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THESIS

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dissertation entitled

MATHEMATICAL MODELS OF THE MANUFACTURING
LEARNING CURVE

presented by

PAUL SPEAKER

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MATHEMATICAL MODELS OF THE
MANUFACTURING LEARNING CURVE

By

Paul Speaker

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Applied Mathematics

2010

ABSTRACT

MATHEMATICAL MODELS OF THE
MANUFACTURING LEARNING CURVE

By

Paul Speaker

The industrial learning curve is the widely observed process whereby, through manu-
facturing experience, improvement in quality metrics improve over time. Such quality
metrics include cost, mean product lifetime, and error rate.

This dissertation motivates, builds, describes, and investigates a learning curve
model based on ideas of information theory and statistical mechanics. We ﬁrst exam—
ine the relationship between the learning curve and learning in other contexts. Then
we create a model of the learning curve based upon the ideas of statistical mechan-
ics. We then build two continuous models for the learning curve. Several additional
features of learning curves, such as multi—dimensional learning and forgetting, are
realized by the model.

The ﬁrst continuous model is based on a new mathematical concept: weak con-
vergence of sequences of sets. The idea of weak convergence of sequences of sets is
developed for both deterministic and random sets. Several functional analytic-like
results are then proven for such weak limits. The weak limits are then used to con-
struct a model of learning. In doing so, the general approach of creating a continuous
model from a discrete model via such a weak limit is described.

The second continuous model to be described is a partial differential equation
model. By making the learning process Poisson in time, it is shown that the learning

model is a discretization of a transport equation. Aspects of this transport equation

are then explored and interpreted.

Further, some simulations of the model are made. It is shown that the model
simulations generate the generally observed form of the learning curve. The disser-
tation concludes with an exploration of further directions to be pursued in future

investigation of the learning curve phenomenon.

Dedication

To Liisa, Isadora, and Evangeline.

iv

Acknowledgements

For a large, interdisciplinary project such as this one, there are many people to
thank. This investigation started with a project on the manufacturing learning curve
for Michigan State University students sponsored by General Motors. I would ﬁrst
like to thank David Field and Randy Urbance of General Motors for their numerous
helpful suggestions on the tOpic. I would also like to thank the other members of my
team, Michael Olson and Rahul Datta for their help in getting me thinking about
the project. Peter Magyar of Michigan State was the project supervisor, and his

guidance both during and after the project time period.

I would also like to thank my dissertation committee members: Roger Calantone,
Gabor Francsics, Baisheng Yan, and Keith Promislow. Roger Calantone pointed me
towards the research on organizational learning by James March, which was very
helpful. Gabor Franciscs and Baisheng Yan had numerous helpful suggestions with
the chapter on the partial differential equation model for the learning curve. Finally,
Keith Promislow was very helpful helping me gain an understanding of some of the

physical analogies for random sets, in particular, with the capacitor example.

I would also like to thank the outside help. I would certainly like to thank Igor
Molchanov. His seminal book on random sets, as well as his suggestions for future
directions, were very helpful in my formulating the learning problem in the context of
random sets. I would also like to thank the people at the Econophysics Colloquium
in Ancona who made helpful suggestions for connecting the learning curve problem

to statistical mechanics.

Finally, I would like to thank most deeply Charles R. MacCluer for the mul-
titude of suggestions, his patience with this work, and his overall focus on what this
dissertation should look like. The suggestions he has provided are too numerous to
list here, but they permeate the entire dissertation. W ithout his tireless efforts, this

dissertation would possess the breath or depth which it has.

vi

Contents

List of Tables .................................. xi
List of Figures .................................. xii

I Introduction 1
1 Literature Review 2
1.1 Introduction ................................ 2
1.2 Deterministic Models ........................... 3
1.3 Stochastic Models ............................. 4

2 Insights from Other Research into Learning 6
2.0.1 The Analogy to Artiﬁcial Intelligence .............. 6

2.0.2 The Application of Statistical Mechanics to Artiﬁcial Intelligence 8
2.1 Statistical Mechanics in Other Areas .................. 10
2.1.1 Differences between Management Theory
and Organizational Economics. ................. 11
2.1.2 Econophysics and Learning. ................... 12
2.2 Further Directions for Statistical

Mechanics in Management Theory .................... 13

vii

II The Mathematical Model

3 The Discrete Case

3.1 The Parameters of the Model ......................

3.2
3.3

3.1.1 Derivation of the Probability ...................
3.1.2 How lesson probabilities are determined ............
3.1.3 Entropy ..............................
3.1.4 Interpretation of Results .....................
Summary .................................
Multi-Dimensional Learning .......................
3.3.1 Introduction and Motivation ...................
3.3.2 Modeling Multi-Dimensional Learning .............

3.4 Learning with Forgetting .........................

4 Continuous Model I: Weak Convergence of Sequences of Sets

4.1
4.2

4.3
4.4

4.5

4.6

Introduction and Motivation .......................

4.2.1

Weak Convergence of Sequences of Sets. ................

Some Functional Analytic Results on Sequences of Sets . . . .

A Distributional Weak Limit of Random Sets .............

Properties of the Principal Dynamic Random Set ...........

4.4.1

The Weak Limit as a Random Functional ...........

Applications of dynamic random sets ..................

4.5.1
4.5.2

Charging a Capacitor .......................
Growth Models ..........................

Creating a Continuous Model of the Learning Curve

via Distributional Random Sets .....................

4.6.1

A Deterministic Model of Learning a Continuum of Lessons

viii

15

16
16
16
19
22
23
27
27
27
29
30

32
32
34
40
42
52
55
59
59
63

64
64

4.6.2 Expected Savings from a Random Set of Lessons. ....... 68

4.6.3 The Smooth Case and Principal Dynamic Random Sets . . . . 69
4.6.4 Prediction of unit cost. ...................... 71
4.6.5 Devolution to the discrete case .................. 72

5 Continuous Model II: A Partial Differential Equation for Learning 75

5.1 The Model as a Limit of a Markov Process ................ 75
5.1.1. Construction of a Master Equation ................ 75

5.1.2. Reduction of Order ......................... 82

5.2 The Continuous Case ........................... 84
5.2.1. The Continuous Case with Equal \Neighting ........... 84

5.2.2. The Continuous Case with Unequal Weighting ......... 91

5.2.3. Properties of the Learning Velocity ................ 93

5.3 Cost Evolution during Learning ...................... 97
5.4 Forgetting .................................. 99
5.5 Conclusion ................................. 100
III Simulations and Applications 101
6 Simulations of Models 102
6.1 Introduction and Methodology ...................... 102
6.2 Results ................................... 104
6.2.1 Regression Comparisons ..................... 107

6.2.2 Comparison of Lesson Structures ................ 108

6.3 Simulations of l\"Iulti-Dimensional Learning ............... 115
6.4 Conclusion ................................. 122

ix

7 Further Directions 124

7.1 Empirical Analysis of Learning Curves ................. 124
7.2 Further Directions for Exploring Dynamic Random Sets ........ 126
128

8 Bibliography

List of Tables

6.1 Pseudocode for the Monte Carlo Experiments .............. 104

6.2 Regression results for learning curves for different managerial effec—
tiveness parameters 3. .......................... 108

xi

List of Figures

4.1

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

6.10

The currents involved in charging a capacitor. ............. 61
Sample learning curve plotting cost per unit produced versus accu-
mulated number of units produced, showing decreasing cost per unit.
during production. ............................ 106
Example of simulated learning curve for a linearly increasing cost
structure with ,3 = 0.8 ........................... 110
Simulated learning curve for a. linearly increasing cost savings space
with a low plateau, with 5 = 0.8. .................... 111
Example of simulated learning curve for a non-uniformly spaced cost
savings space, with [3 = 0.8. ....................... 113
Simulated learning curve for a non—uniformly distributed cost savings
space with a plateau very small relative to initial cost, with x3 = 0.8. . 114
Learning curve for two metrics where lesson set has no correlation
between metrics (the solid line represents improvement in cost, mea-
sured in 104 dollars, with ﬁl = 0.4, and the dotted line represents the
defect rate (in percent)), with ,82 = 0.6 .................. 117
Learning curve for two competing metrics with negative correlation
(solid line is cost, dotted line is defect rate), with 731 = 0.8 and 52 = 0.2.119
Learning Curve for two equally preferred metrics. ........... 120
Learning Curve for two competing metrics with switch between pref-
erence for quality metrics. ........................ 121
Learning Curve for Two Competing Metrics with Switch from no Pref-
erence to Strong Preference for Quality Metrics. ............ 122

xii

Part I

Introduction

Chapter 1

Literature Review

1 . 1 Introduction

It is generally observed that manufacturing measurements of quality (quality met-
rics) exhibit an improvement over time. The improvement given by this learning
curve is generally described as a function of a single parameter—the total quan-
tity produced (called the accumulated production) [Dutton 1984]. Various models
for this learning curve have generally been expressed as either a power law of the

accumulated production q:

C(q) = Coq‘“, (1.1)

or as an exponential law of the form
C(q) = Coe‘iq, (1.2)

where C(q) is the metric associated with the learning curve and q is the accumu—
lated quantity produced [Kantor Zangwill 1998]. This metric C(q) is a quantitative

measure that a manager would wish to minimize, such as manufacturing cost or

production errors. Past models have been little more than a choice between which
function is a better ﬁt to the data, rather than a causal explanation for why this
might be observed. The goal of this work is to provide such an explanation

Several deviations from these analytic forms (1.1) and (1.2) have also been ob-
served. For example, some initial rates of learning occur less quickly than either
model would suggest. Initial downward concavity has been observed in a number
of studies [Garg Milliman 1961]]Muth 1986]. A second, more signiﬁcant deviation
from the standard forms (1.1) and (1.2) is that, after a period of time, improve-
ment ceases, so that the learning curve asymptotically approaches a non-zero level,
rather than approaching inﬁnitesimal costs, as predicted by (1.1) and (1.2) [Con-
way Schultz 1959]. This plateau phenomenon is a. fundamental characteristic of any
learning curve over the long term, as costs are never expected to approach zero, even
asymptotically, a fact ignored by many practitioners of the learning curve.

Attempts to explain the learning curve fall into two broad categories: deter-
ministic and stochastic. Moreover, there have been observed deviations from both

deterministic models and stochastic models.

1.2 Deterministic Models

Several deterministic models have been proposed in the past, such as in [Levy 1965].
The leading deterministic models were proposed by Kantor and Zangwill in a pair of
papers [Zangwill Kantor 1998] [Kantor Zangwill 2000]. These papers propose that

the improvement in a metric C follows the intuitively plausible differential equation

-— = ‘CEUDCUIL (1.3)

where E (q) is a quantitative measure of management effectiveness and where q is the
accumulated production. Therefore, the rate of improvement is proportional both
to the management effectiveness and to the amount by which the metric may still
be decreased. To generate both power law and exponential models for the learning
curve, the authors make the further assumption that E (q) should itself have a power

law form as a function of C(q), that is,

E((I) O< (0(9))0- (1-4)

The choice between a power law and an exponential law then depends on a, the
power of C (q), since a = 0 yields an exponential while a positive a yields a power

law for C.

1 .3 Stochastic Models

Several stochastic models have been proposed for the learning curve. For example in
[Levy 1965] a model was proposed wherein the learning curve was solely dependent
on the remaining amount by which improvement can be made. This idea yielded the

differential equation
d ( 1 / :r)
dn

 

= #(P- 1/113), (1-5)

where :r is the quality metric, P is the eventual plateau level, and p is a parameter

of the equation. This differential equation has the solution

23(72.) = [P — (P — i) e-W] 4, (1.6)

$0

where 1:0 is the initial level for the quality metric. However, as stated in [Muth 1986]
this model is not capable of reproducing the power laws typically seen in actual
industrial settings.

An inﬂuential stochastic model for the learning curve is given in [Muth 1986]. In
this paper, Muth takes a “search theory” approach whereby “[c]ost reductions are
realized through independent random sampling from a. space of . . . alternatives.”
In this model, the distribution of such alternatives is assumed to approach a power
law in some lower bound. The probability of picking an alternative depends only
on its distribution, that is, all choices are equally likely, and the evolution is driven
merely by the distribution of choices.

Muth then proceeds to explain deviations from a power law relationship. He
explains the plateau effect as being caused by an increase in the search cost over
time, so that at some point searching for cost savings ceases being cost effective. He
explains the initial deviation from a power law as being caused by searching prior to
production, in line with what is argued in [Garg and Milliman 1961]. It is this type
of learning curve model that will serve as a springboard for both the discrete and

continuous models discussed below in this work.

Chapter 2

Insights from Other Research into

Learning

Both the deterministic and stochastic models have their appeal. The deterministic
model resembles the sort of scientiﬁc law that leads to predictive modeling. Muth’s
stochastic model on the other hand uses the intuitive idea of sampling of all possible
cost reductions. However, both theories require the assumption of a power law dis-
tribution for some part of the theory, and hence neither are based on ﬁrst principles.

A more foundational approach would incorporate the underlying causes that lead
to the observed learning curve. As the very term implies, the primal cause of this
evolution is learning. Workers on the plant floor learn to do their tasks better over
time. Managers learn how better to allocate labor and capital. Over time, this ,

learning allows the manufacturing to be done in a more efficient. way.

2.0.1 The Analogy to Artiﬁcial Intelligence

It is reasonable then to turn to learning theory for sources of a complete model of

the learning curve. The ﬁeld of Artiﬁcial Intelligence (AI) has a robust model of

learning. While many acts of innovation have occurred via AI [Russell Norvig 2002],
the innovation process itself has not been viewed through the AI lens. Applying AI,
which takes place inside a computer, to a system of humans and technology might.
seem too simplistic to analogize to the ﬁeld of manufacturing, which employs many
different resources—both human and mechanica.l——in many different ways.

On the other hand, a distributed computer network also has many distinct com-
ponents, networked over long distances, working on different aspects of a problem.
Thus, an AI network has such distinct parts learning diverse tasks to improve per-
formance.

In A1 a crucial distinction is made between supervised and unsupervised learning
[Engel and Van den Broeck 2001]. Supervised learning occurs when a machine at-
tempts to generalize examples to ﬁt a general rule. Examples of supervised learning
are regression [Hastie et a1. 2001] and classiﬁcation [Sperduti and Starita 1997]. In
regression analysis, example data is given to a system, and the system then makes
a best ﬁt of the data to an assumed form such as linear or logistic models. The sys-
tem can use the resulting model to make predictions of future data (extrapolations).
Similarly, classiﬁcation (which can be understood as discrete regression) uses a set of
examples to make the best ﬁt for future data by determining to which classiﬁcation
they belong.

Unsupervised learning does not assume a single preconceived model of the output
for the data. Instead, the learning system has to decide on a model to be used, as
Well as a way to fit the given data to that model. In this way, unsupervised learning
is much less structured or dependent. on ﬁxed probabilities than structured learning.

An example of unstructured learning occurs in the study of clustering of points,
Which is very important in image recognition. For clustering, there may be many

different categories of clusters, such as dense (many points clustered together), in-

teracting (for example, two clusters which appear to be attracted to each other), or
uniform (constant density throughout a cluster). The AI system has to decide which
of the models has best ﬁt, or even create a new category to ﬁt existing data, as well
as make predictions based upon this choice on the location of additional features.
The unstructured learning AI system can also modify these categories based upon
experience (how dense does a cluster have to be to be considered is dense, for ex-
ample?) and reassign probabilities for which a possible clustering pattern might be
met.
This dichotomy between supervised and unsupervised learning is directly anal-
ogous to the distinction between exploitation and exploration [March 1991]. Ex-
ploitation is associated with mechanistic structures, routinization, bureaucracy, and
control [He and Wong 2001]. Similarly, supervised learning uses a ﬁxed, mecha-
nistic structure to build and reﬁne outputs. On the other hand, both exploration
and unstructured learning are more open-ended processes, in that the output goal
can change and be modiﬁed over time, based upon, for example, external inputs

(instructions from a manager).

2.0-2 The Application of Statistical Mechanics to Artiﬁcial
Intelligence

TWO aspects of artiﬁcial intelligence theory of learning merit close attention: en-
ergy minimization and the connection with information theory. Learning in the AI
COntext has been described as a stochastic minimization of energy in an information-
t3heoretic setting [Seung 1992] [Watkin 1993]. Similarly, in the manufacturing setting,
One attempts for example to attain a minimization of cost or errors in the manufac-

turing process.

Information theory is undoubtedly relevant to understanding the learning curve.
The evolution of the learning curve is driven by learning, and any learning in the
formal sense is the acquisition of information. The landmark work done by Claude
Shannon demonstrated that the principles of statistical mechanics are applicable to
the act of learning [Shannon 1948]. Later studies demonstrate that the rate of learn-
ing is in fact very predictable using the principles of statistical mechanics [Seung
1992] [Watkin 1993]. Statistical mechanics is now the standard model for explaining
learning in the AI context [Engel Van den Broeck 2001].

March’s model of learning [March 1991] and its reﬁnement over the years have
all the outward appearance of modeling in statistical mechanics. March’s theory has

four key aspects [Miller 2006]:

i. The environment of an organization is modeled as an m—dimensional belief vector
with each coordinate randomly assigned a value of 1, 0, or —1, with m equal to

the number of individuals in the network.

ii. Each individual’s belief about reality is represented by 1 (positive belief), 0 (no

belief), or —1 (disbelief).
iii. Learning within the ﬁrm causes change in the values of these belief values.

iV- The organizational code adjusts over time to reflect the dominant belief among

better-performing members of the organization.

Therefore, according to March’s theory, learning occurs via interaction between
IneInbers of a ﬁrm and the organizational code, and these interactions are proba-
biliStic in nature. One reﬁnement of this theory is described in [Miller 2006], where
learming is also allowed to take place between individuals in a ﬁrm, with a proba-

blllstic bias toward learning by and between “proximate neighbors.”

This type of modeling is exactly how physicists use statistical mechanics because
it attempts to treat a large number of discrete objects in a mechanical, yet proba—
bilistic, way. In statistical mechanics the ﬁrst order approximation for interactions
between individual particles is with a mean ﬁeld. A mean ﬁeld is an averaged effect
of all the interactions a particle and the system. For example, in studying the grav-
itational ﬁeld of the earth, one does not look at the gravitational ﬁeld of each atom,
but one instead looks at the mean ﬁeld averaged over the entire earth.

In the instance of March’s theory, the organizational code plays the role of the
mean ﬁeld. After this level of approximation has been examined, one can look at.
higher orders of approximation, which generally focus on interactions between indi—
viduals. In these higher orders of approximation, it is the interaction between nearest
neighbors that is typically important, while other interactions are negligible.

This hierarchical approach in statistical mechanics can be seen, for example,
in the modeling of magnetic domains [Kodama Berkowitz 1999], protein molecules
[Schilk 2000], and galaxies [Binney Tremaine 1988], but. these steps are ubiquitous
in all modeling problems involving statistical mechanics. Therefore, since modeling
learning within a ﬁrm has already aped the process of statistical mechanics, it is

worth exploring how much further statistical mechanics can explain learning.

2.1 Statistical Mechanics in Other Areas

As noted in [Demsetz], the ﬁrm occupies a anomalous position in standard economic
theory. Speciﬁcally, the ﬁrm is considered to be a “black box” which acts as a sin-
gle economic actor (an agent) and competes with other actors in the marketplace.
Management theory on the other hand looks at the forces of different actors (agents)

within a ﬁrm and seeks to ﬁnd ways that their utility may be maximized. Such a

10

disconnect is puzzling since it is apparent that the utility maximization point of a
ﬁrm need not coincide with the utility maximization point of actors within the firm.

While the problems studied in management theory have not entered the realm
of general economic modeling, the last twenty years have witnessed an explosion of
attempts to apply economic theory to management problems [Donaldson I 1990]. As
shown in [Donaldson I 1990] [Barney 1990], this spread of economic ideas has been
met with some skepticism. This skepticism has been attributed to concerns about
the methodologies of economics [Barney 1990] [Donaldson II 1990], as well as to pro-
tection by an academic group of its own turf [Barney 1990]. However, this diffusion
of economic ideas into management theory has not engendered a theory of the effects

that such organizational economics might have on the overall macroeconomic picture.

2.1.1 Differences between Management Theory

and Organizational Economics.

In 1990 Lex Donaldson published a pair of papers outlining the connection between
management theory and economics [Donaldson I 1990][Donaldson II 1990]. Don-
aldson argued that while traditional economic theory had something to add to the
theory of managerial organization, organizational economics was overly concerned
with the actions and motivations of the individual within the ﬁrm. In contrast, Don-
aldson explained that management theory incorporated action on the collective level
as well as individual motivations.

Central to creation of the emergent. collective level is the issue of trust. An sim-
ple example of how trust creates a collective scale can be seen with the standard

form of the prisoners’ dilemma. Without. trust, the two prisoners act as individu-

11

DJ

 

als, and there is no collective level of action. On the other hand, if the prisoners
trust each other, they work as a single unit, and motivations have to be interpreted
as a type of aggregate behavior. Donaldson states that, in the standard theory of
organizational economics, trust within an organization only exists through enforce-
ment mechanisms—“It is ’trusting’ someone only after you have them ﬁrmly under
control” [Donaldson II 1990]. In contrast, management theory allows for trust of
managers even in the absence of these type of controls.

In general, organizational economists discount the idea that an organization is an
entity unto itself—“organizations, as ﬁction, have been invented by humans to make
sense of the world.” [Donaldson I 1990] In contrast, as Donaldson says, “... systems
analysis is at the heart of management theory because managers are responsible for
the conduct and performance of organizational systems.” [Donaldson II 1990] These
type of collective models are the central aspect of any organizational theory that

does treats the organization itself as more than a convenient ﬁction.

2.1.2 Econophysics and Learning.

Physicists readily recognize and understand the issues brought up by Donaldson and
March. This problem is one of scale formation. Economics, in particular neoclassical
economics, examines issues related to actions occurring on the single scale of the
individual. In contrast, organizational theorists have shown that many actions are
best viewed as occurring at larger scales, or even multi-scales [March 1991].

Since the ideas of statistical mechanics have been successfully used to model
learning in some contexts, it is natural to investigate whether it may be used to

model learning in other contexts. In both artiﬁcial intelligence and an organizational

12

structure, learning typically occurs as through a coupled interaction of subsystems
to external stimuli. Further, this coupling between subsystems is tighter than what
one might see between individuals in a general economic context. Finally, since the
principles of management have become more uniform and standardized over the years

[Dillard 2004], more predictability should be evident.

2.2 Further Directions for Statistical
Mechanics in Management Theory

Just as management theory might learn something from econophysics, econophysics
might gain some breadth from venturing into management theory. As we have argued
above, the regularity of relations within a ﬁrm should make it a more natural ﬁeld
for a study that depends on this regularity. Perhaps more important in the long run,
however, is the centrality of learning in management theory. As some have noted,
one severe deﬁciency in most of econophysics is that nearly all of the models given
are exchange-only models, with no discussion of increases in wealth [Gallegati 2006].
This limitation is in contrast to the idea—a. central to economics since at least Adam
Smith—that exchanges occur to increase the wealth of each participant. Learning,
on the other hand, necessarily involves an increase in a. valuable commodity. Since
learning has already been extensively explored within the framework of statistical
mechanics, the learning paradigm might be a natural avenue through which to in-
corporate growth.

Management theory is a natural ﬁt to the ideas of econophysics. In fact, as de—
scribed above, some management theorists have already anticipated these ideas. It

is hoped that this work will spur the development of the application of econophysics

13

to management theory.

14

Part II

The Mathematical Model

15

Chapter 3

The Discrete Case

3.1 The Parameters of the Model

3.1.1 Derivation of the Probability

Suppose there are n lessons that a manager or production worker can learn that will
improve the quality metric. Let. us ﬁx ideas by supposing this quality metric to be
unit cost, so that learning the i-th lesson will—from that moment forward—yield a
cost saving of 0,- dollars per unit. The ideas developed in this way will carry over
naturally to improvements in other quality metrics.

While in some cases learning one lesson might. affect the probability of learning
some other lessons, in general, most lessons are independent of all other speciﬁc
lessons. Learning a lesson at one stage of the manufacturing process will not. in
general affect decisions made regarding another stage of the manufacturing process.
While in practice many lessons will depend on others, the probability of one given
decision relying on a second given decision is low. Therefore, for purposes of this

model there is no correlation between lesson choices.

16

Also, any improvement, once made, cannot be made a second time. But how
does this removal of a decision choice affect the probabilities of choosing decisions
not already made? One possible answer would be to have the probabilities of the
other choices increase, so that the partition function is only taken over the remain-
ing choices. This idea (the elimination hypothesis) has an intuitive basis, since some
managerial choices will become more obvious once other choices are taken.

However, the elimination hypothesis does not survive closer scrutiny. The elimi—
nation hypothesis guarantees that, at each increment of manufacturing, learning will
take place. In contrast, learning the last lessons will take more steps than the ﬁrst
lessons. It becomes harder and harder to learn the lessons later. The methods of
manufacturing change less and less over time. Therefore, we reject the elimination
hypothesis.

So the question becomes how to reconcile the two above rules. If improvement.
cannot be made a second time, how can the probabilities of learning the lesson stay
the same the manufacturing lifecycle? The resolution is by way of scoring the learn-
ing. We will keep the lesson space, and their probabilities, ﬁxed over time. However,
if a lesson which has been previously learned is chosen a second time, no additional
improvement in the quality metric will be achieved. Over time, as more and more
lessons are learned, the probability of choosing a lesson learned previously will in-
crease. This rule implies that improvements to the process will decrease over time,
and that fewer and fewer new lessons will be learned.

Some cost-saving ideas are more obvious than others; let p,- be the likelihood that
the i-th lesson will be learned. Again, we will assume that there is no correlation
between lesson choices. Therefore, a lesson learned at one step will be statistically
independent from a. lesson learned at the previous or subsequent step.

Finally, we assume that the rate that lessons are drawn “om the pool of possible

17

lessons is proportional to production rate. Learning proceeds through production
experience. So for example, as the k-th hundredfold unit is completed, the k-th draw
is taken from the lesson pool. This assumption is borne out in the learning curve
literature, as learning patterns are observed over production cycles, instead of the
length of time making products.

Based on these rules, we may make the following conclusion regarding the struc-
ture of learning.
Theorem 3.1 The expected cost savings C(k) after It draws from the pool of n
lessons is given by the rule

it
Cw) = Z (1 — (1 whim. (3.1)

i=1

Proof. We proceed via induction. After one draw the expected savings are clearly
T1.
0(1) = Z: PiCi- (3-2)
i=1

The total expected savings after It draws is the expected total savings from the
previous k — 1 draws plus the expected savings for the k-th draw (which assumes

individual lessons have not been drawn in the previous k — 1 draws). In symbols,

C(k) = elk—1) + Emu—mk—lc. (3.3)
izl
= Z (1 —<1—p.>’<>c.:. (3.4)
i=1

Corollary 3.1 The cost U per unit can be expected to decrease with production by

18

the rule
71

UW=%—Zﬂ%bm%. (m

i=1

where U0 is the initial cost per unit when production commences.

Example 3.1. Suppose that. all savings c,- = 1 and that each lesson. is equally
probable: p,- = 1 / n. Then the evolution of the expected unit cost U is given from
(10) by

U(k) = U0 — n(l— ("71)k). (3.6)

72.

 

Thus as one would expect, for large k, well along in the production, cost per unit
is falling and approaching its plateau U0 — n. Because U ” / U ’ = — loglc for lesson
draws k = 1, 2, . . . , initial relative upward concavity is small in contrast to the abrupt

empirical models (3.1) and (3.2).

3.1.2 How lesson probabilities are determined

Let us now uncover why some production lessons are more likely to be learned than
others. The form of the learning curve across industries has been remarkably uniform,
suggesting that such a determination must exist, and that it is likely to be very
simple. Learning is driven by the push by managers to improve some quality metric,
such as production cost. In artiﬁcial intelligence, a fairly strong correlation has been
noted between the likelihood of learning a lesson and the beneﬁt accrued from that
lesson (Russell and Norvig, 2002). These observations indicate that. the decision
probability can be treated solely as a function of the corresponding cost savings, that
is

Pi = 17(021), (3.7)

19

where p = p(c) is some smooth function of the nonncgative real variable c.

Theorem 3.2 The probability p,- that the i-th lesson will be drawn is given by the
rule
43C"
—e—__. 2
Pi (3.8)
:2}: 163C

where the constant ,[3 is determined by the eﬂectiveness of the manager.

Proof. The following argument. will be familiar from statistical mechanics: Suppose
the n lessons 331,172, . . . ,xn deliver the respective cost savings c1,c2, . . . ,cn, and
suppose each individual lesson x,- will be chosen with probability p,. The probabilities
p,- should not depend on the currencies used to value the cost savings ci, so let us ﬁx

once and for all the value of the average cost savings after one draw
0(1) = 21727013 (3.9)

Let us draw one lesson after another, returning each to the lesson pool after each
draw. The relative frequency of drawing the i-th lesson will of course asymptotically
approach p,- as the trials proceed. By the Boltzmann H ~theorem (Feynman, 1972),
the largest number of distinct trials—the freest and most realistic experiment—

occurs simultaneously with the maximum value of the entropy

n.
— — E p; log p, (3.10)
i=1
subject to the obvious constraint
P1+P2+"'+Pn = 1 (3.11)

20

and subject as well to
P1C1+P262 + ' - ° +PnC-n = C(U- (3-12)

Applying the method of Lagrange multipliers to Q = H + A E p,- + {3 Z pici, we see

that H subject to its two constraints will maximize when

8H

dpi

which is p,- = aexp(f3c,~) with a = exp(1 + A). Because of (3.13), the quantity a is
the reciprocal of the denominator of right side of (3.9).
Observation. The expected cost savings (3.12) after one draw is similar to a familiar

thermodynamic quantity:
1 n 1
0(1) = ,7, ZPiI-Ogm — 510M, (314)
. 2'21 l

where l/a is the partition function (Feynman, 1972).

Let us combine the previous two theorems.

Theorem 3.3 As production of a good proceeds, lessons are drawn (with replace—
ment) independently from the pool of it possible lessons. Suppose the i-th lesson,
once learned, yields an ongoing cost saving of C, per unit manufactured. Once this
saving is realized, it is unavailable for future improvement. Then the expected cost
per unit must decrease by the rule
it
U(k) = U0 — Z [1 — (1 — aeﬂCi)k]c,-, (3.15)
i=1

where U0 is the initial cost per unit, where k is proportional to accumulated production

21

q, where
n 3
CJ, (3.16)

1

_ = e

a .
3:1

and where ,8 is the eﬁectiveness of the manager.

Remark 3.1: To understand how the model (3.15) applies to learning, one can look
at the various limits of the eﬁectiveness factor [7’ , which can be considered a measure
of managerial focus. In the limit of a small effectiveness factor (which corresponds
to a high temperature in thermodynamics) the probability factors will be very close
together. Therefore, in this limit, which corresponds to low managerial efﬁciency, a
manager would be equally likely to choose an option that yields little improvement
or one that yields great. improvement. Therefore, in the limit of low managerial
efficiency, the rate of improvements is just a matter of luck, rather than any skill on
the part of the managers.

On the other hand, in the limit of a high effectiveness factor (which corresponds
to a low temperature setting in thermodynamics), the manager is very likely to make
the decisions which yield the best improvements first. Therefore, a high managerial
efﬁciency yields a much more rapid rate of improvement than does a lower managerial
efficiency. Rather than being a. matter of luck, the rate of improvement comes directly
from the skill of the managers and all others who are making the decisions that cause

any improvements in a plant.

3.1.3 Entropy

The concept of entropy is central to statistical mechanics and information theory.
While entropy is often qualitatively considered to be a measure of the disorder of
a system, it is more accurately expressed to be the tendency of a system towards

dispersion of the possible states. The equation for the entropy H of a system is given

22

by the following deﬁnition.

Deﬁnition 3.1 The entropy H of a system of states with corresponding probabilities

p,- is given by

n
H: —k:pilnpi, (3.17)
i=1

where value A: is called the Stefan-Boltzmann constant.

The Stefan-Boltzmann constant is ubiquitous in statistical mechanics. In a thermo-
dynamic system (3.9), the parameter [3’ = 1/kT, where T is the temperature of the
system.

Using the form (3.9) for the probabilities given for our system and ﬂ = l/kT, the

entropy for our learning system becomes
1 Tl.
H = ET 2; AC, p(ACi) + Q, (3.18)
z:

where Q is constant. Equation 3.11 shows that a change in entropy for learning one
lesson is directly proportional to expected cost savings for this lesson. According
to Boltzmann’s H-Theorem, changes that increase the entropy of the system will
be statistically favored. Therefore, this model predicts that learning will tend to

increase the expected value of cost savings.

3.1.4 Interpretation of Results

Equation (3.15) for the expected quality metric C (k) yields some important conse-

quences. The form is not. a simple power law or exponential. It is in fact a sum of

23

exponentials, which in fact is a form suggested in [Zangwill and Kantor 1998]. But
can it accurately match observed learning curves?

As noted in section 1.1, one characteristic of learning curves missing from most
standard models is the plateau effect. In contrast, the plateau effect is present in
our model (3.15). This can be seen by looking at the asymptotic limit of the cost

evolution formula as It becomes very large.

24

The:

P101

Rm

RH

Theorem 3.4 As It becomes very large, the expected total cost C0 — E(k) has the
asymptotic behavior

n
11111 CO — —C—(l€) = C0 + 2 A022 (3.19)

[HOG i=1

Proof. Clear.

Remark 3.2: After many decisions, all options of cost savings have been ex-
hausted, and the cost of production levels out to a ﬁnite, non-zero value. This is
consistent both with what is seen in long term learning curves and what is intu-
itively expected. The cost of production is never expected to go to zero, since the
production of a product will always have, if anything, the commodity costs of raw
materials in making the product. Thus our model (3.15), in contrast all previous

models, incorporates a plateau effect.

Remark 3.3: The second characteristic of many observed learning curves is the
initial lack of an upward concavity. While both the exponential and power law func-
tional forms maintain an upward concavity throughout the curve, many observed
learning curves either have no initial concavity (linear) or have a downward concav-
ity. This phenomenon is sometimes described as an initial rate of teaming which
was slower than what would be expected from the overall shape of the curve [Muth
1986}

One can examine the second derivative (with respect to k) of equation (3.15)
to determine whether the graph of equation (3.15) has the proper characteristics
of concavity. The power-law forms of the learning curves of previous authors have
asymptotically large second derivatives for small values of the amount of production.

In contrast, the function of (3.15) has a ﬁnite derivative that becomes very small as

25

the number of choices n is increased. This fact can be seen in the case where all
improvements have equal weighting, which, as will be proven later in Theorem 5.3,

is maximal for this type of expression.

Lemma 3.1 When probabilities are equal, in the limit of a large number of lessons

learned the second derivative 0 the learnin curve
,

C0 — C(13)

is 0(n—1).
Proof. The second derivative is given by

(125 n k 2
Err—2 = [new —p.:) In (1 —p.:)- (3.20)
‘ ' i=1

The equation for the second derivative of E(k) for equal probabilities becomes

(126 1 k 2 1

For large n, this becomes the asymptotic relation

(120 1 _
if}? a n,AC;_2_=O(n 1). (3.22)

Remark 3.4: Thus, the value of the second derivative vanishes as n goes to inﬁnity.
Furthermore, this phenomenon of a vanishingly small second derivative can be intu-
itively explained by the fact. that when there are a. large number of choices, removing
a few does not signiﬁcantly change the expected value of the cost savings for each

step. This leads to a approximately linear form at. the outset of production. Thus,

26

1

ll

‘)

v .‘m\

this second defect of previous models is corrected by our model.

The results of this chapter are summarized in the following theorem.

Theorem 3.5 For a learning system with n lessons {11, . . . ,l,,} to be learned with
corresponding cost savings {AC1, . . . ,ACn}, the evolution of the learning curve for
a quality metric C as function of the accumulated production .1; with managerial

effectiveness parameter ,6 is given by

71

C-Eg.) = 2(1 — (1 —p,:)k)AC,j, (3.23)

where

 

(3.24)

3.2 Summary

We have derived a model for the learning curve with a discrete number of decisions.
As important as this step is, it is likely to be intractable in practice given the large
number of decisions which might come into play. Therefore, our next step will be to
develop a continuous model which will be more tractable. In fact, two different such
models are developed in the next two chapters. But before departing the discrete

model, we will examine several possible augmentations.

3.3 Multi—Dimensional Learning

3.3.1 Introduction and Motivation

Any analysis of learning curves is complicated by the fact that managers typically

have more than one quality metric that they seek to improve. For example, improv—

27

ing manufacturing cost alone might not. be the only goal; decreasing a defect rate
would also be a desirable goal.

To date there has been really no analytical explanation of how this trade-oﬁr eﬁect
occurs. Several empirical studies in fact have been done. For example in [Gino 2006]
a study was made of learning of a new technology by cardiac surgical units. This
study found a tradeoff effect between efﬁciency and innovation.

This trade-off effect. is especially familiar for anyone who researches mathematics
education. In performing arithmetic, students display a natural trade-off between
speed and accuracy. While both speed and accuracy are desirable goals, it appears
throughout that improving one goal often hampers development the other, if not
making the other goal worse.

In manufacturing, March’s exploration / exploitation distinction is very important.
Some of the research talks about a trade-off between exploration and exploitation.
However, this trade-off is analytically different from the other trade-offs discussed
above, since in the trade-off between exploration and exploitation there is only a
single quality metric considered.

Since there has been no analytical explanation of how the trade-off effect works,
the trade-off effect has been purely an empirical observation. Furthermore, confu-
sion is caused within the literature by referring to two somewhat distinct phenomena
as a trade-off. A trade-off between exploration and exploitation is different from a
trade-off between striving to improve two different quality metrics, since the explo-
ration/ exploitation trade—off still only involves the improvement. of a single quality
metric. Henceforth the trade-off effect will be restricted to the effect of a. multi-

dimensional quality metric.

28

 

 

3.3.2 Modeling Multi—Dimensional Learning

The learning curve model based on (3.13) is able to incorporate the trade—off effect

in a very natural manner.

Theorem 3.6 For a learning curve where two quality metrics are being simultane-
ously improved then the probability of a lesson to be learned is given by an exponential

of a linear combination of the improvements of the metrics for that lesson, that is

p, oc e_(31AQ'i+f32ARi). (3.25)

Proof. By Theorem 3.2, the probability of learning the i-th lesson with improve—

ments in the two quality metrics is equal to Q, and R, has to satisfy both
m 0< et’lAQi (3.26)

and

Pi oc 632”” (3.27)

for some constant values 131 and [32.

Remark 3.5: In the one—dimensional model (3.13) the quality metric took the place
of energy in statistical mechanics. The present model (3.25) results in the most logical
extension to a multidimensional setting—that a linear combination of the quality
metrics take the place of energy. In both the one—dimensional and multi-dimensional
models a minimization of the energy analogue, but additional features are present.
in the multi-dimensional case. In the one-dimensional case, the minimum of the
energy is independent of ,6. However, in the multi-dimensional case the minimum

will depend on the relative values of the positive constants denoted by (13,-.

29

 

 

 

3.4 Learning with Forgetting

It is well known that lessons which have been learned can also be forgotten. This
forgetting can occur in two formats: while production is occurring, and during breaks
in production [Bailey 1989].

It is in fact very simple to model both incarnations of forgetting in the context
of the learning curve. We ﬁrst examine the case of forgetting during suspensions
of production. Rather than the “forgetting” curve being driven the accumulated
production, the forgetting curve will be a function of time t. Since we are now using
a continuous rather than discrete variable, it. is natural to expect that the forgetting
of previously learned lessons will be a Poisson process with rate Af, a forgetting rate.
For any lesson learned, the probability p,- that a lesson will be forgotten then will
obey the rule:

e—AfAC,

Pi O< . (3.28)

In this expression it is important to note that the change in the quality met-
ric AC,- for a forgotten lesson will be equal in magnitude but opposite in sign for
a learned lesson. This gives rise to a reversal of probabilities, with respect to the
probabilities in learning. In learning, the most. valuable lessons were the ones most
likely to be learned; in forgetting, however, it is the most valuable lessons are the
ones least likely to be forgotten, and vice versa. This idea. for forgetting has been
explored for example in [Bailey 1989].

This type of forgetting process can be extended to the situation of forgetting
intermingled with learning. Two simultaneous events occur; learning occurs as a
function of accumulated production, while forgetting is a Poisson process as a. func-
tion of time. The faster the production rate is, the more that learning will relatively

outstrip production. And for learning, the limit of the production rate going to zero

30

 

will yield the forgetting process described previously.

31

 

Chapter 4

Continuous Model I: Weak

Convergence of Sequences of Sets

One approach to making the discrete model of Chapter 3 into a continuous model
leads to some interesting new mathematics. We will ﬁrst develop this mathematics,

and then show how it relates to the model.

4.1 Introduction and Motivation

The basic idea of a dynamic random set can be illustrated with the following mat-
aphor.

Thought Experiment: Consider raindrops hitting a sheet of paper. The distri-
bution of raindrops in time over the sheet is random and is therefore given by some
probability distribution. Over time, the water from the raindrops will completely
cover the sheet of paper. In this model, the raindrops all have a minimum positive
size, and only a ﬁnite number of raindrops are needed to cover the sheet of paper.

However, the diversity of shapes of the possible raindrops makes it. difficult to de—

32

 

 

termine the amount of the sheet of paper that is covered by raindrops. If a limit is
taken so that the number of raindrops increases, while the size of the raindrops goes
to zero (with the product of the number of raindrops and the size of each raindrop

stays constant), we might arrive at a continuous model of the problem.

However, the approach has fundamental problems. This analysis is done as a
limit of the sequence of sets. However, in the limit, this sequence of sets converges
(in the Hausdorff metric) only to a countable number of singletons. At each stage in
the limit, we have a positive measure, and the limit of the measures is also positive.
However, the measure of the limit is zero. In the limit, therefore, the measure dis-
appears. Therefore, an alternative approach is needed.

If these problems can be surmounted, this type of model could have numerous
applications. For example, this problem can be viewed as a type of random growth
model, explored in many publications, such as [Witten 1981]. Many manifestations
of solid state physics deal with the growth of a speciﬁc characteristic in a material
over time, such as with the Ising model [Griffiths 1969]. In mathematical biology,
the modeling of tumor growth is also done according to random growth models. The
standard approach to such problem involves the use of partial differential equations.
However, an alternate approach, based upon the theory of random sets and an ap-
propriate limit, might yield some new insights.

The theory of random sets was rigorously laid out by [h’latheron 1975]. Since
then, several works have presented examples of set—valued random processes, such
as in [Harris 1976] and in [Papageorgiou 1989]. Physical applications of random sets
have been explored, for example in [Koukiou 1990].

However, it was only until very recently that time-dependent random sets have

been explored [Capasso and Villa 2006]. Perhaps because of this fact, no physi-

33

cal applications of dynamic random sets have been investigated. Time-dependent
sets are often seen in very simple applications. One example is modeling the region
of a capacitor plate that is covered with charge—a time-dependent set. However,
there has been little-to—no exploration of time-dependent random sets. A set-based
approach, rather than the conventional function-based approach, might yield some
fruitful results.

Another important area. in which such a theory might. be applied is information
theory. Information theory deals with the quantiﬁcation of data and measures the
rate at which such data may be communicated from a transmitter to a receptor.
When such information is transmitted through a noisy channel, each piece of infor-
mation has the potential to be lost. The signal that is received by a receptor can then
be characterized by a random set. Over time, if the signal is repeated, the receptor
will receive more and more of the actual message, so that over time the complete
message might be ascertained by the receptor. Again, the amount received at any
time can then be characterized as a random set.

A continuous-time model of such a random set could be very useful as a limiting
process. This paper will solve this problem by developing a distributional View of
this limit. This process of deﬁning a distributional set is exactly analogous to the
limit which yields a delta function in a distributional sense. Based upon this view,
which is applicable to sequences of both deterministic and random sets, a rigorous

mathematical model of a dynamic random set will be deﬁned and explored.

4.2 Weak Convergence of Sequences of Sets.

For purposes of this construction the random sets under consideration will be closed

subsets of a ﬁxed compact subset of the real numbers IR. However, the idea and

34

results described below can be easily generalized to any compact subset of R".

In order to build a different concept of convergence, we will make an analogy to
generalized functions, for which the notions of a weak convergence were ﬁrst explored.
A delta function, for example, can be seen as a linear functional which is the weak
limit of a sequence of classical functions fn. For our weak limit, we must also start
with a concept of an inner product. Weak convergence in the ordinary context only

requires that the real—valued sequence of inner product converges, that is,

(friaQ) _’ (fag) (4-1)

for all suitable test functions g.

We therefore start by deﬁning an “inner product” between two measurable sets.

Deﬁnition 4.1 Given two measurable sets A and B, the pseudo-inner product (A, B)
is given by the rule

(A, B) = p(A r) B), (4.2)

where u is Lebesgue measure.

Remark 4.1: This pseudo-inner product can also be expressed as

(A, B) = /B XA dl‘, (4.3)

where X A is the characteristic function of A.

Deﬁnition 4.2 Given a measurable set A, the norm of A is

llAll = WW1). (4.4)

35

 

Lemma 4.1 For the pseudo-inner product of (4.2) and norm (4.4), the following

properties are true for any measurable sets A, B, C:

Z') IIAII 2 0. (4-5)
a) ”A” = 0 4:» A = i (1.6., (4.6)
iii) (A, B) = (3,4), (4.7)
iv) IIAU B|| S IIAII + llBlla (4-8)
v) (AB) S llAllllBll- (4.9)

Proof. Clear.

Remark 4.2: In particular, the triangle inequality property (iv) directly follows
from the subadditivity property of measures. Also, the Schwarz inequality (v) is a
straightforward application of Hélder’s inequality to the characteristic function of
the sets A and B. Two measurable sets are orthogonal if they are disjoint up to a

set of measure zero.

Remark 4.3: This pseudo—inner product is sublinear:

(A o B, C) 3 (.4, C) + (B, C), (4.10)

with equality only if the three sets have an intersection of measure zero. Furthermore,
a concept of scalar multiplication is missing. The lack of these properties prevent
the set of measurable sets from being considered a Hilbert space. Nevertheless, this
pseudo—inner product provides an example of a. sublinear functional on the set of

measurable sets.

36

Deﬁnition 4.3 A sublinear functional is a function f* : Z —> [0, 00), where E is

the set of sets ofﬁnite measure, such that for any two sets E1 and E2 in 2,
f*(E1UE2) S f*(E1)+f*(E2)- (4-11)

For example, for any measurable set A, we can deﬁne the associated sublinear func-

tional f A such that

fA(B) = <A, 43> = [Bx/144: (4.12)

for any set B of bounded measure.

In this example, (4.12) suggests a way of deﬁning a functional without regard
to a measurable set. For any integrable function f, we can deﬁne a sublinear func-

tional f such that.

f(B) z [B fdax (4.13)

This functional is sublinear for the same reason as the pseudo—inner product (4.12)
is sublinear. Now we consider a. concept. of a weak convergence with respect to this

pseudo-inner product.

Deﬁnition 4.4 A sequence {An} of measurable subsets is said to converge weakly

to a sublinear functional f* if
(A7123) —* f*(B), (4-14)

for every set B of ﬁnite measure.

37

 

The set of sets of ﬁnite measure then takes the place of the set of test functions
in this analog of conventional distributions. It is also evident that if a sequence of
measurable sets {An} converges in the normal sense, i.e. the limit inﬁmum equals
the limit supremum, then the sequence also weakly converges, by the Lebesgue dom-
inated convergence theorem.

This weak limit makes sense in the context of sequences of both the deterministic

and the random sets to be deﬁned in Section 4.3.

Example 4.2: Consider the following sequence of subsets of [0,1]. Let A1 = [0, 1 / 2],
A2=[0,1/4]U[1/2,3/4], A3 = [0,1/8]U[1/4,3/8]U[1/2,5/8]U[3/4,7/8], and so forth.
The measure of each set in the sequence is 1 / 2. Also, a limit of the sequence does not
exist, since the limit inﬁmum of the sequence is a subset of the rationals, while the
limit supremum of the sequence is the interval [0, 1]. The sequence fails to converge
in the Hausdorff sense as well.

However, it is evident that
(An, B) ——> 1/2,u.(B F) [0,1]) (4.15)

for every measurable subset B. Therefore, the sequence {An} weakly converges to

the sublinear functional f * as in (4.12), where

f*(B)=1/2/B><[o,1)d$ =4<Br110.11)/2 (4.16)

for every measurable set B, even though it does not converge in any usual sense.

This construction may be generalized as follows.

Lemma 4.2 Given any step function of the form ua = ax E with a in [0,1], E a

38

bounded measurable set, there exists a weakly convergent sequence of measurable sets

{An} with limit u* such that for any set B of ﬁnite measure,

u*(B)=/Buadx. (4.17)

Proof. We make the proof by explicit construction. First, given a set E of nonzero

but. ﬁnite measure we deﬁne a. continuous nondecreasing function

., T =ii(l—OO»I10E)
n(. ) p(E) . (4.18)

 

For any 0 S c _<_ 1 let rn_1(c) denote the least value x for which m(x) = c. Deﬁne

the sequence

A1 = E ﬂ[—oo,mm_1(a)],

A2 = Eﬂ([—1(._a/2)]U[m1(1/2),m—1(1/2+a/2)]),

An = E(ﬂ U:[m_1/(in,m_1(i/n+a/n)]),

Every term in the sequence has measure au(E). Furthermore, it is clear that for any

measurable set B,

(A7,,B) —> f*(B) = /BaXE dx, (4.19)

completing the proof.

39

Examining the density function

,. _ - [1(Bﬂ[iL‘—h,$+h])
”03(1) _ i136 2h.

 

(4.20)

sheds further light on weak convergence of a sequence of sets. It is a result of analysis

that one can use the density function to determine the measure of the set by the rule

MB) = [3419mm (4.21)

which is directly analogous to (4.12) above. For these reasons, the weak limit can be
viewed as generating a generalized density function.
Remark 4.4: The relation (4.21) of course holds for characteristic functions as

well, that is,

MB) = /3 X303) 42:. (4.22)

However, it is more natural to employ the density function to realize weak limits
because characteristic functions can only take the values 0 or 1, whereas a density
function can take any intermediate value, albeit only on a set of measure zero. There-
fore, the density function is closer in spirit to the idea of the weak limit, as illustrated

in Example 4.2 above.

4.2.1 Some Functional Analytic Results on Sequences of Sets

We now extend several familiar results from functional analysis to the context of
sequences of sets. First we associate a function with the weak limit of a weakly

convergent sequence of sets.

Theorem 4.1 For every sequence of measurable sets {An} that weakly converges

to a sublinear functional f*, there exists a measurable function. p* : R —> [0,1] {a

40

generalized density function) such that

f*(B)=/Bp*(.r)dx (4.23)

for every set B of ﬁnite measure.

Proof. Suppose that a sequence of sets {An} weakly converges to the sublinear func—
tional f *. Since the terms in the corresponding sequence of characteristic functions
{ X An} are bounded in the L00 sense, they possess a weak* convergent subsequence,
which we will denote {X41} Denote its weak* limit by p*. Then for every set B of

ﬁnite measure,

/p*dx=/p*Xde (4.24)
B

— I _ O ,

_ "151100 XAngBd/l" — nlgrci)O(An, B) (4.25)
— . _ *
_ Riggs/41ml?) — f (B). (4.26)

Corollary 4.1 Let {An} be a sequence of measurable sets. Then {An} contains a

weakly convergent subsequence.

Proof. As seen in (4.24)—(4.26), the weak* convergence of the {X1441} to p* implies
the weak convergence of {A;,} to a sublinear functional with density p*

Next we prove that the set of all possible measurable density functions, whose
range is contained in [0,1], occupies a large portion of the space of functions, the

range of Which is contained in [0, 1].

Theorem 4.2 Given a compact set C, let Ill be the set of generalized density func-

tions p* guaranteed by Theorem 4.1 as the representation of the weak limit f* of

41

a sequence of sets {An}, with all An C G. Then AI is dense with respect to the
L1 norm in the set of absolutely integrable functions with support G and with range

contained in [0, 1].

Proof. Without loss of generality, we may assume C = [0,1]. Recall that from
Lemma 4.2, that any constant real-valued function ua with support E and range in
[0,1] has a corresponding sequence of measurable sets which weakly converge to a
sublinear functional with density ua. Combining such a construction using n disjoint

sets E,- of ﬁnite measure, one can construct a simple function s(x) given by
71.
5a) = Ema, (4.27)
i=1

where 0 g a,- S 1, and where s(x) is the representation of the sublinear functional
of the weak limit of a. sequence of sets. But any integrable function 9 supported on
C with range in [0,1] is clearly well approximated by simple functions of the type

(4.29).

4.3 A Distributional Weak Limit of Random Sets

Let us next carry over the idea of weak convergence of sets to the regime of ran-
dom sets. A random set is a. random variable X whose values lie in the family of

measurable subsets of IR. More formally, we state the following deﬁnition.

Deﬁnition 4.5 Let f be a family of measurable sets of IR". Let E(f) be a sigma
algebra on .7: and it be a probability measure on 2(7), giving that (f, E(f'),7r) is a
probability space. A random measurable set ( or simply random set ) X of R" is the

identity function X : J: —> .73,

42

Example 4.5: Consider the collection N = {{1}, {2}, . . . , {n2}} of singletons of the
ﬁrst m natural numbers, each corresponding to a decision at,- that can be made in
learning. Assign to each singleton (decision) {i} its probability p,- of being learned.
Let .7: equal the collection of measurable sets of R and let E(f) equal the smallest
sigma algebra containing every element of .7. For any set S of E(f), deﬁne the

probability measure it by the rule

«(8) = Z 102'-

{i}ENﬂS

Our random set X is then simply the identity map on 3:.

Example 4.6: Let p be a measurable probability density on R. Let .7 equal the
collection of measurable sets of R and let E(T) equal the smallest sigma algebra
containing every element of .7. We deﬁne a function A! such that for any collection
of measurable sets S, A! (S) equals the minimal set of real numbers which has all of
the singleton members of S as a subset. We deﬁne 7r(S) = f M( S) p(x) dx. Then the

identity map X : .7: —> .7: is a random set of R.

Example 4.7: Let p be a measurable probability density on R and let b be a
ﬁxed positive real number. Let f equal the collection of measurable sets of R. For
any collection S of sets in 7:, deﬁne a function M such that Ill (S) equals the set
of left endpoints of those members of S which are closed intervals of length b. Let
E(f') equal the collection of all collections S of sets such that .M (S) is Lebesgue
measurable. We deﬁne 7r(S) = L121(S)p($)d$' Then the identity map X : .7: —> .7:

is a random set of R.

43

Ex

[1

 

Example 4.8: Let p be a measurable probability density on R2. Let. f equal
the collection of all measurable set of R. For any collection S of sets in .7, deﬁne
a function .M such that 111(S) equals the set of ordered pairs (x, y) where x is the
left endpoint of any member of S that is a closed interval with length y. Let E(f)
equal the collection of all collections S of sets such that 1")! (S) is Lebesgue measur-
able. Deﬁne 7r( =ffM(S )p( .1:, y )dy dx. Then the identity map X: .7: ——> 7:18 a

random set of R.

Example 4.9: Let p be a measurable probability density on Rn“. Let .7: equal
the collection of all measurable sets of R" and let E(f) equal the smallest sigma
algebra containing every element of T. We deﬁne a function 114 such that for any
S of E(f), M (S) equals the set of points in R"+1 where the ﬁrst. n coordinates
locate the center of an n—ball in S and where the last coordinate is the radius of that
n-ball. Deﬁne 7r(S) = f...L(I(S)p(:r1,...,;r,,+1)d.r,,+1... (1x1. Then the identity

map X : .7: -—> .7: is a random set of R”.

Example 4.10: Let p be a measurable probability density on R. Let .7: equal
the collection of all measurable sets of R and let E(f) equal the smallest sigma
algebra containing every element of f. For any S of E(F) we deﬁne 81 to equal the
set of all members of S which are the union of three closed intervals, each of length
1 / 3. For each such member of S1 we associate a point 31 = (x1,x2, 1:3) in R3 where
the coordinates are the left endpoints of these three intervals, with x1 3 x2 S x3.
We deﬁne a function At such that M (S) equals the set of all such points 31 associ-
ated with S in R3. Deﬁne 71( =fffu<S) p(.r1)p(.1:2)p(:1:3) dx3dx2dx1. Then the

identity map X : .7: —> .7: IS a random set of R”.

44

 

Henceforth we will adopt a standard abuse of notation: X will denote both the
random set itself as well as the map. The context will indicate which is meant. Now

we introduce the fundamental idea. of a capacity functional.

Deﬁnition 4.6 Given a random set X, the capacity functional TX (.1:) is the proba-
bility that the point x is contained in the random set X. That is, under the notation

of Deﬁnition 4.5,
TX(x) = 7r(X E fgx E X) = ffo(x)d7r(X). (4.28)

The capacity functional is a basic tool for characterizing random sets. For example,
it can be used to calculate the expected value of the measure of any random set.
To do this, it is important to realize that the measure of a random set is itself a
real-valued random variable [IVIolchanov 2005]. we now quote a fundamental result

due to Robbins published in 1944.

Theorem 4.3 (Robbins’s Theorem) Let X be a random set in R" and u the

Lebesgue measure on R". Then p(X) is a random variable with

B(12(X)) = A,” Tm >444: ) (429)

Proof sketch. By Fubini’s Theorem and (4.30),

E(iI(X)) = [#(XM /X)=/Rnxx(I )de( I)dI(X) (430)

= [Rn/M1016 I)dII( (X)dII(I )= / Tx(I )dII(I ) (431)

Corollary 4.2 For a random closed set in R” and u the Lebesgue measure, the

45

second moment of p(X) is given by

E(u2(X)) = [Rn kn Imam)divisive). (4.32)

Higher order moments are given analogously.

Given a measurable set A and random set X we can deﬁne an “intersection” random

set, X 0 A.

Deﬁnition 4.7 Let f A be the collection of measurable subsets of A. For each a in
E(f'), let

0.4: {f4 efA;3f€0.fA={yﬂA;i/Ef}}.

Let SALT/4) equal the set of all such 0A. Then EA(fA) is a sigma algebra given
for the intersection random set. The probability measure it A of S A is now given

by the rule 7r A(o A) = 7r(o). The intersection random set will be the identity map

X : .7: A -—2 fA.
Given this deﬁnition, the following is evident.

Corollary 4.3 For a random set X of R" and measurable set A in R", the expected

value of the measure of the intersection of X with A is given by

E(11.(X F) A)) = LTX(17)d#(T)- (4.33)

The expected value of the measure of a random set is therefore the same in form
as the integral formula for the inner product in (4.21), with the generalized density
function being replaced by a probability. In other words, the deterministic case
given in (4.21) is the same as (4.32), with the points contained in the deterministic

set having probability one. Therefore, the integral formulation (4.32) for a weak limit.

46

of a sequence of random sets closely matches what. has been previously presented.
Thus Robbin’s Theorem motivates the exploration of the weak limit of a sequence
of random sets. First, we extend the deﬁnition from Section 4.2 of a. sublinear

product to an inner product between a random set and a set of ﬁnite measure.

Deﬁnition 4.8 The sublinear product between a random set X and a set B of ﬁnite

measure is deﬁned to be
(X, B) = E(p.(X H B)) =/ TX(x) du(x). (4.34)
B

Next, we deﬁne weak convergence of a sequence of random sets.

Deﬁnition 4.9 A sequence S = {Xvi} 0f random sets weakly converges to a sublin—

ear functional X * if for every set B of ﬁnite measure we have (Xn, B) —> X *(B).

Corollary 4.4 For every weakly convergent sequence S = {Xn} of random sets
converging to the sublinear functional X * there exists a measurable function Ts :

R —+ [0,1] such that

X*(B)=/BTS(;1:)du(.r). (4.35)

Proof. The weak* convergence argument from the proof of Theorem 4.1 gives that
for the sequence of corresponding capacity functionals TX", there exists a. bounded

measurable function T5(x) of modulus at. most 1 such that

X*(B)= lim (Xn,B)= lim /BTXn(x)dp(x)=/BTS(x)du(x). (4.36)

n—IOO 71-430

Remark 4.5: Thus, the weak limit. of (4.37) gives rise to the concept of generalized

random sets X that are determined by their corresponding capacity functional T3.

47

 

As will be seen in an example below, it is possible to construct a generalized random
set which is not itself a random set.

Finally, a dynamic random set will be deﬁned as follows.

Deﬁnition 4.10 A dynamic random set is a triple (S0.D,S), where S0 = {Xn}
is a ﬁxed weakly convergent sequence of random sets, S is a collection of weakly
convergent sequences of random sets that contains the sequence S0, and where D(t)
is a one-parameter family of operators on S, where t is a non-negative real number,

such that for any element of S in S with the limiting capacity functional TS,

ii. S = D(t)S0 for some t E [0, 00),
iii. D(t + s)S == D(t)D(s)S, and

iv. D(t) is continuous at t = 0, that is

Egg, [3 IID(.)s(i)—Ts(v)141v<v) = 0 (4.37)

for every set B ofﬁnite measure.

The parameter t will be interpreted later as time. This deﬁnition is directly parallel
to that of a dynamical system or a. semigroup. As in dynamical systems, for every
dynamic random set and value t, there exists a weakly convergent sequence S (t),

hence its limit X*(t).

We now deﬁne what will be shown to be an example of a Dynamic Random Set,

and which will be applied to learning.

48

Deﬁnition 4.11 (The Principal Dynamic Random Set) Let f be a diﬁ’eren-
tiable probability density on R with compact support. Let .7: equal the collection
of all measurable sets of R and let E(f) equal the smallest sigma algebra containing
every element off. Fix n. Then for any Sf”) on(.’F) we deﬁne S]n)(t) to equal the
set of all members of S(") which are the union of n closed intervals, each of length
At/ n, with /\ a ﬁxed positive number and t a nonnegative real parameter. For each
such member of S[71)(t) we associate a point sf”) = (x1,x2,...,x,-,,) in R” where
the coordinates are the left endpoints of these n intervals which are ordered so that

x1 3 x2 3 S xn. We deﬁne a function .Mt such that llvft(S(")) equals the set of

all points SE") in R". Deﬁne

Tl
«(’I)(s("))=/.../ ( f(x,j))dxn...dx1. (4.38)
Mt(S(")) .:1

’t

Then the identity map X.n(t) : .7: ——> .7: is a random set of reals with respect to
(II)

the probability measure it, . Let So be the sequence of random sets X1105), with
t = 0, and where D(t’)(S0) is the corresponding sequence with t = t’. In general,

D(t’)({Xn(t)}) = {Xn(t + t’)}. Finally, let S denote all D(t)S0.

Theorem 4.4 Let (S0, D(t),S) be deﬁned as Deﬁnition 4.10. Then for any t Z 0,
the sequence D(t)S0 = S (t) will weakly converge to the limit X *, where this limit has

the property that for any set of ﬁnite measure B,
x;(B) = f (1 — e—f‘ff (2)41. (4.39)
B

Furthermore,

thug] ITD(t)s(I) - Ts(I)| dII(I) = 0 (4-40)
_. B

and therefore the triple (S0, D(t), S) forms a dynamic random set.

49

 

Proof. We need to show the existence of the sublinear functional X; such that for

any measurable set B

X;(B)= lim /TXn(,)(x)du(x). (4.41)
n B

In order to show this existence, we examine the limiting behavior of the hitting
probability for the given sequence of random sets. Fix x and t. Recall that Xn(t)
takes only values which are the union of n closed intervals which are each of length
At/n. For each Xn(t) in the sequence, the probability of being in an interval of the

type comprising X 71(t) is equal to

/: f(x’)du(x’). (4.42)

—At/n

The probability of not being in the union of n. such intervals is therefore given by

(1— ft f(:c’)d4(v’))". (4.43)

—)\t/n.

and so
(13

TXn(t)(1’) = 1 “ (1 — [PM/11

For large n, the differentiability and the compact support. of f ensures that.

Hawaii)". (2144)

 

(I f(1‘-’)dlt($’)= W“) + 0(1/222) (445)
x—At/n 72.

An expression of the form

(1— % + 00/1120” (4.46)

I.

 

converges pointwise to e“. Therefore, from (4.44)-(4.46) we have

11111 TX,,(,)(1:) = 1 — e_’\ff($). (4.47)

71—300

By our deﬁnition of the measure of the limit of a weakly convergent. sequence
of random sets and the use of the Lebesgue Dominated Convergence Theorem, we
obtain a sublinear functional which represents the limiting measure of the sequence

such that for any measurable set B,

x;(B) = /B (1 — (”617544. (4.48)

Hence, at t = 0, we have a capacity function equal to 0 for any singleton. In contrast,

for A > 0, the capacity functional for any given point approaches 1 for large t.

To show that the Principal Dynamic Random Set is indeed a Dynamic Ran-
dom Set, we may observe that Properties (i)-(iii) for a Random Dynamic Set are

apparent. Property (iv) follows from (4.48). To see this we examine the limit

tl/iE10/BITD(,/)S(x)—TS(x)|dx. (4.49)

In the present. case, we obtain that this limit is equal to
lim / )(1 — BMW” (1‘)) — (1 — eWIm 41; = 0. (4.50)
B

tI—IO

Therefore, the Principal Dynamic Random Set is indeed a Dynamic Random Set.

51

 

Remark 4.6: We can now observe that the sequence X", does not strongly con-
verge to any random set X (t) for a ﬁxed nonzero value of At. To see this, suppose
some such random set exists, X (t). For any interval E which contains a set of posi—
tive measure on which f is nonzero the hitting probability TX(,.)(E) is 1. However,
consider a sequence of intervals E, which converge to a singleton Ego which is in the
support of f. The limit of the hit ting probabilities will be 1, while the hitting proba-
bility of the limiting set will be 1 — e‘Mfl‘f) . This contradicts the result of Choquet’s
Theorem (Matheron 1975) which requires the upper semicontinuity property

lim T(En) g T(nli_1’1;O E") (4.51)

71—*OO

for every random set. It is unknown whether the sequence weakly converges to a

random set.

4.4 Properties of the Principal Dynamic Random

Set

Given a set B which is the minimal compact support of f (x), deﬁne Gf(t) as

cf(t)=X,:':(B)=/B(1—e‘-A’f<r>)41~. (4.52)

Theorem 4.5 Cf(t) has the following properties:

, dC (t)
i- ’Tlff‘lt20 = A#(B);

dC 1‘
ii. ——atL(-) > 0 for all t; and

(120 (t
iii. 7% is negative for all t.

Proof. Clear.

Remark 4.7: Of course A need not be a constant, but. could instead depend of
position or even time. If however A = A(x), the dependence would be indistinguish-
able from a position dependence of the probability distribution. Therefore, one could
redeﬁne the probability distribution to incorporate this dependence, and leave A as
a constant. On the other hand, if A depends on time, the derivative properties from
Theorem 4.5 might no longer hold. Therefore, we will assume A to be a constant.
Another very important property is that. Cf(t) is maximized when f (x) is uni-
form. This result is intuitively plausible since a uniform probability minimizes the

chances of overlap over time.

Theorem 4.6 (Maximizing Distribution) If f is a probability distribution den-
sity with compact support B, a subset of R, with A not dependent on time, then the
maximum value for C f(t), t is positive and ﬁxed, is obtained when is f is constant

with respect to :13.

Proof. To maximize Cf is to minimize
/ exp(—Atf)dx.
B
This result is a special case of the following. Deﬁne a functional J such that
4(1) = [B G(f(I))dI.

Where G is non-increasing and f satisﬁes

/B f(ﬂdx =1,

53

Then
J(/) 2 /B G(smpf(I))dI~.

and the lower bound is realized when

f(I) E 1/II(B)

almost everywhere.

Remark 4.8: Given a principal random dynamic set X t(t), the complement Xt(t)
has a nice pseudo-semigroup property. Recall that for standard semigroup R we have

the properties

Let us deﬁne an operator R(t) acting on measurable subsets E by the rule R(t)E
yields the subset E \ X (t). It is clear from the properties of a Dynamic Random Set.
that property (1) is met. However, property (ii) must be understood in a different

way than is standard for a semigroup. The second property can be rewritten as
R(t + s)E = (R(t)E) ﬂ (R(s)E). (4.53)

However, this expression is at least correct. with the expectation value of the mea-

sures. It is trivial to show that

)1.(B(1+ s)E) = )1.((11(1)E) r1 (R(s)E)) (4.54)

54

and therefore much of the content of the semigroup concept is seen in the properties

of Dynamic Random Sets.

4.4.1 The Weak Limit as a Random Functional

A weak limit X * of a sequence of random sets {X71} (see Deﬁnition 4.8) can alter-
natively be thought of as a random functional that acts upon measurable sets to
produce a real-valued random variable. To obtain this result, we will ﬁrst deﬁne a
real-valued random variable arising from a single random set. Second, we will then
derive a random functional from this real-valued random variable. Third, we will use

a sequence of such random functionals to create a limit random functional.

Deﬁnition 4.12 Let X be a random set of the triple (f, E(f),rr) with .7: equal to
the set of all measurable sets of R, E(ZF) a sigma algebra on F, and it a probability
measure on E(f). Further suppose that, for any measurable set M of non-negative

real numbers, the set D of all sets which have measure m for some m E M is in

E(f). Consider the probability space (9, 2(9), ﬁx) where
i. Q is the set of real numbers,
ii. E(Q) is the sigma algebra of all Lebesgue measurable subsets of R, and

iii. for any measurable set S E 7:, the probability measure 7rX of S equals 7r(D),

where D is the set of all sets with measure m as m ranges over all of S.

Then the identity map X l : it —> Q is a real-valued random variable of the triple
(Q,Z(§2),er). We will call this random variable X l the measure random variable

associated with X .

55

We next deﬁne a functional with a domain consisting of all measurable subsets

of R and range equal to a subset of the collection of all real-valued random variables

on R.

Deﬁnition 4.13 Given a random set X suppose that for any measurable subset B
of R, there exists a measure random variable (X F) B)Jr for the intersection random
set X 0B (see Deﬁnition 4.7). Then we deﬁne the measure random functional f X to
be a function with a domain consisting of all measurable subsets of R and range equal
to a subset of the set of real-valued random variables, such that for any measurable

set B,

fX(B) = (X r1 8)]. (4.55)

Thus this random functional f X operates on a measurable set B such that f X(B) is

a real—valued random variable.

Example 4.11: Choose the random set X to be that given in Example 4.7, where
p(x) is a uniform distribution on [0, 1], and b = 1 / 2. Then we have a random variable

f X([0, 1 / 2]) with cumulative distribution function

0 x<0

0(4): §+x 03443;- (4-56)
1
1 IE>§

In general, the relationship between the random set X, the intersection random
set X n B, and the linear functional is captured by the commutative diagram of

Figure 4.1.

56

intersection .
X ————> X H B

functional [ [functional

fX action of functional} fX(B) = (X H 3)]

 

Fig. 4.1. The commutative diagram for intersection

random sets and the measure functional.
Now we deﬁne the limit random functional X (:0

Deﬁnition 4.14 Assume we are given a sequence of random sets {X11} with a cor-
responding sequence of measure random functionals { f Xn} such that the sequence of
real-valued random variables { f X,,(B)} converges in probability for every measurable
set B. Then we deﬁne the limit random functional X3;O so that for every measurable

set B,
XL,(B)= lim fX,,(B). (4.57)

71—430

Since expected values pass in the limit for convergence in probability, the expected
value of X ;(B ) then will be given by the sublinear product X *(B) given above in
(4.35):

B(Xl,,(B)) = X*(B) = f3 lim TX,,,(4,~)411(1), (4.58)

n—ioo

again directly analogous to the expected value given in Robbin’s Theorem. Higher
order moments will also be given with analogous formulas. If P(x 6 X and y E y)
is the probability that x and y are both in random set X, then we obtain for the

second moment

n——>oc- 771,—? 00

E((X;O(B))2) = [8/8 lim lim P(x E X and y E X)d/i(x)d,u,(y). (4.59)

57

Finally, the variance of X 00(B ) will be given by
42(Xto(B)) = (B(Xl.(B)))2 — B((Xl.(B))2) (4.60)

In the context of dynamic random sets, we can similarly deﬁne a one-parameter set.
of random functionals Xg0(t). When we examine the random functional associated
with the Principal Dynamic Random Set given in Deﬁnition 4.10, one diseerns a very

import ant property.

Theorem 4.7 Let X;o(t) be the random functional associated with a Principal Dy—
namic Random Set of Deﬁnition 4.11 with f and a measurable set B that is the

minimal compact support off, the variance of X;o(t)(B) is zero.

Proof. Using the formulas (4.58)--(4.59) for the ﬁrst and second moment of X {10(1) (B )

we obtain

B((Xl..<t)<B))2) = lim IX..(:2:)TX,,(4)dvdy (4.61)

m,n—+oo

l.
f (1 — e-Mflflﬂl — e‘f‘tﬂyl) dydx (4.62)
B

(1—e_’\ff(fl)dx/ (1—e-W<I>)dy (4.63)
B

(Xi.(2)<B)))2. (4.64)

H
3106\m\to\

Therefore, the random variable X go(t)(B ) will have zero variance.

Since the random variable X ;C(t)(B) is a limit of measures within of random sets,
the result can be interpreted to mean that. the measure of the principal random set
has zero variance is therefore deterministic. As shown by Theorem 4.4, its expected
value will be given by Gf(t). Therefore, the measure of X 3;, is deterministic, even

though the random sets in the sequence X ”(t) do not have a deterministic measure.

58

Remark 4.9: Theorem 4.7 is a special case of the more general case that if the prob-
abilities of the random set containing two distinct points x and y are uncorrelated,
then the variance of its measure will be zero. For the Principal Dynamic Random
Set, given A and t, ﬁnd an no such that for all n > no implies that At / n < |x — y].
Therefore, after the ﬁrst n0 terms in the sequence, x and y will never be in the same
interval, but instead in uncorrelated intervals. This lack of correlation between the
probability of containing two distinct points is an important motivation for the use

of the principal dynamic random set to model learning.

4.5 Applications of dynamic random sets

Many of the possible applications to this problem can be seen in the ﬁeld of statistical

mechanics. However, the simplest application is from a beginning physics course.

4.5.1 Charging a Capacitor

Consider a constant source of voltage V, across which is connected a resistor of R
Ohms in series with a parallel-plate capacitor of C farads. When this circuit is ﬁrst
connected, electrons ﬂow from the battery onto one plate of the capacitor, giving
that plate a negative charge. Electrostatic repulsion pushes electrons from the other
plate into the other segment the circuit, giving the second plate a positive charge
which is equal in magnitude to the charge on the ﬁrst plate. Over time, the current i,
which equals the product of the charge of an electron and the rate that the electrons
are collecting on the ﬁrst plate, decreases, and the magnitude of the charge on the
plate asymptotically approaches a maximum value Qmag; 2 CV.

The standard explanation for the eventual slowing down of charge ﬂow is that

59

there is effectively a smaller applied voltage across the resistor R in the circuit,
since the charge accumulated on the plates creates a potential difference between
the plates. lV’Iacroscopically, this is certainly true. However, the use of random sets
affords us a microscopic view of what is actually happening at the capacitor plates.

Rather than a single current i we will consider two currents: an incoming current
iz-nc, and a current ire fl reﬂected back from the capacitor towards the electron
source. The existence of two separate currents is merely heuristic and is not meant
to represent two separate, distinct ﬂows of electrons. The incoming current will

depend only on the applied voltage V and the resistance R:

_ V
Zinc = E. (4.65)

The macroscopic current i will be the difference between these two, since they are

traveling in opposite directions:

7: 2 2line "‘ z.refl° (466)

The incoming current imc will be constant. The reflected current ire f) on the other
hand will not be constant. Reﬂected current will be created when charge attempts
to go to a part of the plate where charge already resides. Since the capacitor will
initially be assumed to be uncharged, in. fl will initially be zero. As the plate ﬁlls
with electrons, the probability that electrons will attempt to travel to parts of the

plate already occupied will increase, and irefl will correspondingly increase.

60

 

Figure 4.1: The currents involved in charging a capacitor.

To make this description more precise, we will use the Principal Dynamic Random
Set given previously in Section 4.4. Electrons will be equally likely to occupy any
area on the capacitor. The amount of space an electron will take up on the capacitor
plate will equal the area of the plate divided by the number of electrons which can
ﬁt on the plate, or

‘16-

A8_ = Aplate’CTV (4-67)

Since this number is vanishingly small in comparison to the size of the plates, this
phenomenon can be modeled well by the principal dynamic random set, which looks
at the limit of a large number of intervals covering a space. We will assume that

f (x), the probability for the random dynamic set to cover a point x, to be constant:

1
(I) = 24—111.; (4.68)

61

In the present case, the Principal Dynamic Random Set will be 2—dimensional rather
than 1-dimensional, but the n-th term of the sequence of random sets will merely be
formed by the union on 11 squares rather than the union of n intervals.

The parameter A represents the coverage rate, the rate at which electrons cover
the plate:

in if I
A = ————’"C = L“ = —— 4.69
62...... CV BC ( )

Given f (x) and A, the area covered by electrons can be found by (4.67)

A(t) = f (1 — e—I‘tf (3)4114 (4.70)
Aplate

: Aplateu _ e—t/RC)2 (4'71)

which yields the charge Q(t) as a function of time to be

_ AU)
Q“) — qe-Ae_

= ova — e-I/RC), (4.73)

 

(4.72)

which exactly matches the rate observed in the charging of a capacitor. The total

current can be found by

1(2) = L9”) = K.a-I/RC, (4.74)

dt R

again exactly matching experiments.

Finally, the current reﬂected from the capacitor will be given by the expression

2....(2) = — 2(2) = —(1 — 84/25, (4.75)

to<

Therefore, in the the random set model, the charging of a capacitor plate slows not

62

because less current is incoming at the plates, but because arriving charge becomes
more and more likely to attempt to go to spaces already occupied by charge, causing

a reﬂected current.

4.5.2 Growth Models

Several scientiﬁc problems involve the growth of a body to ﬁll up a space. The best
example is that of a tumor [Holash et al..] A tumor starts as a very small body with
one or very few cells. Over time, it can grow to macroscopic sizes, even hundreds of
cubic centimeters. The typical cluster model involves discrete steps of adding parts
on the cluster, with blood vessel proximity increasing the probability of additional
tumor cells at a site.

This type of problem seems to be particularly well suited to dynamic random
sets. A dynamic random set can explicitly deal with the problem of tumor growth
in continuous time, rather than the discrete time as in the model in [Holash et al.
1999]. The fact that different places have different growth probabilities is naturally
handled in the case of random dynamic sets. This potential application for random

sets is a topic I certainly hope to explore in the future.

63

4.6 Creating a Continuous Model of the Learning
Curve

via Distributional Random Sets

4.6.1 A Deterministic Model of Learning a Continuum of

Lessons

We will now use random sets to build a continuous model of learning. To this end we
must overlay a cost. structure over a space. Suppose that each lesson to be learned
has been labeled by exactly one real number :1: so that the cost savings realized by
learning the lessons labeled by points lying in a measurable set. B is given by the

Stieltjes integral
AC(B)=/ dC(.r). (4.76)
B

where the cumulative cost savings function G (1:) is an everywhere deﬁned, nonde-
creasing, right—continuous function that is continuously differentiable between iso-

lated jumps.

Example 4.12: Suppose there are exactly 11 lessons to be learned, labeled by
x,- = 1,x2 = 2,. . . ,x,, = 11.. Each lesson xi = i yields a cost improvement of ci.

Then the total cost improvement obtained by learning the lessons in a set B is

AC(B)= / dC(.r) :2... (4.77)

B 1768

Example 4.13: Suppose cumulative cost function C is everywhere continuously

differentiable on the open set G. Then the cost. savings contributed by an inﬁnitesimal

64

neighborhood of x E G is dC(x) = C’(x) dx = c(.1:)dx, giving that the cost savings

for lessons in B C G is

AC(B) = f

B

dC(x) = ch(:r) dx = /GXB(x)c(x) (1.1:. (4.78)

Remark 4.10: Points of differentiability of the cumulative cost. function C should
be thought of as incremental improvements, while jump discontinuities are paradigm
shifts. For the incremental movements. learning is done incrementally, and improve-
ment happens continuously over time. As discussed in Section 2.1, this type of
learning is what March called “exploitation” [March 1991] and is considered “super-
vised learning” in the artiﬁcial intelligence context [Russell and Norvig 2002].

On the other hand, the paradigm shift corresponds to March’s idea of the “ex—
ploration” aspect of learning. which is considered “unsupervised learning” in the
artiﬁcial intelligence context. Learning under the category of exploration includes,
for example, new technology, new worker insight. to production, new cost structure,
material replacement, etc. Rather than continuous improvement, this type of learn-
ing takes the form of sudden leaps forward. While continuous improvement may be
modeled with a continuum of learning choices, explorative learning takes the form
of a ﬁnite number of breakthroughs.

As was argued for the discrete case in Chapter 3, the probability prob(X) of
choosing the lessons forming the measurable set X is a smooth function p = p(y) of

the cost savings y 2 AC (X ) resulting from these lessons:

prob(X) = p( [X 40(2)). (470)

65

 

Further, from Chapter 3’s use of the Boltzmann’s H—theorem [Feynman 1972], it
follows that p(y) = aexp(/3y), where the normalization 01 will be determined below.
In particular, if F is the cumulative probability that all lessons indexed by x or

less will be chosen, then

.17
F(x) 2 aexp (3/ dC(y)) = aexp(/3C(x)), (4.80)
—00
where
1 00
g. = ex13(23/ (10(4)) = BXP(2’3C(oo)). (4.81)
—oo

Note that the corresponding probability density function must be
f(x) = F’(x) = (113 exp(,1‘3C(x))c(x) = )‘3F(x)c(x), (4.82)

where

((1') = C’(x) (4.83)

at points x of differentiability of C. Jumps of C of height Co will yield jumps in F
of height Ore/300.

But now suppose X is a random set of lessons; that is, X is a random set. Each
random set X possesses a hitting probability T X(x), which is the probability that
x lies in X. This hitting probability is obtained by integrating the characteristic
function X X(x) over the probability space of the set-valued random variable X.

For any integrable real-valued function g of a real variable x using Robbin’s

theorem:

E[/Xg(x)d;1:] = [30 g(x)TX(x) .1... = E[g(x)|xEX]. (4.84)

—00

66

Remark 4.11: At ﬁrst glance, the expected cost savings from learning all the lessons

from this random sample of lessons X would from (4.76) be

AC(X) = E[/ dC(x)] = /00 TX (1:) dC(x). (4.85)

X —00
The second equality of (4.84) is indeed a correct statement of probability. How-
ever, lesson choice must be independent for this to be the learning observed in man-
ufacturing, as shown in Chapter 3. Moreover, this independence is necessary to
obtain the probability structure (4.84). Unfortunately, a random set X may possess

correlated points, that is, where x1 324 x2 yet
prob(x1 E X and x2 6 X) aé TX(x1)TX(x2).

But when correlation is absent, (4.85) holds for random lesson choice.

Theorem 4.8 If lesson choice from X is uncorrelated, then the expected cost savings

from learning the random set of lessons of X is indeed given by

AC(X) = E[/X dC(x)] = [00 TX(x)dC(x). (4.86)

m

Our task then is to construct an evolving random lesson sampling technique that
grows over time to eventually cover all lessons, a process suggested by Robbins’s
1944 modeling of airﬁeld carpet bombing. This dynamic model of learning must
allow uncorrelated sampling of lessons as well as additivity of the resulting cost savings
when compound lessons are reﬁned into independent sublessons. Both properties are

necessary if we hope to retain the probability structure (4.80)—(4.81).

67

4.6.2 Expected Savings from a Random Set of Lessons.

We now exploit the machinery of the Principal Dynamic Random Set to build a
continuous model of learning. There are two reasons why the Principal Dynamic
Random Set gives the right framework for a continuous model of learning:

First, using the Principal Dynamic Random Set allows for inﬁnite uncorrelated
sampling of lessons. Recall that one of the central premises of Chapter 3 was that
the lesson learning had to be uncorrelated (Assumption 3.4). The formation of the
Principal Dynamic Random Set involves a sequence with a limit which qualitatively
resembles an inﬁnite number of inﬁnitesimally small intervals. However, at each term
of the sequence, At is the measure of what is sampled. Because of the structure of the
construction, there is no correlation of the probability of whether two distinct points
are within the sampled space. There is no other process that samples an inﬁnite
number of points (indeed, a set of positive measure) but does so in a completely
uncorrelated way.

Second, sampling using the Principal Dynamic Random Set naturally yields inﬁ-
nite additivity. A useful property of the construction of the probabilities is that the
probabilities were additive. Take two lessons L1 and L2. Additivity means that if L1
and L2 are conjoined, the probability of picking the L1 + L2 lesson scales uniformly
with to the product of the probabilities of learning L1 and L2 individually. With the
Principal Dynamic Random Set we can subdivide lessons inﬁnitely many times (this
relates to my construction of the discrete set of lessons with each lesson comprising
an interval). In other words, the Principal Dynamic Random Set carries additivity
to its most extreme limit. Also, as will be shown later, a side beneﬁt. of the inﬁnite

additivity is that it allows for partial learning of lessons.

68

In short, we will sample via. the Principal Dynamic Random Set. because it allows
i. Inﬁnite Uncorrelated Learning, and

ii. Inﬁnite Additivity.

4.6.3 The Smooth Case and Principal Dynamic Random

Sets

We present here only the ﬁnite incremental learning case, where cumulative cost
saving C = C(x) is everywhere continuously differentiable and where C’ (x) = C(x)
has compact support. Suppose that learning is taking place at the ﬁxed Poisson rate
A so that the lessons that comprise the measurable set L require t = ,u( L) / A seconds

to learn, where u is Lebesgue measure.

The construction. For each natural number k let X k(t) denote the random set
of lessons that is the union of k intervals of type [a, a + At/ k), where the It left end
points a are the result of k independent draws with probability given by the density

function f of (4.86). Intuitively, and as the next result shows,
E122(Xv.(t))] s )2. (4.87)

that is, the lessons of X k(t) require on average at most t seconds to learn. Thus

X k(t) is a random sample of k compound lessons that can be learned in time t.

Lemma 4.3 Let TXk(t)(x) denote the hitting probability of Xk(t), namely the prob-

69

 

 

 

ability that x will lie in Xk(t). Then

(13

TX,(.)(2:) = 1— [1 —/_ka(21)441". (488)

Proof. The point x will lie in the random interval [a, a + At / k) exactly when x 2

a > x — At with probability

£13

prob(a E (x — At/k,x]) = /—At/k f(y) dy. (4.89)

Thus x lies in none of the k intervals [a, a + At) forming X k(t) with probability

[1 — f1: f(y) 4211’? (4.90)

41/1.

Lemma 4.4 Lesson choice is asymptotically independent: Let
TXk(t)(x1,x2) = prob(x1 E Xk(t) and x2 E Xk(t)). (4.91)
If x1 55 x2, then for all sufﬁciently large k,

Txk(t)(171.1‘2) = TXk(1)(IFl) “TXk(t)(l‘2)- (492)

Proof. W'hen At/k < ng — x1] it is impossible for both x1 and x2 to belong to the
same one of the k random intervals [a, a + At/ k) that comprise X k(t).

Thus as k increases, the sample X k(t) consisting of k ever—shortening compound
lessons begins to reassemble an independent sample of individual lessons, all of which

can be learned in time t.

Lemma 4.5 The hitting probabilities converge pointwise. In fact, for each x and

70

t 2 0, we have

11111 TXk(,)(x) = 1—e_’\ff(‘f). (4.93)

k—ioo

Proof. Since the probability density f has compact support,

 

f1: f(x)dx = ”Ufa” +O(1/A:2). (4.94)
.1:—wk 1»

Therefore,

 

. x , k _ . _AthE) ,2 k
khmm[1—L_At/kf(y)dy] — khmoo [1 k + 0(1/k )] (4.95)

= exp(—Atf(x)). (4.96)

It is worth noting that this construction matches what is present above in Deﬁnition

4.11. Therefore, we may say the following.

Theorem 4.9 The construction {4.87)-(4.96) yields a Principal Dynamic Random
Set whose limiting hitting probability has the form (4.96). The resulting expected
costs savings (4 .98 ) asymptotically reduce to the discrete case (3.13), as shown below

in Section 4.6. 5.

4.6.4 Prediction of unit cost

As seen in Lemma B, our random sample X k(t) of k compound lessons, learnable
in t seconds, approaches for large k an independent sampling of individual lessons.
By Lemma C, the corresponding hitting probability TX]: (t) approaches the limiting
simple expression (4.93). Therefore, it is more than plausible that in the limit we

have obtained a prediction of the decrease in cost-per—unit during manufacturing.

Theorem 4.10 Assume cumulative cost saving C = C(x) is continuously differen-

71

tiable and that its derivative has compact support. If learning is proceeding at the

Poisson rate A, then the expected production cost U per unit is given by the rule
30
11(2) = U0 — f (1 — .I‘If (1‘)) (10(1), (4.97)
—oc-

where U0 is the initial cost per unit at the onset of production and where the proba-

bility density f is given by
f(x) = aexp(,l3C(x))C’(x). (4.98)

Proof. The unit cost is decreased from its initial cost by the expected cost saving

per unit from learning during time t, i.e.,
U(t) = U0 — A502,). (4.99)
But as argued above,

AC(t) = lim E[ dC(x)]. (4.100)
(“—00 XMI)

Hence (4.93) will then yield (4.96).

4.6.5 Devolution to the discrete case

As further evidence that Theorem B gives the correct model for unit cost evolution
resulting from incremental learning, let us demonstrate that it cuts back to the
discrete case of Theorem A. Let I = [a, b] be the smallest closed interval containing
the compact support of f. We may, as always, translate and rescale our lesson labeling

system at will, so that in this case I = [0,1]. Let us partition I in the usual way

72

into n congruent subintervals [x 1,-_ 1, x,], with 0 = x0 < x1 < < .13., = 1 and
Ax,- = x,- — x7;_1 = 1/n. Then as in (4) the cost saving accrued by the compound
lesson [xi_1, x,-) is

c,- = f 1 40(1), (4.101)

 

p,- 2 [ti f(x)dx = “337). (4.102)

We take snapshots in time tk under the time scaling where one compound lesson
[x,-_1, x,) is learned on average per second. Then because A is the average rate of
learning per lesson length, tkA = k / n. Therefore the Riemann-Stieltjes sum approx-

imation of the integral of (4.96) takes on the form

00 1
/ 1 — e_)‘tkf(‘f) dC(x) = —/ [1 — e—kf(x)/n] dC(x) (4.103)
—00 0
=[1: e—kﬂ‘r [fl/n] c,- + 0(1/n2) (4.104)
i=1

_f(Ii)

7t

 

)k] Ci + 004/112) (4.105)

i=1

[1_(1
2::[1—-(1—p,-))k]c,j + 0(k/n2), (4.106)

which is consistent with the discrete result in Chapter 3.

This random set model for learning has features which are not. possible in other
models. As will be shown in Chapter 5, the only viable learning model which uses
a. partial differential equation is one that. aggregates states, so it measures only the
fraction of lessons learned, not which lessons might be learned. The random set

model allows for an inﬁnite uncorrelated sampling of an inﬁnite number of lessons.

73

 

As mentioned in the previous paragraph, the random set model also naturally allows
for the partial learning of a ﬁnite number of lessons.

As discussed in Remark 4.6, the sequence of random sets associated with the
construction of the dynamic random set does not converge strongly to any random
set. It is unknown whether it converges weakly to a random set, however. It seems
doubtful that this weak convergence occurs either. Therefore, we make the following

conjecture:

Conjecture 4.1: The sequence of random sets associated with the construction

of the dynamic random set does not converge weakly to any random set.

74

 

 

Chapter 5

Continuous Model II: A Partial

Differential Equation for Learning

5.1 . The Model as a Limit of a Markov Process

5.1.1 . Construction of a Master Equation

Let us build intuition by ﬁrst. examining a discrete toy model of learning with a
small number n of lessons to be learned. For the moment we will focus on counting
the probable number m(k) of distinct lessons learned after k steps (draws) with
replacement. The central idea of discrete learning models is that at each step either
one lesson is learned or nothing is learned. Of any of the particular m lessons (say
the i-th lesson), the probability p, of learning this i-th lesson was found in Chapter
3 to be

12.- IX 6,13AC,’ (5.1)

 

where )3 is a parameter of the learning process and AC,- is the improvement in the
quality metric when the i—th lesson is learned 1.

The relationship (5.1) means that the more that. the choice improves the quality,
the more likely it is to be chosen. This fact is supported by much in the artiﬁcial
intelligence literature, as several studies in artiﬁcial intelligence have demonstrated
that the most valuable rules are learned ﬁrst, and any less important rules are learned
later [Russell and Norvig 2002]. If at any stage the roll of the dice yields a lesson
that has already been learned, no learning takes place during that step. Thus it is
harder to ﬁnd the lessons that have not already been learned if many lessons have
been learned. The probabilities used in the model do not change over time, which is
characteristic of a Markov chain.

In order to create a Markov chain model, we must ﬁrst deﬁne states. In Chapter
3, it was assumed that a state is described by particular coordinates of an n-tuple v
in which the entries are either 0 or 1. Each entry represents a discrete lesson which
can be learned to improve the metric. The initial state vo = (0,0, . . . ,0) is the 0
vector, while the desired state 112n_1 = (1, 1, . . . , 1) is the state where every possible

lesson has been learned. For n lessons, there are of course 2" possible states.

Deﬁnition 5.1 Assume there are 71 lessons to be learned. Let q(0) be the 2n-tuple

(1,0,. . .,0)T. Then deﬁne qu') = (qlkl0, . . . ,qéf)_1) by the recursion

q(k+1) = q(k) + AqUI) = (1 + A)kq(0) (5.2)

where A is a 2” by 2" matrix governing the change of probabilities between iterations

(k)

and q, is the probability for the system to be in state v,- after k iterations of the

 

1Recall that AC is not necessarily the change in the magnitude of the quality metric. For
example, if the quality metric is cost, than a. decrease AC in the quality metric is an improvement.
On the other hand, if the quality metric is mean life of the product, that an increase AC in the
quality metric is an improvement.

76

equation.

Recall that at each step in the discrete process, one of the it possible lessons is
chosen. If the chosen lesson has not. been previously learned then a transition occurs.
Otherwise, there is no change to the state. Recursion (5.2) is the master equation
governing the evolution of the probabilities of the states of learning.

The master equation formulation of a Markov chain is preferable over a. transition
matrix formulation since we wish to pass to an inﬁnite number of possible lessons
to be learned; a master equation formulation passes to the limit as a differential

equation, as we will soon see. In any case, given a master equation

q(I+1) = qa) + Aqa)

with resulting state probabilities q<kl = {q]k), . . .,q$,)}T at step k, and matrix A
from (5.2), the traditional transition matrix B for the system of learning states is
given by

B = (A+1)T,

a matrix with nonnegative entries where each row sums to 1.

Example 5.1: Let us examine what the model would look like for n = 4 lessons,
with the i-th lesson having probability p, to be learned at each step. In this case,
there are 16 possible states, each with distinct probabilities qj to account for each of

the 4 lessons which could be turned off or on:

(k)

1. q0 is the probability that the system after k steps is in the state where no

lessons have been learned.

2. q)“ is the probability that the system after k steps is in the state where only

77

 

10.

11.

12.

lesson 1 has been learned.

(I)

. q2 is the probability that the system

lesson 2 has been learned.

(I)

. q3 is the probability that the system

lesson 3 has been learned.

qik) is the probability that the system

lesson 4 has been learned.

qék) is the probability that the system

lessons 1 and 2 have been learned.

(1g) is the probability that the system

lessons 1 and 3 have been learned.

qgk) is the probability that the system

lessons 1 and 4 have been learned.

(I)

. q8 is the probability that the system

lessons 2 and 3 have been learned.

qék) is the probability that. the system

lessons 2 and 4 have been learned.

q]? is the probability that the system

lessons 3 and 4 have been learned.

(’1')

q11 is the probability that the system

lessons 1, 2, and 3 have been learned.

78

after k steps °

after A: steps '

after k steps '

after k steps '

after k steps '

after k steps '

after k steps '

after k steps '

after k. steps '

after 1.: steps '

in the

in the

in the

in the

in the

in the

in the

in the

in the

in the

state

state

state

state

state

state

state

state

state

state

where

where

where

where

where

where

where

where

where

where

only

only

only

only

only

only

only

only

only

only

 

 

 

13. gig) is the probability that the system after k steps is in the state where only

lessons 1 , 2, and 4 have been learned.

14. (1&3) is the probab111ty that the system after k steps 18 111 the state where only

lessons 1, 3, and 4 have been learned.

15. q)? is the probability that the system after k steps is in the state where only

lessons 2, 3, and 4 have been learned.

16. q)? is the probability that the system after k steps is in the state where all

lessons have been learned.

The initial probability for the system to be in the ‘ground state’ qéO) will be 1;

(0)

all other q, will thus be zero. In this example, the evolution of probability of states

will be given by the iteration

where the matrix A is given by the block matrix

(—1 0 0 0 0)
A1 A2 0 0 0
0 143.44 0 0 . (5-4)
0 0 A5 A6 0
\0 0 0 A7 of

 

 

79

 

 

 

where

 

{ -(1—P1)
0
0
( 0
.43-

(2.)

 

 

141—— P2
P3
(24)
0 0
—(1—P2) 0
0 (—1—P3)
0 0

(P2 1010 0)
P3 0 P10
P4 0 0 P1
0 113 P2 0
0 P4 0 p2
\0 0 P4 P3f

 

 

80

 

(5.5)

 

 

(5.7)

 

 

 

and

 

 

 

0 0
0 -(P2+P4) 0 0
0 0 —(P2+P3) 0
0 0 0 —(P1+P4)
0 0 0 0
( 0 0 0 0
(P3 P2 0 P1 0 0)
0 0 0
A5: P4 P2 P1
P4 P3 0 0 P1 0
( 0 0 p. 223 222 0 )
(—p4 0 0 0 \
0 —])3 0 0
A6—

0 0 —pg 0

 

 

(0 0 0—p1}

A7 = (P4123~P2~P1)

any one ‘pure state’ (i.e. elements of the natural basis).

81

(5.9)

(5.10)

(5.11)

The block entries (5.4)——(5.11) follow from exai‘nining the one step evolution of

The system of equations (5.3)2—(511) trivially meets the mass conservation re-

 

 

quirement that the columns of A sum to zero,

272.

Z A,,- : 0. (5.12)
i=1

A system such as (5.2)-(5.10) is given by a lower triangular matrix A and is thus
explicitly solvable. However, it is apparent that problems arise when n is increased
without limit. The number of equations is 0(2"), and the somewhat irregular struc-
ture of A makes it extremely difﬁcult to obtain even qualitative results for the system.

In the general case, with n lessons to be learned, the states and their probabilities
will then be ordered in the following way: the ‘ground state’ in which no lessons have
been learned will come ﬁrst. The next n states are where just one lesson has been
learned. Then the next n(n + 1)/ 2 states are all the possible states where 2 lessons

have be learned, and so on.

5.1.2 . Reduction of Order

This problem can be better analyzed once we reduce the order of the system. A
natural approach is to amalgamate the states. Examining the master equation, one
can see that there is a block relationship among the state probabilities. Speciﬁcally,
transitions between states only occur between those states which have one more or
one fewer lesson to be learned.

Therefore, one can redeﬁne the states in the following way: again let it be the

number of lessons.

There are n + 1 aggregated states, where the i-th aggregate state

has had exactly i lessons learned, and where i can run from 0 to n.

7

Deﬁnition 5.2 Given 11 lessons to be learned, let qlk) = (mg/Chg)“ ...,qg]? )T be

82

      

the 2"—tuple for the state probabilities according to Deﬁnition 5.1. Then n(k‘) =

(ugk),...,u$,k))T is to be the (11+ 1)-tuple with n(k) equal to the probability that

i

exactly i lessons have been learned at the k-th iteration.

(k) (12‘)

Lemma 5.1 In the formulation of Deﬁnition 5. 2, 110 = q0 . Moreover, fori > 0,

.r::'°_':;.‘-;’-...'_‘.a.14.. Lyman-1f; 1: 17';:7_;a‘;;ips.:a}aé\nuwpﬁn'gsqﬁig;x.;fm‘z:rr'v_nmv

the i-th component of u<kl will be given by

I . ,-
. ".110."-
l IA -

w+(7,,l)—l
112(k): Z q)“, (5.13)
j=w

where

w = 1i (Z) (5.14)

(’1‘)

Proof: Clear, since the the number of q,- aggregated into the state will be (n).

2

To determine the evolution equation for these aggregated states and their asso-
ciated probabilities, we need an expression which yields the probability of transition
from the aggregated state i to the aggregated state i+ 1. For the rest of this chapter
we will only consider aggregated states. we will ﬁrst examine the simple case, where
the probabilities p,- are all equal, and make the model continuous in time. Then we

will examine the case of unequal probabilities.

"- 'I , '. atrvzvi'" :2..- . . l. 0
I"“7T'zf‘ ”Phi... :HN ...-dél'i' ‘d *"P In! ‘

83

. . ’ Mama's-Mum!“ 2-.~:22221.~.1.-1‘1.-15,-«: . ;- -‘.'.

 

5.2 The Continuous Case

5.2.1 . The Continuous Case with Equal Weighting
Making the Model Continuous in Time

Let us examine the special case of the aggregated state model where all lessons have
an equal probability of being learned. If lessons have already been learned, there is
a likelihood that a transition will not occur. If i out of n. lessons have already been

learned, then the probability of a transition due to a new lesson is

 

 

 

7’ — 2. (5.15)
n.
This transition probability yields the transition equation
u(k+1) = ulk) + Uu<kl = (I + U)u(k), (5.16)
where
l _1 0 0 )
—1
1 "n 0 0
0 "—1 —1’:—2 0 0
U = n 7‘ . (5.17)
0 " " 0

 

 

Let us now derive a. model which is continuous in time. We do this by having
learning occur as a Poisson process with rate A in place of the step variable k. A
Poisson process is a model of a random occurrence process, characterized by a rate

constant A, where the probability of exactly k occurrences during a time period At

84

 

 

is given by .
9%t—lie—AAI. (5.18)

Lemma 5.2 Given the transition process
n(kH) = ulk) + Mulk) (5.19)
= (I + M)u(k>, (5.20)

where the transitions are instead made via a Poisson process with rate A, the expected

state after a time At, starting from the initial state 11, will be given by

E(u(At)) = u + AAtMu + 0(At2). (5.21)

Proof. The expected value will be given by

 

 

,0 .
=sz e_’\At (I + M)u (5.22)
Fe
AA (1 + AAt(I + M)).. + v, (5.23)
where
,0 .
12:??? 12")“ (I + M)u (5.24)

The 1—norm of v is easily estimated.

 

,0 .
))v)). s .—AAI1149|)|)1+.11)h Z); ’,))I+M)){))u)). (5.25)

 

,0 .

g e—W (1A1)2 ((14-111)), QZMA (11+ 111191111), (5.26)
1 _ .

= 5. Am(.\A1,)2))u)|1))1+111)|§:IAI||1+M||1. (5.27)

85

 

 

The ﬁrst term in the expression (5.23) can be reduced as follows

e—AAIU + AAIU + IlI))u (5.28)
00 .
—AAt J
= Z (__'._) (1 + AAt(I + 111))11 (5-29)
1:0 J'
= u + Mlu + W. (530)

where ||w||1 is clearly 0(At2).
We have thus proved that. in the Poisson limit the iteration matrix changes from
I + AI to the inﬁntesimal generator AAI. Therefore, the discrete—time system (5.19)

becomes the continuous-time system 11. = A111 11.

Theorem 5.1 When lessons are learned at a Poisson stochastic rate, the aggregated

state probabilities evolve by the rule

it = AUu, (5.31)

where U is given by

 

 

 

—1
1 "-1. 0 0
0 71—1 _ n—2 0 0 ...
U = 7" " . (5.32)
E 0 0
.-—"+1 —1
0 0 0 "—,’,— "1. 0

 

 

Proof: Clear.

86

 

Making the Model Continuous in Space

We next pass to a limit of a continuous spatial variable. We deﬁne the spatial variable
x to be the fraction of all lessons which have been learned. Thus the domain of this
problem will be the unit interval [0,1], where x. = 0 represents the state where no
lessons have been learned and x = 1 represents the state where all lessons have been
learned. With such continuous learning, the number of lessons to be learned become
inﬁnite. Before, at any time t, the mean number of steps taken will be At. To reach
more than an inﬁnitesimal fraction x, we take a limit where A goes to inﬁnity, so
that k, i, and n also go to inﬁnity, and deﬁne a new parameter A’ according to the

rules

A —> oo, (5.33)

n —> 00, (5.34)

i —-+ 00, (5.35)

A’ = 5. (5.36)
n

The variable u will then become a function of the spatial variable x as well as of t.
The probability of having learned a fraction of lessons between x and x + dx then is

equal to udx.

Lemma 5.3 Deﬁne a differential operator P as follows

Pu = —A’;—I ((1 — x)u), (5.37)

where (1 —x)u belongs to lHl1([0, 1]). Then (5.32) is the backwards Euler discretization

0f the operator P.

87

 

 

Proof: W’e discretize the operator P given in (5.37) according to a backwards Euler
method the spatial derivative in P over 11 intervals, so that
(1- 1‘1)III - (1 - -T1—1)III—1

8
5; ((1 — 2122)).- ~ ,/ . (5.38)

 

so that on the interval [0, 1] where x, = i / n. we obtain

 

, _ . 1 .— .
Pu],- % nA’ (n—Z—Lu,_1 — n 211,-) , (5.39)
n n

01'

 

,1 _ . 1 '1_ .
Pu], z A (um-4 — 71 2vi) , (5.40)
n. n

from (5.29). Thus the approximation for Pu, yields the right hand side of (5.32).

Theorem 5.2 Taking the limits ( 5.33)—( 5. 36 ) yields the continuous aggregated states

model of the fraction x of lessons learned,

- ,8 ,.
11.: —.\ a—Iai —.1)11). (5.41)

We also note that the initial condition of the system will be given by

n(x, 0) = (5(1). (5.42)

This initial condition is dictated from the modeling requirement that the initial
state of the system has no lessons which have been learned and that the probability

of ﬁnding the system in some state x E [a, b] is given by
b
/ u(x, t) dx. (5.43)
Cl

88

 

 

Remark 5.1. Model (5.41) has the form of a transport equation with a singu-
lar point at x = 1. As with any standard transport equation, the term A’ (1 — x)
represents the velocity of the state. This velocity is positive and monotonically de-
creasing on [0,1], reﬂecting the fact that learning happens continuously for as long
as there are lessons to be learned, and that the learning rate decreases as learning

progresses.

Properties of the Transport Model of Learning

Lemma 5.4 Given the impulsive condition u(x,0) = 6(x), the solution of (5.41)
becomes

~ —A’t
11(x,t)= 0(x — (1 — e )). (5.44)
Proof. We will now solve for the characteristics:

dt

_ = 1 .4
ds , (5 5)
d
—f = A’(1—.1:), (546)
ds
él—L = A'u. (5.47)
(15

This system when solved yields the following

t = s, (5.48)

x = 1— (1 — x0)e)‘,t, (5.49)
1

u = u(x0)e)‘ t. (5.50)

89

 

 

When these equations are solved to eliminate x0 and 110, we obtain

11(1, 1) = .A'tsn — (1 — 1).“), (5.51)
which equals
, _ ~ . _ _ —A’t
u(.r, t) — 0(x (1 e )), (5.52)
via the identity
a26(ax + b) = (5(x + b/a). (5.53)

When the system is within the initial state, (110 improvements have been made
at the initial time),

u(x, 0) = (5(x). (5.54)

Remark 5.2: The discrete case for at least two lessons to be learned involves some
variance in the probabilities at any positive time (i.e., for all iterations, all u, are
strictly less than 1). However, the continuous model has no variance in the time
evolution of learning, as n(x, t) retains the form of a delta function for all t. The
disappearance of the variance matches the result obtained in the application of weak
convergence of random sets to the problem in Chapter 4, where the variance vanished,
as guaranteed by the examination of the associated random functionals in Section

4.4.1.

The results of this section are summarized in the following theorem.

Theorem 5.3 Suppose all lessons have an equal probability of being chosen. Then
the evolution of the fraction x of the lessons learned by time t is governed by the

partial diﬂerential transport equation

11: —A’(,—% ((1 — x)u), (5.55)

90

 

 

where u = u(x, t) is for each t > 0 the probability density

b

prob(a _<_ x g b) = / u(.r.t)d.r. (5.56)
a
The initial condition of the system will be given by
u(x, 0) = 6(x), (5.57)

which is dictated from the modeling requirement that the initial state of the system

 

has no lessons which have been learned.

5.2.2 . The Continuous Case with Unequal Weighting

Suppose now that the lessons are no longer equally weighted, but instead each lesson
has its own distinct probability p of being learned. We will now examine more
closely the set of differential equations which govern this unequally-weighted case.
We again replace the discrete steps with a Poisson process of rate A. The transitions

are governed by:

110 = 4110, (5.58)
121 = A(II()—f(1)II1). (5-59)
(5.60)
= A(1(222—1)22..-1—1(222)22..). (5.61)
(5.62)
= Ian—nun-.. (5.63)

91

 

or in vector form

11 = Vu, (5.64)

where

V=A , (5.65)

 

 

and where f (2) equals the probability that a lesson not previously learned will be
learned at the next iteration, given that i lessons have already been learned.

We next examine a. limit with a continuous spatial variable. At any time t, the
mean number of steps taken will be At. We take a limit where A goes to inﬁnity, so

that i and n also go to inﬁnity, and A’ is deﬁned as follows:

A ——> oo, (5.66)
n —-2 oc, (5.67)
i —-> oo, (5.68)
A 1

The function f will now have a domain of [0,1], with x 6 [0,1] representing the
fraction of the total number of lessons which have already be learned.
We can then again see that. the matrix V is a backwards Euler discretization of
the operator Q, where
_ 251

Qu = 8x (f(x)u). (5.70)

92

 

 

Thus our learning curve model becomes:
ii = Qu. (5.71)

with Q given by (5.69).
Thus the limiting case of the discrete system (5.65)-(5.68) when lessons are not
equally weighted leads to the modiﬁed transport equation of the form

all _ A, a

E — — 870(55th (5-72)

where the product A’ f (1‘) represents the characteristic velocity of transport for the

system, as will be proven in Theorem 5.4.

Theorem 5.4 In a continuous learning model where there are unequal probabilities
for each lesson to be learned, the learning will follow the equation

(9’21 _ A, a

671‘ — — 5;(f(17)’u)» (5-73)

with initial condition u(;r, 0) = 6(1) and learning velocity A’f(:r).

5.2.3. Properties of the Learning Velocity

One important property of the resulting partial differential equation (5.80) is that
the learning velocity is maximized in the case when the probabilities of learning each
lesson are equal. In the ﬁnite learning case, the probabilities of all the lessons to
be learned together constitute an n-dimensional parameter in the function m(k).

In more detail, the probability of learning the i-th lesson after k attempts (with

93

 

replacement) is

1— (1 —p,)"‘. (5.74)

Summing this expression over all lessons gives the expected number of distinct

lessons learned after I: chances:

71.

712(k): :1 — (1 — pi)k, (5.75)

i=0

as given by (3.15).

This expression is strictly monotonic and therefore invertible, even though it
cannot. be expressed in closed form. In general, the inverse of an expression for an
expected value of a random variable is not invertible; however, if the variance is
zero, the random variable is almost surely a single value, and thus deterministic.
Therefore, one can find the inverse relation simply by inversion. The inverse k(m)
therefore asymptotically yields the expected number of draws it takes to have m

distinct objects.

77.
Lemma 5.5 . The expression m(k) = :1 — (1 — pi)k is maximized at each k 2 1

i=0
when the probabilities p,- are equal.

Proof. This Lemma is a special case of the following.

Lemma 5.6 Suppose H = H (x) is a symmetric convex: real-valued function on the
convex set K C R" such that H has a minimum on K at a point x. Suppose further
that K is closed under any permutation of the coordinates of the points in K. Then

this minimum is taken at. a point of K with all coordinates equal.

Proof. Suppose H takes on a global minimum value m at some point a? = (r1, . . . ,1”)

of K. Since H is synnnetric, H ((7:17) 2 m where 0:1: 2 (I01, . . . ,a‘ml) is the vector

94

 

obtained by permuting the coordinates of a: by the permutation a. Let

1
I = 5'- : 01‘ (5.76)
' o

where the summation is over all 72.! possible permutations. Clearly (II = I But
because H is convex, H (I) = m.

The proof of Lemma 5.5 then is the special case where H is given by the expression
in Lemma 5.5 with domain 191 + - -- + pn = 1, p,- Z 0. As a corollary, since the

expression
n

29'“ = 2(1- Pilk (5-77)
i=1

i=1

is minimized when the probabilities are equal, the expression
71.
m(k) = 21 — (1 — p,)k (5.78)
i=1

is maximized for equal probabilities.

Remark 5.2: For the present investigation into learning, this Lemma 5.2 guarantees
that the inverse function m(k) will be maximized when the probabilities are all equal.
It should be pointed out that while having equal probabilities does maximize the
number of lessons learned, it does not necessarily maximize the beneﬁts of learning.
As discussed previously, an artiﬁcial intelligence model of learning implies a direct
correlation between the beneﬁt of learning a lesson and the probability of learning

that lesson, such as rule 5.1 given previously, namely

P(AC) oc e"3AC. (5.79)

95

 

 

As can be seen by this rule, the limit as 1'3 goes to zero leads to the case of equal

probability, and consequently leads to a. maximal rate to learning.

Remark 5.3: In the preceding form (5.72) of the transport. equation for an arbitrary
distribution of choices, f (:r) is determined by the distribution of decisions. As with
any transport equation the factor A’ f (1) represents the velocity for the evolution of
the state. As described in Section 5.2, f (:r.) = 1 — :1: for a uniform distribution. For
a non-uniform distribution, f (51:) < 1 — 1‘, since Lemma 5.5 requires that the max-
imum learning rate occurs in the case of equal probabilities. Speciﬁcally, since the
inequality is true in the discrete case for every n, the inequality passes through the
limit to the continuous case. Therefore, the equal—probability case places an upper

limit on the learning rate for every case.

Lemma 5.7 Characteristic curves for the partial differential equation ( 5. 72) move
with velocity v given by

v = A'f(:c). (5.80)

Proof. Since (5.56) is linear and of ﬁrst order, the characteristics are given by

dt

—— = 1, 5.81
d8 I ( )
dm

g; — Affl”), (5-82)
du

Zs— — 0. (5.83)

This system of differential equations yields that solutions of (5.72) are of the form

u(;r, t) = u(;1: — vt), (5.84)

96

 

 

where

5=A7@) (5%)

Since f (0) = 1, f (1) = 0, and f (.1:) is monotonic on [0,1], the model reﬂects the
intuition that the rate at which lessons will be learning is monotonically decreasing,

and will only asymptotically approach :1: = 1.

Remark 5.4: Since (5.41) is a standard transport equation, it also obeys a conserva—
tion law. Speciﬁcally, the characteristic equations imply that the equation preserves
mass, that is,

d
aHun, = 0. (5.86)

5.3 . Cost Evolution during Learning

The ultimate goal of any learning curve model is to predict the evolution of an
associated given quality metric, such as cost. This determination can be made by
examining the expected value of cost savings at each state 31:. In the continuous case,

the expected cost savings would then be given by

1
5(1),)2/0 u(;I:,t)dP(a:), (5.87)

97

 

where dP(:1:) = p(.1:)d.‘1: is the differential cost savings between :2? and :1: + (1.1:. By

employing our model (5.55), we see that for t > 0

d“. _d 1 ,. _ 1... , , r
&C(11)——[) u(.l.t)dlD(.1:)_/0 (”(133419“) (0.88)

.I.

1 (9 1 1,. ,
2/0 —Ay(f(;r)u)dP(a:) = A/() f(1)u(;17)p(.r)d.r (5.89)

1
= A / p’(r)f(r)u(r) (12:. = AE(p’(r)f(:r)) (5.90)
0

since u(.r) gives the probability of fraction :2: at each 1‘. Note that the boundary
terms in the integration by parts disappear since u(0) = u(1) = O for t > 0. The

expected value is over the probabilities u, and is therefore time-dependent.

Remark 5.5: While having the probabilities equal to each other may maximize
the total number of options taken, it will not necessarily result in a maximal im-
provement in the quality metric. Early in the run, when the probability of choosing
a new decision is very high, it is wise to make the probabilities so that the options
with the best cost savings are most likely to be chosen. Later on, when the best
options have been chosen, the improvement in the quality metric is optimized by
having the probabilities more equal. The point at which a transition is made be-
tween the two regimes will depend on the underlying distribution of available options
for improvements to the quality metric. Finally, it is also worth noting the similarity

of (5.85) to the model of cost savings (4.86) from Chapter 4.

98

5.4 . Forgetting

We now examine the phenomenon of forgetting in learning. Forgetting can take
place in two forms: during lapses of production or intermingled with learning during
production. The first type of forgetting is modeled well as negative learning, with
evolution determined by

021.

Bu

where A f is a forgetting rate that is (typically) smaller than the learning rate. In this
situation, f (O) = 0 (if everything is forgotten, then no more forgetting will occur)
and f (1) = —1 (if every lesson has been learned, something will surely be forgotten).

The more interesting type of forgetting is forgetting intermingled with learn-
ing during production. In this case, there are simultaneous possibilities of going
backwards or forwards. Although a comprehensive theory of how these ideas might
interlink does not. exist, we can examine some qualitative aspects of such a model.
We deﬁne az- to represent a forgetting rate if i lessons have already been learned. The

state space equations have the form

{ —1 0,1 0 \
1'1 2 A 1 —a1 — f(1) a2 0 . .. (5.92)
0 f(1) ~02 —f(2) a3

With three distinct. terms for almost every equation, it is apparent that a ﬁrst order

 

 

partial differential equation will not form a continuous model. Instead, the continu-
ous model should have the form

it, = Pu, (5.93)

99

where

Pu = a(:1:)u.l7 + b(zr)um. (5.94)

From even this simple qualitative treatment, several facts are evident. First,
we have a parabolic model, where more complicated behaviors are to be expected.
Second, the introduction of a second derivative introduces a dispersive effect on the
evolution of the state. That the model is parabolic also implies that some variance
in the expected value of the state will exist—in contrast. to the model which does not
incorporate forgetting (which becomes deterministic in the continuous limit).

In fact, some other characteristics of this spreading can be determined through
qualitative analysis of the coefficients a and b. For example, it is evident that b(O) = 0
and b(1) = 0, since otherwise there would be a nonzero probability of reaching either
a. negative value of :17 or a value which is greater than 1. Also, a(1) _<_ 0 (again, no
transport past :1: = 1). However, unlike the previous case, a does not have to be 0 at
the right endpoint, although a will be monotonic on [0,1]. What is likely to occur
is that on (0, 1), there will exist some state I in which it is equally likely to learn a

lesson as it is to forget a lesson.

5.5 . Conclusion

A partial differential equation model of learning has been presented. Instead of
having to deal with an impossibly large set of parameters to analyze, this continuous
limit involves only a few functional parameters, namely A' and f(1) that govern
the process. The analysis of these parameters will yield accurate predictions in the

analysis of real—life learning curves.

100

Part III

Simulations and Applications

101

Chapter 6

Simulations of Models

6.1 Introduction and Methodology

In this chapter several different simulations of the model will be presented. The
ﬁrst two sections will concentrate on the standard formulation of the model, and the
multi-dimensional quality metrics will be discussed in the subsequent section. The
simulations will be exclusively that of the discrete model, since we have shown in
Sections 4.6 and 5.2 that the continuous models has a ﬁrst order discretization equal
to the discrete model

These simulations were conducted using a program written by the author in the
C language. For each simulation several inputs must be speciﬁed. One such input
is the managerial effectiveness parameter 1’3. The second input is a ﬂoor level, the
best obtainable value for the quality metric. The last (and most complex) input is
the set of the quality improvements accrued by each lesson to the quality metric.
Typical sizes for the set of possible lesson improvement values range between 50 to
500. When running these simulations, it is generally noticed that overall results

were not sensitive to the size of the set. of possible improvement values, although, as

102

expected, more variance was observed for those smaller choice spaces.
To obtain the evolution of the quality metric, a Monte Carlo sampling of the
choice space was performed, according to the speciﬁed probabilities. Although for

the simple learning curve model (3.23)—(3.24),

n

o—oa) = 2(1— (1 —p,)k)Ao,, (6.1)
i:1

where
ef3ACi

 

Pi (6-2)

= 2321 (,6ch .
an analytical solution can be expressed functionally, there are three reasons for pro—
ceeding with Monte Carlo experiments. First, such a stochastic format allows the
exploration of variances in learning rates (The analytical expressions (6.1)—(6.2) are
merely expressions of the expected learning rates). Second, it can be observed that
the expression (6.1) involves the sum of a large number of components, each of which
involve multiplication of pairs of small numbers, typically much less than one, all
yielding a likelihood of signiﬁcant rounding errors being introduced. In contrast, the
Monte Carlo simulation avoids these large sums and therefore avoids the problems
with rounding errors. Finally, we will extend the Monte Carlo simulations to certain
situations, such as where a varying managerial effectiveness parameter is present and
multi-dimensional learning, where no analytical expression yet exists. Monte Carlo

simulations to prepare for the more complex future analyses.
The process for the Monte Carlo simulation is as follows: once a possible lesson

is learned, a subsequent choice of the same lesson would yield no further improve-

ment. The experiment was run several times, with the results averaged. Since the

103

 

Table 6.1: Pseudocode for the Monte Carlo Experiments.

variance was smaller for larger sample spaces (and the computational needs of the
program were not extensive), most of the experiments were performed with larger
sets of improvement values.

The data thus produced were then exported to Excel for graphing and data anal-

ysis.

6.2 Results

First we present a pseudocode for the simulation algorithm used in the one-dimensional

context:

Table 6.1: Pseudocode for the Monte Carlo Experiments.

70 Initialize lesson set (the possible values of the improvement in the quality

metric, A0,)
initialize 1’3
initialize initial value of the quality metric {AC1, AC2, ...ACn}
assign a pre-probability weight to lessons by the rule p,- = e‘ﬂACi
determine all cumulative sums s,- = 101 + p2 + - - ' + 1),: of pre-probability weights

pj, 1 g i g n
repeatedly simulate a manufacturing learning experiment { typically 10 times )

{

pick a random number :1:, uniformly between 0 and sn

70 ﬁnd lesson between 0 and 8,, associated with the random number :1:

104

loop i = 1 to n
If8i_1< :13 < 3i
then i-th lesson is chosen

}

if lesson just found previously learned
{
then no change in the quality metric is found

else, change value of quality metric based on value of lesson learned.

}
}

average the learning curves over all experiments

end

Experiment 6.1: Below in Figure 6.1 is a graph of the results of a typical sim-
ulation of decreasing cost per unit. In this graph, the horizontal axis represents ac-
cumulated production; in this case, the number of lessons attempted to be learned.
The initial cost. was set at $200, [3 = 3, and the AC,- were increasing in uniform

steps: {0, 01,02, ...,6.9, 7.0}, that is, AC,- 2 .1(i — 1),1 g i g 71.

105

400 . 1 . .

 

W W
0 U1
0 O
1 r
/
/
/
1 1

N
U1
0
1
’1
/
i

 

N
O
O
l
1

dollars per unit produced
G
O

I

100 "

l

50 _

 

 

O ._____ , ‘Ial__ ____ _, l _ l l

0 200 400 600 800 1000
accumulated number of units produced

Figure 6.1: Sample learning curve plotting cost per unit produced versus accumulated

number of units produced, showing decreasing cost per unit. during production.

Figure 6.1 has many of the typical results seen in all such simulations. An analy-
sis of an initial segment indicates that this part is almost perfectly linear. Later on,

once the most profitable cost decisions have been taken, the learning rate slows down.

106

6.2.1 Regression Comparisons

A regression analysis of the data displayed in Figure 6.1 and other, similar curves
are ﬁt very well with a power law. For example, the data displayed in Figure 6.1
ﬁts a power law with R2 = 0.988. In fact, as discussed in Chapter 2, the best ﬁt
seems to be a two—piece curve, where initial stages of learning are modeled with an
exponential law, and later stages are modeled with a power law. This provides again
an explanation why some empirical curves are exponential and some are power laws
— the exponential is indicative of a learning process that has a substantial amount
of savings decisions still possible, while a power law is indicative of a more mature

learning process.

Experiment 6.2: A suite of simulations was performed to see how varying the
efficiency parameter )3 changed the power law ﬁt. In these experiments the cost
of production started at 400 with a plateau of 100. There were 251 lessons to be
learned, with individual improvements to the quality metric ranging from 0 to 2.2 in
increasing, equally—spaced intervals. Table 6.2 below shows the results for different

managerial effectiveness parameters.

107

Table 6.2: Regression results for learning curves for different managerial effec—

tiveness parameters 1‘3.

 

 

 

 

 

 

 

 

 

 

13 Power Law Exponent. R2

0 —0.2597 0.9836
0.005 —0.2687 0.9907
0.01 -0.2897 0.9884
0.015 -0.2961 0.9836
0.02 —0.3442 0.9848
0.025 -0.3362 0.9885
0.03 -0.3758 0.9828
0.04 -0.4083 0.9869
0.05 -0.4785 0.976

 

 

 

 

 

 

As can be seen in Table 6.2, the larger ,3 is, the more negative the power law ex-
ponent becomes. This demonstrates the general rule discussed above, increasing 13

increases the learning rate.

6.2.2 Comparison of Lesson Structures

Now we compare the results of a unform structure of cost savings to that of a non-

uniform structure.

Experiment 6.3: Uniform Structure of Cost Savings.

For this simulation, the following assumptions were in force:

i. The initial cost of production is 350 dollars per unit.

108

ii. The set of decisions have an associated set of cost savings which are uniformly
spaced over an interval. Speciﬁcally, the cost savings ranged from 0 to 3 dollars,

at increments of 10 cents between the potential cost savings.
iii. The time to implement decisions is approximately constant. for all decisions.

Multiple Monte Carlo simulations of the model (6.1)—(6.2) under these assumptions
were averaged for Figure 6.2 below. As can be seen from Figure 6.2, the overall
shape does indeed match what is commonly seen in a learning curve. Furthermore,
the common deviations from the analytic forms of the learning curve are all readily
observed. In contrast to the exponential or power law curves, the learning curve in
Figure 6.2 exhibits the observed lack of signiﬁcant upward concavity at the begin-
ning of production. Also, this simulated learning curve can be seen to plateau at a
non—zero value, which indicates that neither the power law nor the exponential form

are appropriate in the long run.

109

360 1 ”r 1 1

 

340 i i
320 — _
300 — 4
280 — —
260 — a
240 — —

220 7 \\\\\ r

200 7 ‘

dollars per unit produced

180 7 “

 

 

 

160 l J l l
0 200 400 600 800 1000

accumulated number of units produced

 

Figure 6.2: Example of simulated learning curve for a linearly increasing cost struc-

ture with 13 = 0.8.

The results shown in Figure 6.2 were found to ﬁt a power law (R2 = 0.960) much
better than an exponential (R2 = 0.741). This R2 value is well within the bounds of
deviation from a power law that actual learning curves exhibit. On the other hand,
if the initial cost is adjusted so that the plateau level is much smaller than the initial

cost, it is found that an exponential ﬁts the curve much better than a power law.

110

 

180 l T T f

160

140

120

100

80

 

60

dollars per unit produced

\

20 7 a

 

 

 

0 I 1 l I
0 200 400 600 800 1000

accumulated number of units produced

 

Figure 6.3: Simulated learning curve for a linearly increasing cost savings space with

a low plateau, with ,8 = 0.8.

The R2 value for an exponential ﬁt to the low-plateau learning data graphed in
ﬁgure 6.3 is 0.979, while for a power law ﬁt R2 is 0.932. This result indicates that
whether a graph can be better ﬁt by a power law or exponential does not depend on
the raw values of the costs savings, but instead depends on how much further cost

savings can push down the cost on a percentage basis.

Therefore, for a relatively new industry, where the space of cost savings
has only begun to be explored, an exponential learning curve is likely to
be observed, while in a more mature industry, a power law decrease is

more likely to be observed.

111

Experiment 6.4: (Non—Uniform Structure of Cost Savings) Let us now simulate a
situation with a non-uniform structure of cost savings options, that is, where the cost.
savings options are not equally spaced, but are instead more concentrated, typically
on the lower end of the saving values. In this simulation, it is again assumed that
the initial cost of production is 40 dollars. However, in this case, the decisions has
associated cost savings which have a cost savings structure such that the spacing
between cost savings is inversely proportional to the value of the cost savings. This
distribution should give a more realistic model, since in an actual manufacturing
setting there will be many decisions that yield a small decrease in cost, while there
are only a few possible decisions that will effect a large decrease in cost. Again, the
time to implement decisions is roughly constant for all decisions. And again, for the
sake of simplicity, the probability of no successful decision is equal to the probability

of all decisions previously made.

 

 

 

 

4O 1 T 1 1
35 — \ -
.. \
OJ
5’
‘0 30 7 x
9
Q.
1‘:
C
3
a; 25 1. .1
Q.
3
2
T)
U
20 7 ‘
15 1 1 1 1
0 100 200 300 400 500

accumulated number of units produced

Figure 6.4: Example of simulated learning curve for a non-uniformly spaced cost

savings space, with 1’3 2 0.8.

As can be seen in Figure 6.4, the evolution of the learning curve ﬁts a power law
exceedingly well. The R2 value for this ﬁt is 0.9957, while an exponential curve ﬁts
the data badly (R2 = 0.7531). However, if the initial cost is adjusted in this case, so
that the plateau level is much lower than the initial level, it is again the exponential

curve that. is a much better ﬁt to the data, as seen in Figure 6.5,

113

 

25 T 1 r 1

157

107

dollars per unit produced

 

 

 

 

O l l l l
O 100 200 300 400 500

accumulated number of units produced

Figure 6.5: Simulated learning curve for a non-uniformly distributed cost savings

space with a plateau very small relative to initial cost, with 6 = 0.8.

For the small-plateau data exhibited in Figure 6.5 the exponential ﬁt had a R2
value of 0.9908, while the power law had one of only 0.8956. Therefore, when the
cost savings can drive down the cost to a small fraction of the original cost, the
exponential curve is a much better ﬁt to the data than the power—law ﬁt.

Therefore, we can see that explicit connections can be made between the struc-
ture of the learning curve model (3.26)——(3.27) and observed learning curves. First,
the form of the learning curve conforms to the level of learning plateau, relative to
the starting point. Also, an explicit relationship can be seen between the manage-
rial effectiveness parameter ,6 and the parameters in the exponential/ power law ﬁts.

That the model presented herein ﬁt observed learning curves so well serves as a par—

114

tial validation of the model.

The results of the theory developed in Chapter 3 have proven to explain many of

the exact details of real-life learning curves. In contrast to standard learning curve
analysis, which treats the exponential and power laws as mere tools of regression,
these simulations demonstrate that the form of learning curves is attributable to
speciﬁc causes in the learning space. The choice of a learning curve form between an
exponential and a power law is attributable to the level of the plateau, relative to
the initial level of cost. The coefﬁcients in the power law or exponential relate to the
structure of the cost savings space; again, rather than being merely outputs of the
regression, these parameters are tied to speciﬁc causes. The most important results
are summarized below.
For a relatively new industry, where the space of cost savings has only
begun to be explored, an exponential learning curve is likely to be ob-
served, while in a more mature industry, a power law de-crease is more
likely to be observed.

Next we examine the predictions of our theory on multi-dimensional learning.

6.3 Simulations of Multi—Dimensional Learning

This section will examine some variations on the standard learning curve model
presented heretofore. First, some simulations of learning with a two-dimensional
quality metric will be examined. Next, some simulations with forgetting will be
examined.

Recall that for multidimensional learning the probability of learning the i-th

lesson with different improvements to the quality metrics Q and R is related to those

115

 

improvements by

pi CX efilAQi-l-HQARZ'. (63)

Experiment 6.5: To keep ideas grounded, we will assume the ﬁrst quality metric to
be cost and the second quality metric to be defect rate. In the ﬁrst set of simulations,
no correlation is assumed between the improvements in the two metrics. Speciﬁcally,
there are 441 lessons arranged in a uniform square grid, from (0,0) to (20, 20), where
the ﬁrst coordinate measures the improvement (in dollars) in the manufacturing
cost, and the second coordinate measures the improvement (in percent) in the defect
rate. Therefore, for example, lesson (1,2) has 1 dollar improvement in the cost
and 2 percent defect improvement. In this experiment, then, the lesson set has no

correlation between metrics. Also, for this experiment, 61 = 0.4 and 62 = 0.6.

116

x10

 

l
l

3.5

I
\
I

2.5

Quality Metric

 

 

 

0 .5 l 1 l l
0 1000 2000 3000 4000 5000

Accumulated Production

Figure 6.6: Learning curve for two metrics where lesson set has no correlation between
metrics (the solid line represents improvement in cost, measured in 104 dollars, with

1'31 2 0.4, and the dotted line represents the defect rate (in percent)), with 62 = 0.6.

What. is apparent from the results in Figure 6.6 is that there is not any trade-off
effect shown—both metrics decrease in a similar form. Improvement in one metric
does not suffer due to improvement in another metric. However, since cost has the
higher ,B-factor, it experiences the more rapid improvement. As expected, since there
is no correlation between payoffs, both cost saving and defect reductions will follow
the form of the learning curves obtained previously in Experiments 6.1-6.2.

However, this lack of correlation is unrealistic. More likely is the situation wherein

choices which improve cost are less likely to improve defect rate. If one hurries the

117

production line to improve cost. the defect rate is likely to suffer. If one is painstak-
ing enough to remove all defects, cost is likely to suffer. This reasoning points to
the fact that there will likely be a negative correlation among the choices between

quality improvement and cost i1111.)rovement.

Experiment 6.6: Next. we examine the more realistic situation where the les-
son space has an inverse correlation between the two quality metrics is introduced;
that is, the larger a lesson improves one quality metric, the less it improves the second
quality metric. Below in Figure 6.7 is one such case. In this case, the lessons have
cost and defect improvement rate components which lie on the line AQ, +AR, = 10,
so that a lesson which brings a larger cost improvement. will have a smaller improve-
ment in the defect rate, and vice versa. Also in this case, the ,ﬁ-factor is much larger
for cost (:31 = 0.8) than for defect rate (132 = 0.2). In a business setting, this dif-
ference in 1:3 represents heavy pressure to improve cost, but very little pressure to

improve quality.

118

 

 

1600 I I I I If I
.‘ \O‘dl0lg’tg'.

 

Cost
----- Quality

 

 

 

1400

1200

1000

800

Quality Metric

600

400

200

 

 

 

1

0 L l l I I
0 1 000 2000 3000 4000 5000 6000 7000
Accumulated Production

 

Figure 6.7: Learning curve for two competing metrics with negative correlation (solid

line is cost, dotted line is defect rate), with 1'31 = 0.8 and [3’2 = 0.2.

Several characteristics can be observed in Figure 6.7. Initially costs plummet
dramatically, while quality is very level. It. is only after costs have begun to plateau
that any improvement is seen in quality. Even at this stage, the improvement rate for
quality is very slow, much slower than is exhibited with cost. The relative weightings
of the [3 factors has forced the cost improvements to occur ﬁrst, and this result is
borne out in the graph.

Experiment 6.7: Contrast this previous simulation with a simulation shown in
Figure 6.8 with the same lesson space but where the two quality metrics are instead

treated equally, i.e., they have the same {3 values (,131 = 132 = 0.08):

119

 

5200 1 I 1 1 . 1

5000

h» h
C) 03
O O
O 0

Quality Metric
t
O
O

4200

4000

 

 

 

 

3800 l l I l I l
0 1 000 2000 3000 4000 5000 6000 7000

Accumulated Production

Figure 6.8: Learning Curve for two equally preferred metrics.

As should be expected, the two quality metrics improved at almost exactly the
same rate.
Experiment 6.8: Next we examine the case where a change in emphasis between
two quality metrics occurs in the middle of the production process—see Figure 6.9.
In this case there is initially a strong preference for improvement in the cost metric
(,61 = 1.5, {32 = 0.1). Also present here are lessons which greatly improve one metric,
while actually making the other metric worse. In this case, as should be expected,
the cost metric initially improves greatly, while little improvement occurs for the
quality metric. Midway through production, the preferences are reversed ([31 = 0.1,

[3% = 1.5). Cost then becomes worse, while quality suddenly improves greatly.

120

 

5200 1 r 1 1 1 I

5000 ... .u "on" o

4800

4600

4400

Quality Metric

4200

4000

3800

 

 

 

 

3600 I L l l l l
0 1 000 2000 3000 4000 5000 6000 7000

Accumulated Production

Figure 6.9: Learning Curve for two competing metrics with switch between preference

for quality metrics.

Experiment 6.9: Now we can look at the when the two quality metric are
initially weighted the same (131 = 132 = 0.08), but in the middle of the production
process a decision is made to weight one much more heavily than the other (,131 = .15,

1'32 = 0.01). Below in Figure 6.10 is a graph of the results:

121

 

5200 1 1 1 1 1 1

5000

4800

4600

4400

Quality Metric

4200

4000

 

3800

 

 

 

3600 I l l l l L
0 1000 2000 3000 4000 5000 6000 7000

Accumulated Production

Figure 6.10: Learning Curve for Two Competing Metrics with Switch from no Pref-

erence to Strong Preference for Quality Metrics.

We can see from Figure 6.10 that the switch causes one of the quality metric to

improve slightly more rapidly. while the second one is allowed to stagnate.

6.4 Conclusion

W hat the above graphs have shown is that the gross form of the predicted learning
curves matches the learning curves seen in manufacturing. Like stated previously,
our learning curve theory explains the observed data in existing learning curves. As

the previous section describes, this theory is also very useful in exploring an area

122

which the literature barely touches—multidimensional learning. The usefulness of
the theory in explaining any possible types of learning goes a long way to validate
the theory, as well as merit further explorations into where the theory can go.
Furthermore, the theory has the capability to explain many more details of man-
ufacturing learning curves than any previous theory. The effect of managers is cap-
tured in a single parameter )3. The theory can explain whether and to what extent
the learning in two different metrics interacts. Finally, it gives a testable equation
for the evolution of learning. The testing of the theory will be explored in the ﬁrst

section of the next chapter.

123

Chapter 7

Further Directions

This thesis has laid out a complete analytical theory for the learning curve, as well
as some practical mathematical formulations for understanding the structure of the
problem. In this ﬁnal chapter a few further directions for the ideas in this paper will

be proposed.

7 .1 Empirical Analysis of Learning Curves

Empirical validation of the learning curve results described herein is an important
goal. Several factors are important in considering the effectiveness of such an effort,
such as data, assumptions of choice space, and choice of statistical model.

First, the data problem is dire. Without reliable data, no proposed model can
be veriﬁed. Many of the published studies with learning curve time series have a
very small number of data points, which makes ﬁtting data to a model problemati-
cal. Fortunately, recent data-tracking techniques which allow a real-time tracking of

internal cost and price data make this scarcity of data. problem less paramount. For

124

example, managerial dashboards allow managers to track data for production cost
and production numbers on a daily basis [Irgens 2008]. The data problem is likely
to be less of an issue as more companies compete with such analytical tools.

Next, we must examine the choice of model. Should we use a discrete or contin-
uous model? If discrete, how many lessons do we provide for? If continuous, what
labeling subset of the reals do we use, and what will be the cumulative cost savings
C (1‘)" At ﬁrst glance, these problems might seem insurmountable. However, recall
that one of the properties for learning with random sets was inﬁnite additivity. This
property allows the choice probability of an aggregate lesson to be directly related
to the probabilities of each of the sub—lessons. If we were to use a discrete case, then,
to some extent it should not matter how many possible lessons we choose. It will
only affect the result to the extent that. the variance of the stochastic process would
be larger or smaller, depending on the size of the lesson space.

Finally, we brieﬂy examine the possible statistical methods available for verifying
our model. For the discrete case, a regression is sufficient. Speciﬁcally, we will have

the cost-per-unit function is given by
6(1) 2 Co + Zn -— (1 — ek'fmCiDACi, (7.1)

where k is the accumulated production and n is the number of cost lessons assumed.
The values which are needed to be found through regression are /3 and the A0,.
Therefore, for 71 lessons, 11 + 2 values must be found through regression.

The continuous case has some very interesting aspects. Recall that the form of

the continuous learning curve is given by

35(1) /G (1 — e7*‘f<C<-'r>>)dC(:1:) (7.2)
= / (1—e7*’f<C<-T)>)c(1)1111~ (7.3)
G
: /c(.r)d.r—/ e_)‘tf(c(‘r))c(x)dx. (7.4)
G G

If we choose f so that f (((1)) = x and deﬁne t' 2 At we obtain

30
AC(t) =/Gc(x)dx—/0 e_‘rt,c(x)XG($)dx. (7.5)

The second integral can be immediately recognized as a Laplace Transform. Thus,
if the cost. is related to a Laplace Transform of the cost space, the cost. space may be
obtained via an inverse Laplace Transform. It is known that numerically inverting
a Laplace Transform is very difﬁcult [Cohen 2007] Nonetheless, some interesting

aspects of this problem may still be explored.

7 .2 Further Directions for Exploring Dynamic Ran-
dom Sets

This section sketches some applications of the dynamic random sets constructed in
Chapter 4. The example given earlier of raindrops falling on a sheet of paper is
prototypical. Although the theory described in chapter 4 is in a one-dimensional
setting, it may be easily generalized to higher dimensions. In this case, f(x) becomes
the probal_)ility distribution of a raindrop hitting a point x 011 the sheet of paper.
The paper is initially dry, but over time, the measure of “wetness” of the sheet.

asymptotically a1_)1.)1'oaches the measure of entire sheet, as it becomes more and more

126

likely that each point has been covered over time.

This random set process can also be viewed as a random growth model. For
example, [Holash et al. 1999] used random sets created via a Poisson model to model
tumor growth. As a Poisson process, however, the growth model was discontinuous.
Our model has no such discontinuities.

The deterministic concept of a weak convergence of sets can also be used to
model the process of coarse graining which is pervasive in statistical mechanics—
see for example [Arnold Avez 1968]. Coarse graining is the method by which one
examines the average properties within small subsets called grains, rather than the
properties at each point. In the example of Arnold and Avez, if we stir vigorously
a solution of 20 percent rum and 80 percent TMCoke, every part of the solution
will consist of 20 percent rum and 80 percent TMCoke. Even though at some level
the components might not be completely mixed, such a level is too small to be
relevant. This process of coarse graining is exactly what is being given in the weak
convergence of deterministic sets and is very similar given in the example of the
sequence of deterministic sets given in section II above.

There are many possible generalizations of the principal dynamic random set.
The strongest assumption we have imposed is the independence of the n intervals
which form each term of the sequence, the weak limit of which is the example process.
Many growth models would likely have some dependence between the intervals. Also,
as is evident from the description above, many possible applications are available for

a modeler.

127

Chapter 8

Bibliography

Argote L. and Epple D., Learning Curves in l\"Ianufacturing, Science 247 (1990) 920-
924.

Arnold V. I. and Avez A. (1968) Ergodic Problems in Statistical Mechanics, W.
A. Benjamin, Inc., New York.

Bailey, C., Forgetting and the Learning Curve: A Laboratory Study, Management
Science 35 (1989) 340-352.

Barney J ., The debate between Traditional Management Theory and Organizational
Economics: Substantive Differences or Intergroup Conﬂict? Academy of Manage-
ment Review 15 (1990) 383—393.

Binney J. and Tremaine S. (1988) Galactic Dynamics. Princeton.

Capasso V. and Villa E., Continuous and Absolutely Continuous Random Sets,
Stochastic Analysis and Applications 24 (2006) 381-397.

Cohen A. (2007) Numerical Methods for Laplace Transform Inversion. Springer,
New York.

Conway R. and Schultz A., The lV’Ianufacturing Progress Function, Journal of In—
dustrial Engineering 10(1) (1959) 39-53.

Demsetz H., The Firm in Economic Theory: a Quiet Revolution, American Eco-
nomic Review 87 (1997) 426—429.

128

Dillard J ., Riggsby, J. T. et al., The Making and Remaking of Organization Context:
Duality and the Institutionalization Process, Accounting, Auditing and Accountabil-
ity Journal 17 (2004) 506-542.

Donaldson L., The Ethereal Hand: Organizational Economics and Management The—
ory, Academy of Management Review 15 (1990) 369-381.

Donaldson L., A Rational Basis for Criticisms of Organizational Economics: A Reply
to Barney, Academy of Management Review 15 (1990) 394-401.

Engel A. and Van den Broeck C. (2001) Statistical Mechanics of Learning, Cam-
bridge.

Feynman, R. P. (1972) Statistical Mechanics. Addison Wesley, New York.

M. Gallegati, S. Keen, T. Lux and P. Ormerod, Worrying Trends in Econophysics,
Physica A 370 (2006) 1-6.

Garg, A. A. and Milliman, P., The Aircraft Progress Curve Modiﬁed for Design
Changes, J. Industrial Engineering 12(1) (1961) 23-27.

Grifﬁths R. B., Nonanalytic Behavior Above the Critical Point in a Random Ising
Ferromagnet, Phys. Rev. Let., 23 (1969) 17-19.

Hastie T. and Tibshirani R. and Hiedman J. (2001) The Elements of Statistical
Learning: Data Mining, Inference, and Prediction. Springer-Verlag, New York.

Haussler et al., Rigorous Learning Curve Bounds from Statistical Mechanics. Ma-
chine Learning, 25(2) (1986) 195-236.

He Zi-Linp and Wong Poh-Kam, Exploration vs. Exploitation: An Empirical Test
of the Ambidexterity Hypothesis, Organization Science 15(4) (2001) 481-494.

Holash J. and Wiegand S. J ., Yancopoulos G. D., New model of tumor angiogenesis:
dynamic balance between vessel regression and growth mediated by angiopoietins

and VEGF, Nature 18(38) (1999) 5356-5362.

Irgens et al., Optimization for Operational Decision Support: The Rig Management
Case. SPE Annual Technical Conference and Exhibition (2008) 116616-MS.

Jaber M. Y. and Bonney M., Production Breaks and the Learning Curve: The

129

Forgetting Phenomenon, Applied Mathematical l\»'lodelling 20(2) (1996) 162-169.

Kodama R. H. and Berkowitz A. E., Atomic—Scale Magnetic Modeling of Oxide
Nanoparticles. Physical Review B 59(9) (1999) 6321-6336.

Levy, F. K., Adaption in the Production Process, he‘lanagement Science 11(6) (1965)
136-154.

March, J. G. (1964) A Primer on Decision Making, hflacmillan, New York.

March, J. G., Exploration and Exploitation in Organizational Learning. Organi-
zation Science 2(1) (1991) 71—87.

Miller K. D., Zhao M., and Calantone R., Adding Interpersonal Learning and Tacit
Knowledge to March’s Exploration-Exploitation Model. Academy of Management
Journal 49(4) (2006) 709-722.

Muth, J ., Search Theory and the l\/Ianufacturing Progress Function. Management
Science 32(8) (1986) 948-962.

Robbins H. E., On the l\-='Ieasure of a Random Set, Annals of Mathematical Statistics
15 (1944) 70-74.

Russell, Stuart; Norvig, Peter (2002) Artiﬁcial Intelligence: A Modern Approach,
Prentice—Hall, New York.

Schilk T. (2000) l\r’Iolecular Modeling and Simulation. Springer, New York.

Seung, H. S., Statistical hrlechanics of Learning from Examples, Physical Review
A 45(8) (1992) 6056-6091.

Shannon C. E., A Mathematical Theory of Communication, Bell Systems Technical
Journal, 27 (1948) 379-423; 623-656.

Speaker P., Using Random Sets to Model Learning in l\/Ia.nufacturing, submitted
to European Journal of Operations Research

Speaker P. and MacCluer C. R., A Distributional Construction of Random Sets,
submitted SIAM Journal of Applied Mathematics.

Sperduti A. and Starita A., Supervised Neural Networks for the Classiﬁcation of
Structures, IEEE Transactions on Neural Networks 8(3) (1997) 714-735.

130

Tushman, M. L. and Anderson, R, Technological Discontinuities and Organizational
Environments, Administrative Science Quarterly 31(3) (1986) 439—465.

VVatkin T., Rau A., and Biehl M., The Statistical Mechanics of Learning a Rule,
Reviews of Modern Physics 65(2) (1993) 499-556.

VVitten, T. A., Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon, Physics
Review Letters 47, (1981) 1400-1403.

Yelle L. E., The Learning Curve: A Comprehensive Survey, Decision Sciences 10
(1979) pp. 302-328.

 

Zangwill, W. I. and Kantor, P. B., Toward a Theory of Continuous Improvement
and the Learning Curve. Management Science 44(7) (1998) 910-920.

Zangwill, W. I. and Kantor, P. B., The Learning Curve: A New Perspective. Inter-
national Transactions in Operations Research 7(6) (2000) 595-607.

131

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