a setting where its solution is can be
obtained through a genetic algorithm, two important steps are taken: encoding the search

space into chromosomes (a string of binary/real values); and defining a fitness function

119

that plays the role of an evaluation function in a heuristic search. The implementation of

a chromosome is typically in the form of bit strings.

4.3.1.1 What is a Simple GA (SGA)?

The main steps of a GA are reproduction, recombination, and mutation in the
following algorithm (Michalski et al., 1998):

1. Construct the initial population as a set of binary strings generated randomly
or by some pre-specified mechanisms.

2. Replicate the specimen in the population into a set of survivors by a
mechanism that ensures that specimens with a higher fitness value have a
higher chance of survival.

3. Pair up all the survivors, such that each has a mate. Next specific chunks of
encoded data are swapped between mates. Mutation arises when a single bit
flip-flops.

4. If the fitness function has not improved through several cycles, stop;

otherwise go to step 2.

4.3.1.2 Specific use of GAs in pattern classification

Genetic Algorithms have been shown to be an effective tool to use in data mining
and pattern recognition (Freitas, 2002; Jain and Zongker, 1997; F alkenauer, 1998; Pei et
al., 1997; Park and Song, 1998; Michalewicz, 1996; De Jong et al., 1993). An important
aspect of GAS in a Ieaming context is their use in pattern recognition. There are two

different approaches to applying GA in pattern recognition:

120

1. Apply a GA directly as a classifier. Bandyopadhyay and Murthy (1995)
applied GA to find the decision boundary in N dimensional feature space.

2. Use a GA as an optimization tool for resetting the parameters in other
classifiers. Most applications of GAs in pattern recognition optimize some
parameters in the classification process.

Many researchers have used GAs in feature selection (Bala et al. 1997; Guerra-
Salcedo and Whitley 1999, Vafaie and De Jong, 1993; Martin-Bautista and Vila, 1999).
GAs have been applied to find an optimal set of feature weights that improve
classification accuracy. First, a traditional feature extraction like Principal Component
Analysis (PCA) is applied, and then a classifier like k-NN is used to calculate the fitness
function for GA (Seidlecki, 1989; Pei et al., 1998). Combined classifiers are another area
that GAs have been used to optimize. Kuncheva and Jain (2000) used a GA for selecting
the features as well as selecting the types of individual classifiers in their design of a
Classifier Fusion System. GA is also used in selecting the prototypes in the case-based
classification (Skalak, 1994).

In this work we focus on the second approach and use a GA to optimize a
combination of classifiers. Our objective is to predict the students’ final grades based on
their web-use features, which are extracted from the homework data. We design,
implement, and evaluate a series of pattern classifiers with various parameters in order to
compare their performance on a data set from LON-CAPA. Error rates for the individual

classifiers, their combination and the GA optimized combination are presented.

121

4.3.2 Implementation of a GA to optimize the prediction accuracy

We use the GAToolBoxl2 from MATLAB to implement a GA to optimize
classification performance. Our goal is to find a population of best weights for every
feature vector, which minimize the classification error rate.

The feature vector for our predictors are the set of six variables for every student:
Success rate, Success at the first try, Number of attempts before correct answer is derived,
the time at which the student got the problem correct relative to the due date, total time
spent on the problem, and the number of online interactions of the student both with other
students and with the instructor.

We randomly initialize a population of six dimensional weight vectors with values
between 0 and 1, corresponding to the feature vector and experimented with different
number of population sizes. We obtained good results using a population with 200
individuals. The GA Toolbox supports binary, integer, real-valued and floating-point
chromosome representations. Real-valued populations may be initialized using the
toolbox function crtrp. For example, to create a random population of 6 individuals with
200 variables each: we define boundaries on the variables in F ieldD which is a matrix

containing the boundaries of each variable of an individual.

FieldD = [ 0 0 0 0 0 0; % lower bound

1 1 1 l 1 1]; % upper bound

An initial population created with Chrom = crtrp (200, FieldD) , An example is as

follows:

 

12 Downloaded from http://www.shcf.ac.uk/~gaipp/ga-toolbox/
1 22

Chrom = 0.23 0.17 0.95 0.38 0.06 0.26
0.35 0.09 0.43 0.64 0.20 0.54
0.50 0.10 0.09 0.65 0.68 0.46

0.21 0.29 0.89 0.48 0.63 0.89

We use the simple genetic algorithm (SGA), which is described by Goldberg (1989).

4.3.2.1 GA Operators

The SGA uses common GA operators to find a population of solutions which

optimize the fitness values.

4.3.2.1.1 Recombination

We use “Stochastic Universal Sampling” (Baker, 1987) as our selection method. A
form of stochastic universal sampling is implemented by obtaining a cumulative sum of
the fitness vector, F ith, and generating N equally spaced numbers between 0 and
sum(Fith). Thus, only one random number is generated, all the others used being
equally spaced from that point. The index of the individuals selected is determined by
comparing the generated numbers with the cumulative sum vector. The probability of an

individual being selected is then given by

 

Fm): f(X.-)

N ind

2 f(x.-)

i = 1
where f(x,-) is the fitness of individual x,- and F (xi) is the probability of that individual

being selected.

123

4.3.2.1.2 Crossover

The crossover operation is not necessarily performed on all strings in the population.
Instead, it is applied with a probability Px when the pairs are chosen for breeding. We
select Pr = 0.7. There are several functions to make crossover on real-valued matrices.
One of them is recint, which performs intermediate recombination between pairs of
individuals in the current population, OldChrom, and returns a new population after
mating, NewChrom. Each row of OldChrom corresponds to one individual. recint is a
function only applicable to populations of real-value variables. Intermediate
recombination combines parent values using the following formula (Muhlenbein and
Schlierkamp-Voosen, 1993).

Offspring = parentl + Alpha X (parent2 — parentl)

Alpha is a Scaling factor chosen uniformly in the interval {-0.25, 1.25]

4.3.2.1.3 Mutation

A further genetic operator, mutation is applied to the new chromosomes, with a set
probability Pm. Mutation causes the individual genetic representation to be changed
according to some probabilistic rule. Mutation is generally considered to be a background
operator that ensures that the probability of searching a particular subspace of the
problem space is never zero. This has the effect of tending to inhibit the possibility of
converging to a local optimum, rather than the global optimum.

There are several functions to make mutation on real-valued population. We used

mutbga, which takes the real-valued population, OldChrom, mutates each variable with

124

given probability and returns the population after mutation, NewChrom =
mutbga(OldChrom, F ieldD, MutOpt) takes the current population, stored in the matrix
OldChrom and mutates each variable with probability by addition of small random
values (size of the mutation step). We considered 1/600 as our mutation rate. The

mutation of each variable is calculated as follows:

 

 

Mutated Var = Var + Muth X range X MutOpt(2) X delta

 

where delta is an internal matrix which specifies the normalized mutation step size;
Muth is an internal mask table; and MutOpt specifies the mutation rate and its
shrinkage during the run. The mutation operator mutbga is able to generate most points
in the hypercube defined by the variables of the individual and the range of the mutation.
However, it tests more often near the variable, that is, the probability of small step sizes

is greater than that of larger step sizes.

4.3.2.2 Fitness Function

During the reproduction phase, each individual is assigned a fitness value derived
from its raw performance measure given by the objective fiinction. This value is used in
the selection to bias towards more fit individuals. Highly fit individuals, relative to the
whole population, have a high probability of being selected for mating whereas less fit
individuals have a correspondingly low probability of being selected. The error rate is
measured in each round of cross validation by dividing “the total number of misclassified
examples” into “total number of test examples”. Therefore, our fitness function measures
the error rate achieved by CMC and our objective would be to maximize this

performance (minimize the error rate).

125

 

4.3.3 Experimental Results of GA Optimization

For GA optimization, we used 200 individuals in our population, running the GA
over 500 generations. We ran the program 10 times and got the averages, which are
shown, in Table 4.12. In every run 500x200 times the fitness function is called in which
we used lO-fold cross validation to measure the average performance of CMC. So every
classifier is called 3x106 times for the case of 2-classes, 3-classes and 9-classes. Thus,
the time overhead for fitness evaluation is critical. Since using the MLP in this process
took about 2 minutes and all other four non-tree classifiers (Bayes, INN, 3NN, and
Parzen window) took only 3 seconds, we omitted the MLP from our classifiers group so
we could obtain the results in a reasonable time.

Figures 4.6-4.8 shows the best result of the ten runs over our data set. These figures
represent the population mean, the best individual at each generation and the best value
yielded by the run. The results in Table 4.12 represent the mean performance with a two-
tailed t-test with a 95% confidence interval. For the improvement of GA over non-GA
result, a P—value indicating the probability of the Null-Hypothesis (There is no

improvement) is also given, showing the significance of the GA optimization.

126

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Best = 0.7%
Mean: 0.72014

 

Figure 4.7. Graph of GA Optimized CMC performance in the case of 3-Classes

127

LON-CAPA' PHY183 Classification Optimization, 9 Classes (2CD individuals)

 

 

 

 

 

 

 

 

 

 

68 I] j
l
64 . {—1 d
82 " .. .4 kl- V .a \,-_I ' ‘.‘ ”"v\("~\‘." . '?'.(N‘i.w‘.,,o'“§ '.‘A'ilv' l.“ :13
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56 -} d
541 Best =055 — Bes‘ —
Mean=062403 Mean
52 L 1 l 1 1 1 n 1 1
0 50 100 150 2011 250 3(1) 350 403 450 500

Generation

Os

Figure 4.8. Graph of GA Optimized CMC performance in the case of 9-Classes

Table 4.12. Comparing the CMC Performance on PHY183 data set Using GA
and without GA in the cases of 2-Classes, 3-Classess, and 9—Classes, 95% confidence

 

 

 

 

 

 

interval.
Performance %

Classifier 2-Classes 3-Classes 9-Classes

CMC 0f. f°“’ Class‘fie’s 83.87 i 1.73 61.86 i 2.16 49.74 i: 1.86
wrthout GA
GA 01"?qu CMC: Mean 94.09 :1: 2.84 72.13 i 0.39 62.25 i 0.63
mdrvrdual

Improvement 10.22: 1.92 10.26i 1.84 12.51 i 1.75

 

 

 

 

All have p<0.000, indicating significant improvement. Therefore, using GA, in all
the cases, we got more than a 10% mean individual performance improvement and about
12 to 15% mean individual performance improvement. Figure 4.9 shows the graph of
average mean individual performance improvement.

128

 

 

 

I CMC Performance without GA :1 GA Optimized CMC '

 

 

 

 

 

 

 

 

 

CMC Performance

 

 

 

 

 

 

 

 

 

 

 

 

2-Classes 3-Classes 9-Classes
Students' Classes

 

 

Figure 4.9. Char t of comparing CMC average performance, using GA and
without GA.

Finally, we can examine the individuals (weights) for features by which we obtained
the improved results. This feature weighting indicates the importance of each feature for
making the required classification. In most cases the results are similar to Multiple Linear
Regressions or tree-based software that use statistical methods to measure feature
importance. Table 4.13 shows the importance of the six features in the 3-classes case
using the Entropy splitting criterion. Based on entropy, a statistical property called
information gain measures how well a given feature separates the training examples in
relation to their target classes. Entropy characterizes impurity of an arbitrary collection
of examples S at a specific node N. In Duda et al. (2001) the impurity of a node N is

denoted by i(N).

Entropy(s)=i(1v)= ‘2 P(a)j)log , 10(0),.)

129

where P(a)j.) is the fraction of examples at node N that go to category co].

Table 4.13. Feature Importance in 3-Classes Using Entropy Criterion

 

 

 

 

 

 

 

 

Feature Importance %
Total_Correct gAnswers l 00.00
Tota1_Number of Submissions 58.61
First;Got;Correct 27.70
Time_SpentAto_§olve 24.60
Total TimeASpent 24.47
Communication 9.2 1

 

 

 

The GA results also show that the “Total number of correct answers” and the “Total
number of submissions” are the most important features for classification accuracy; both
are positively correlated to the true class labels. The second column in Table 4.13 shows
the percentage of feature importance. One important finding is that GAs determine
optimal weights for features. Using this set of weights we can extract a new set of
features which significantly improve the prediction accuracy. In other words, this

resultant set of weights transforms the original features into new salient features.

130

4.4 Extending the work toward more LON-CAPA data sets

We selected 14 student/course data sets of MSU courses, which used LON-CAPA as

shown in Table 4.14 and 4.15.

Table 4.14. 14 of LON-CAPA courses at MSU

 

 

Course Term Title
ADV 205 8803 Principles of Advertising
BS 111 8802 Biological Science: Cells and Molecules
BS 111 8803 Biological Science: Cells and Molecules
CE 280 8803 Civil Engineering: Intro Environment Eng.
F I 414 8803 Advanced Business Finance (w)
LBS 271 F802 Lyman Briggs School: Physics I
LBS 272 8803 Lyman Briggs School: Physics 11
MT 204 8803 Medical Tech.: Mechanisms of Disease
MT 432 8803 Clinic Immun. & Immunohematology
PHY 183 8802 Physics Scientists & Engineers I
PHY 183 8803 Physics Scientists & Engineers I
PHY 231c 8803 Introductory Physics I
PHY 232C F803 Introductory Physics 11

PHY 232 F803 Introductory Physics 11

131

Table 4.15 Characteristics of 14 of MSU courses, which held by LON-CAPA

 

 

Course Nursiber Number of Size of Size of Number of
Students Problems Activrty log useful data Transactions
ADV205_SSO3 609 773 82.5 MB 12.1 MB 424,481
B8111_8802 372 229 361.2 MB 34.1 MB 1,112,394
881 l l_8803 402 229 367.6 MB 50.2 MB 1,689,656
CE280_8803 178 19 6 28.9 MB 3.5 MB 127,779
FI414_8803 169 68 16.8 MB 2.2 MB 83,715
LB8271_F 802 132 174 119.8 MB 18.7 MB 706,700
LB8272_8803 102 166 73.9 MB 15.3 MB 585,524
MT204_8803 27 150 5.2 MB 0.7 MB 23,741
MT432__8803 62 150 20.0 MB 2.4 MB 90,120
PHY183_8802 227 184 140.3 MB 21.3 MB 452,342
PHY183_8803 306 255 210.1 MB 26.8 MB 889,775
PHY231c_SSO3 99 247 67.2 MB 14.1 MB 536,691
PHY232c_8803 83 194 55.1 MB 10.9 MB 412,646
PHY232_F803 220 259 138.5 MB 19.7 MB 981,568

 

For example, the third row of the Table 4.16 shows that B8111 (Biological Science:
Cells and Molecules) was held in spring semester 2003 and contained 229 online
homework problems, and 402 students used LON-CAPA for this course. The B8111
course had an activity log with approximately 368 MB. Using some Perl script modules
for cleansing the data, we found 48 MB of useful data in the B8111 8803 course. We
then pulled from these logged data 1,689,656 transactions (interactions between students

and homework/exam/quiz problems) from which we extracted the following nine features

(Having revised six features that were explained in 4.1):

1. Number of attempts before correct answer is derived

2. Total number of correct answers

3. Success at the first try

4. Getting the problem correct on the second try

5. Getting the problem correct between 3 and 9 tries

132

6. Getting the problem correct with a high number of tries (10 or more tries).
7. Total time that passed from the first attempt, until the correct solution was
demonstrated, regardless of the time spent logged in to the system
8. Total time spent on the problem regardless of whether they got the correct answer
or not
9. Participating in the communication mechanisms
Based on the above extracted features in each course, we classify the students, and
try to predict for every student to which class he/she belongs. We categorize the students
with one of two class labels: “Passed" for grades higher than 2.0, and ”Failed” for grades
less than or equal to 2.0 where the MSU grading system is based on grades from 0.0 to

4.0. Figure 4.10 shows the grade distribution for the B8111 fall semester 2003.

 

3.5

2.5

Grades

1.5

0.5

 

 

 

1 I

o 10 20 3o 40 50 6'0

# of Students

Figure 4.10. LON-CAPA: BSIII SS03, Grades distribution

133

4.4.1 Experimental Results

Without using GA, the overall results of classification performance on our datasets
for four classifiers and classification fusion are shown in the Table 4.16. Individual
classifiers, INN and kNN have mostly the best performance. However, the classification
fusion improved the classification accuracy significantly in all data sets. That is, it

achieved in average 79% accuracy over the given data sets.

Table 4.16 Comparing the average performance% of ten runs of classifiers on
the given datasets using 10-fold cross validation, without GA

 

Data sets Bayes INN kNN Parzen Classrficatron

 

Window Fusion

ADV 205, 03 55.7 69.9 70.7 55.8 78.2
BS 111, 02 54.6 67.8 69.6 57.6 74.9
BS 111, 03 52.6 62.1 55.0 59.7 71.2
CE 280, 03 66.6 73.6 74.9 65.2 81.4
F1414, 03 65.0 76.4 72.3 70.3 82.2
LBS 271, 02 66.9 75.6 73.8 59.6 79.2
LBS 272, 03 72.3 70.4 69.6 65.3 77.6
MT 204, 03 63.4 71.5 68.4 56.4 82.2
MT 432, 03 67.6 77.6 79.1 59.8 84.0
PHY 183, 02 73.4 76.8 80.3 65.0 83.9
PHY 183, 03 59.6 66.5 70.4 54.4 76.6
PHY 231c, 03 56.7 74.5 72.6 60.9 80.7
PHY 232C, 03 65.6 71.7 75.6 57.8 81.6
PHY 232, 03 59.9 73.5 71.4 56.3 79.8

 

For GA optimization, we used 200 individuals (weight vectors) in our population,
running the GA over 500 generations. We ran the program 10 times and got the averages,

which are shown, in Table 4.17.

134

Table 4.17 Comparing the classification fusion performance on given datasets,
without-GA, using-GA (Mean individual) and improvement, 95% confidence

 

 

 

interval
Data sets Without GA GA gptimized Improvement
ADV 205, 03 78.19i 1.34 89.11 i 1.23 10.92 i 0.94
BS 111, 02 74.93 i 2.12 87.25 i 0.93 12.21 i 1.65
B8111,03 71.19i'l.34 81.09i2.42 9.82i1.33
CE 280, 03 81.43i2.13 92.61i2.07 11.36i 1.41
F1414, 03 82.24i1.54 91.73i1.21 9.50i 1.76
LBS 271, 02 79.23 i 1.92 90.02 i 1.65 10.88 i 0.64
LBS 272, 03 77.56i 0.87 87.61 i 1.03 10.11 i 0.62
MT 204, 03 82.24i 1.65 91.93 i 2.23 9.96 i 1.32
MT 432,03 84.03i2.13 95.21i 1.22 11.16i 1.28
PHY 183, 02 83.87i 1.73 9409—1 2.84 10.22 11‘- 1.92
PHY183, 03 76.56i 1.37 87.14i 1.69 9.36i 1.14
PHY 231C, 03 80.67311 1.32 91.41 i 2.27 10.74i 1.34
PHY 232C, 03 81.55 i 0.13 92.39i 1.58 10.78 i 1.53
PHY 232, 03 79.771: 1.64 88.61 i 2.45 9.13 i 2.23
Total Average 78.98 i 12 90.03 i 1.30 10.53 i 56

 

The results in Table 4.17 represent the mean performance with a two-tailed t-test
with a 95% confidence interval for every data set. For the improvement of GA over non-
GA result, a P-value indicating the probability of the Null-Hypothesis (There is no
improvement) is also given, showing the significance of the GA optimization. All have
p<0.000, indicating significant improvement. Therefore, using GA, in all the cases, we
got approximately more than a 10% mean individual performance improvement and
about 10 to 17% best individual performance improvement. Fig. 4.11 shows the results of
one of the ten runs in the case of 2-Classes (passed and failed). The doted line represents
the population mean, and the solid line shows the best individual at each generation and

the best value yielded by the run.

135

LON-CAPA' 88111 Classification Optimization, 2 Classes (2E1) individuals)

 

 

CMC Performance

 

 

 

 

 

 

 

 

75% 1
, Mean=080535 __ Best
741' Best=0.82353 Mean 7
73 4 L I I l I l I l
0 50 11]] 150 200 250 300 350 4(1) 450 500
Generation

Figure 4.11 GA-Optimized Combination of Multiple Classifiers’ (CMC) performance in the case of
2-Class labels (Passed and Failed) for 881 II 2003, 200 weight vectors individuals, 500 Generations

Finally, we can examine the individuals (weights) for features by which we obtained
the improved results. This feature weighting indicates the importance of each feature for
making the required classification. In most cases the results are similar to Multiple Linear
Regressions or some tree-based software (like CART) that use statistical methods to
measure feature importance. The GA feature weighting results, as shown in Table 4.18,
state that the “Success with high number of tries” is the most important feature. The
“Total number of correct answers” feature is also the most important in some cases; both
are positively correlated to the true class labels.

If we use one course as the training data and another course as the test data we again
achieve a significant improvement in prediction accuracy for both using the combination
of multiple classifiers and applying genetic algorithms as the optimizer. For example,

using B8111 from fall semester 2003 as the training set and PHY231 from spring

136

semester 2004 as the test data, and using the weighted features in the training set, we

obtain a significant improvement for classification accuracy in the test data.

Table 4.18 Relative Feature Importance%, Using GA weighting for 881 II

 

 

2003 course

Feature Importance %
Average Number of Tries 18.9
Total number of Correct Answers 84.7
# of Success at the First Try 24.4
# of Success at the Second Try 26.5
Got Correct with 3-9 Tries 21.2
Got Correct with # of Tries 2 10 91.7
Time Spent to Solve the Problems 32.1
Total Time Spent on the Problems 36.5
# of communication 3.6

 

Table 4.19 shows the importance of the nine features in the BS 111 8803 course,
applying the Gini splitting criterion. Based on Gini, a statistical property called
information gain measures how well a given feature separates the training examples in
relation to their target classes. Gini characterizes impurity of an arbitrary collection of
examples 8 at a specific node N. In Duda et a1. (2001) the impurity of a node N is

denoted by i(N) such that:

Gini(8) = i(N) = 210(8) ,. )P(a),.) = 1 — 2122(0)!)

1‘1 j

where P(a)j) is the fraction of examples at node N that go to category 60].. Gini

attempts to separate classes by focusing on one class at a time. It will always favor

working on the largest or, if you use costs or weights, the most important class in a node.

137

 

Table 4.19 Feature Importance for BSIII 2003, using decision-tree software
CART, applying Gini Criterion

 

 

 

 

 

 

 

 

 

 

Variable
Total number of Correct Answers 10000 111111111111111111111111111111111111111111111111111111111111
Got Correct with # ofTries 2 10 93.34 11111111111111111111111111111111111111111111111111111111
Average number OfU'ieS 58-61 11111111111111111111111111111111111
# of Success at the First Try 37.70 111111111111111111
Got Correct with 3—9 Tries 30.31 11111111111111
# of Success at the Second Try 23.17 1111111L
Time Spent to Solve the Problems 16.60 11111
Total Time Spent on the Problems 14.47 111L
# of communication 2.21 1

 

 

 

Comparing results in Table 4.18 (GA-weighting) and Table 4.19 (Gini index
criterion) shows the very similar output, which demonstrates merits of the proposed

method for detecting the feature importance.

4.5 Summary

We proposed a new approach to classifying student usage of web-based instruction.
Four classifiers were used to segregate student data. A combination of multiple classifiers
led to a significant accuracy improvement in all three cases (2-, 3- and 9-Classes).
Weighting the features and using a genetic algorithm to minimize the error rate improved
the prediction accuracy by at least 10% in the all cases. In cases where the number of
features was low, feature weighting was a significant improvement over selection. The
successful optimization of student classification in all three cases demonstrates the value
of LON-CAPA data in predicting students’ final grades based on features extracted from
homework data. This approach is easily adaptable to different types of courses, different

population sizes, and allows for different features to be analyzed. This work represents a

138

 

rigorous application of known classifiers as a means of analyzing and comparing use and
performance of students who have taken a technical course that was partially/completely
administered via the web.

For future work, we plan to implement such an optimized assessment tool for every
student on any particular problem. Therefore, we can track students’ behaviors on a
particular problem over several semesters in order to achieve more reliable prediction.
This work has been published in (Minaei-Bidgoli & Punch, 2003; Minaei-Bidgoli, et al.

2003; Minaei-Bidgoli et al., 2004c-e).

139

Chapter 5 Ensembles of Multiple Clusterings

Since LON-CAPA data are distributed among several servers and distributed data
mining requires efficient algorithms form multiple sources and features, this chapter
represents a framework for clustering ensembles in order to provide an optimal
framework for categorizing distributed web-based educational resources. This research
extends previous theoretical work regarding clustering ensembles with the goal of
creating an optimal framework for categorizing web-based educational resources.
Clustering ensembles combine multiple partitions of data into a single clustering solution
of better quality. Inspired by the success of supervised bagging and boosting algorithms,
we propose non-adaptive and adaptive resampling schemes for the integration of multiple
independent and dependent clusterings. We investigate the effectiveness of bagging
techniques, comparing the efficacy of sampling with and without replacement, in
conjunction with several consensus algorithms. In our adaptive approach, individual
partitions in the ensemble are sequentially generated by clustering specially selected
subsamples of the given data set. The sampling probability for each data point
dynamically depends on the consistency of its previous assignments in the ensemble.
New subsamples are then drawn to increasingly focus on the problematic regions of the

input feature space. A measure of data point clustering consistency is therefore defined to

140

guide this adaptation. Experimental results show improved stability and accuracy for
clustering structures obtained via bootstrapping, subsampling, and adaptive techniques. A
meaningful consensus partition for an entire set of data points emerges from multiple
clusterings of bootstraps and subsamples. Subsamples of small size can reduce
computational cost and measurement complexity for many unsupervised data mining
tasks with distributed sources of data. This empirical study also compares the
performance of adaptive and non-adaptive clustering ensembles using different consensus
functions on a number of data sets. By focusing attention on the data points with the least
consistent clustering assignments, one can better approximate the inter—cluster boundaries
and improve clustering accuracy and convergence speed as a function of the number of
partitions in the ensemble. The comparison of adaptive and non-adaptive approaches is a
new avenue for research, and this study helps to pave the way for the useful application

of distributed data mining methods.

5.1 Introduction

Exploratory data analysis and, in particular, data clustering can significantly benefit
from combining multiple data partitions. Clustering ensembles can offer better solutions
in terms of robustness, novelty and stability (Fred & Jain, 2002; Strehl & Ghosh, 2002;
Topchy et al., 2003a). Moreover, their parallelization capabilities are a natural fit for the
demands of distributed data mining. Yet, achieving stability in the combination of
multiple clusterings presents difficulties.

The combination of clusterings is a more challenging task than the combination of
supervised classifications. In the absence of labeled training data, we face a difficult

correspondence problem between cluster labels in different partitions of an ensemble.

141

Recent studies (Topchy et al., 2004) have demonstrated that consensus clustering can be

found outside of voting-type situations using graph-based, statistical or information-

theoretic methods without explicitly solving the label correspondence problem. Other

empirical consensus functions were also considered in (Dudoit & Fridlyand, 2003; Fisher

& Buhmann, 2003, Fern & Brodley, 2003). However, the problem of consensus

clustering is known to be NP complete (Barthelemy & Leclerc, 1993).

Beside the consensus function, clustering ensembles need a partition generation

procedure. Several methods are known to create partitions for clustering ensembles. For

example, one can use:

1.

2.

different clustering algorithms (Strehl & Ghosh, 2002),
different initializations — parameter values or built-in randomness of a specific

clustering algorithm (Fred & Jain, 2002)

. different subsets of features (weak clustering algorithms) (Topchy et al.,

2003L
different subsets of the original data (data resampling) (Dudoit & Fridlyand,

2003; Fisher & Buhmann, 2003, Minaei etal., 2003).

The focus of this study is the last method, namely the combination of clusterings

using random samples of the original data. Conventional data resampling generates

ensemble partitions independently; the probability of obtaining the ensemble consisting

of B partitions {7a, 75,. . .,7r3} of the given data, D, can be factorized as:

B
p({7ri,rr2,...,7r3} | D) =gpoz, 1D) (5.1)

142

Hence, the increased efficacy of an ensemble is mostly attributed to the number of
independent, yet identically distributed partitions, assuming that a partition of data is
treated as a random variable 7r. Even when the clusterings are generated sequentially, it is

traditionally done without considering previously produced clusterings:

p(7r, 1n,_l,n,_2,...,7rl;D) = p(rr, 1D) (5.2)

However, similar to the ensembles of supervised classifiers using boosting algorithms
(Brieman 1998), a more accurate consensus clustering can be obtained if contributing
partitions take into account the previously determined solutions. Unfortunately, it is not
possible to mechanically apply the decision fusion algorithms from the supervised
(classification) to the unsupervised (clustering) domain. New objective functions for
guiding partition generation and the subsequent decision integration process are
necessary in order to guide further refinement. Frossyniotis et a1. (2004) apply the general
principle of boosting to provide a consistent partitioning of a data set. At each boosting
iteration, a new training set is created and the final clustering solution is produced by
aggregating the multiple clustering results through a weighted voting.

We propose a simple adaptive approach to partition generation that makes use of
clustering history. In clustering, ground truth in the form of class labels is not available.
Therefore, we need an alternative measure of performance for an ensemble of partitions.
We determine clustering consistency for data points by evaluating a history of cluster
assignments for each data point within the generated sequence of partitions. Clustering

consistency serves for adapting the data sampling to the current state of an ensemble

143

during partition generation. The goal of adaptation is to improve confidence in cluster
assignments by concentrating sampling distribution on problematic regions of the feature
space. In other words, by focusing attention on the data points with the least consistent
clustering assignments, one can better approximate (indirectly) the inter-cluster
boundaries.

The main objectives of this chapter are four-fold:

1. to present a detailed taxonomy of clustering ensemble approaches (section
5.2),

2. to expose critical and unaddressed issues in applying resampling methods
(section 5.5),

3. to provide a detailed comparison of bootstrap versus subsampling ensemble
generation (section 5.7),

4. and finally to study adaptive partitioning ensembles (section 5.6).

The remainder of the chapter is devoted to different consensus functions used in our
experiments (section 5.4), algorithms for resampling schemes (sections 5.3 and 5.6),
addressing the problems of estimation of clustering consistency and finding a consensus
clustering (section 5.6). Finally, we evaluate the performance of adaptive clustering
ensembles (Section 5.8) on a number of real-world and artificial data sets in comparison
with non-adaptive clustering ensembles of bootstrap partitions (Dudoit & Fridlyand,

2003; Fisher & Buhmann, 2003, Minaei-Bidgoli et al., 2003b).

5.2 Taxonomy of different approaches

A growing number of techniques have been applied to clustering combinations. A
co-association consensus function was introduced for finding a combined partition in

(Fred & Jain, 2002). The authors further studied combining k-means partitions with

144

random initializations and a random number of clusters. Topchy et al. (2003) proposed
new consensus functions related to intra-class variance criteria as well as the use of weak
clustering components. Strehl and Ghosh (2002) have made a number of important
contributions, such as their detailed study of hypergraph-based algorithms for finding
consensus partitions as well as their object-distributed and feature-distributed
formulations of the problem. They also examined the combination of partitions with a
deterministic overlap of points between data subsets (non-random).

Resampling methods have been traditionally used to obtain more accurate estimates
of data statistics. Efron (1979) generalized the concept of so—called “pseudo—samples” to
sampling with replacement — the bootstrap method. Resampling methods such as bagging
have been successfully applied in the context of supervised Ieaming (Breiman 1996). Jain
and Moreau (1987) employed bootstrapping in cluster analysis to estimate the number of
clusters in a multi-dimensional data set as well as for evaluating cluster tendency/validity.
A measure of consistency between two clusters is defined in (Levine & Moreau, 2001).
Data resampling has been used as a tool for estimating the validity of clustering (Fisher &
Buhmann, 2003; Dudoit & Fridlyand, 2001; Ben-Hur et al., 2002) and its reliability (Roth
et al., 2002).

The taxonomy of different consensus functions for clustering combination is shown
in Figure 5.2. Several methods are known to create partitions for clustering ensembles.
This taxonomy presents solutions for the generative procedure as well. Details of the

algorithms can be found in the listed references in Figure 5.1.

145

 

Generative mechanisms (How to obtain different components?)

1. Apply various clustering algorithms (Strehl & Ghosh, 2002)
2. Use a single algorithm
2.1. Different built-in initialization (Fred & Jain, 2002; Topchy et al. 2003b)
2.2. Different parameters (Fred & lain, 2002)
2.3. Different subsets of data points
2.3. 1. Deterministic subsets (Strehl & Ghosh, 2002)
2.3.2. Resampling (Dudoit & Fridlyand,2001;Monti et al. 2003; Fisher & Buhmann, 2003,
Minaei-Bidgoli et al., 2003b)
2.3.2.1. Bootstrap (Sampling with replacement)
2.3.2.2. Subsampling (Sampling without replacement)
2.3.2.3. Adaptive scheme (Topchy et al., 2004; F rossyniotis et al., 2004)
2.4. Projecting data onto different subspaces (Topchy et al., 2003a; Zhang&Brodely, 2003)

2.5. Different subset of features (Strehl & Ghosh, 2002)

Consensus functions (How to integrate cluster ensemble?)
1. Using Co-association Matrix (Fred & Jain, 2002; Monti et al., 2003)

1.1. Single Link (SL)/ Minimum Spanning Tree (MST)
1.2. Complete Link (CL)
1.3. Average Link (AL)
1.4. Ward, or other similarity based algorithms
2. (Hyper) Graph Partitioning (Strehl & Ghosh, 2002)
2.1. Hyper Graph Partition Algorithm (HGPA)
2.2. Meta CLustering Algorithm (MCLA)
2.3. Clustering Similarity Partition Algorithm (CSPA)
3. Information-theoretic methods, e.g. Quadratic Mutual lnfonnation (Topchy et al. 20038)

4. Voting Approach (Dudoit & Fridlyand,2001)

5. Mixture Model (Topchy et al. 2003b)

 

Figure 5.1 Different approaches to clustering combination

146

 

        

       

Clustering Ensembles Approaches

 

Generative mechanism

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Different One _\ Single hnk
318m algorithm ——q Co-association-based Comp. hnk
Different built-in initialimtion Avg Ink
1-—' Voting approach Others
Different parameters
___. Inforrmtion
Different subsets of features A Theoretic approach CSPA
Projecting data onto difi‘erent subspaces —' Hyper (8731311) “GPA
. . . pa g MCLA
Determrmstic
Different subsets _d Mixture Model (EM)
Resampling ./ of obiects

 

 

 

 

 

 

Figure 5.2 Taxonomy of different approaches to clustering combination

It is a long-standing goal of clustering research to design scalable and efficient
algorithms for large datasets (Zhang et al., 1996). One solution to the scaling problem is
the parallelization of clustering by sharing processing among different processors (Zhang
et al., 2000; Dhillon & Modha, 2000). Recent research in data mining has considered a
fusion of the results from multiple sources of data or from data features obtained in a
distributed environment (Park & Kargupta, 2003). Distributed data clustering deals with
the combination of partitions from many data subsets (usually disjoint). The combined
final clustering can be constructed centrally either by combining explicit cluster labels of
data points or, implicitly, through the fusion of cluster prototypes (e.g., centroid-based).
We analyze the first approach, namely, the clustering combination via consensus

functions operating on multiple labelings of the different subsamples of a data set. This

147

study seeks to answer the question of the optimal size and granularity of the component

partitions.

5.3 Non-adaptive algorithms

Bootstrap (sampling with replacement) and subsampling (without replacement) can
discern various statistics from replicate subsets of data while the samples in both cases
are independent of each other. Our goal is to obtain a reliable clustering with measurable
uncertainty from a set of different k—means partitions. The key idea of the approach is to
integrate multiple partitions produced by clustering of pseudo-samples of a data set.

Clustering combinations can be formalized as follows. Let D be a data set of N data
points in d-dimensional space. The input data can be represented as an N x d pattern
matrix or N x N dissimilarity matrix, potentially in a non-metric space. Suppose that X =
{X1,...,XB} is a set of B bootstrap samples of size N or subsamples of size S < N. A
chosen clustering algorithm is run on each of the samples in X, which results in B
partitions II={7r1, 7t2,..., 7:3}. Each component partition in H is a set of non-overlapping

and exhaustive clusters with iti={ C; ,C5,...,C,:m }, X,- = C,’ U...U C; We, where k(i) is

(i) ,
the number of clusters in the i-th partition.

The problem of combining partitions is to find a new partition 0 ={C1,. . .,CM} of the
entire data set D given the partitions in 11, such that the data points in any cluster of o are
more similar to each other than to points in different clusters within 0'. We assume that
the number of clusters, M, in the consensus clustering is predefined and can be different

from the number of clusters, k, in the ensemble partitions. In order to find the target

partition 0', one needs a consensus function utilizing information from the partitions {m}.

148

Several known consensus functions (Fred & Jain, 2002; Strehl & Ghosh, 2002; Topchy et
al., 2003a) can be employed to map a given set of partitions II={711, 7:2,..., itg} to the
target partition, 0, in our study.

The similarity between two objects, x and y, is defined as follows:

1, if a=b

0, if a i b (53)

. 1 B
swarm) = 3325(7f.(x),7f, 0», 5(a, b) =1

i=1
Similarity between a pair of objects simply counts the number of clusters shared by the
objects in the partitions {n1,..., n3}. Under the assumption that diversity comes from

independent resampling, two families of algorithms can be proposed for integrating

clustering components (Minaei-Bidgoli eta1., 2004a,b).

5.3.1 Similarity-based algorithm

The first algorithm family is based on the co-association matrix, and employs a group
of hierarchical clustering algorithms to find the final target partition. In this type,
similarity-based clustering algorithms are used as the consensus function, F. Hierarchical
clustering consensus functions with single-, complete-, and average-linkage criteria were
used to obtain a target consensus clustering, o. The pseudocode of these algorithms is

shown in Figure 5.3.

149

 

Input: D — the input data set N points
B — number of partitions to be combined
M — number of clusters in the final partition, 0
k — number of clusters in the components of the combination
F - a similarity-based clustering algorithm
for j=l to B
Draw a random pseudosample X,-
Cluster the sample Xj: 7r(i)(—k-means( {/Yj})
Update similarity values (co-association matrix) for all patterns in X,-
end

Combine partitions via chosen F: o<—F (P)
Validate final partition, o(optional)
return 0' // consensus partition

 

Figure 5.3 First algorithms for clustering ensemble, based on co-association
matrix and using different similarity-based consensus functions

5.3.2 Algorithms based on categorical clustering

The second family of algorithms for achieving clustering combination is based on
new features extracted through the partitioning process. In this approach, one can view
consensus clustering as clustering in a space of new features induced by the set, II. Each
partition, 715-, represents a feature vector with categorical values. The cluster labels of each
object in different partitions are treated as a new feature vector, a B-tuple, given B

different partitions in II.

150

Table 5.1 (a) Data points and feature values, N rows and d columns. Every row
of this table shows a feature vector corresponding to N points. (b) Partition labels
for resampled data, n rows and B columns.

 

 

 

 

 

 

(a)
Data Features
X1 X11 X12 X1 j Xid
X2 X21 X22 ij X28
x. "” X12 xii Xid
XN XNi 1092 KM Km
(1))
Data Partitions Labels
X1 7:1(x1) n2(x.) nj(x1) 7:3(x1)
X2 711012) 7120(2) 790(2) 7180(2)
Xi 41(Xi) 1t2(Xi) 71100) “800)
XN HIQ‘N) fizQXN) 711(1‘18) TCBO‘N)

 

 

Therefore, instead of the original d attributes, which are shown in Table 5.1(a), the
new feature vectors from a table with N rows and B columns (Table 5.1(b)) are utilized,
where each column corresponds to the results of mapping a clustering algorithm (k-
means) onto the resampled data and every row is a new feature extracted vector, with
categorical (nominal) values. Here, III-(xi) denotes the label of object x,- in the j-th partition
of 11. Hence the problem of combining partitions becomes a categorical clustering

problem.

151

 

Input: D — the input data set N points
B - number of partitions to be combined
M — number of clusters in the final partition 0
k — number of clusters in the components of the combination
F - consensus function operating with categorical features
Reference Partition (— k-means(D)
for i=1 to B
Draw a random pseudo-sample Xj
Cluster the sample Xj: 7r(i) (— k-means( {/Yj})
Store partition 7:,-
end
Re-label (if necessary)
Apply consensus function F on the set of partition labels, II, to find final partition 0'
Validate final partition 0(optional)
return 0' // consensus partition

 

Figure 5.4 Algorithms for clustering ensemble based on categorical clustering

The parameter k in both algorithms is the number of clusters in every component
partition. If the value of k is too large then the partitions will overfit the data set, and if k
is too small then the number of clusters may not be large enough to capture the true
structure of data set. In addition, if the total number of clusterings, B, in the combination
is too small then the effective sample size for the estimates of distances between co-
association values is also insufficient, resulting in a larger variance. In the case of the
subsampling algorithm (without replacement), the right choice of sample size S is closely
related to the value of k and the value of B and proper setting of S is required to reach
convergence to the true structure of the data set. The algorithmic parameters will be
discussed in section 6. In the rest of this chapter “k” stands for number of clusters in
every partition, “B” for number of partitions/pseudosamples (in both the bootstrap and

the subsampling algorithms), and “S " for the sample size.

152

5.4 Consensus functions

A consensus function maps a given set of partitions Fl = {n1,..., 7:3} to a target

partition 0'. In this experiment we have employed four types of consensus firnctions:

5.4.1 Co-association based functions

This consensus function operates on the co-association matrix. Similarity between
points (co-association values) can be estimated by the number of clusters shared by two
points in all the partitions of an ensemble. Then, numerous hierarchical agglomerative
algorithms (criteria) can be applied to the co-association matrix to obtain the final
partition, including Single Link (8L), Average Link (AL) and Complete Link (CL) (Jain
& Dubes, 1988). There are three main drawbacks to this approach.

I First, it has a quadratic computational complexity in the number of patterns and
features 0(kN2d'z) (Duda et al., 2001), where k is the number of clusters, N is the
number of data points, and d is the number of features.

' Second, there are no established guidelines for which clustering algorithm should be
applied, e. g. single linkage or complete linkage.

. Third, an ensemble with a small number of partitions may not provide a reliable

estimate of the co-association values (Topchy et al. 2003b).

5.4.2 Quadratic Mutual Information Algorithm (QMI)

Assuming that the partitions are independent, a consensus function based on k-
means clustering in the space of standardized features can effectively maximize a
generalized definition of mutual information (Topchy et al., 2003a). The complexity of

this consensus function is 0(kNB), where k is the number of clusters, N is the number of

153

items, and B is the number of partitions. Though the QMI algorithm can be potentially
trapped in a local optimum, its relatively low computational complexity allows the use of
multiple restarts in order to choose a quality consensus solution with minimum intra-

cluster variance.

5.4.3 Hypergraph partitioning

The clusters could be represented as hyperedges on a graph whose vertices
correspond to the data points to be clustered. The problem of consensus clustering is then
reduced to finding the minimum-cut of a resulting hypergraph. The minimum k—cut of
this hypergraph into k components gives the required consensus partition (Strehl &
Ghosh, 2002). Hypergraph algorithms seem to work effectively for approximately
balanced clusters. Though the hypergraph partitioning problem is NP-hard, efficient
heuristics to solve the k-way min-cut partitioning problem are known, i.e. the complexity
of CSPA, HGPA and MCLA is estimated in Strehl & Ghosh (2002) as 0(szB), 0(kNB),
and 0(k2NB2), respectively. These hypergraph algorithms are described in Strehl &
Ghosh (2002) and their corresponding source codes are available at
http://www.strehl.com. A drawback of hypergraph algorithms is that they seem to work

the best for nearly balanced clusters (Topchy et al., 2003b).

5.4.4 Voting approach

In the previous algorithms there is no need to explicitly solve the correspondence
problem between the labels of known and derived clusters. The voting approach attempts
to solve the correspondence problem and then uses a majority vote to determine the final

consensus partition (Dudoit & F ridlyand, 2003). The main idea is to permute the cluster

154

labels such that the best agreement between the labels of two partitions is obtained. All
the partitions from the ensemble must be re-labeled according to a fixed reference
partition. The complexity of this process is 0(kl), which can be reduced to 0(k3) if the
Hungarian method is employed for the equivalent minimal weight bipartite matching
problem.

All the partitions in the ensemble can be re-labeled according to their best agreement
with some chosen reference partition. A meaningful voting procedure assumes that the
number of clusters in every given partition is the same as in the target partition. This
requires that the number of clusters in the target consensus partition is known (Topchy et
al. 2003b).

The performance of all these consensus methods is empirically analyzed as a
function of two important parameters: the type of sampling process (sample redundancy)

and the granularity of each partition (number of clusters).

5.5 Critical issues in resampling

Let us emphasize the challenging points of using resampling techniques for

maintaining diversity of partitions and estimation of co-association values.

5.5.1 Variable number of samples

There is no essential difference between bootstrap and subsampling algorithms with
regard to the diversity of the data samples. In both cases the pseudosample can be
incomplete (missing some objects). In bootstrap one has no control over the number of

distinct objects in a sample, while in subsampling the size of a pseudosample can be

155

much smaller than the original sample size. On the average, 37% of objects are not

included into a bootstrap sample (Hastie et al., 2001).

5.5.2 Repetitive data points (objects)

In resampling with replacement (bootstrap) some of the data points can be drawn
multiple times. However, in computing the co-association matrix, only one COpy of a data
point should contribute to the co-association similarity. Hence, for repeated objects, we
count each pair of objects only once. This problem does not appear in the case of

subsampling

5.5.3 Similarity estimation

Co—association values require adjustment when bootstrap partitions are used.
Typically the similarity value between the objects x and y, sim(x,y), is calculated by
counting the number of shared clusters in all the given partitions (Eq. 5.4). This is
justified when each possible pair of objects appears the same number of times in these
partitions. However, in bootstrap, different objects can appear in a different number of
samples. Therefore, the effective sample size for a pair of objects may no longer be equal

to B. Hence, co-association values should be computed as following:

Sim (xay):%ZR5(Pi(x)9Pl(y)). (5'4)

where R is the number of bootstrap samples containing both x and y, and the sum is taken

over such samples.

156

5.5.4 Missing labels

In both bootstrap and subsampling, some of the objects are missed in drawn samples.
When one uses co-association based methods, this poses no difficulty because the co-
association values are only updated for existing objects. However, missing labels can
cause a number of problems for other consensus functions. For example, when an object
is missing in a bootstrap sample, there will be no label assigned to it afier running the
clustering algorithm. Thus, special consideration of the missing labels is necessary during

the process of re-labeling, before running a consensus function.

5.5.5 Re-labeling

We must consider how to re-label two bootstrap samples with missing values. When
the number of objects in the drawn samples is too small, this problem becomes harder.

For example, consider four partitions, P1, _,_,P4 for five data points x1, ...,x5 as shown in

Table 5.2.

Table 5.2 An illustrative example of re-Iabeling difficulty involving five data
points and four different clusterings of four bootstrap samples. The numbers
represent the labels assigned to the objects and the “?” shows the missing labels of
data points in the bootstrapped samples.

 

 

P1 P2 P3 P4
X] I ? 2 3
XZ I 2 3 I
X3 ? 2 ? 2
x4 ? ? 1 ?
X5 2 I 3 I

 

One can re-label the above partitions in relation to some reference partition.
However, the missing labels should not be considered in the re-labeling process.

Therefore, if the reference partition is P1, and we want to re-label P2, then only the data

157

points x; and x5 participate in the re-labeling process. Similarly, if P; is re-labeled based
on the reference P1, then x1, x2 and x3 are used in the Hungarian algorithm to find the best
match. Once the best agreement among the given labels is found then all the objects in

the partition, except those with missing labels, are re—labeled.

5.5.6 Adaptation of the k-means algorithm

The core of the QMI consensus function is the k-means algorithm, which utilizes a
special transformation of the space of N x B extracted labels. Since no labels are assigned
to the missing objects using the clustering algorithm, the missing objects should not be
considered in the k—means algorithm that calculates the mutual information among the
obtained labels and the target partition. A revised and extended version of the k-means
algorithm was developed such that it can handle the missing coordinates, as described in
Jain & Dubes (1988) and Dixon (1979). The k-means algorithm calculates the Euclidean
distances between every object and the cluster centers. If some coordinates are missing
then those coordinates are ignored in the calculation. The experiments in this study used

this modification of the k-means for the QMI algorithm.

5.6 Adaptive sampling scheme

While there are many ways to construct diverse data partitions for an ensemble, not
all of them easily generalize to adaptive clustering. The adaptive approach (Topchy et al.,
2004) extends the studies of ensembles whose partitions are generated via data
resampling (Dudoit & Fridlyand 2003; Fisher & Buhmann, 2003; Minaei et al, 2004).
Though, intuitively, clustering ensembles generated by other methods also can be

boosted. The adaptive partition generation mechanism (Brieman, 1998) is aimed at

158

reducing the variance of inter-class decision boundaries. Unlike the regular bootstrap
method that draws subsamples uniformly from a given data set, adaptive sampling favors
points from regions close to the decision boundaries. At the same time, the points located
far from the boundary regions are sampled less frequently. It is instructive to consider a
simple example that shows the difference between ensembles of bootstrap partitions with
and without the weighted sampling. Figure 5.5 shows how different decision boundaries
can separate two natural classes depending on the sampling probabilities. Here we
assume that the k-means clustering algorithm is applied to the subsamples.

Initially, all the data points have the same weight, namely, the sampling

probability p, = X, , i e[1,...,N]. Clearly, the main contribution to the clustering error is

due to the sampling variation that causes inaccurate inter-cluster boundaries. Solution
variance can be significantly reduced if sampling is increasingly concentrated only on the
subset of objects at iterations t; > t. > to, as demonstrated in Figure 5.5.

The key issue in the design of the adaptation mechanism is the updating of
probabilities. We have to decide how and which data points should be sampled as we
collect more and more clusterings in the ensemble. A consensus function based on the co-
association values (Jain & Fred, 2002) provides the necessary guidelines for adjustments
of sampling probabilities. Remember that the co-association similarity between two data
points, x and y, is defined as the number of clusters shared by these points in the

partitions of an ensemble, Fl:

159

A

A t0 A t1 ‘
A‘A‘ 4559‘ . A§A 1
A t i’ 46 § ,’ mfi ,1;

‘ A ‘1 - A ‘.’. AA “I4.

12

A o e A‘ ’4 0 0 do 0 C
‘ I” ‘1‘“... 34.0..900 [50:000
I, .. 3". .. ’I . . g a 00 I .: . ~§® CO

5. I )J C p": 015% O
: I. .6: . ' e .360 ’ C100

Figure 5.5 Two possible decision boundaries for a 2-cluster data set. Sampling
probabilities of data points are indicated by gray level intensity at different
iterations (to < t1< t;) of the adaptive sampling. True components in the 2-class
mixture are shown as circles and triangles.

Table 5.3. Consistent re-labeling of 4 partitions of 12 objects.

 

I I I I a t x I a I j I .
1 2 3 4 1 2 3 4 Consrstency
X 1 2 e x 0'- 2 1 2 1 0.5
X 2 2 A x °‘ 2 2 2 1 0.75
x 2 A Y 13 2 2 1 2 0.75
3
x 2 a x 13 2 1 2 2 0.75
4
x 1 A x 15 1 2 2 2 0.75
5
x s 2 A Y 1” 2 2 1 2 0.75
x 7 2 e x °1 2 1 2 1 0.5
x 1 a Y 01 1 1 1 1 1
8
x 9 1 a Y 1” 1 1 1 2 0.75
x 1 A Y 01 1 2 1 1 0.75
10
X 11 2 a Y 1" 2 1 1 1 0.75
x 1 a Y 01- 1 1 1 1 1
12

 

 

m_. ____a....._

A consensus clustering can be found by using an agglomerative clustering algorithm

(e.g., single linkage) applied to such a co-association matrix constructed from all the

points. The quality of the consensus solution depends on the accuracy of similarity values

as estimated by the co-association values. The least reliable co-association values come

from the points located in the problematic areas of the feature space. Therefore, our

160

adaptive strategy is to increase the sampling probability for such points as we proceed
with the generation of different partitions in the ensemble.

The sampling probability can be adjusted not only by analyzing the co-association
matrix, which is of quadratic complexity 0(N2), but also by applying the less expensive
0(N + K3) estimation of clustering consistency for the data points. Again, the motivation
is that the points with the least stable cluster assignments, namely those that frequently
change the cluster they are assigned to, require an increased presence in the data
subsamples. In this case, a label correspondence problem must be approximately solved
to obtain the same labeling of clusters throughout the ensemble’s partitions. By default,
the cluster labels in different partitions are arbitrary. To make the correspondence
problem more tractable, one needs to re-label each partition in the ensemble using some
fixed reference partition. Table 5.3 illustrates how four different partitions of twelve
points can be re—labeled using the first partition as a reference.

At the (t+1)-th iteration, when some t different clusterings are already included in the
ensemble, we use the Hungarian algorithm for minimal weight bipartite matching
problem in order to re-label the (t+1)—th partition. As an outcome of the re-labeling
procedure, we can compute the consistency index of clustering for each data point.
Clustering consistency index C] at iteration t for a point x is defined as the ratio of the
maximal number of times the object is assigned in a certain cluster to the total number of

partitions:

CI (x) = 119—max {i 5(77, (x), L (5.5)

i=1 }1Leclusrer _ labels 1

The values of consistency indices are shown in Table 5.3 after four partitions were

161

generated and re-labeled. We should note that clustering of subsamples of the data set, D,
does not provide the labels for the objects missing (not drawn) in some subsamples. In
this situation, the summation in Eq. (5.5) skips the terms containing the missing labels.
The clustering consistency index of a point can be directly used to compute its
sampling probability. In particular, the probability value is adjusted at each iteration as

follows:

191.101) = Z(a' p,(x)+1— CI(x)), (5.6)

where or is a discount constant for the current sampling probability and Z is a
normalization factor. The discount constant was set to 01:03 in our experiments. The

proposed clustering ensemble algorithm is summarized in pseudocode in Figure 5.6:

 

Input: D — data set of N points

B — number of partitions to be combined

M — number of clusters in the consensus partition 0'

K — number of clusters in the partitions of the ensemble

F — chosen consensus function operating on cluster labels

p — sampling probabilities (initialized to UN for all the points)

Reference Partition (— k-means(D)

for i=1 to B
Draw a subsample X,- from D using sampling probabilities p
Cluster the sample X,~: n(i) (— k-means(X,~)
Re-label partition n(i) using the reference partition
Compute the consistency indices for the data points in D
Adjust the sampling probabilities p

end

Apply consensus function F to ensemble II to find the partition 0'

Validate the target partition 0' (optional)

return 0' // consensus partition

 

Figure 5.6 Algorithms for adaptive clustering ensembles

162

5.7 Experimental study on non-adaptive approaches

The experiments were performed on several data sets, including two challenging
artificial problem, the “Halfrings” data set, and the “2-Spiral” data set, two data sets from
UCI repository, the “Iris” and “Wine” and two other real world data set, the “LON” and

“Star/Galaxy” data sets. A summary of data set characteristics is shown in Table 5 .4.

 

1'
I

 

 

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Figure 5.7 “Halfrings” data set with 400 patterns (100-300 per class) , “2-
Spirals” dataset with 200 patterns (100-100 per class)

163

Table 5.4. A summary of data sets characteristics

 

 

No. of No. of No. of Patterns per
Classes Features Patterns class
Star/Galaxy 2 14 4192 2082-2110
Wine 3 13 178 59-71-48
LON 2 6 227 64-163
Iris 3 4 150 50-50-50
3-Gaussian 3 2 300 50-100-1 50
Halfrings 2 2 400 100-300
Z-Spirals 2 2 200 1 00-1 00

 

5.7.1 Data sets

The Halfrings and 2-Spiral data set, as shown in Figure 5 .7, consist of two clusters,
though the clusters are unbalanced with 100- and BOO-point patterns in the Halfrings data
set and balanced in the 2-Spiral. The k—means algorithm by itself is not able to detect the
two natural clusters since it implicitly assumes hyperspherical clusters. 3-Gaussian is a
simulated data set that includes three unbalanced classes with 50, 100, and 150 data
points. The Wine data set described in Aeberhard et al. (1992) contains the value of the
chemical composition of wines grown in the same region but derived from three different
cultivars. The patterns are described by the quantities of thirteen constituents (features)
found in each of the three types of wines. There are 178 samples in total.

The LON data set (Minaei & Punch, 2003) is extracted from the activity log in a
web-based course using an online educational system developed at Michigan State
University (MSU): the Learning Online Network with Computer-Assisted Personalized
Approach (LON-CAPA). The data set includes the student and course information on an
introductory physics course (PHY183), collected during the spring semester 2002. This

course included 12 homework sets with a total of 184 problems, all of which were

164

completed online using LON-CAPA. The data set consists of 227 student records from
one of the two groups: “Passed” for the grades above 2.0, and “Failed” otherwise. Each
sample contains 6 features.

The Iris data set contains 150 samples in 3 classes of 50 samples each, where each
class refers to a type of iris plant. One class is linearly separable from the other two, and
each sample has four continuous-valued features. The Star/Galaxy data set described in
Odewahn (1992) has a significantly larger number of samples (N=4192) and features
(d=14). The task is to separate observed objects into stars or galaxies. Domain experts
manually provided true labels for these objects.

For all these data sets the number of clusters, and their assignments, are known.
Therefore, one can use the misassignment (error) rate of the final combined partition as a
measure of performance of clustering combination quality. One can determine the error
rate after solving the correspondence problem between the labels of derived and known
clusters. The Hungarian method for solving the minimal weight bipartite matching

problem can efficiently solve this label correspondence problem.

5.7.2 The role of algorithm's parameters

The bootstrap experiments probe the accuracy of partition combination as a function
of the resolution of partitions (value of k) and the number of partitions, B (number of
partitions to be merged).

One of our goals was to determine the minimum number of bootstrap samples, B,
necessary to form high-quality combined cluster solutions. In addition, different values of

k in the k-means algorithm provide different levels of resolution for the partitions in the

165

combinations. We studied the dependence of the overall performance on the number of
clusters, k. In particular, clustering on the bootstrapped samples was performed for the
values of B in the range [5, 1000] and the values of k in the interval [2, 20].

Analogously, the size of the pseudosample, S, in subsampling experiments is another
important parameter. Our experiments were performed with different subsample sizes in
the interval [N/20, 3N/4], where N is the size of the original data sample. Thus, in the
case of the Halfrings, S was taken in the range [20, 300] where the original sample size is
N=400, while in the case of the Galaxy data set, parameter S was varied in the range [200,
3000] where N=4l92. Therefore, in resampling without replacement, we analyzed how
the clustering accuracy was influenced by three parameters: number of clusters, k, in
every clustering, number of drawn samples, B, and the sample size, S. Note that all the
experiments were repeated 20 times and the average error rate for 20 independent runs is
reported, except for the Star/Galaxy data where 10 runs were performed.

The experiments employed eight different consensus functions: co-association based
functions (single link, average link, and complete link), hypergraph algorithms (HGPA,

CSPA, MCLA), the QMI algorithm, as well as a Voting-based function.

5.7.3 The Role of Consensus Functions (Bootstrap algorithm)

Perhaps the single most important design element of the combination algorithm is
the choice of a consensus function. In the Halfrings data set the true structure of the data
set (100% accuracy) was obtained using co-association based consensus functions (both

single and average link) in the case of k=15 and number of partitions taking part in the

166

combination where B2100. None of the other six consensus methods converged to an
acceptable error rate for this data set.

For the Wine data set an optimal accuracy of 73% was obtained with both the
hypergraph-CSPA algorithm and co—association based method using average link (AL)
with different parameters as shown in Table 5.6. For the LON data set the optimal
accuracy of 79% was achieved only by co-association—based (using the AL algorithm)
consensus function. This accuracy is comparable to the results of the k-NN classifier,
multilayer perceptron, naive Bayes classifier, and some other algorithms when the “LON”

data set is classified in a supervised framework based on labeled patterns (Minaei &

 

 

 

 

 

 

 

 

Punch,2003).
+k=2
Iris, MCLA +k=3
80 +k=4 _.
+k=5
g 70 ” —e—k=10
60 F W
E 50 _ .
g... -
g 30 ~ °
.E 20 r c
E 10 __ .‘_2-___ ‘4
0 l. + I + 1* +-
5 10 20 50 100 250

Number of Partitions, B

Figure 5.8 “Iris” data set. Bootstrapping for fixed consensus function MCLA,
different B, and different values of k.

For the “Iris” data set, the hypergraph consensus function, HPGA algorithm led to
the best results when k 2 10. The AL and the QMI algorithms also gave acceptable

167

results, while the single link and average link did not demonstrate a reasonable
convergence. Figure 5.8 shows that the optimal solution could not be found for the Iris
data set with k in the range [2, 5], while the optimum was reached for k 2 10 with only
8210 partitions.

For the Star/Galaxy data set the CSPA function (similarity based hypergraph
algorithm) could not be used due to its computational complexity because it has a
quadratic complexity in the number of patterns 0(szB).

The HGPA function and SL did not converge at all, as shown in Table 5.5. Voting
and complete link also did not yield optimal solutions. However, the MCLA, the OM]
and the AL functions led to an error rate of approximately 10%, which is better than the
performance of an individual k-means result (21%).

The major problem in co-association based functions is that they are
computationally expensive. The complexity of these functions is very high (0(kN2a‘?))
and therefore, it is not effective to use the co-association based functions as a consensus

function for the large data sets.

168

Table 5.5 “Star/Galaxy” data experiments. Average error rate (% over 10
runs) of clustering combination using resampling algorithms with different number
of components in combination B, resolutions of components, k, and types of
consensus functions.

 

 

K B QMI MCLA SL AL CL Voting
2 5 18.5 19.4 49.7 49.7 49.7 20.4
2 10‘ 18.7 18.8 49.6 49.6 49.6 19.5
2 20 18.5 18.9 49.6 24.4 49.7 19
2 50 18.7 18.8 49.6 18.8 49.7 18.9
2 100 18.8 18.8 49.7 18.8 18.8 18.9
3 5 13.4 15.5 49.7 49.7 49.7 -

3 10 17.8 15.6 49.6 49.6 49.6 -

3 20 11.5 15.3 49.7 18.8 42.9 -

3 50 13.3 15.4 49.7 11 35.9 -

3 100 11 15.4 49.7 11 48.2 -
4 5 15.2 13.1 49.7 49.7 49.7 -
4 10 11.4 14.5 49.6 49.7 49.7 -
4 20 14 13.7 49.6 24.3 48.7 -
4 50 22.2 11.9 49.7 10.7 48 -
4 100 11 11.9 49.7 10.7 47.9 -

5 5 14.9 13.8 49.7 49.7 49.7 -

5 10 14.9 13.1 49.7 47.9 49.6 -

5 20 10.7 13.4 49.6 11 49.7 -

5 50 11.4 13.4 49.7 10.8 48.7 -

5 100 11 12.5 49.7 10.9 48 -

 

Note that the QMI algorithm did not work well when the number of partitions
exceeded 200, especially when the value of k was large. This might be due to the fact that
the core of the QMI algorithm operates in ka—dimensional space. The performance of
the k-means algorithm degrades considerably when B is large (>100) and, therefore, the

QMI algorithm should be used with smaller values of B.

5.7.4 Effect of the Resampling method (Bootstrap vs.

Subsampling)

In subsampling the smaller the S the lower the complexity of the k-means clustering,

which therefore results in much smaller complexity in the co-association based consensus

169

functions, which is super-linear N. Comparing the results of the bootstrap and the
subsampling methods shows that when the bootstrap technique converges to an optimal
solution, that optimal result could be obtained by the subsampling as well, but with a
critical size of the data points. For example, in the Halfrings data set the perfect
clustering can be obtained using a single-link consensus fimction with k=10, B=100 and
S=200 (US data size) as shown in Figure 5.9 (compare to the bootstrap results in table
5.6) while this perfect results can be achieved with k=15, B = 50, and S = 80 (US data
size). Thus, there is a trade off between the number of partitions B and the sample size S.
This comparison shows that the subsampling method can be much faster than the

bootstrap (N=400) in relation to the computational complexity.

 

 

 

 

 

 

 

. _ _ —'—Single.link
250 Halfrmgs, k—lO,B—100 I Avg.link _
—*—Comp.link
—O—HPGA
a) 200 r ' 'X' " MCLA
g —°—CSPA 1
E —l-—QMI J1
150 * \
E ‘\ ~
.9 w
a 100 ’
m “\
.2
E 50
0
=13:
0

 

 

20 40 80 100 200 300

Sample size. S

Figure 5.9 “Halfrings” data set. Experiments using subsampling with k=10 and
B=100, different consensus function, and sample sizes S.

The results of subsampling for "Star/Galaxy" data set as shown in Figure 5.10, shows

170

that in resolution k=3 and number of partitions B=100, with only sample size S = 500 (1/8
of the entire data size) one can reach 89% accuracy, the same results with entire data set
in the bootstrap method. It shows that for this large data set, a small fraction of data can
be representative of the entire data set, and this would be computationally very interesting

in distributed data mining.

Star/Galaxy, k =4, B =50

 

 

 

 

 

 

 

 

2500
g 2000 1
1i 1500 —
8
C
.99
g 1000 ~ _
.12
g 500 .
11:
O 1 1 1 L l

200 500 1000 1500 2000 3000

Sample size, 5

Figure 5.10 “Star/Galaxy” data set. Experiments using subsampling, with k = 4
and B = 50 and different consensus function and sample sizes S.

Note that in both the bootstrap and the subsampling algorithms all of the samples are
drawn independently, and thus the resampling process could be performed in parallel.
Therefore, using the B parallel processes, the computational process becomes B times

faster.

171

 

Table 5
this research
100 runs. wi
the true num

in Table 5.7.

Tablr

average

\
Data set

Iris
Wine
LON

Stan/G .
\

Data set

Halfr‘in 25

wine

%

Table 5.6 shows the error rate of classical clustering algorithms, which are used in
this research. The error rates for the k-means algorithm were obtained as the average over
100 runs, with random initializations for the cluster centers, where value of k was fixed to

the true number of clusters. One can compare it to the error rate of ensemble algorithms

in Table 5.7.

Table 5.6 The average error rate (%) of classical clustering algorithms. An
average over 100 independent runs is reported for the k-means algorithms

 

 

Data set k-means Single Link Complete Link Average Link
Halfrings 25% 24.3% 14% 5.3%

Iris 15.1% 32% 16% 9.3%
Wine 30.2% 56.7% 32.6% 42%
LON 27% 27.3% 25.6% 27.3%

Star/Galaxy 21% 49.7% 44.1% 49.7%

 

Table 5.7 Summary of the best results of Bootstrap methods

 

 

 

 

 

 

Data set Best Consensus Lowest Error Parameters
function(s) rate obtained

Hal frings Co—association, SL 0% k _>_ 10, B. 2 100
Co-assomation, AL 0% k 2 15, B Z 100

Iris Hypergraph-HGPA 2.7% k 2 10, B Z 20
Wine Hypergraph-CSPA 26.8% k 2 10, B Z 20
Co-association, AL 27.9% k 2 4, B 2 100

LON Co-association, CL 21.1% k 2 4, B Z 100
Hypergraph-MCLA 9.5% k 2 20, B Z 10
Galaxy/ Star Co-association, AL 10% k _>_ 10, B _>_ 100

Mutual Information 11% k _>, 3, B Z 20

 

172

 

Table
of partition
siz

Data set

\

Halfrings
\
Iris
\
Wine

\
LON

Table 5.8 Subsampling methods: trade-off among the values of k, the number
of partitions B, and the sample size, S. Last column denote the percentage of sample
size regarding the entire data set. (Bold represents most optimal)

 

 

 

 

 

 

Best Lowest % of

Data set Consensus Error k B S entire
function(s) rate data
SL 0% 10 100 200 50%

Hal frings SL 0% 10 500 80 20%
AL 0% 15 1000 80 20%
AL 0% 20 500 100 25%
Iris HGPA 2.3% 10 100 50 33%
HGPA 2.1% 15 50 50 33%
AL 27.5% 4 50 100 56%

Wine HPGA 28% 4 50 20 11%
CSPA 27.5% 10 20 50 28%
LON CL 21.5% 4 500 100 44%
CSPA 21.3% 4 100 100 44%
Galaxy / MCLA 10.5% 10 50 1500 36%
Star MCLA l 1.7% 10 100 200 5%
AL 1 1% 10 100 500 12%

 

The optimal size S and granularity of the component partitions derived by
subsampling are reported in Table 5.8. We see that the accuracy of the resampling
method is very similar to that of the bootstrap algorithm, as reported in Table 5.6. This
level of accuracy was reached with remarkably smaller sample sizes and much lower
computational complexity! The trade-off between the accuracy of the overall clustering
combination and computational effort for generating component partitions is shown in
table 8, where we compare accuracy of consensus partitions. The most promising result is
that only a small fraction of data (i.e., 12% or 5% for the “Star/Galaxy” data set) is
required to obtain the optimal solution of clustering, both in terms of accuracy and
computational time.

The question of the best consensus function remains open for fiirther study. Each
consensus function explores the structure of data set in different ways, thus its efficiency

greatly depends on different types of existing structure in the data set. One can suggest

173

having several consensus functions and then combining the consensus function results
through maximizing mutual information (Strehl & Ghosh; 2002), but running different

consensus functions on large data sets would be computationally expensive.

5.8 Empirical study and discussion of Adaptive approach

The experiments were conducted on artificial and real-world data sets (“Galaxy”,
“half-rings”, “wine”, “3-gaussian”, “Iris”, “LON”), with known cluster labels, to validate
the accuracy of consensus partition. A comparison of the proposed adaptive and previous
non-adaptive (Minaei et al. 2004) ensemble is the primary goal of the experiments. We
evaluated the performance of the clustering ensemble algorithms by matching the
detected and the known partitions of the datasets. The best possible matching of clusters
provides a measure of performance expressed as the misassignment rate. To determine
the clustering error, one needs to solve the correspondence problem between the labels of
known and derived clusters. Again, the Hungarian algorithm was used for this purpose.
The k-means algorithm was used to generate the partitions of samples of size N drawn
with replacement, similar to bootstrap, albeit with dynamic sampling probability. Each
experiment was repeated 20 times and average values of error (misassignment) rate are

shown in Figure 5.11.

174

 

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110

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Si

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# of mlsasslgned patterns
0
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-l— Non-Adaptive

  
   

+Adaptive

 

 

 

 

Halfrings data set, CSPAconsensus, k=3

 

 

 

 

 

 

 

 

%1
% a
25 50 75 100 125 150
to! Patitions.B
(@
35
Wine data set. Ml consensus, k=6,
n30 .
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25 50 75 100 125 150
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9 3- Gaussian data set k: 4 CSPA

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3&51 --m- ,
on
'3 520 h N
g 515
E 510 Galaxy data set. MIA consensus lubction. k=6
0 505 -
1.. 500 . 4 .

 

 

25 50 75 100
I of Petitions, B
((0

Figure 5.11 Clustering accuracy for ensembles with adaptive and non-adaptive
sampling mechanisms as a function of ensemble size for some data sets and selected
consensus functions.

Consensus clustering was obtained by four different consensus functions:

hypergraph-based MCLA and CSPA algorithms (Strehl & Ghosh; 2002), quadratic

mutual information (Topchy et al., 2003a) and EM algorithm based on mixture model

(Topchy et al., 2003b). Herein, we report only the key findings. The main observation is

that adaptive ensembles slightly outperform the regular sampling schemes on most

175

 

 

benchmarks
depends on
ensembles v
lor 2. With
ensembles V
algorithms. 1
that points w
Most si‘;
partitions. La
in clustering
(except for th

01 the CODSCns

benchmarks. Typically, the clustering error decreased by 1-5%. Accuracy improvement
depends on the number of clusters in the ensemble partitions (k). Generally, the adaptive
ensembles were superior for values of k larger than the target number of clusters, M, by
lor 2. With either too small or too large a value of k, the performance of adaptive
ensembles was less robust and occasionally worse than corresponding non-adaptive
algorithms. A simple inspection of probability values always confirmed the expectation
that points with large clustering, uncertainty are drawn more frequently.

Most significant progress was detected when combination consisted of 25-75
partitions. Large numbers of partitions (B>75) almost never lead to further improvement
in clustering accuracy. Moreover, for B>125 we often observed increased error rates
(except for the hypergraph-based consensus functions), due to the increase in complexity

of the consensus model and in the number of model parameters requiring estimation.

5.9 Concluding remarks

A new approach to combine partitions is proposed by resampling of original data.
This study showed that meaningful consensus partitions for the entire data set of objects
emerge from clusterings of bootstrap and subsamples of small size. Empirical studies
were conducted on various simulated and real data sets for different consensus functions,
number of partitions in the combination and number of clusters in each component, for
both bootstrap (with replacement) and subsampling (without replacement). The results
demonstrate that there is a trade-off between the number of clusters per component and
the number of partitions, and the sample size of each partition needed in order to perform

the combination process converges to an optimal error rate.

176

The bootstrap technique was recently applied in (Dudoit & Fridlyand, 2003; Fisher
& Buhmann, 2003; Monti et al., 2003) to create a diversity in clusterings ensemble.
However, our work extends the previous studies by using a more flexible subsampling
algorithm for ensemble generation. We also provided a detailed comparative study of
several consensus techniques. The challenging points of using resampling techniques for
maintaining diversity of partitions were discussed in this chapter. We showed that there
exists a critical fraction of data such that the structure of entire data set can be perfectly
detected. Subsamples of small sizes can reduce costs and measurement complexity for
many explorative data mining tasks with distributed sources of data.

We have extended clustering ensemble framework by adaptive data sampling
mechanism for generation of partitions. We dynamically update sampling probability to
focus on more uncertain and problematic points by on-the-fly computation of clustering
consistency. Empirical results demonstrate improved clustering accuracy and faster
convergence as a function of the number of partitions in the ensemble.

Further study of alternative resampling methods, such as the balanced (stratified) and
recentered bootstrap methods are critical for more generalized and effective results. This

work has bee published in (Minaei et al., 2004a; Minaei et al. 2004b, Topchy et al. 2004).

177

Chapter 6 Association Analysis in LON-CAPA

A key objective of data mining is to uncover the hidden relationships among the
objects in a data set. Web-based educational technologies allow educators to study how
students learn and which Ieaming strategies are most effective. Since LON-CAPA
collects vast amounts of student profile data, data mining and knowledge discovery
techniques can be applied to find interesting relationships between attributes of students,
assessments, and the solution strategies adopted by students. This chapter focuses on the
discovery of interesting contrast rules, which are sets of conjunctive rules describing
interesting characteristics of different segments of a population. In the context of web-
based educational systems, contrast rules help to identify attributes characterizing
patterns of performance disparity between various groups of students. We propose a
general formulation of contrast rules as well as a framework for finding such patterns.

We apply this technique to the LON-CAPA system.

6.1 Introduction

This chapter investigates methods for finding interesting rules based on the
characteristics of groups of students or assignment problems. More specifically, our
research is guided and inspired by the following questions: Can we identify the different
groups of students enrolled in a particular course based on their demographic data?

Which attribute(s) best explain the performance disparity among students over different

178

sets of assignment problems? Are the same disparities observed when analyzing student
performance in different sections or semesters of a course?

We address the above questions using a technique called contrast rules. Contrast
rules are sets ~of conjunctive rules describing important characteristics of different
segments of a population. Consider the following toy example of 200 students who
enrolled in an online course. The course provides online reading materials that cover the
concepts related to assignment problems. Students may take different approaches to solve
the assignment problems. Among these students, 109 students read the materials before
solving the problems while the remaining 91 students directly solve the problems without
reviewing the materials. In addition, 136 students eventually passed the course while 64

students failed. This information summarized in a 2 X 2 contingency table as shown in

Table 6.1.

Table 6.1 A contingency table of student success vs. study habits for an online

 

 

 

 

course _
Passed Failed Total
Review materials 95 14 109
Do not review 41 50 91
Total 136 64 200

 

 

 

 

 

 

The table shows that there are interesting contrasts between students who review the
course materials before solving the homework problems and students who do not review

the materials. The following contrast rules can be induced from the contingency table:

 

Review materials: Passed, s = 47.5%, c = 87.2%

Review materials: Failed, s = 7.0%, c = 12.8%

 

 

 

Figure 6.1 A contrast rule extracted from Table 6.1

179

where s and c are the support and confidence of the rules (Agrawal et al., 1993).
These rules suggest that students who review the materials are more likely to pass the
course. Since there is a large difference between the support and confidence of both rules,
the observed contrast is potentially interesting. Other examples of interesting contrast

rules obtained from the same contingency table are shown in Figures 6.2 and 6.3.

 

 

Passed: Review materials, 5 = 47.5%, c = 69.9%

Failed : Review materials, 5 = 7.0%, c = 15.4%

 

Figure 6.2 A contrast rule extracted from Table 6.1

 

 

Passed: Review materials, 5 = 47.5%, c = 69.9%

Passed: Do not review, 5 = 20.5%, c = 30.1%

 

Figure 6.3 A contrast rule extracted from Table 6.1

Not all contrasting rule pairs extracted from Table 6.1 are interesting, as the example

 

 

in Figure 6.4 shows.
Do not review : Passed, s = 20.5%, c = 45.1%
Do not review : Failed, s = 25.0%, c = 54.9%

 

 

Figure 6.4 A contrast rule extracted from Table 6.1

The above examples illustrate some of the challenging issues concerning the task of
mining contrast rules:

1. There are many measures applicable to a contingency table. Which

measure(s) yield the most significant/interesting contrast rules among

different groups of attributes?

180

 

 

2. Many rules can be extracted from a contingency table. Which pair(s) of rules
should be compared to define an interesting contrast?

This chapter presents a general formulation of contrast rules and proposes a new
algorithrn for mining interesting contrast rules. The rest of this chapter is organized as
follows: Section 6.2 provides a brief review of related work. Section 6.3 offers a formal
definition of contrast rules. Section 6.4 gives our approach and methodology to discover
the contrast rules. Section 6.5 describes the LON-CAPA data model and an overview of

our experimental results.

6.2 Background

In order to acquaint the reader with the use of data mining in online education, we
present a brief introduction of association analysis and measures for evaluating
association rules. Next, we explain the history of data mining in web-based educational

systems. Finally, we discuss previous work related to contrast rules.

6.2.1 Association analysis

Let I = {i1, i2, im} be the set of all items and T = {t1, t2, ..., m} the set of all
transactions where m is the number of items and N is the number of transactions. Each
transaction t,- is a set of items such that t,- g 1. Each transaction has a unique identifier,
which is referred to as TID. An association rule is an implication statement of the form X
2 Y, where X C I, Y c I, and X and Y are disjoint, that is, X (I Y = Q. X is called the

antecedent while Y is called the consequence of the rule (Agrawal et al., 1993; Agrawal

& Srikant, 1994).

181

Support and confidence are two metrics, which are often used to evaluate the quality
and interestingness of a rule. The rule X :> Y has support, 3, in the transaction set, T, if
s% of transactions in T contains X U Y . The rule has confidence, c, if c% of
transactions in T that contain X also contains Y. Formally, support is defined as shown in
Eq. (6.1),

MXUY)

s(X:>Y)= N

, (6.1)

where N is the total number of transactions, and confidence is defined in Eq. (6.2).

ctr:r)=“i¥fx mm

 

Another measure that could be used to evaluate the quality of an association rule is

presented in Eq. (6.3).

s(X)

RuleCoverage =
N

 

m»

This measure represents the fraction of transactions that match the left hand side of a
rule.

Techniques developed for mining association rules often generate a large number of
rules, many of which may not be interesting to the user. There are many measures
proposed to evaluate the interestingness of association rules (F reitas, 1999; Meo, 2003).
Silberschatz and Tuzhilin (1995) suggest that interestingness measures can be categorized
into two classes: objective and subjective measures.

An objective measure is a data-driven approach for evaluating interestingness of

rules based on statistics derived from the observed data. In the literature different

182

objective measures have been proposed (Tan et al., 2004). Examples of objective
interestingness measure include support, confidence, correlation, odds ratio, and cosine.
Subjective measures evaluate rules based on the judgments of users who directly
inspect the rules (Silberschatz & Tuzhilin, 1995). Different subjective measures have
been addressed to discover the interestingness of a rule (Silberschatz & Tuzhilin, 1995).
For example, a rule template (Fu & Han, 1995) is a subjective technique that separates
only those rules that match a given template. Another example is neighborhood-based
interestingness (Dong & Li, 1998), which defines a single rule’s interestingness in terms

of the supports and confidences of the group in which it is contained.

6.2.2 Data mining for online education systems

Recently, several researchers have worked on the application of data mining to
examine or classify students’ problem-solving approaches within web-based educational
systems. For example, we previously developed tools for predicting the student
performance with respect to average values of student attributes versus the overall
problems of an online course (Minaei et al., 2003). Zaiane (2001) suggested the use of
web mining techniques to build an agent that recommends on-line learning activities in a
web-based course. Ma et al. (2000) focused on one specific task of using association rule
mining to select weak students for remedial classes. This previous work focused on
finding association rules with a specific rule consequent (i.e. a student is weak or strong).
Herein, we propose a general formulation of contrast rules as well as a framework for

finding such patterns.

183

6.2.3 Related work

An important goal in data mining is the discovery of major differences among
segments of population. Bay and Pazzani (2001) introduced the notion of contrast sets as
a conjunction of attributes and values that differ “meaningfully” in their distribution
across groups. They used a chi-square test for testing the null hypothesis that contrast-set
support is equal across all groups. They developed the STUCCO (Search and Testing for
Understandable Consistent Contrast) algorithm to find significant contrast sets. Our work
represents a general formulation for contrast rules using different interestingness
measures. We show that alternative measures allow for different perspectives on the
process of finding interesting rules.

Liu et a1. (2001) have also used a chi-square test of independence as a principal
measure for both generating the association rules and identifying non-actionable rules.
Below, we briefly discuss the chi-square test of independence and one of its
shortcomings.

Chi-square testing is used as a method for verifying the independence or correlation
of attributes. The chi-square test compares observed frequencies with the corresponding
expected frequencies. The greater the difference between observed and expected
frequencies, the greater is the power of evidence in favor of dependence and relationship.
Let CT be a contingency table with K rows and L columns. The chi-square test for
independence is shown in Eq. (6.4) where 195K, and ISjSL, and degree of freedom is

(K-l)(L-1).

 

1-‘51-2 6.4
12:22 1E J) ()
,-

. . 2 . . .
However, a drawback of this test is that the I value rs not invariant under the row-

column scaling pr0perty (Tan et al., 2004). For example, consider the contingency table

shown in Table 6.2(a). If lzis higher than a specific threshold (e.g. 3.84 at the 95%

significance level and degree of freedom 1), we reject the independence assumption. The

chi-square value corresponding to Table 6.2(a) is equal to 1.82. Therefore, the null

hypothesis is accepted. Nevertheless, if we multiply the values of that contingency table '
by 10, a new contingency table is obtained as shown in Table 6.2(b). The zzvalue

increases to 18.2 (>384). Thus, we reject the null hypothesis. We expect that the

 

relationship between gender and success for both tables as being equal, even though the

sample sizes are different. In general, this drawback shows that 12 is proportional to N.

Table 6.2 A contingency table proportional to table 6.1

 

 

 

(a)
Passed Failed Total
Male 40 49
Female 60 51 1 1 1
Total 100 100 200

 

 

6.3 Contrast Rules

 

 

 

 

 

0))
Passed Failed Total
Male 400 490 890
Female 600 510 l l 10
Total 1000 1000 2000

In this section, we introduce the notion of contrast rules. Let A and B be two itemsets

whose relationship can be summarized in a 2X2 contingency table as shown in Table 6.3.

Table 6.3 A contingency table for the binary case

 

 

 

 

 

 

B E Total
A fn f12 f1+
A— f21 f22 f2+
T0131 f +] flz N

185

Let Q be a set of all possible association rules that can be extracted from such a

contingency table (Figure 6.5).

 

 

A:B,A:§,Z:B,T:§,B:A,§:A,B:7, 3:7

 

Figure 6.5 Set of all possible association rules for Table 6.3.

We assume that B is a target variable and A is a conjunction of explanatory
attributes. Let ,u be a set of measures that can be applied to a rule or contingency table.

Examples of such measures include support, confidence, chi-square, odds ratio,
correlation, cosine, Jaccard, and interest (Tan et al., 2004). Below we provide a formal

definition of “contrast rule.”

 

Definition (General Formulation of Contrast Rules):

 

A contrast rule, cr, is a 4-tuple <br, u(br), M, A> where:

 

1. br c Q , is the base rule,

2. 0(br) c Q is a neighborhood to which the base rule br is compared,

3. M=<mbase, mneighbo? is an ordered pair of measures where mbase, mneighbo, e ,u, and
mbm measures the rules in br and mneighbo, measures the rules in u(br),

4. A(mbm(br), mneighbor(u(br))) is a comparison function between mba,e(r) and

mneighbor(v(br))

 

 

A contrast rule, or, is interesting if and only if A(mbm(br), mneighbo,(0(br))) 2 o,
where o is a user defined threshold, which implies that there is a large difference

between br and its neighborhood with respect to M.

 

Figure 6.6 Formal definition of a contrast rule

186

 

 

As shown in Figure 6.6, the contrast rule definition is based on a paired set of rules,
base rule br and its neighborhood u(br). The base rule is a set of association rules with
which a user is interested in finding contrasting association rules. Below are some

examples that illustrate the definition.

Example 1: cr; (Difference of confidence)

The first type of contrast rules examines the difference between rules
A :> B andA : B. An example of this type of contrast was shown in Figure 6.1. Let
confidence be the selected measure for both rules. Let absolute difference be the

comparison function. We can summarize this type of contrast as follows:

- br: {A : B}
- u(r): {A : B}
- M: <confidence, confidence>

- A: absolute difference

The evaluation criterion for this example is shown in Eq. 6.5. This criterion can be

used for ranking different pairs of contrast rules

A = 1 CU) - C(00)) 1
=1 c(A:>B)-C(A:>B) I

L_;f_lg_ = If” - ftzl (6.5)
f1. f|+ f1. ’

 

 

 

where f,j corresponds to the values in the i-th row and j-th column of Table 6.3.

Since c(A : B) + C(A :B) = 1, therefore,

187

A = |C(A:B)—C(A:>B)I
= |2c(A:>B)-—l|
oc C(ADB).

Thus, the standard confidence measure is sufficient to detect an interesting contrast of

this type.

Example 2: er; (Difference of proportion)

An interesting contrast could be considered between rules B :> A and B :> A. An
example of this contrast was shown in Figure 6.2. Once again, let confidence be the
selected measure for both rules. Let absolute difference be the comparison function. We

can summarize this type of contrast as follows:

- br: {B : A}
- u(br): {B :> A}
- M: <confidence, confidence>

- A: absolute difference

The evaluation criterion for this example is shown in Eq. 6.6, where A is defined as

follows:

A = 1 C(r) -6 (”(0)1

= | c(B=>A)-C(B:A)I
(6.6)

 

:1;11—;12|=1p(/1 :> B)—-p(A :> B)

 

where p, is the rule proportion (Agresti, 2002) and is defined in Eq. 6.7.

188

_ P(AB) _
p(A :> B) - —P(B) -— c(B => A) (6-7)

Example 3: cr; (Correlation and Chi-Square)
Correlation is a broadly used statistical measure for analyzing the relationship
between two variables. The correlation between A and B in Table 6.3 is measured as

follows:

: f11f22_f12f21 (6.8)
\/fl+f+lf2+f+2

 

 

0077‘

The correlation measure compares the contrast between the following set of base rules

and their neighborhood rules:

- bris{A : 3,3: A,A: 3,3: 2}
-u(br)is{A:>B,B:> A,A:> 3,8: A}
- M: <confidence, confidence>,

- A: The difference in the square root of confidence products (see Eq. 6.9).

A : Jclczcztc‘: _ JC5C6C7C8 (6'9)

 

 

where c1, 62, C3, C4, C5, C6, C7, and C3 correspond to c(A:B), c(B: A), 42:3),
c(B : A) , C(A : B), c(3 : A), C(A : 3), and c(B : A) respectively. Eq. 6.10 is obtained

by expanding Eq. 6.9.

 

 

 

 

=JP(AB)P(AB)P(3§)P(Z§)_\/P(A3)P(A3)P(73)P(73)

_ _ _ _ (6.10)
P(A) 3(3) P(A) 3(3) P(A) 3(3) P(A) P(B)

189

Z P(AB)P(A—B—)— P(AB)P(AB)
JPtA)P(B)P(Z)P(§)

 

A

 

(6.11)

Eq. 6.11 is the correlation between A and B as shown in Eq. 8. Chi-square measure is

related to correlation in the following way:

2
= 2' 6.12
corr A ( )

Therefore, both measures are essentially comparing the same type of contrast.

Contrast rules and interestingness measures

Different measures have different perspectives on finding interesting rules.
Specifically, each measure defines a base rule and a neighborhood of rules from which
interesting contrast rules can be detected. In our proposed approach a user can choose a
measure and detect the corresponding contrast rules. In addition, the user has flexibility
to choose a base rule/attribute according to what he or she is interested in. Then he or she
selects the neighborhood rules as well as the measures to detect the base rule and its
neighborhood. This is similar to rule template approaches (see 6.2.1). We implemented

examples 1-3 for LON-CAPA data sets, which will be explained in section 6.5.

6.4 Algorithm

In this section we propose an algorithm to find surprising and interesting rules based
on the characteristics of different segments of students/problems. The difficulty with

190

algorithms such as Apriori is that when the minimum-support is high, we miss many
interesting, but infrequent patterns. On the other hand if we choose a minimum-support
that is too low the Apriori algorithm will discover so many rules that finding interesting

ones becomes difficult.

 

Mining Contrast Rules (M CR) Algorithm:

 

Input: D — Input set of N transactions
B — Target variable, the basis of interesting contrasts
a— Minimum (very) low support
m — A measure for ranking the rules
k — Number of the most interesting rules
Divide data set D based on the values of the target variable
foreach j in 3
Select 00'), a subset of transactions including j
Find the set of closed frequent itemsets, L(/') within DU)
foreach e e L( j)
Generate rule I : j
Compute measure m(/.’ : j)
end
end
Find common rules among the different groups of rules
foreach br and u(br) pair compute difference in measures, A
Sort the rules with respect to A

Select top k rules
return R

 

Figure 6.7 Mining Contrast Rules (MCR) algorithm for discovering interesting
candidate rules

In order to employ the MCR algorithm, several steps must be taken. During the
preprocessing phase, we remove items whose support is too high. For example, if 95% of
students pass the course, this attribute will be removed from the itemsets so that it does
not overwhelm other, more subtle rules. Then we must also select the target variable of
the rules to be compared. This allows the user to focus the search space on subjectively

interesting rules. If the target variable has C distinct values, we divide the data set, D, into

191

C disjoint subsets based on the elements of the target variable, as shown in Figure 6.7.
For example, in the case where gender is the target variable, we divide the transactions
into male and female subsets to permit examination of rule coverage.

Using Borgelt’s implementation” of the Apriori algorithm (version 4.21), we can
find closed itemsets employing a simple filtering approach on the prefix tree (Borgelt,
2003). A closed itemset is a set of items for which none of its supersets have exactly the
same support as itself. The advantage of using closed frequent itemsets for our purposes
is that we can focus on a smaller number of rules for analysis, and larger frequent
itemsets, by discarding the redundant supersets.

We choose a very low minimum support to obtain as many frequent itemsets as is
possible. Using perl scripts, we find the common rules between two contrast subsets.
Finally, we rank the common rules with all of the previously explained measures, and
then the top k rules of the sorted ranked-rules are chosen as a candidate set of interesting

rules. Therefore an important parameter for this algorithm is minimum support, 0'; the

lower the o; the larger the number of common rules. If the user selects a specific ranking

measure, m, then the algorithm will rank the rules with respect to that measure.

6.5 Experiments

In this section we first provide a general model for data attributes, data sets and their
selected attributes, and then explain how we handle continuous attributes. Finally, we

discuss our results and experimental issues.

 

13 The code for this program is available at http://fuzzy.cs.uni-magdeburg.de/~borgelt/apriori.html.
192

 

6.5.1 Data model

In order to better understand the interactions between students and the online
educational system, a model is required to analyze the data. Ideally, this model would be
both descriptive and predictive in nature.

As shown in Figure 6.8, each student is characterized by a set of attributes which are

static for any particular analysis (GPA, gender, ethnicity, etc.) and can be easily
quantized. The u-tuple (S,.('),S,.(2’, ..., Sf”) describes the characteristics of the i-th
student. The set of problems is determined by the scope of the analysis — at this time,

single courses over individual terms, but with future possibilities for multi-term analysis

— and characterized by a set of attributes, some of which are fixed (Bloom’s taxonomic

categorization, content type, simulation-dependent, etc.). The v-tuple (Pf”,Pj‘2), ...,

P11” ) describes the characteristics of the j-th problem.

 

 

S 1(1) Student

 

 

 

 

 

  

 

 

   

 

 

 

 

 

 

 

 

(53.931322), 53,3“) Problem

3:") '
hala‘ ‘

(u)
52 \
S") P—r
(56.2.9.5? , SP1?)
5(a)

 

 

 

 

 

 

 

 

 

 

 

Figure 6.8 Attribute mining model, Fixed students’ attributes, Problem
attributes, and Linking attributes between students and problem

193

 

The interaction of these two sets becomes a third space where larger questions can be

asked. The k-tuple (SR}”,SP;2), ..., SPJ'”) describes the characteristics of the i-th

student linking to the j-th problem. LON-CAPA records and dynamically organizes a
vast amount of information on students' interactions with and understanding of these
materials.

The model is framed around the interactions of the two main sources of interpretable
data: students and assessment tasks (problems). Figure 6.9 shows the actual data model,
which is frequently called an entity relationship diagram (ERD) since it depicts categories

of data in terms of entities and relationships.

 

 

 

 

 

 

 

 

 

 

 

STUDENT

Student ID 4 COURSE

Name x r / Cousre ID

Birth date 7’0 EN ROLLS 'N 0‘“: Name

Address Schedule

Ethnicity ‘ Credits

GPA ‘~ ,

Lt_GPA Q

Department '

Gender I

\——HA
GENERATES 3—1
l
/1 0

ACTIVITY LOG , A
Stu ID (is ID Prb ID 1 _ 'ASSESSNENTTA—SR “7
#of Tries ._o__ ?er Ib‘_""” * r
Success ; ‘f, Open date !
Time BELONGSTO Due date ’
aide . . , ,, .‘ K\—O< Type i

Degree of Difficulty 1
Degree of Discrimination .

 

 

Figure 6.9 Entity Relationship Diagram for a LON-CAPA course

194

 

The attributes selected for association analysis are divided into four groups within
the LON-CAPA system:

a) Student attributes: which are fixed for any student. Attributes such as Ethnicity,
Major, and Age were not included in the data out of necessity — the focus of this work is
primarily on the LON-CAPA system itself, so the demographics of students is less
relevant. As a result, the following three attributes are included:

GPA: is a continuous variable that is discretized into eight intervals between zero and
four with a 0.5 distance.

Gender: is a binary attribute with values Female and Male.

LtGPA (Level Transferred (i.e. High School) GPA): measured the same as GPA

b) Problem attributes: which are fixed for any problem. Among several attributes for
the problems we selected the four following attributes:

DoDifl (degree of difficulty): This is a useful factor for an instructor to determine
whether a problem has an appropriate level of difficulty. DoDiff is computed by the total
number of students’ submissions divided by the number of students who solved the
problem correctly. Thus, DoDiff is a continuous variable in the interval [0,1] which is
discretized into terciles of roughly equal frequency: easy, medium, and hard.

DoDisc (degree of discrimination): A second measure of a problem’s usefulness in
assessing performance is its discrimination index. It is derived by comparing how
students whose performance places them in the top quartile of the class score on that
problem compared to those in the bottom quartile. The possible values for DoDisc vary

from —1 to +1. A negative value means that students in the lower quartile scored better

195

on that problem than those in the upper. A value close to +1 indicates the higher
achieving students (overall) performed better on the problem. We discretize this
continuous value into terciles of roughly equal frequency: negatively-discriminating, non-
discriminating, and positively-discriminating.

Angries (average number of tries): This is a continuous variable which is
discretized into terciles of roughly equal frequency: low, medium, and high.

c) Student/Problem interaction attributes: We have extracted the following attributes
per student per problem from the activity log:

Succ: Success on the problem (YES, NO)

Tries: Total number of attempts before final answer.

Time: Total time from first attempt until the final answer is derived.

d) Student/Course interaction attributes: We have extracted the following attributes
per student per course from the LON-CAPA system.

Grade: Student’s Grade, the nine possible labels for grade (a 4.0 scale with 0.5
increments). An aggregation of “grade” attributes is added to the total attribute list.

Pass-Fail: Categorize students with one of two class labels: “Pass” for grades above

2.0, and “Fail” for grades less than or equal to 2.0.

6.5.2 Data sets

For this research we selected three data sets from the LON-CAPA courses as shown
in Table 6.4. For example the second row of the table shows that BSlll (Biological
Science: Cells and Molecules) integrated 235 online homework problems, and 382
students used LON-CAPA for this course. BS] 11 had an activity log with approximately

239 MB of data. Though 881 11 is a larger course than LBSZ71 (first row of the table), a

196

 

physics course, it is much smaller than CEM141 (third row), general chemistry 1. This
course had 2048 students enrolled and its activity log exceeds 750MB, corresponding to

more than 190k student/problem interactions when students attempting to solve

homework problems.

Table 6.4 Characteristics of three MSU courses which used LON-CAPA in fall
semester 2003

 

 

. # of # of Size of # of
Data set Course Tltle Students Problems Activity log Interactions
LBS 271 Physics_l 200 174 152.1 MB 32,394

BS 11 1 BiologicalScience 382 235 239.4 MB 71,675
CEM141 Chemistry_1 2048 1 14 754.8 MB 190,859

 

For this chapter we focus on two target variables, gender and pass-fail grades, in
order to find the contrast rules involving these attributes. A constant difficulty in using
any of the association rule mining algorithms is that they can only operate on binary data
sets. Thus, in order to analyze quantitative or categorical attributes, some modifications
are required — binarization — to partition the values of continuous attributes into discrete

intervals and substitute a binary item for each discretized item. In this experiment, we

mainly use equal-frequency binning for discretizing the attributes.
6.5.3 Results

This section presents some examples of the interesting contrast rules obtained from
the LON-CAPA data sets. Since our approach is an unsupervised case, it requires some
practical methods to validate the process. The interestingness of a rule can be subjectively

measured in terms of its actionability (usefulness) or its unexpectedness (Silberschatz &

197

 

 

Tuzhilin, 1995; Piatetsky-Shapiro & Matheus, 1994; Liu et al., 1999; Silberschatz &

Tuzhilin, 1996).

One of the techniques for mining interesting association rules based on

unexpectedness. Therefore, we divide the set of discovered rules into three categories:

1.

Expected and previously known: This type of rule confirms user beliefs, and
can be used to validate our approach. Though perhaps already known, many
of these rules are still useful for the user as a form of empirical verification of
expectations. For our specific situation (education) this approach provides
opportunity for rigorous justification of many long-held beliefs.

Unexpected: This type of rule contradicts user beliefs. This group of
unanticipated correlations can supply interesting rules, yet their
interestingness and possible actionability still requires further investigation.
Unknown: This type of rule does not clearly belong to any category, and
should be categorized by domain-specific experts. For our situations,
classifying these complicated rules would involve consultation with not only
the course instructors and coordinators, but also educational researchers and

psychologists.

The following rule tables present five examples of the extracted contrast rules

obtained using our approach. Each table shows the coded contrast rule and the “support”

and “confidence” of that rule. Abbreviations are used in the rule code, and are

summarized as follows: Succ stands for success per student per problem, LtGPA stands

for transfer GPA, DoDiff stands for degree of difficulty of a particular problem, and

198

 

 

DoDisc stands for degree of discrimination of a problem. In our experiments, we used

three measures to rank the contrast rules:

6.5.3.1 Difference of confidences

The focus of this measure is on comparing the confidences of the contrast rules
(A :> B andA 2 23"). Therefore, top rules found by this measure have a high value of
confidence ratio (cl/oz). Contrast rules in Table 6.5 suggest that students in LBS 271 who

are successful in homework problems are more likely to pass the course, and this comes

with a confidence ratio cl/cz=12.7.

Table 6.5 LBS_271 data set, difference of confidences measure

 

 

 

 

 

Contrast Rules Support & Confidence
(Succ=YES) ==> Passed (s=86.1%, c=92.7%)
(Succ=YES) ==> Failed (s=6.8%, c=7.3%)-

 

 

This rule implies a strong correlation among the student’s success in homework

problems and his/her final grade. Therefore, this rule belongs to the first category; it is a

known, expected rule that validates our approach.

Table 6.6 CEM_141 data set, difference of confidences measure

 

 

Contrast Rules Support & Confidence
(Lt_GPA=[l .5,2)) ==> Passed (s=0.6%, c=7.7%)
(Lt GPA=[1.5,2)) ==> Failed

 

 

 

 

(s=7.l%, c=92.3%)

Contrast rules in Table 6.6 could belong to the first category as well; students with
low transfer GPAs are more likely to fail CEM 141 (cz/c1=12). This rule has the
advantage of actionability; so, when students with low transfer GPAs enroll for the

course, the system could be designed to provide them with additional help.

199

6.5.3.2 Difference of Proportions

The focus of this measure is on comparing the rules (3 :> A and B :> A). Contrast
rules in Table 6.7 suggest that historically strong students that take long periods of time
between their first (incorrect) solution attempt and subsequent attempts tend to be female.
Though this rule may belong to the second category, there is some empirical evidence
that female students have better performances over long periods of time than males
(Kashy D.; 2001). We found this interesting contrast rules using the difference of
confidences to discover the top significant rules for BS 111. Though the support of the

rules is low, that is the result would be of an interesting rule with low-support.

Table 6.7 BS_111 data set, difference of proportion measure

 

 

 

 

 

 

 

Contrast Rules Support & Confidence
Male ==> (Lt_GPA=[3.5,4] & Time>20 hours) (s=0.l%, c=26.3%)
Female ==>(LtLGPA=[3.5,4] & Time>20_hours) (s=0.6%, c=89.7%)

 

 

6.5.3.3 Chi-square

It is a well-known condition in chi-square testing for contingency tables that cell
expected values need to be above 5 to guarantee the veracity of the significance levels
obtained (Agresti, 2002). We do pruning if this limitation is violated in some cases, and

this usually happens when the expected support corresponding to f” or f. 2 in Table 6.3 is

 

 

 

 

low.
Table 6.8 CEM_141 data set, chi-square measure
Contrast Rules Support & Confidence
(Lt_GPA=[3,3.5) & Sex=Male & Tries=1) ==> Passed (s=4.4%, c=82.7%)
(Lt_GPA=[3,3.5fic Sex=Male & Tries=l) ==> Failed (s=0.9%, c=17.3%)

 

 

 

200

 

Contrast rules in Tables 6.8 suggest that students with transfer GPAs in the range of
3.0 to 3.5 that were male and answered homework problems on the first try were more
likely to pass the class than to fail it. (Cl/62:41.8). This rule could belong to the second

category. We found this rule using the chi-square measure for CEM 141.

Table 6.9 LBS_271 data set, difference of confidences measure

 

[ Contrast Rules Support & Confidence

 

& Succ=YES & Tries=l) (s=28.9%, c=94.l%)

[DoDiffimedium & DoDisc=non_discriminating
=> Passed

 

& Succ=YES & Tries=1) (5:1 .8%, c=6.9%)

(DoDiff=medium & DoDisc=non_discriminating
==> Failed

 

 

 

Contrast rules in Table 6.9 show more complicated rules for LBS 271 using
difference of proportion (cl/cz=15.9); these rules belong to the third (unknown) category
and fimher consultation with educational experts is necessary to determine whether or not

they are interesting.

6.6 Conclusion

LON-CAPA servers are recording students’ activities in large logs. We proposed a

general formulation of interesting contrast rules and developed an algorithm to discover a
set of contrast rules investigating three different statistical measures. This tool can help
instructors to design courses more effectively, detect anomalies, inspire and direct fiirther
research, and help students use resources more efficiently. An advantage of this
developing approach is its broad firnctionality in many data mining application domains.

Specifically, it allows for contrast rule discovery with very low minimum support,

201

 

therefore permitting the mining of possibly interesting rules that otherwise would go
unnoticed.

More measurements tend to permit discovery of higher coverage rules. A
combination of measurements should be employed to find out whether this approach for
finding more interesting rules can be improved. In this vein, we plan to extend our work

to analysis of other possible kinds of contrast rules. This work has been published in

(Minaei-Bidgoli et al., 2004g).

202

 

____‘

 

Chapter 7 Summary

This dissertation addresses the issues surrounding the use of a data mining
framework within a web-based educational system. We introduce the basic concepts of
data mining as well as information about current online educational systems, a
background on Intelligent Tutoring Systems, and an overview of the LON-CAPA system.
A body of literature has emerged, dealing with the different problems involved in data
mining for performing classification and clustering upon web-based educational data.
This dissertation positions itself to extend data mining research into web-based
educational systems — a new and valuable application. Results of data mining tools help
students use the online educational resources more efficiently while allowing instructors,

problem authors, and course coordinators to design online materials more effectively.

7.1 Summary of the work

This dissertation provides information about the structure of LON-CAPA data, data
retrieval processes, and representation of student statistical information including
problem and solution strategies. We explain how we provide assessment tools in LON-
CAPA on various aspects of teaching and learning. The LON-CAPA system is used for
both formative and summative assessment. Feedback from numerous sources has

improved the educational materials considerably, a continuous and cyclic task which data

mining has the opportunity to impact.

203

 

 

7.1.1 Predicting Student Performance

The first aim of this dissertation is to provide a data mining tool for classifying
student characteristics based on features extracted from their logged data. We can use this
tool to predict the group to which any individual student will belong with reasonable
precision. Eventually, this information will help students use course resources better,
based on the usage of that resource by other students in similar groups. Four tree-based
(C5.0, CART, Quest, and Cruise) and five non tree-based classifiers (k-nearest neighbor,
Bayesian, Parzen window and neural network) are used to segregate student data. Using a
combination of multiple classifiers leads to a significant accuracy improvement for
various LON-CAPA courses. Weighting the features and using a genetic algorithm to
minimize the error rate improves the prediction accuracy by at least 10% in all the cases
tested.

The successful implementation of student classification to predict their performance,
demonstrates the merits of using the LON-CAPA data for pattern recognition in order to
predict the students’ final grades based on features extracted from their homework data.
We design, implement, and evaluate a series of pattern classifiers with various parameters
in order to compare their performance in analyzing a real dataset from the LON-CAPA
system.

This approach is easily adaptable to different types of courses, different population
sizes, and allows for different features to be analyzed. This work represents a rigorous
application of known classifiers as a means of analyzing and comparing usage and
performance of students who have taken a technical course that was partially/completely

administered via the web.

204

 

7.1.2 Clustering ensembles

A second objective of this research is to extend previous theoretical work regarding
clustering ensembles with the goal of creating an optimal framework for categorizing
web-based educational resources. We propose non-adaptive and adaptive resampling
schemes for the integration of multiple clusterings (independent and dependent).
Experimental results show improved stability and accuracy for clustering structures
obtained via bootstrapping, subsampling, and adaptive techniques. This study shows that
meaningful consensus partitions for an entire data set of objects can emerge from
clusterings of bootstrap (with replacement) and subsamples (without replacement) of
small size.

Empirical studies are conducted on several data sets for different consensus
functions, number of partitions in the combination and number of clusters in each
component. The results demonstrate that there is a trade-off between the number of
clusters per component and the number of partitions, and that the sample size of each
partition needed in order to perform the combination process converges to an optimal
error rate. These improvements offer insights into specific associations within the data
sets. The challenging points of using resampling techniques for maintaining the diversity
of partitions are discussed. We show that a critical fraction of data exists such that the
structure of an entire data set can be perfectly detected. Subsamples of small sizes can
reduce computational costs and measurement complexity for many explorative data
mining tasks with distributed sources of data. This empirical study also compares the
performance of adaptive and non-adaptive clustering ensembles using different consensus

functions on a number of data sets. By focusing attention on the data points with the least

205

 

consistent clustering assignments, one can better approximate the inter-cluster boundaries
and improve clustering accuracy and convergence speed as a function of the number of
partitions in the ensemble. The comparison of adaptive and non-adaptive approaches is a
new avenue for research, and this study helps to pave the way for the useful application

of distributed data mining methods.

7.1.3 Interesting association rules

Finally, this dissertation proposes techniques for discovering interesting associations
between student attributes, problem attributes, and solution strategies. We develop an
algorithm for the discovery of “interesting” association rules within a web-based
educational system. The main focus is on mining interesting contrast rules, which are sets
of conjunctive rules describing interesting characteristics of different segments within a
population. In the context of web-based educational systems, contrast rules help to
identify attributes characterizing patterns of performance disparity between various
groups of students. This dissertation presents a general formulation of contrast rules as
well as a new algorithm for mining interesting contrast rules.

We address the issue of choosing different measures for the discovery of contrast
rules. Different measures have different perspectives on finding interesting rules.
Specifically, each measure defines a base rule and a neighborhood of rules fi'om which
interesting contrast rules can be detected. In our proposed approach a user can choose a
measure and detect the corresponding contrast rules. In addition, the user has flexibility
to choose a base rule/attribute according to what he or she is interested in. Then the user

selects the neighborhood rules as well as the measures to detect the base rule and its

neighborhood.

206

 

 

 

Examining these contrasts can improve the online educational systems for both

teachers and students — allowing for more accurate assessment and more effective

evaluation of the Ieaming process.

7.2 Future work

There are several promising directions to extend the work presented in this thesis:

1.

Develop a tool to find the effects of different types of problems on student
achievement. These problems will be classified to find patterns in which students
are successful.

Develop techniques that apply student information in helping individuals to use
resources more efficiently (recommendation system). As an example, the
following suggestion might be made by the system: “You are about to start a test.
Other students similar to you, who succeeded in this test, have also accessed
Section 5 of Chapter 3. You did not. Would you like to access it now before
attempting the test?”

Find clusters of learners with similar browsing behavior, given students ’ browsing
data and course contents. Though the implications of this clustering are not
completely known at this time, it seems a valid question amidst the other solid
and useful applications of this work.

Identify those students who are at risk of failure, especially in very large classes.
This will help the instructor provide appropriate advising in a timely manner.
Identify sequences of strategies that students use in solving homework problems.
This may help in detecting anomalies in designed problems and assist instructors

in developing more effective homework.

207

 

 

 

 

Appendix A: Tree Classifiers Output

5.0

Using C5.0 for classifying the students: This result shows the error rate in each fold
in lO-fold cross-validation, and confusion matrix.

In 2-classes (PJassed, Failed)

Fold Rules

No Errors

18.2%
22.7%
27.3%
30.4%
17.4%
21.7%
13.0%
17.4%
17.4%
21.7%

H
GDDCDOQQU‘NQKO

\OCOxlONU‘ubL-JNr-‘O

20.7%
1.6%

Mean
SE 0.

\roo
O

In 3-classes (High, Middle, Low) we got the following results:

Fold Decision Tree
Srze Errors

0 7 36.4%
1 12 45 5%
2 6 45.5%
3 7 47 8%
4 10 34 8%
5 9 34 8%
6 6 47 8%
7 8 43.5%
8 10 47 8%
9 9 47.8%
Mean 8.4 43.2%
St 0.6 1 8%

208

In 9-classes we got the following results:

 

Fold Decision Tree
Size Errors
0 57 81 8%
1 51 63 6%
2 55 63 6%
3 61 78 3%
4 48 73.9%
5 56 73 9%
6 58 69 6%
7 53 87 0%
8 56 78.3%
9 56 73 9%
Mean 55.1 74.4%
SE 1.2 2.4%
(a) (b) (c) (d) (e) (f) (g) (h) (i) <-classified as
1 1 (a): class 1 t
i i L L 2 (b): class 2 -

1 7 t 1 7 3 (c): Class 3

4 L 5 r 4 3 3 (d): class 4

a 5 F 12 11 9 2 (e): class 5

T S i 15 9 7 (f): Class 6

4 6 3 15 5 8 (g): class 7

l 1 3 2 14 (h): class 8

(1): class 9

Here, there are a sample of rule sets resulted the from C50 in 3-class classification

209

Rule

Rule

Rule

Rule

Rule

Rule

1: (8, lift 2.9)
FirstCorrect > 64
FirstCorrect <= 112
TotalCorrect > 181
Angries > 1270
TotalTimeSpent <= 87.87
Discussion <= 0
-> class High [0.900]

2: (5, lift 2.8)
FirstCorrect > 93
FirstCorrect <= 99
TotalCorrect > 181
Angries <= 1270
Discussion <= 14
-> class High [0.857]

3: (15/2, lift 2.7)
FirstCorrect <= 112
TotalCorrect > 181
Discussion > 0
Discussion <= 14
-> class High [0.824]

4: (8/1, lift 2.6)
FirstCorrect <= 112
TotalCorrect > 174
TotalCorrect <= 180
Angries <= 1768
Discussion <= 0
-> class High [0.800]

5: (3, lift 2.6)
FirstCorrect > 112
FirstCorrect <= 117
TotalCorrect > 180
TotalTimeSpent > 14.01
Discussion <= 1
—> class High [0.800]

15: (3/1, lift 2.2)
FirstCorrect <= 112
TotalCorrect > 180
TotalCorrect <= 181
Discussion <= 0
—> class Low [0.600]

Here, there are a sample of rule sets resulted the from C50 in 2-class classification

210

 

 

Rules:

Rule 1: (158/25, lift 1.2)
TotalCorrect > 165
-> class Passed [0.838]

Rule 2: (45/8, lift 1.1)
Discussion > 1
-> class Passed [0.809]

Rule 3: (7, lift 3.2)
FirstCorrect <= 78
TotalCorrect <= 165
-> class Failed [0.889]

Rule 4: (2, lift 2.7)
TotalCorrect <= 165
AVgTries > 669
Discussion > 1
Discussion <= 4
-> class Failed [0.750]

Rule 5: (47/15, lift 2.4)
TotalCorrect <= 165
-> class Failed [0.673]

Default class: Passed

Evaluation on hold-out data (22 cases):

Rules
No Errors
5 3(13.6%) <<

And a sample of tree, which is produced by C50 in one of the folds in 3 classes:

21]

TotalCorrect <= 165:
:...Angries > 850: Low (13/2)
Angries <= 850:
:...Discussion > 2:
:...TotalTimeSpent <= 20.57: Low (2)
TotalTimeSpent > 20.57: Middle (3/1)
Discussion <= 2:
:...TotalTimeSpent > 22.63: Low (8)
TotalTimeSpent <= 22.63:
:...Angries <= 561: Low (7/1)
Angries > 561:
:...TotalCorrect > 156: Low (2)
TotalCorrect <= 156:
:...TotalCorrect <= 136: Low (3/1)
: TotalCorrect > 136: Middle (6)
TotalCorrect > 165:
:...Angries <= 535:
:...TotalCorrect <= 177: Low (5)
TotalCorrect > 177: High (5/2)
Angries > 535:
:...FirstCorrect > 112:
:...TotalCorrect <= 172: Middle (6)
TotalCorrect > 172:
:...TotalCorrect > 180: Middle (38/13)
TotalCorrect <= 180:
:...TimeTillCorr <= 23.47:
:...TotalCorrect > 178: High (2)
TotalCorrect <= 178:
:...TotalCorrect <= 174: High (4/2)
: TotalCorrect > 174: Middle (8/1)
TimeTillCorr > 23.47:
:...FirstCorrect > 129: Middle (2)
FirstCorrect <= 129:
:...TotalCorrect > 175: Low (7/1)
TotalCorrect <= 175:
:...FirstCorrect <= 118: Low (2)
. FirstCorrect > 118: High (2)
FirstCorrect <= 112:
:...TotalTimeSpent > 87.87: Middle (5/1)
TotalTimeSpent <= 87.87:
:...TotalCorrect <= 169: High (5/1)
TotalCorrect > 169:
:...TotalCorrect <= 174: Middle (8)
TotalCorrect > 174:
:...Discussion > 7: Middle (5/1)
Discussion <= 7:
:...TotalCorrect <= 177: High (5/1)
TotalCorrect > 177:
:...TotalCorrect <= 181:
:...Angries <= 1023: High (3)
Angries > 1023: Middle (9/2)
TotalCorrect > 181:
:...Discussion > 0: High (15/2)
Discussion <= 0:
:...FirstCorrect > 99: Middle (5/1)
FirstCorrect <= 99:
:...FirstCorrect > 89: High (7/1)
FirstCorrect <= 89:
:...Angries <= 1355: Middle (4)
Angries > 1355: [S1]

Evaluation on hold-out data (22 cases):
Decision Tree
Size Errors
32 7(31.8%) <<

212

CART

Some of CART report for 2-Classes using Gini criterion:

File:PHY183.XLS

Target Variable: CLASSZ

Predictor Variables: FIRSTCRR, TOTCORR, TRIES, SLVDTIME,
TOTTIME, DISCUSS

Tree Sequence

 

 

 

 

 

 

 

 

 

 

 

Tree Termin Cross- Resubstitution Complexity
Numbe al Validated Relative Cost

r Nodes Relative Cost

1 23 0.873 :1: 0.099 0.317 -1.000

2 22 0.984 3: 0.104 0.317 1.00E-005

3 15 1.016 d: 0.104 0.397 0.003

4 9 0.762 :t 0.089 0.476 0.004

5 7 0.778 i 0.091 0.508 0.004

6 5 0.841 i 0.093 0.556 0.007
7** 3 0.667 :t 0.090 0.619 0.009

8 2 0.714 d: 0.088 0.683 0.018

9 1 1.000 t 6.73E- 1.000 0.088

005

 

 

 

 

 

 

* Minimum Cost
** Optimal

Classification tree topology for: CLASSZ

213

 

 

 

Error Curve

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

214

l 1;; 1.5
I 8
‘ .j‘>_’ 1.0 ~~-l - - -_ — -I-..__
i m0_5nerv t'rrrr; Y;.4. ¢.,T, ;-_A, +,,Afir+1|+g .‘et
" o1234567891011121314151617181920212223
Number of Nodes
Gains for2
i“ _ _
i ‘ 100
i
:80
i g 160
l g {40
lezo
0 -r— t +«—~—+——-—+ t-- inieo
0 20 40 60 80 100
%Population
Gains Data for 2
Node Cases % of % Cum% Cum% % Cases Cum Lift
Class Node Class Class 2 Pop Pop in Node lift Pop
2 Class2 2
1 35 70.00 55.56 55.56 22.03 22.03 50 2.522 2.522
2 7 70.00 11.11 66.67 26.43 4.41 10 2.522 2.522
3 21 12.57 33.33 100.00 100.00 73.57 167 1.000 0.453
Variable Importance
Variable
TOTCORR 100-00 IIHIIIIIIIIIIHIIIIIIIIIIIIHIIIIIIIIIIll
TRIES 56-32 Illllllllllllllllllllll
FIRSTCRR 4.58 I
TOTTIME 0.91
SLVDTIME 0.83
DISCUSS 0.00

 

 

Misclassification for Learn Data

 

Class N NMis- Pct Cost
Cases Classed Error
1 164 18 10.98 0.11
2 63 21 33.33 0.33

 

 

 

 

 

 

 

 

 

Misclassification for Test Data

 

Class N NMis- Pct Cost
Cases Classed Error
1 164 21 12.80 0.13
2 63 21 33.33 0.33

 

 

 

 

 

 

 

 

 

Some of CART report for 3-Classes using Twoing criterion: (1 O-fold Cross-Validation):

215

Tree Sequence

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Tree Termin Cross- Resubstitution Complexity
Numbe a1 Validated Relative Cost
r Nodes Relative Cost

1 42 0.802 3: 0.050 0.230 -1.000

2 38 0.808 t 0.050 0.236 0.001
3 37 0.808 t 0.050 0.238 0.001
4 36 0.794 d: 0.050 0.242 0.003
5 35 0.786 at 0.050 0.247 0.003
6 27 0.778 i 0.050 0.289 0.004
7 24 0.762 3: 0.050 0.311 0.005
8 23 0.762 t 0.050 0.319 0.005
9 22 0.761 :t 0.050 0.327 0.005
10 21 0.731 :t 0.049 0.336 0.006
11 18 0.734 :t 0.049 0.366 0.007
12 14 0.727 i 0.049 0.407 0.007
13 13 0.740 :1: 0.049 0.418 0.007
14 11 0.732 3: 0.049 0.444 0.009
15** 10 0.694 :t 0.049 0.457 0.009
16 8 0.720 i 0.050 0.500 0.014
17 6 0.743 :t 0.050 0.545 0.015
18 5 0.741 :t 0.050 0.574 0.019
19 4 0.728 t 0.050 0.605 0.021
20 3 0.745 1 0.050 0.661 0.037
21 2 0.758 :t 0.035 0.751 0.060
22 1 1.000 i 0.000 1.000 0.166

 

 

 

 

 

Classification tree topology for: CLASS3

 

Error Curve

216

 

 

0.90

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0.80 _ ll: --—l
g 0.70 '.'-.11.-.. ’ ' '.‘-Hi. i.
C‘ 0.60 7 . 7+ —-—--+ —-- 7+ "+7 —7———-77+— 7————++ 7+
0 10 20 30 40 50
Number of Nodes
Gains for 1
I
0v r 7 + 0
0 20 40 60 80 100
%Populalion
Gains Data for 1
Node Cases % of % Cum % Cum % Cases Cum Lift
Class 1 Node Class 1 Class 1 % Pop in lift Pop
Class 1 Pop Node
9 4 80.00 5.80 5.80 2.20 2.20 5 2.632 2.632
2 4 80.00 5.80 11.59 4.41 2.20 5 2.632 2.632
6 31 62.00 44.93 56.52 26.43 22.03 50 2.138 2.040
4 8 57.14 11.59 68.12 32.60 6.17 14 2.090 1.880
7 13 27.08 18.84 86.96 53.74 21.15 48 1.618 0.891
5 1 12.50 1.45 88.41 57.27 3.52 8 1.544 0.411
10 2 11.76 2.90 91.30 64.76 7.49 17 1.410 0.387
8 2 11.11 2.90 94.20 72.69 7.93 18 1.296 0.366
1 4 8.00 5.80 100.00 94.71 22.03 50 1.056 0.263
3 0 0.00 0.00 100.00 100.0 5.29 12 1.000 0.000

 

 

 

 

 

 

 

 

 

 

 

217

Variable Importance

 

Variable

 

TOTCORR 100.00

llllllllllllllllllllllllllllllllllllllllll

 

TRIES 40.1 1

llllllllllllllll

 

FIRSTCRR 24.44

llllllllll

 

TOTTIME 23.22

lllllllll

 

SLVDTIME 21.67

llllllll

 

 

 

DISCUSS 14.44

 

lllll

 

Misclassification for Learn Data

 

 

 

 

 

 

 

 

 

Class N NMis- Pct Cost
Cases Classed Error

1 69 22 31.88 0.32

2 95 34 35.79 0.36

3 63 15 23.81 0.24

 

Misclassification for Test Data

 

 

 

 

 

 

 

 

 

Class N NMis- Pct Cost
Cases Classed Error

1 69 35 50.72 0.51

2 95 52 54.74 0.55

3 63 21 33.33 0.33

 

218

 

 

 

 

Different tree topologies for: CLASS-9

Entropy Gini Twoing

Descriptive Statistics in CART for 3-Classes

 

Variable N Mean SD Min Max
Su
Overall

FIRSTCRR 227.00 106.242 20.462 47.000 150.000
24117.000

TOTCORR 227.00 171.678 18.155 80.000 184.000
38971.000

TRIES 227.00 977.987 450.898 265.000 3095.000
222003.000

SLVDTIME 227.00 36.620 24.837 2.590 130.870
8312.700

TOTTIME 227.00 37.948 25.434 3.000 130.870
8614.170

DISCUSS ‘227.00 1.330 3.034 0.000 23.000
302.000
CLASSS * 1

FIRSTCRR 69.00 103.145 19.598 57.000 149.000
7117.000

TOTCORR 69.00 179.290 7.900 141.000 184.000
12371.000

TRlES 69.00 1088.406 439.742 487.000 2227.000
75100.000

SLVDTIME 69.00 39.060 22.209 2.590 98.840
2695.130

TOTTIME 69.00 39.764 22.797 3.000 99.200
2743.720

DISCUSS 69.00 1.493 2.988 0.000 14.000
103.000
CLASS3 = 2

FIRSTCRR 95.00 108.505 20.973 54.000 150.000
10308.000

TOTCORR 95 00 175.453 12.412 118.000 184.000
16668.000

TRIES 95.00 984.937 443.874 392.000 3095.000
93569.000

SLVDTIME 95.00 36.866 27.353 4.100 130.870
3502.240

TOTTIME 95.00 37.916 27.865 4.130 130.870
3602.010

DISCUSS 95.00 1.537 3.596 0.000 23.000
146.000
CLASS3 = 3

219

 

FIRSTCRR
6692.000
TOTCORR
9932.000
TRIES
53334.000
SLVDTIME
2115.330
TOTTIME
2268.440
DISCUSS
53.000

63.

63.

63.

63.

63.

63.

00

00

00

00

00

00

106.

157.

846.

33.

36.

222

651

571

577

007

.841

20.

24

23.

24

482

.763

.210

605

.561

.953

47.

80.

000

000

.000

.870

.920

.000

184

2623.

107.

.000

.000

000

100

.210

.000

A Sample of CART tree for 3-Classes using Entropy criterion: (10-fold
Cross-validation)

     

Node 1
TOTCO RR

 

 

Terminal
Node 1
W = 50.000
lIl‘lfrrttt-l

 

 

 

   

> 10 .500

Node 2
FIRST CRR

   

 

<= 1+ .500

 

7 Node 3
TOTCO RR

= ”[500

Node 4
TOTCORR

 

i

l

l

      

> 18 .500

Node 5
SLVDTME

 

l

 

 

 

 

Terminal
Node 6
W = 9.000
.m. 3‘!

Terminal
Node 7
W = 61 .000
l_—

 

 

 

 

 

Terminal
Node 2
W = 5.000
—I

 

 

 

Terminal
Node 3
W = 45.000
_—

 

 

Terminal
Node 4
W = 48.000
_-

 

 

Terminal
Node 5
W = 9.000
._I

 

 

 

220

 

QUEST

Summary of
Size

226

Summary of
Size

226

Summary of
Size

226

Summary of
Size

226

Summary of
Size

226

Summary of
Size

226

numerical

Obs

226

numerical

Obs

226

numerical

Obs

226

numerical

Obs

226

numerical

Obs

226

numerical

Obs

226

variable: FirstCorr
Min Max
0.470E+02 0.150E+03
variable: TotCorr
Min Max
0.960E+02 0.184E+03
variable: Angries
Min Max
0.193E+01 0.169E+02
variable: TimeCorr
Min Max
0.249E+01 0.942E+02
variable: TimeSpent
Min Max
O.260E+01 0.942E+02
variable: Discuss
Min Max
0.000E+00 0.140E+02

221

Mean

.106E+03

Mean

.172E+03

Mean

.551E+01

Mean

.2805+02

Mean

.281E+02

Mean

.912E+00

Sd

.204E+02

Sd

.171E+02

Sd

.246E+01

Sd

.1855+02

Sd

.185E+02

Sd

.201E+01

Result for 2-classes

Summary of response variable: ClassZ

class frequency
Failed 62
Passed 164

2 226

Options for tree construction
Learning sample
estimated priors are
Class prior
Failed 0.27434
Passed 0.72566
Size and CV misclassification cost and SE of subtrees:

Tree #Tnodes Mean SE(Mean)
l 26 0.1947 0.2634E-01
2' 16 0.1903 0.26llE—01
3 13 0.1947 0.2634E-01
4 12 0.2035 0.2678E—01
5 6 0.1947 0.2634E-01
6H 0.1947 0.2634E-01
7 3 0.2301 0.2800E—01
8 2 0.2434 0.2854E-01
9 1 0.2743 0.2968E-01

CART O-SE tree is marked with *
CART SE-rule using CART SE is marked with **

use lO—fold CV sample pruning
SE-rule trees based on number of 885 = 1.00

subtree # Terminal complexity current

number nodes value cost
1 26 0.0000 0.0885
2 16 0.0022 0.1106
3 13 0.0029 0.1195
4 12 0.0044 0.1239
5 6 0.0052 0.1549
6 4 0.0111 0.1770
7 3 0.0177 0.1947
8 2 0.0354 0.2301
9 1 0.0442 0.2743

Classification tree:

Node 1: TotCorr <= 156.9
Node 2: Failed
Node 1: TotCorr > 156.9
Node 3: TotCorr <= 168.8
Node 18: Discuss <= 1.279
Node 20: Failed
Node 18: Discuss > 1.279
Node 21: Passed
Node 3: TotCorr > 168.8
Node l9: Passed

Classification matrix based on learning sample
predicted class
actual class Failed Passed
Failed 35 27
Passed 13 151

Classification matrix based on 10-fold CV
predicted class
actual class Failed Passed
Failed 33 29
Passed 15 149

222

 

 

Result for 3-classes

use lO-fold CV sample pruning
SE-rule trees based on number of SEs

1.00

Size and CV misclassification cost and SE of subtrees:

Tree #Tnodes Mean SE(Mean)
1 47 0.5354 0.3318E-01
2 30 0.5265 0.3321E-01
3 24 0.5354 0.3318E—01
4 10 0.4735 0.3321E-01
5‘ 9 0.4425 0.3304E-01
6 8 0.4513 0.3310E~01
7 6 0.4602 0.3315E-01
8** 4 0.4735 0.3321E-01
9 2 0.4823 0.3324E~01

10 1 0.5796 0.3283E-01

CART O-SE tree is marked with '
CART SE-rule using CART SE is marked with '*

Following tree is based on **

 

Structure of final tree

Node Left node Right node Split variable Predicted class

1 2 3 TotCorr

2 * terminal node * Low

3 26 27 lstGotCrr
26 28 29 TotCorr
28 ' terminal node * Middle
29 * terminal node ‘ High
27 * terminal node * Middle

Number of terminal nodes of final tree = 4

Total number of nodes of final tree 7

Classification tree:

Node 1: TotCorr
Node 2: Low
Node 1: TotCorr > 165.5
Node 3: lstGotCrr <= 117.5
Node 26: TotCorr <= 181.5
Node 28: Middle
Node 26: TotCorr > 181.5
Node 29: High
Node 3: lstGotCrr > 117.5
Node 27: Middle

<= 165.5

Classification matrix based on learning sample
predicted class

actual class High Low Middle

High 34 4 31

Low 2 34 26

Middle 21 11 63
Classification matrix based on lO-fold CV

predicted class

actual class High Low Middle

High 27 10 32

Low 4 37 21

Middle 25 15 55

Bagging (Leeve-one-out method)

estimated priors are

Class prior
High 0.30531
Low 0.27434

223

 

Middle
minimal node size: 2
use univariate split
use (biased) exhaustive search for variable and split selections
use the divergence famliy
with lambda value: 0.5000000

0.42035

use 226-fold CV sample pruning
SE-rule trees based on number of SEs = 1.00

Size and CV misclassification cost and SE of subtrees:

Tree #Tnodes Mean SE(Mean)
1 67 0.5354 0.3318E—01
2 61 0.5354 0.3318E-01
3 31 0.5177 0.3324E—01
4 24 0.5000 0.3326E-01
5* 10 0.4204 0.3283E—01
6 9 0.4425 0.33048-01
7** 8 0.4469 0.3307E-01
8 6 0.5000 0.3326E-01
9 4 0.4956 0.3326E-01

10 2 0.4823 0.3324E—01
11 1 0.5796 0.3283E-01

CART O—SE tree is marked with *
CART SE-rule using CART SE is marked with **

Following tree is based on **

Structure of final tree

Node Left node Right node Split variable Predicted class

1 2 3 TotCorr
2 * terminal node * Low
3 32 33 lstGotCrr
32 34 35 TotCorr
34 36 37 TotCorr
36 * terminal node * High
37 terminal node * Middle
35 70 71 TimeCorr
70 terminal node * High
71 terminal node * Middle
33 104 105 TimeCorr
104 terminal node * Middle
105 132 133 TotCorr
132 terminal node * Low
133 terminal node * Middle
Number of terminal nodes of final tree = 8
Total number of nodes of final tree = 15
Number of terminal nodes of final tree = 10

Total number of nodes of final tree = 19

Classification tree:

Node 1: TotCorr <= 165.5
Node 2: Low
Node 1: TotCorr > 165.5
Node 3: lstGotCrr <= 117.5
Node 26: TotCorr <= 181.5
Node 28: TotCorr <= 169.5
Node 30: High
Node 28: TotCorr > 169.5
Node 31: Middle
Node 26: TotCorr > 181.5
Node 29: TimeCorr <= 52.44
Node 56: High
Node 29: TimeCorr > 52.44

224

Node 57: Middle
Node 3: lstGotCrr > 117.5
Node 27: TimeCorr <= 24.49
Node 80: Middle
Node 27: TimeCorr > 24.49
Node 81: TotCorr <= 180.5
Node 104: Low
Node 81: TotCorr > 180.5
Node 105: lstGotCrr <= 130.0
Node 110: TimeCorr <= 27.16
Node 112: Low
Node 110: TimeCorr > 27.16
Node 113: Middle
Node 105: lstGotCrr > 130.0
Node 111: High

Classification matrix based on learning sample
predicted class

actual class High Low Middle
High 38 5 26

Low 3 47 12

Middle 13 13 69

Classification matrix based on 226-fold CV
predicted class

actual class High Low Middle
High 31 9 29

Low 5 43 14

Middle l9 19 57

225

‘.

 

Result for 9-classes

Classification tree:

Node 1: TotCorr <= 104.0
Node 2: 0
Node 1: TotCorr > 104.0
Node 3: TotCorr <= 165.5
Node 4: 3
Node 3: TotCorr > 165.5
Node 5: TotCorr <= 181.5
Node 44: TimeCorr <= 2.565
Node 46: 8
Node 44: TimeCorr > 2.565
Node 47: TimeCorr <= 71.06
Node 48: Angries <= 3.145
Node 50: 4
Node 48: Angries > 3.145
Node 51: lstGotCrr <= 77.00
Node 56: 5
Node 51: lstGotCrr > 77.00
Node 57: Angries <= 12.52
Node 58: TimeCorr <= 40.50

Node 60: 5
Node 58: TimeCorr > 40.50
Node 61: 6
Node 57: Angries > 12.52
Node 59: 3
Node 47: TimeCorr > 71.06
Node 49: 3

Node 5: TotCorr > 181.5
Node 45: Angries <= 2.630
Node 108: 4
Node 45: Angries > 2.630
Node 109: lstGotCrr <= 111.5
Node 110: lstGotCrr <= 55.50
Node 112: 5
Node 110: lstGotCrr > 55.50
Node 113: TimeCorr <= 57.44
Node 114: Angries <= 5.330
Node 116: 6
Node 114: Angries > 5.330
Node 117: 8
Node 113: TimeCorr > 57.44
Node 115: 6
Node 109: lstGotCrr > 111.5
Node 111: 6

predicted class

actual class 0 2 3 4 5
0 1 0 0 0 0
2 1 0 8 0 0
3 0 0 21 2 2
4 0 0 7 5 8
5 0 0 6 0 32
6 0 0 5 0 19
7 0 0 2 1 16
8 0 0 2 0 3
elapsed time: 386.25 seconds (user: 386.14, system: 0.11)

226

t—IN
.nmowNwr—rom

OOOOOOOOQ

._s

\Odet—‘OOOW

CRUISE

Here, some output results of CRUISE for 3-classes: CV misclassification cost and

SE of subtrees:
Subtree CV R(t)
(largest) 0.549823 0
1 0.539823 0
2 0.544248 0
3 0.539823 0
4 0.526549 0
5 0.544248 0
6 0.553097 0
7 0.553097 0
8 0.561947 0
9 0.557522 0
10 0.535398 0
11 0.504425 0
12* 0.460177 0
13 0.504425 0
14 0.579646 0

* denotes 0-SE Tree
** denotes given-SE Tree
* tree is same as ** tree

Following tree is based on **
Splits of the Tree:

Node Split variable
1 TotCorr
2 * terminal *
3 TotCorr
8 TotCorr
24 ‘ terminal *
25 * terminal '
9 TotCorr
27 * terminal *
28 TimeCorr
84 * terminal *
85 * terminal *

Tree Structure:

CV SE

.3313E-01
.33158-01
.3313E-01
.33155‘01
.33218-01
.33138-01
.3307E-01
.3307E-01
.33008-01
.3304E-01
.33188-01
.3326E-01
.33155-01
.3326E-Ol
.3283E-01

4 Terminal Nodes

227

82
70
67
59
56
41
38
23
21
15

9

HNmCD

Node 1: TotCorr <= 163.156
Node 2: Terminal Node, predicted class = Low
Class label : High Low Middle
Class size : 3 28 11

Node 1: TotCorr > 163.156
Node 3: TotCorr <= 171.059
Node 8: TotCorr <= 168.639

Node 24: Terminal Node, predicted class = Low
Class label : High Low Middle
Class size : 2 7 1

Node 8: TotCorr > 168.639
Node 25: Terminal Node, predicted class = Middle
Class label : High Low Middle
Class size : 3 4 9

Node 3: TotCorr > 171.059
Node 9: TotCorr <= 183.206
Node 27: Terminal Node, predicted class = Middle
Class label : High Low Middle
Class size : 34 18 53

Node 9: TotCorr > 183.206

Node 28: ABS(TimeCorr - 35.4849 ) <= 19.1162
Node 84: Terminal Node, predicted class = High
Class label : High Low Middle
Class size : 22 4 11
Node 28: ABS(TimeCorr - 35.4849 ) > 19.1162
Node 85: Terminal Node, predicted class = Middle
Class label : High Low Middle
Class size : 5 1 10

Detailed Description of the Tree:

Nodes No. Subnode Split Split Split
label cases label stat. variable value
1 226 2 F TotCorr <= 163.16
3 < infinity
# obs mean/mode of TotCorr
Class High : 69 179.290
Class Low : 62 158.903
Class Middle : 95 175.453
2 42 *'** terminal, predicted class: Low
4 obs
Class High : 3
Class Low : 28
Class Middle : ll
3 184 8 F TotCorr <= 171.06
9 < infinity
# obs mean/mode of TotCorr
Class High : 66 180.576
Class Low : 34 175.176
Class Middle : 84 179.226
8 26 24 F TotCorr <= 168.64
25 < infinity
# obs mean/mode of TotCorr
Class High : 5 167.600
Class Low : 11 167.182
Class Middle : 10 169.800
24 10 ***' terminal, predicted class: Low
# obs
Class High : 2

228

Class Low 7
Class Middle 1
25 16 *'** terminal, predicted class: Middle
# obs
Class High 3
Class Low 4
Class Middle 9
9 158 27 F TotCorr <= 183.21
28 < infinity
# obs mean/mode of TotCorr
Class High 61 181.639
Class Low 23 179.000
Class Middle 74 180.500
27 105 **** terminal, predicted class: Middle
1 obs
Class High 34
Class Low 18
Class Middle 53
28 53 84 Levene ABS(TimeCorr - 35.5 ) <=19.116
85 < infinity
# obs mean/mode of TimeCorr
Class High 27 34.7652
Class Low 5 36.5700
Class Middle 21 36.1519
84 37 **** terminal, predicted class: High
# obs
Class High 22
Class Low 4
Class Middle 11
85 16 '**‘ terminal, predicted class: Middle
# obs
Class High 5
Class Low 1
Class Middle 10
Number of nodes in maximum tree = 153
Number of nodes in final tree = 11
Number of terminal nodes in final tree = 6
Classification Matrix Predicted class
High Low Middle
Actual class #obs Prior ---------------------
High 69 0.305 22 5 42
Low 62 0.274 4 35 23
Middle 95 0 420 11 12 72
Total obs = 226, # correct = 129
Resubstitution misclassification cost = 0.4292
S.E. of resubstitution misclassification cost = 0.3055E-01
Cross-validation error cost from pruning = 0.4602
S.E. of CV misclassification cost = 0.3315E-01

Elapsed system time in seconds

: 5.54

229

Bibliography

[1]

[2]

[51

[6]

[7]

[8]

[9]

Aeberhard, S., Coomans D., and de Vel, O. (1992) “Comparison of Classifiers in
High Dimensional Settings”, Tech. Rep. no. 92-02, (1992), Dept. of Computer
Science and Dept. of Mathematics and Statistics, James Cook University of North
Queensland.

Agrawal, R., Imielinski, T.; Swami A. (1993), "Mining Associations between Sets
of Items in Massive Databases", Proc. of the ACM-SIGMOD 1993 Int?
Conference on Management of Data, Washington DC, May 1993.

Agrawal, R.; Srikant, R. (1994) "Fast Algorithms for Mining Association Rules",
Proceeding of the 20th International Conference on Very Large Databases,
Santiago, Chile, September 1994.

Agrawal, R. and Srikant. R. (1995) “Mining Sequential Patterns”. In Proceeding

of the 1 1“ International Conference on Data Engineering, Taipei, Taiwan, March
1 995.

Agrawal, R., Shafer, J.C. (1996) "Parallel Mining of Association Rules", IEEE
Transactions on Knowledge and Data Engineering, Vol. 8, No. 6, December
1996.

Agresti, A. (2002), Categorical data analysis. 2"d edition, New York: Wiley,
2002.

Albertelli, G., Minaei-Bigdoli, B., Punch, W.F., Kortemeyer, G., and Kashy, E.,
(2002) “Concept Feedback In Computer-Assisted Assignments”, Proceedings of
the (IEEE/ASEE) Frontiers in Education conference, 2002

Albertelli, G. Sakharuk, A., Kortemeyer, G., Kashy, E., (2003) “Individualized
examinations for large on-campus courses”, Proceedings of the (IEEE/ASEE)
Frontiers in Education Conference 2003, vol 33.

Aming, A., Agrawal, R., and Raghavan, P. (1996) “A Linear Method for
Deviation Detection in Large Databases” Proceeding of 2"d international
conference on Knowledge Discovery and Data Mining, KDD I 996.

230

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

Azevedo, R, Bernard, R, M, “A Meta-analysis of the Effects of Feedback in
Computer-based Instruction”, J. Educational Computing Research 13, 111-127.
(1995)

Baker, J. E. (1987). Reducing bias and inefficiency in the selection algorithm,
Proceeding ICGA 2, pp. 14-21, Lawrence Erlbuam Associates, Publishers, 1987.

Baker, E. (1978). Cluster Analysis by Optimal Decomposition of Induced Fuzzy
Sets. Delfl University Press, Delfl, The Netherlands.

Baker, E., and Jain, A. K. (1981). “A Clustering performance measure based on
fuzzy set decomposition.” IEEE Transactions on Pattern Analysis and Machine
Intelligence PA MI 3, 66-75.

Bala J ., De Jong K., Huang J ., Vafaie H., and Wechsler H. (1997). Using Ieaming
to facilitate the evolution of features for recognizing visual concepts.

Evolutionary Computation 4(3) - Special Issue on Evolution, Learning, and
Instinct: 100 years of the Baldwin Eflect. 1997.

Bandyopadhyay, S., and Muthy, CA. (1995). “Pattern Classification Using
Genetic Algorithms”, Pattern Recognition Letters, Vol. 16, pp. 801-808.

Barchman R. J., and Anand T. (1996), The process of Knowledge Discovery in
Database, 37-57, AAAI/MIT Press.

Barthelemy J .-P. and Leclerc, B. (1995) “The median procedure for partition”, In
Partitioning Data Sets, A MS DIMA CS Series in Discrete Mathematics, I.J. Cox et
al eds., 19, pp. 3-34, 1995.

Bay, S. D. and Pazzani, M. J., (2001) “Detecting Group Differences: Mining
Contrast Sets. Data Mining and Knowledge Discovery, 2001, Vol 5, No 3 213-
246.

Ben-Hur, A., Elisseeff, A., and Guyon, I. (2002) "A stability based method for
discovering structure in clustered data," in Pac. Symp. Biocomputing, 2002, vol.
7, pp. 6-17.

Bloom, B. S. (1956). Taxonomy of Educational Objectives. New York, NY,
McKay.

Bloom, B. S. (1984). “The 2-Sigma Problem: The Search for Methods of Group
Instruction as Effective as One-to-one Tutoring.” Educational Researcher 13: 4-
l6.

Bonham, S, W, Beichner, R, J, and Deardorff, D. L, (2001). “Online Homework:
Does It Make a Difference?” The Physics Teacher.

231

[23]

[24]

[25]

[261

[27]

[28]

[29]

[30]

[311

[321

[33]

[341

[35]

[36]

Borgelt, C. (2003), “Efficient Implementations of Apriori and Eclat”, Workshop
of Frequent Item Set Mining Implementations (FIMI) 2003.

Bransford, J. D., A. L. Brown, et al. (2000). How People Learn. Washington,
DC, National Academy Press.

Breiman, L. (1998) “Arcing classifiers”, The Annals of Statistics, 26(3), 801-
849,1998.

Breiman, L., (1996) “Bagging Predictors”, Journal of Machine Learning, Vol 24,
no. 2, 1996, pp 123-140.

Breiman, L., F reidman, J .H., Olshen, R. A., and Stone, P. J. (1984). Classification
and Regression Trees. Belmont, CA: Wadsworth International Group.

Bruggemann-Klein, A. and Wood, D. (1988). Drawing Trees Nicely with TEX.

Chan, K.C.C. Ching, J.Y. and Wong, A.K.C. (1992). “A Probabilistic Inductive
Learning Approach to the Acquisition of Knowledge in Medical Expert Systems”.
Proceeding of 5th IEEE Computer Based Medical Systems Symposium. Durham
NC.

CAPA, See http://capa.msu.edu/

Cestnik, B. Konenenko, I. Bratko, I. (1987). ASSISTANT 86: A Knowledge
Elicitation T 001 for Sophisticated Users, in Bratko, I. and Navrac, N. (eds),
Progress in Machine Learning, Sigma Press, UK.

CLEAR Software, Inc. (1996). allCLEAR User ’s Guide, CLEAR Software, Inc,
199 Wells Avenue, Newton, MA.

Chernoflf H. (1973), “The use of Faces to represent Points in k-Dimensional

Space Graphically”, Journal of American Statistical Association, Vol. 68, page
361-368, June 1973.

Ching, J.Y. Wong, A.K.C. and Chan, CC. (1995). “Class Dependent
Discretisation for Inductive Learning from Continuous and Mixed-mode Data”.
IEEE Transaction. PAM, 17(7) 641 - 645.

Comon, P. (1994) "Indepenent Component Analysis — a New Concept?" Signal
Processing, Vol. 36, No. 3, page 287-314.

Courselnfo TM, Blackboard Inc., Washington, DC. (Courselnfo is a trademark of

Blackboard, Inc., see http://product. blackboard. net/courseinfo/;
http://www. blackboard. com).

232

[37]

[38]

[39]

[40]

[41]

[42]

[43]

[44]

[451

[46]

[47]

[48]

[49]

Dan, S.; and Colla. P., (1998) CART--Classification and Regression Trees. San
Diego, CA: Salford Systems, 1998.

De Jong K.A., Spears W.M. and Gordon DP. (1993). Using genetic algorithms
for concept learning. Machine Learning 13, 161-188, 1993.

Dempster, A.P., Laird, N. M., and Rubin, D. B. (1977) Maximum Likelihood
From Incomplete Data Via the EM Algorithm, Journal of the Royal Statistical
Society B, 39: 1-22, 1977.

Ding, Q., Canton, M., Diaz, D., Zou, Q., Lu, B., Roy, A., Zhou, J., Wei, Q.,
Habib, A., Khan, MA. (2000), “Data mining survey”, Computer Science Dept.,
North Dakota State University. See

http://midas. cs. ndsu. nodak. edu/~ding/datamining/dm-survey. doc

Dixon, J.K. (1979) “Pattern recognition with partly missing data” IEEE
Transaction on Systems, Man and Cybernetics SMC 9, 617-621

Dhillon, I. S. and Modha, D. S., (2000)“A Data-clustering Algorithm on
Distributed Memory Multiprocessors”, In Proceedings of Large-scale Parallel
KDD Systems Workshop, ACM SIGKDD, in Large-Scale Parallel Data Mining,
Lecture Notes in Artificial Intelligence, 2000, 1759: 245-260.

Dong, G., Li, J ., (1998) “Interestingness of Discovered Association Rules in terms
of Neighborhood-Based Unexpectedness”, Proceedings of Pacific Asia
Conference on Knowledge Discovery in Databases (PAKDD), pp. 72-86.
Melbourne, 1998.

Duda, R.O., Hart, RE. (1973). Pattern Classification and SenseAnalysis . John
Wiley & Sons, Inc., New York NY.

Duda, R.O., Hart, PE, and Stork, no. (2001). Pattern Classification. 2"d
Edition, John Wiley & Sons, Inc., New York NY.

Dudoit S. and Fridlyand J., (2003) “Bagging to improve the accuracy of a
clustering procedure”, Bioinformatics, 19 (9), pp. 1090-1099, 2003

Dumitrescu, D., Lazzerini, B., and Jain, LC. (2000). Fuzzy Sets and Their
Application to Clustering and Training. CRC Press LLC, New York.

Efron, B. (1979) "Bootstrap methods: Another Look at the Jackknife". Annals of
Statistics, 1979, 7: 1-26.

El-Sheikh, E. M. (2001). An Architecture for the generation of intelligent tutoring
systems from reusable components and knowledge-based systems, (Ph.D.
Dissertation), 2001, MSU.

233

 

 

 

[50]

[51]

[52]

[53]

[54]

[55]

[56]

[57]

[58]

[59]

[60]

[61]

[62]

Ester, M., Kriegel, H.-P., Xu. X. (1995) "A Database Interface for Clustering in
Large Spatial Databases", Proceedings of the Knowledge Discovery and Data
Mining Conference, pages 94-99, Montreal, Canada, 1995.

eCollege, See http://convene.com; http://eCollege.com (real education);
http://www.e-education.com (jones knowledge)

F alkenauer E. (1997). Genetic Algorithms and Grouping Problems. John Wiley &
Sons, 1998.

Fayyad, U. M., Pitatesky-Shapiro, G., Smyth, P., and Uthurasamy, R. (1996).
Advances in Knowledge Discovery and Data Mining, AAAI/MIT Press.

Fern, X., and Brodley, C. E., (2003) “Random Projection for Hi Dimensional
Data Clustering: A Cluster Ensemble Approach”, In Proc. 20’ Int. conf on
Machine Learning, [CM 2003.

Fischer, B. and Buhmann, J.M. (2002) “Data Resampling for Path Based
Clustering”, in: L. Van Gool (Editor), Pattern Recognition - Symposium of the
DA GM 2002, pp. 206 - 214, LNCS 2449, Springer, 2002.

Fischer, B. and Buhmann, J.M. (2003) “Path-Based Clustering for Grouping of
Smooth Curves and Texture Segmentation”, IEEE Trans. on PAM, 25 (4), pp.
513-518, 2003.

Frawley, W., Piatetsky-Shapiro, G., and Matheus, C. (1992). Knowledge
Discovery in Databases: An Overview. AI Magazine, Fall 1992, pgs 213-228.

Fred, A.L.N. (2001) “Finding Consistent Clusters in Data Partitions”. In Proc. 3d
Int. Workshop on Multiple Classifier Systems. Eds. F. Roli, J. Kittler, LNCS,
2001, 2364: 309-318

Fred, A.L.N. and Jain, AK. (2002) “Data Clustering Using Evidence
Accumulation”, In Proc. of the 16'“ International Conference on Pattern
Recognition, ICPR 2002 ,Quebec City: 276 — 280.

Freitas, A.A. (2002) “A survey of Evolutionary Algorithms for Data Mining and
Knowledge Discovery”, In: A. Ghosh and S. Tsutsui. (Eds) Advances in
Evolutionary Computation, pp. 819-845. Springer-Verlag, 2002.

Freitas, A.A. (1999) “On rule interestingness measures”, Knowledge-Based
Systems journal 12 (5-6), 309-315. Oct. 1999.

Freitas, A.A. (1999) Data Mining and Knowledge Discovery with Evolutionary
Algorithms. Berlin: Springer-Verlag, 2002.

234

 

 

 

[63]

[64]

[65]

[66]

[67]

[68]

[69]

I70]

[71]

[72]

[73]

Frossyniotis, D., Likas, A., and Stafylopatis, A., “A clustering method based on

boosting”, Pattern Recognition Letters, Volume 25, Issue 6, 19 April 2004, Pages
641-654

Fridlyand, J. and Dudoit, S. (2001) “Applications of resampling methods to
estimate the number of clusters and to improve the accuracy of a clustering
methods,” Division of Biostattistics, University of California, Berkley, Tech. Rep.
600, 2001.

Fu Y. and Han, J., (1995) “Meta-rule-guided mining of association rules in
relational databases”. Proc. 1995 Int ’1 Workshop. on Knowledge Discovery and
Deductive and Object-Oriented Databases (KDOOD '95), Dec. 1995, pp. 39-46.

Garofalakis, M., Rastogi, R. and Shim K. (1999). “SPIRIT: Sequential Pattern
Mining with Regular Expression Constraints”. In Proc. of the 25”' Very Large

Database Conference, Edinburgh, Scotland, September 1999.

Goldberg, DE. (1989). Genetic Algorithms in Search, Optimization, and
Machine Learning, MA, Addison-Wesley.

Goossens, M, Rahtz, S. and Mittelbach, F. (1997). The LaTeX Graphics
Companion, Addison Wesley.

Guerra-Salcedo C. and Whitley D. (1999). “Feature Selection mechanisms for
ensemble creation: a genetic search perspective”. In: Freitas AA (Ed.) Data

Mining with Evolutionary Algorithms: Research Directions - Papers from the
AAA] Workshop, 13-17. Technical Report WS-99-06. AAAI Press, 1999.

Guha, S., Rastogi, R., and Shim, K. (1998) “CURE: An efficient clustering
algorithm for large databases” In Proceeding of the I 998 ACM SIGMOD

International Conference on Management of Data Engineering, Pages 512-521
Seattle, WA USA.

Hall, M A., and Smith, LA. (1997), “Feature subset selection: a correlation
based filter approach", Proceeding of the 1997 International Conference on

Neural Information Processing and Intelligent Information Systems, Springer,
Page 855-858.

Han, J.; Kamber, M.; and Tung, A. 2001. Spatial Clustering Methods in Data
Mining: A Survey. In Miller, H., and Han, J., eds., Geographic Data Mining and
Knowledge Discovery. Taylor and Francis. 21

Hand, D. J. (1987), Discrimination and Classification, John Wiley & Sons,
Chichester UK.

235

[74]

[75]

[76]

[77]

[78]

[79]

[80]

[81]

[82]

[83]

H34}

[85]

[86]

Hastie, T., Tibshirani, R., Friedman, J.H. (2001) The Elements of Statistical
Learning, Springer-Verlag, 2001

Heckerman, D. (1996), The process of Knowledge Discovery in Database, 273-
305, AAAI/MIT Press.

Harrell, Jr., Frank E. (2001), Regression modeling strategies: with applications to
linear models, logistic regression, and survival analysis, Springer- Verlag, New
York.

Hoppner, F.; Klawnn, F .; Kruse, R.; Runkler, T.; (2000). Fuzzy Cluster Analysis:
“Methods for classification, data analysis and image recognition ”, John Wiley &
Sons, Inc., New York NY.

Houtsma M., and Swami A., (1993) "Set-oriented mining of association rules".
Research Report RJ 9567, IBM Almaden Research Centers, San Jose California,
1993

Hume, D. An Enquiry Concerning Human Understanding, Oxford Philosophical
Texts, Tom L. Beauchamp (Editor) Oxford University Press, 1999

Jain, A.K.; and Dubes; R. C. (1988). Algorithms for Clustering Data. Englewood
Cliffs, NJ: Prentice-Hall.

Jain, A.K.; Mao, J.; and Mohiuddin, K.; (1996). "Artificial Neural Networks: A
Tutorial," IEEE Computer, March. 1996.

Jain AK. and Moreau (1987), “The Bootstrap Approach to Clustering”, in
Pattern Recognition Theory and Applications, P.A. Devijver and J. Kittler (eds.),
Springer-Verlag, 1987, pp. 63-71.

Jain, A.K.; Duin R. P.W.; and Mao, J. (2000). “Statistical Pattern Recognition: A
Review”, IEEE Transaction on Pattern Analysis and Machine Intelligence, Vol.
22, No. 1, January 2000.

Jain, A.K.; Mutty, M.N.; Flynn, R]. (1999) “Data clustering: a review”, ACM
Computing Surveys, Vol. 31, N3, pp 264-323, 1999

Jain, A.K.; Zongker, D.; (1997). "Feature Selection: Evaluation, Application, and
Small Sample Performance" IEEE Transaction on Pattern Analysis and Machine
Intelligence, Vol. 19, No. 2, February 1997.

John, G. H.; Kohavi R.; and Pfleger, K.; (1994). “Irrelevant Features and Subset
Selection Problem”, Proceeding of the Eleventh International Conference of

Machine Learning, page 121-129, Morgan Kaufrnann Publishers, San Francisco,
CA.

236

[87]

[88]

[89]

[90]

[91]

[92]

[93]

[94]

195]

[96]

[97]

[98]

Karhunen J., Oja B, Wang L., Vigario R., and Joutsenalo P. (1997) "A class of
neural networks for independent component analysis", IEEE Transactions on
NeuRAL Networks. Vol. 8, No. 3, pp486—504, 1997.

Kashy, D A, Albertelli, G, Kashy, E, and Thoennessen, M, (2001), “Teaching
with ALN Technology: Benefits and Costs”, Journal of Engineering Education,
Vol. 90, No. 4, October 2001, pp. 499-505

Kashy, E, Gaff, S, J, Pawley, N, H, Stretch, W, L., Wolfe, S, L., Morrissey, D.J.,
Tsai, Y., (1995) "Conceptual Questions in Computer-Assisted Assignments",
American Journal of Physics, Vol, No 63, 1995, pp. 1000-1005.

Kashy, E., Thoennessen, M., Tsai, Y., Davis, N. E., and Wolfe, S. L. (1998),
“Using Networked Tools to Promote Student Success in Large Classes”, Journal
of Engineering Education, ASEE, Vol. 87, No. 4, 1998, pp. 385-390

Kashy, E., Thoennessen, M., Tsai, Y., Davis, NE, and Wolfe, S.L. (1997),
"Using Networked Tools to Enhance Student Success Rates in Large Classes, "

Frontiers in Education Conference, Teaching and Learning in an Era of Change,
IEEE CH 36099.

Kaufman, L., and Rousseuw P. (1990) Finding Groups in Data- An Introduction

to Cluster Analysis, Wiley Series in Probability and Mathematical Sciences,
1990.

Klosgen, W., and Zytkow, J ., (2002) Handbook of Data Mining and Knowledge
Discovery, Oxford University Press, 2002.

Kluger, A, N, DeNisi, A, (1996). “The Effects of Feedback Interventions on
Performance: Historical Review, a Meta-Analysis and a Preliminary Feedback
Intervention Theory”. Psychological Bulletin, 119, 254-284.

Kluger, A, N, DeNisi, (1998). A, “Feedback interventions: Toward the
understanding of a double-edged sword’, Current Directions in Psychological
Science, 7, 67, 1998

Kohavi, Ron (1995) “A Study of Cross-Validation and Bootstrap for Accuracy
Estimation and Model Selection”, Proceeding of International Joint Conference
on Artificial intelligence (IJCIA), 1995.

Kontkanen, P., Myllymaki, P., and Tirri, H. (1996), Predictive data mining with
finite mixtures. In Proceeding 2"“ International Conference “Knowledge
Discovery and Data Mining (KDD ’96)”, pages 249-271, Portland, Oregon,
August 1996.

Kortemeyer, G., MSU LectureOnline software package, 1997-2000.
237

[99]

[100]

[101]

[102]

[103]

[104]

[105]

[106]

[107]

[108]

[109]

[110]

[111]

Kortemeyer, G., and Bauer, W. (1999), "Multimedia Collaborative Content
Creation (mc3) - The MSU LectureOnline System", Journal of Engineering
Education, 88 (4), 421.

Kortemeyer, G., Albertelli, G., Bauer, W., Berryman, F., Bowers, J., Hall, M.,
Kashy, E., Kashy, D., Keefe, H., Minaei-Bidgoli, B., Punch, W.F., Sakharuk., A.,
Speier, C. (2003): “The LeamingOnline Network with Computer-Assisted
Personalized Approach (LON-CAPA)”. PGLDB 2003 Proceedings of the I PGL
Database Research Conference, Rio de Janeiro, Brazil, April 2003.

Kotas, P, (2000) “Homework Behavior in an Introductory Physics Course”,
Masters Thesis (Physics), Central Michigan University (2000)

Kuncheva , LI, and Jain, LC, (2000), "Designing Classifier Fusion Systems by
Genetic Algorithms”, IEEE Transaction on Evolutionary Computation, Vol. 33
(2000), pp 351-373.

Mason, B, J, Bruning, R, “Providing Feedback in Computer-based instruction:
What the Research Tells Us.” http://dwb.unl.edu/Edit/MB/MasonBruning.html
(2003)

McLachlan, G]. and Krishnan, T. (1997) The EM Algorithm and Extensions.
Wiley series in probability and statistics.

R. Meo, “Replacing Support in Association Rule Mining”, Rapporto Tecnico
RT70-2003, Dipartimento di Inforrnatica, Universita di Torino, April, 2003

Murty, M. N. and Krishna, G. (1980). “A computationally efficient technique for
data clustering”. Pattern Recognition. 12, pp 153-158.

Lange, R. (1996), An empirical test of weighted effect approach to generalized
prediction using neural nets. In Proceeding 2"“ International Conference
“Knowledge Discovery and Data Mining (KDD'96)”, pages 249-271, Portland,
Oregon, August 1996.

LectureOnline, See http://www. lite. msu. edu/kortemeyer/lecture.html

Levine, E. and Domany E., (2001) “Resampling method for unsupervised
estimation of cluster validity”. Neural Computation, 2001, 13, 2573--2593.

Loh, W.-Y. & Shih, Y.-S. (1997). Split Selection Methods for
Trees, Statistica Sinica 7: 815-840.

Classification

Liu, B., Hsu, W., Mun, LP. and Lee, H., (1999) “Finding Interesting Patterns
Using User Expectations”, IEEE Transactions on Knowledge and Data
Engineering, Vol 11(6), pp. 817-832, 1999.

238

 

 

 

[112]

[113]

[114]

[115]

[116]

[117]

[118]

[119]

[120]

[121]

[122]

Lin, B., Hsu, W. and Ma, Y, (2001) “Identifying Non-Actionable Association
Rules”, Proc. 0f seventh ACM SIGKDD International conference on Knowledge
Discovery and Data Mining(KDD-2001) San Francisco, USA.

Lim, T.-S., Loh, W.-Y. & Shih, Y.-S. (2000). “A Comparison of Prediction
Accuracy, Complexity, and Training Time of Thirty-three Old and New
Classification Algorithms”. Machine Learning, Vol. 40, pp. 203-—228, 2000. See
http://www.stat.wisc.edu/~limt/mach13 17.pdf)

LON-CAPA, See http://www. lon-capa. org

Lu, S. Y., and Fu, K. S. (1978). “ A sentence-to-sentence clustering procedure for
pattern analysis.” IEEE Transactions on Systems, Man and Cybernetics SMC 8,
381-389.

Lu, H., Setiono, R., and Liu, H. (1995). Neurorule: A connectionist approach to
data mining. Proceeding 21“ International Conference: Very Large Databases,
pages 478-489, Zurich, Switzerland, September 1995.

Ma, Y., Lin, B., Kian C., Wong, Yu, PS, and Lee, S.M., (2000) “Targeting the
Right Students Using Data Mining”. Proceedings of the ACM SIGKDD
International Conference on Knowledge Discovery & Data Mining (KDD-2000,
Industry Track), Aug, 2000, Boston, USA

Martin—Bautista MJ and Vila MA. (1999) A survey of genetic feature selection in

mining issues. Proceeding Congress on Evolutionary Computation (CEC-99),
1314-1321. Washington DC, July 1999.

Masand, B. and Piatetsky-Shapiro, G. (1996), A Comparison of a proaches for
maximizing business payoff of prediction models. In Proceeding International
Conference “Knowledge Discovery and Data Mining (KDD ’96)”, pages 195-201,
Portland, Oregon, August 1996.

Matheus, OJ. and Piatetsky-Shapiro, G. (1994), An application of KEFIR to the
analysis of healthcare information. In Proceeding “AAAI ’94 Workshop
Knowledge Discovery in Database (KDD '94)”, pages 441-452, Seattle, WA, July
1994.

Michalewicz Z. (1996). Genetic Algorithms + Data Structures = Evolution
Programs. 3rd Ed. Springer-Verlag, 1996.

Michalski, R.S., Bratko, 1., Kubat M. (1998), Machine Learning and Data
Mining, Methods and applications, John Wiley & Sons, New York.

239

[123]

[124]

[125]

[126]

[127]

[128]

[129]

[130]

[131]

[132]

[133]

Michalski, R.S., Stepp, RE, 111 (1983), "Automated construction of
classification: conceptual clustering versus numerical taxonomy" IEEE
Transactions on Pattern Analysis and Machine Intelligence PAMI 5, 396-410.

Mitchell, Tom M. (1997), Machine Learning, McGraw-Hill.

Minaei-Bidgoli, 8., Tan, P-N., Punch, W.F., (2004g) “Mining Interesting Contrast Rules
for a Web-based Educational System”, International Conference on Machine Learning
Applications (ICMA 2004), Louisville, KY, USA, Dec 2004.

Minaei-Bidgoli, B., Kortemeyer G., Punch, W.F., (2004i) “Association Analysis for an
Online Education System”, IEEE International Conference on Information Reuse and
Integration (IRI-2004), Las Vegas, Nevada, USA, Nov 2004.

Hall, M., Parker, J., Minaei-Bidgoli, B., Albertelli, G., Kortemeyer, G., and Kashy, E.,
“Gathering and Timely Use of Feedback from Individualized On-line Work”
(IEEE/ASEE) FIE 2004 Frontier In Education, Savannah, Georgia, USA, Oct. 2004.

Minaei-Bidgoli, B., Kortemeyer, G., Punch, W.F., (2004c) “Optimizing Classification
Ensembles via a Genetic Algorithm for a Web-based Educational System”, IAPR
International workshop on Syntactical and Structural Pattern Recognition (SSPR 2004)
and Statistical Pattern Recognition (SPR 2004), Lisbon, Portugal, Aug. 2004.

Minaei-Bidgoli, B., Kortemeyer, G., Punch, W.F., (2004d) “Enhancing Online Learning
Performance: An Application of Data Mining Methods”, The 7th IASTED International
Conference on Computers and Advanced Technology in Education (CA TE 2004) Kauai,
Hawaii, USA, August 2004.

Minaei-Bidgoli, B., Kortemeyer, G., Punch, W.F., (2004c) “Mining Feature Importance:
Applying Evolutionary Algorithms within a Web-Based Educational System”,

International Conference on Cybernetics and Information Technologies, Systems and
Applications: CIT SA 2004, Orlando, Florida, USA, July 2004.

Minaei-Bidgoli, B., Punch, W.F., (2003) “Using Genetic Algorithms for Data
Mining Optimization in an Educational Web-based System”, GECCO 2003

Genetic and Evolutionary Computation Conference, Springer-Verlag 2252-2263,
July 2003 Chicago, IL.

Minaei-Bidgoli, B., Kashy, D.A., Kortemeyer G., Punch, W.F., (2003)
“Predicting Student Performance: An Application of Data Mining Methods with
an educational Web-based System”, (IEEE/ASEE) FIE 2003 Frontier In
Education, Nov. 2003 Boulder, Colorado.

Minaei-Bidgoli, B., Topchy A., and Punch, W.F., (2004a) “Ensembles of
Partitions via Data Resampling”, to be appear in proceeding of IEEE
International Conference on Information Technology: Coding and Computing,

ITCC 2004, vol. 2, pp. 188-192, April 2004, Las Vegas, Nevada.
240

 

[134]

[135]

[136]

[137]

[138]

[139]

[140]

[141]

[142]

[143]

[144]

[145]

Minaei-Bidgoli, B., Topchy A., and Punch, W.F., (2004b) “A Comparison of
Resampling Methods for Clustering Ensembles”, in Proc. Intl. Conf. Machine
Learning Methods Technology and Application, MLMTA 04, Las Vegas, 2004.

McLachlan, G. (1992), Discrimination Analysis and Statistical Pattern
Recognition, John Wiley & Sons, New York.

McQueen, J. B. (1967). “Some methods of classification and analysis of
multivariate observations.” Proceedings of Fifth Berkeley Symposium on
Mathematical Statistics and Probability, pp. 281-297.

Mori, Y.; Kudo, M.; Toyama, J.; and Shimbo, M. (1998), “Visualization of the
Structure of Classes Using a Graph”, Proceeding of International Conference on
Pattern Recognition, Vol. 2, Brisbane, August 1998, page 1724-1727.

Montgomery, Douglas C., Peck, Elizabeth A., and Vining, Geoffrey G. (2001)
Introduction to linear regression analysis. John Wiley & Sons, Inc., New York
NY.

Monti, S., Tamayo, P., Mesirov, J., Golub, T., (2003) “Consensus Clustering: A
reamlping-Based Method for Class Discovery and Visualization of Gene

Expression Microarray Data”, Journal on Machine Learning, Volume 52 Issue 1-
2, July 2003.

Moore, Geoffrey, A, “Crossing the Chasm”, HarperCollins, 2002. Users of LON-
CAPA in its very early beta versions would appreciate the statement on p.31, as
these early adopters also forgave “... ghastly documentation...” as well as “...
bizarrely obtuse methods of invoking needed functions ...”

Muhlenbein and Schlierkamp—Voosen D., (1993). Predictive Models for the
Breeder Genetic Algorithm: 1. Continuous Parameter Optimization, Evolutionary
Computation, Vol. 1, No. 1, pp. 25-49, 1993

Murray, T. (1996). “From Story Boards to Knowledge Bases: The First Paradigm
Shifi in Making CAI, Intelligent”, ED—A/EDIA 96 Educational Multimedia and
Hypermedia Conference, Charlottesville, VA.

Murray, T. (1996). “Having It All, Maybe: Design Tradeoffs in ITS Authoring
Tools”. ITS '96 Third Intl. Conference on Intelligent Tutoring Systems, Montreal,
Canada, Springer-Verlag.

Murthy, S. K. (1998), "Automatic construction of decision trees from data: A

multidisciplinary survey”, Data Mining and Knowledge Discovery, vol. 4, pp.
345--389, 1998.

MMP, Multi-Media Physics, See http.'//lecture. lite. msu. edu/~mmp/
241

[146]

[147]

[148]

[149]

[150]

[151]

[152]

[153]

[154]

[155]

[156]

[157]

[158]

Ng, R.T., Han, J. (1994) “Efficient and Effective Clustering Methods for Spatial
Data Mining”, In 20th International Conference on Very Large Data Bases, Pages
144-145, September 1994, Santiago, Chile.

Odewahn, S.C., Stockwell, E.B., Pennington, R.L., Humphreys, RM. and
Zumach WA. (1992). Automated Star/Galaxy Discrimination with Neural
Networks, Astronomical Journal, 103: 308-331.

PhysNet, See http://physnet2. pa. msu. edu/

Park, B.H. and Kargupta, H. (2003) “Distributed Data Mining”. In The Handbook
of Data Mining. Ed. Nong Ye, Lawrence Erlbaum Associates, 2003

Park Y and Song M. (1998). A genetic algorithm for clustering problems. Genetic
Programming I998: Proceeding of 3rd Annual Conference, 568-575. Morgan
Kaufmann, 1998.

Pascarella, A, M, (2004) “The Influence of Web-Based Homework on
Quantitative Problem-Solving in a University Physics Class”, NARST Annual
Meeting Proceedings (2004)

Pei, M., Goodman, ED, and Punch, W.F. (1997) "Pattern Discovery from Data
Using Genetic Algorithms", Proceeding of I” Pacific-Asia Conference
Knowledge Discovery & Data Mining (PAKDD-97). Feb. 1997.

Pei, M., Punch, W.F., Ding, Y., and Goodman, ED. (1995) "Genetic Algorithms
For Classification and Feature Extraction", presented at the Classification Society
Conference , June 95.

See http://garage.cse.msu.edu/papers/papers-index.html

Pei, M., Punch, W.F., and Goodman, ED. (1998) "Feature Extraction Using
Genetic Algorithms", Proceeding of International Symposium on Intelligent Data
Engineering and Learning ’98 (IDEAL '98), Hong Kong, Oct. 98.

Piatetsky-Shapiro G. and Matheus. C. J., (1994) “The interestingness of
deviations”. In Proceedings of the AAAI-94 Workshop on Knowledge Discovery
in Databases, pp. 25-36, 1994.

Punch, W.F., Pei, M., Chia-Shun, L., Goodman, E.D., Hovland, P., and Enbody
R. (1993) "Further research on Feature Selection and Classification Using Genetic

Algorithms", In 5'" International Conference on Genetic Algorithm , Champaign
IL, pp 557-564, 1993.

Quinlan, J. R. (1986), Induction of decision trees. Machine Learning, 1:81-106.

242

 

[159]

[160]

[161]

[162]

[163]

[164]

[165]

[166]

[167]

[168]

[169]

[170]

[171]

[172]

Quinlan, J. R. (1987), Rule induction with statistical data, a comparison with
multiple regression. Journal of Operational Research Society, 38, 347-352.

Quinlan, J. R. (1993), C4.5: Programs for Machine Learning, San Mateo, CA:
Morgan Kaufrnann.

Quinlan, J. R. (1994), Comparing connectionist and symbolic learning methods,
MIT Press.

Roth, V., Lange, T., Braun, M., Buhmann, J .M. (2002) “A Resampling Approach
to Cluster Validation”, in:, Proceedings in Computational Statistics: 15th
Symposium COMPSTAT 2002, Physica-Verlag, Heidelberg, 123-128.

Ruck, D.W. , Rogers, S.K., Kabirsky, M., Oxley, ME, and Suter, B.W. (1990),
“The Multi-Layer Perceptron as an Approximation to a Bayes Optimal
Discriminant Function”; IEEE Transactions on Neural Networks, vol. 1, no. 4.

Russell, 8.; and Norvig P. (1995). Artificial Intelligence: A Modern Approach.
Prentice-Hall.

Shih, Y.-S. (1999), “Families of splitting criteria for classification trees”,
Statistics and Computing, Vol. 9, 309-315.

Shute, V. and J. Psotka (1996). Intelligent Tutoring Systems: Past, Present, and

Future. Handbook of Research for Educational Communications and Technology.
D. Jonassen. New York, NY, Macmillan.

Shute, V. J. (1991). “Who is Likely to Acquire Programming Skills?” Journal of
Educational Computing Research 1: 1-24.

Shute, V. J., L. A. Gawlick-Grendell, et a1. (1993). An Experiential System for
Learning Probability: Stat Lady. Annual Meeting of the American Educational
Research Association, Atlanta, GA.

Shannon, CE, (1948) “A mathematical theory of communication”, Bell System
Technical Journal, vol. 27, pp. 379-423 and 623-656, July and October, 1948.

Shute, V. J. and J. W. Regian (1990). Rose garden promises of intelligent tutoring
systems: Blossom or thorn? Proceedings from the Space Operations, Applications
and Research Symposium, Albuquerque, NM.

Shute, V. R. and R. Glaser (1990). “A Large-scale Evaluation of an Intelligent
Discovery World: Smithtown.” Interactive Learning Environments 1: 51-76.

Skalak D. B. (1994). Using a Genetic Algorithm to Learn Prototypes for Case
Retrieval an Classification. Proceeding of the AAAI-93 Case-Based Reasoning

243

[173]

[174]

[175]

[176]

[177]

[178]

[179]

[180]

[181]

[182]

[183]

[184]

[185]

Workshop, pp. 64-69. Washigton, DC, American Association for Artificial
Intelligence, Menlo Park, CA, 1994.

Siedlecki, W., Sklansky J., (1989), “A note on genetic algorithms for large-scale
feature selection”, Pattern Recognition Letters, Vol. 10, Page 335-347, 1989.

Silberschatz A. and Tuzhilin, A., (1995) “On subjective measures of
interestingness in knowledge discovery”. Proceeding of KDD, 275-281, 1995.

Silberschatz A. and Tuzhilin, A. (1996) “What makes patterns interesting in
Knowledge discovery systems”. IEEE Transactions on Knowledge and Data
Engineering, 8(6):970-974, December, 1996.

Singley, K. and J. R. Anderson (1989). The Transfer of Cognitive Skill.
Cambridge, MA, Harvard University Press.

Sleeman, D. (1987). PIXIE: A Shell for Developing Intelligent Tutoring Systems.
Artificial Intelligence and Education, R. W. Lawler and M. Yazdani. Norwood,
NJ, Alex Publishers: 239-265.

Sleeman, D. H. and J. S. Brown (1982). Intelligent Tutoring Systems, London,
UK, Academic Press.

Srikant, R. and Agrawal R. (1996). "Mining Quantitative Association Rules in
Large Relational Tables", In Proc. of the ACM-SIGMOD I 996 Conference on
Management of Data, Montreal, Canada, June 1996

Strehl, A. and Ghosh, J. (2002) “Cluster ensembles - a knowledge reuse
framework for combining multiple partitions”. Journal on Machine Learning
Research, 2002, 3: 583-617

Tan, P.N., Kumar V., and Srivastava, J. (2004), “Selecting the Right Objective
Measure for Association Analysis”, Information Systems, 29(4), 293-313, 2004.

Tobias, 8., Raphael, J., (1997) “The Hidden Curriculum”, contribution by E.
Kashy and D. J. Morrissey, pp 165-167, Plenum Press, New York 1997.

Topchy, A., Minaei-Bidgoli, B., Jain, A.K., Punch, W.F., (2004) “Adaptive Clustering
Ensembles”, International Conference on Pattern Recognition (ICPR 2004), Cambridge,
UK, Aug. 2004.

Topchy, A., Jain, A.K., and Punch W.F. (2003a), “Combining Multiple Weak
Clusterings”, In proceeding of IEEE Intl. Conf on Data Mining 2003.

Topchy, A., Jain, A.K., and Punch W.F. (2003b), “A Mixture Model of Clustering
Ensembles”, submitted to SIAM Intl. Conf. on Data Mining 2003

244

 

[186]

[187]

[188]

[189]

[190]

[191]

[192]

[193]

[194]

[195]

[196]

[197]

[198]

TopClass TM, WBT Sytems, San Fransisco, CA. (TopClass is a trademark of WBT
Systems, see http://www.wbtsystems.com).

Toussaint, G. T. (1980). “The relative neighborhood graph of a finite planar set.”
Pattern Recognition 12, 261-268.

Trunk, G.V. (1979), “A problem of Dimensionality: A Simple Example”, IEEE
Transaction on Pattern Analysis and Machine Intelligence, Vol. PAMI-I, No. 3,
July 1979.

Urban-Lurain, M. (1996). Intelligent Tutoring Systems: An Historic Review in the
Context of the Development of A rtificial Intelligence and Educational Psychology,
Michigan State University.

Urquhart, R. (1982). “Graph theoretical clustering based on limited neighborhood
sets.” Pattern Recognition 15, 173-187.

Vafaie H and De Jong K. (1993). Robust feature Selection algorithms. Proceeding
1993 IEEE Int. Confon Tools with AI, 356-363. Boston, Mass, USA. Nov. 1993.

VU, (Virtual University), See http://vu.msu.edu/; http://www. vu. org.

Wu, Q., Suetens, and Oosterlinck, A. (1991) Integration of heuristic and Bayesian
approaches in a pattem-classification system. In Piatetsky-Shapiro G., and
Frawely W.J., editors, “Knowledge Discover yin Database”, pages 249-260,
AAAI/MIT press.

WebCT TM, University of British Columbia, Vancouver, BC, Canada. (WebCT is
a trademark of the University of British Columbia , see http://www.webct.com/)

Weiss, S. M. and Kulikowski C. A. (1991), Computer Systems that Learn:
Classification and Prediction Methods from Statistics, Neural Nets, Machine
Learning, and Expert Systems, Morgan Kaufman.

Woods, K.; Kegelmeyer Jr., W.F.; Bowyer, K. (1997) “Combination of Multiple
Classifiers Using Local Area Estimates”; IEEE Transactions on Pattern Analysis
and Machine Intelligence, Vol. 19, No. 4, 1997.

Yazdani, M. (1987). Intelligent Tutoring Systems: An Overview. Artificial
Intelligence and Education: Learning Environments and Tutoring Systems. R. W.
Lawler and M. Yazdani. Norwood, NJ, Ablex Publishing. 1: 183-201

Za‘iane, Osmar R. (2001) “Web Usage Mining for a Better Web-Based Learning

Environment”, in Proc. of Conference on Advanced Technology for Education, pp
60-64, Banff, Alberta, June 27-28, 2001.

245

[199]

[200]

[201]

Zhang, B., Hsu, M., Fonnan, G. (2000) "Accurate Recasting of Parameter
Estimation Algorithms using Sufficient Statistics for Efficient Parallel Speed-up
Demonstrated for Center-Based Data Clustering Algorithms", Proc. 4th European
Conference on Principles and Practice of Knowledge Discovery in Databases, in
Principles of Data Mining and Knowledge Discovery, D. A. Zighed, J.
Komorowski and J. Zytkow (Eds), 2000.

Zhang Fern, X., and Brodley, C. E. (2003) “Random Projection for High
Dimensional Data Clustering: A Cluster Ensemble Approach”, in Proc. of the 20”1
Int. conf. on Machine Learning ICA/H. 2003.

Zhang, T; Ramakrishnan, R., and Livny, M. (1996), “BIRCH: An Efficient Data
Clustering Method for Very Large Databases”, Proceeding of the ACM SIGMOD
Record, 1996, 25 (2): 103-114

246

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