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ACCURACY OF POPULATION PROJECTION MODELS

FOR LOCAL GOVERNMENTS IN MICHIGAN

BY

IKKI KIM

Sumitted to
Michigan State University

in partial fulfillment of the requirements

MASTER OF SCIENCE

School of Urban Planning and Landscape Architecture

College of Social Science

1985

ABSTRACT

ACCURACY OF POPULATION PROJECTION METHODS
FOR LOCAL GOVERNMENTS IN MICHIGAN

BY
IKKI KIM

Trend extrapolation is a popular method for population
projection in local planning offices because it is simple and
inexpensive. However, its accuracy is often open to question
because users have no way to determine the quality of their
projections. There is a need to identify the conditions and
models that make trend extrapolation methods acceptable for
planning practice.

Six extrapolation models are tested with data for 172
cities and towns in Michigan. The models were calibrated on
population data from 1940 to 1970, and projections were made
for 1980. To test accuracy, the projected populations were
then compared with actual 1980 census data. Based on this
accuracy testing, recommendations are proposed for selecting
the most accurate models for various types of cities,
according to their growth rate change.

The findings of the research have been used to develop a
composite method that comprises the models yielding the
minimum mean error for each of the various types of cities.
In addition, a computer program has been developed using the

composite method.

 

ACKNOWLEDGEMENTS

I wish to express warm thanks to Dr. Rene Hinojosa, my
major projessor, for his encouragement and comments. He
contributed much appreciated counsel and assistance in
directing this study.

Many thinks go also to the remaining members of my
graduate committee, Dr. Roger Hamlin and Dr. Robert Thomas
for their advice and assistance.

To my wife SoonYoung, especial thanks for her continued
patience, support and encouragement. Finally, I wish to
express my affection and respect to my parents. Their love
and encouragement provided the motivation which made this

achievement possible.

ii

 

 

TABLE OF CONTENTS

List of Tables

List of Figure

List of Appendix

CHAPTER

I.

II.

III.

VI.

V.

INTRODUCTION

Introduction
Purposes of the Study
Data and Methoodology

POPULATION TREND EXTRAPOLATON MODELS

Linear Extrapolation .
Exponential Extrapolation

Double Logarithmic Extrapolation
Modified Exponential Curve
S-Shape Curve . . .
Polynomial Curve

ACCURACY TEST

Accuracy for Overall Cities and Towns

City Types Accoding to Growth Rate Change
Great Increase in Growth Rate
Medium Increase in Growth Rate
Moderate Increase in Growth Rate
Moderate Decrease in Growth Rate
Medium Decrease in Growth Rate
Great Decrease in Growth Rate
Extreme Decrease in Growth Rate

Cities Classified According to 1970 Population

Summary . . . . . . . .

DEVELOPMENT OF COMPUTER PROGRAM

Criteria Basis to Select Best Model .
Micro—Computer Program with BASIC Language

CONCLUSION

LIST OF REFERENCES

iii

iv

vi

10

11
13
14
15
18
20

22

22
25
28
33
37
43
51
58
63
69
72

76

76
77

81

107

TABLE

10.

11.

 

LIST OF TABLE

Frequency of Percentage Error for 172
Cities in Michigan

Frequency of Percentage Error for Cities
of Great Increase Type in Michigan

Frequency of Percentage Error for Cities
of Medium Increase Type in Michigan

Frequency of Percentage Error for Cities
of Moderate Increase Type in Michigan

Frequency of Percentage Error for Cities
of Moderate Decrease Type in Michigan

Frequency of Percentage Error for Cities
of Medium Decrease Type in Michigan

Frequenciy of Percentage Error for Cities
of Great Decrease Type in Michigan

Frequency of Percentage Error for Cities
of Extreme Decrease Type in Michigan

Mean Percentage Error of Each Model for City
Types Based on 1970 Population

Population Projection Model for Cities
Based on Growth Rate Change

Composite Method

iv

PAGE

23

29

34

38

45

52

59

64

7O

74

76

 

LIST OF FIGURES

FIGURE

1. Linear Extrapolation Model

2. Modified Exponential Curve

3. Gompertz Curve

4. Second Degree Polynomial Curve

5 Example of Great Increase Type
( City of Saline )

6. Example of Medium Increase Type
( City of Flushing )

7. Example of Moderate Increase Type
( City of Ann Arbor )

8. Example of Moderate Decrease Type
( City of Pontiac )

9. Example of Medium Decrease Type
( City of Adrian )

10. Example of Great Decrease Type
( City of Holland )

11. Example of Extreme Decrease Type

( City of Midland )

PAGE

11

17

19

20

31

36

41

49

56

62

67

LIST OF APPENDICES

APPENDIX

1.

Results of Accuracy Test for Each Type
Based on Past Growth Rate Change

Results of Accuracy Test for Each Type

Based on the Combination of Past Growth Rate
Change and Future Grate Rate Change

Computer Program Flowchart

Computer Program with BASIC

Example of Projecting Population With
Computer Program . . . .

Population Data

vi

PAGE

84

85

89

90

100

102

CHAPER 1

INTRODUCTION

Introduction

 

Information about future population is essential to the
functioning of a local planning office, because population
projections provide quantitative yardstick of future
demandes for public facilities and services. Reliable
projections thus provide a foundation for sound planning
decisions.

Population trend extrapolation techniques are popular
methods of that extend past and present trends of population
growth or decline against time into the future for
projecting the total population. Although population trend
extrapolation has weak theoretical foundations and lacks of
component detail, projections are ppularly used as simple
and inexpensive techniques of forecasting local population
in the short-term (Greenberg, Michael and Krueckeberg,
1978).

The accuracy of such projections, however, is often
open to question because users have few ways to determine
the quality of their projections. Therefore, there is a
need to identify the accuracy of widely used trend
extrapolation techniques with certain city characteristics,

to make them acceptable as planning models.

To meet this need, Isserman (1977) carried out accuracy
tests of a wide range of extrapolation methods and
communities. He used census data from 1930 to 1950 to
project 1960 population and data from 1930 to 1960 to
project 1970 population for townships in Illinois and
Indiana. To reduce projection errors, he used different
methods for different townships. Two characteristics were
used to define the township types ; namely their population
levels and growth rates (Isserman, 1977 : 253). The
population growth pattern of cities in the United States,
however, was changed over the 19705 in many places. The
trend of migration into cities started to decelerate. Many
Michigan cities, especially, showed of decrease in
population growth rates during the 19703. It is, therefore,
necessary to retest and determine the accuracy of popular
extrapolation techniques for use with the changed trends.

This study tested the accuracy of six popular trend
extrapolation models with data from 172 cities and towns in
Michigan. Three of the six models were selected from
Isserman's study (1977) : the linear model, the exponential
model and the double logarithmic regression model. The
Gompertz model, the modified exponential model, and the
second degree polynomial model were added to test whether or
not they are appropriate for the changed population trends.
Growth rate change was used to identify and designate types

of cities.

The six models were calibrated on population data from
1940 to 1970, and projections made for 1980. The projections
were subsequently compared with actual 1980 census data.
The accuracy of each model was then examined using the
cities' growth rate change classifications. Based on
accuracy tests, a proposal is made to create a projection
method by selecting specific models for different types of
cities and towns, as no one model yielded the minimum mean
error across city types. This method is called the

composite method in this study.

Purposes gf the Study

 

Usually, the local planning offices have a
responsibility to provide population projection data for
various purposes (Goodman and Freund, 1968 : 51). In many
cases, however, a local planning office lacks data and
funds, and has a limited amount of the time and expertise
needed to project future population through sophisticated
techniques (Isserman, 1977). In some cases, all that is
riquired only a quick and rough future population
projection, in which a certain range of projection error is
acceptable (Pittenger, 1976 : 29).

To satisfy the limits, trend extrapolation models have
been popular in local planning office because the models are
simple and inexpensive. Local planners do not have any

information about the accuracy of trend extrapolation models

because there is no way to measure the quality of their
projections by mathematical functions.

In two previous studies, using linear and exponential
regression models,Schmitt and Crosetti (1953), and Greenberg
(1972) provided information about projection accuracy for
communities in general, but not for specific types of
communities, (Isserman, 1977 : 253). Only Isserman (1975)
did accuracy test with a wide range of extrapolation methods
and communities. The hybrid methods which he developed are
a mixture of extrapolation techniques to fit communities
with particular characteristics called composite methods in
this study.

Isserman tested accuracy using 1930 to 1970 census
data. Until 1970, the trend of migration into the cities
was seen throughout the United States. This trend of
population migration, however slowed and changed, during the
19708 in Michigan (Rathge, Wang and Beegle, 1981 : 2). Only
19 out of 172 cities in this study show a net population
decrease from 1940 to 1950. Population losses in 25 cities
from 1950 to 1960, in 52 cities from 1960 to 1970, and in
101 out of 172 cities from 1970 to 1980 show net population
decrease. These changes probably affected the accuracy of
the extrapolation models, because trend extrapolation
techniques are dependent on past population changes. One
objective of this study is to deternube whether or not the

accuracy of each model assessed by Isserman's test is still

reliable in Michigan under changed trends, and, if not, to
find a model which is appropriate changed trends.

In addition to providing information about the accuracy
of various extrapolation models for each type of city,
another purpose of this study is to develop a micro-computer
program for local planners. The program will make
population projection easier, faster, less expensive, and
probably more accurate than arbitrarily selecting an
extrapolation model.

In summary, the three major purposes of this study are
as follows

(1) To provided information about the accuracy of the six
major extrapolation models;

(2) To test the accuracy of each of the six models and
developing the composite methods for minimizing error.

(3) To develop a micro-computer program of population
projection for local planners to project population

easily, fast, and inexpensively.

Data and Methodology

 

Cities and towns with a population over 2,500 were
selected from 1970 census data. Data for 172 cities and
towns were available, back to 1940. All of those cities were
used for testing the accuracy of the six trend extrapolation
models instead of using a sample of cities and towns in

Michigan. The number of 172 excluded Detroit, because of its

size inequity and probability that Detroit has different
migration trends than other cites.

Population projections for the accuracy tests were
carried out through unadjusted projection, rather than
adjusted projection as in stepdown projections. For
regression models, the point projection, which involves a
parallel shift of line passing the last observed population
point, were used as Isserman (1977) used it. He argued that
the point projection may correct for accumulated errors in
the regression lines.

The census data from 1940 to 1970 were used to project
1980 population for 172 cities and towns, using six trend
extrapolation models. These projected populations were then
compared to actual 1980 census data in order to estimate the
percentage error as a measure of the accuracy of each
model. The percentage error was calculated by the following
equation.

Percent Error = [(projected - actual) / actual] x 100

Average percentage errors were calculated for all
cities and for a certain city type with the percentage
errors between the actual population and the projected
population Projected calculated by the six models. The mean
percentage error for all of cities or for a city type were
then compared among the six models to determine which model
yielded the minimum mean percentage error for a certain type

of city.

Because minimizing the mean percentage error is not
adequate as a decision-making rule for choosing among
alternative methods, standard deviation was used as the
secondary criterion.

In order to develop the composite method, which is the
method where different models are applied to different types
of citis to minimize error, two characteristics were used to
define the city or town types : growth rate change and
population size. The growth rate change was calculated by
the growth rate from 1960 to 1970, minus the growth rate
from 1950 to 1960 in this study. The growth rate change was
used as a major characteristic to classify cities, instead
of the growth rate that Isserman (1977) used to classify the
township types, because the growth rate change implies
change of the past three point observations with one figure,
while the growth rate represents just change of the past two
point observations. For example, the growth rate and the
growth rate change for each city can be calculated with
census data up to 1970 by the following equation.

R1

(P2 - Pl) / P1

R2 (P3 - P2) / P2
RC = R2 — R1

where P1 1950 population

P2 = 1960 population

P3 = 1970 population

R1 = growth rate from 1950 to 1960

R2 = growth rate from 1960 to 1970

RC = growth rate change between '505 and '608

If the growth rate is used to classify city types, the
value of R2 will be used as represent the populaton trend in
1960 and 1970. If the growth rate change is used, the value
of RC will represent the value of R1 and R2. Consequently,
the growth rate change implies the population trend in 1950,
1960 and 1970. Because population extrapolation models
merely extend past population trends into the future, the
depth of the historical periodic population data may
influence projections. The deeper historical data may be
the better indicator of city types. Therefore, in this
study, seven types of cities were identified according to
growth rate change.

Population size was considered a less important
characteristic than growth rate change, because population
size gives a less clear distinction of accuracy among the
models. Isserman (1977 : 256) also came to the same
conclusion about populatin size. Other characteristics were
not considered, because Isserman tested with several
characteristics of cities (Greenberg, Krueckeberg and
Mautner : 16) and found that growth rate was the better
criterion for discriminating between city types (Isserman,
1977).

The mean percentage error and standard deviation of
percentage error drived from the six extrapolation models
were compared for each type of city. Certain models were

chosen as the best models for particular types of cities,

according to minimum mean percentage error. If the mean
error of two or more models was very close, the model that
had the smallest standard deviation was considered best. A
composite method was, then, developed comprising the best
model for each type of city meaning that different methods

are applied to different types of cities.

CHAPTER 2

POPULATION TREND EXTRAPOLATION MODELS

The basic notion of population extrapolation models and
models used for extrapolation provide the theoretical
foundation for this research. Therefore, basic notions of
extrapolation and fundamental equations for each model are
described briefly in this chapter.

As mentioned before, population trend extrapolation
models are simply the placement of lines or curves on a
graph from past observed population points to the future.
Therefore, extrapolation models involve nothing more than
the extension of past trends over time. The lines or curves
used to depict the past population trends can be interpreted
by certain mathematical equations. The equation is a kind
of extrapolation model.

All of these models deal with total population against
time. They never deal with any other factors. Social,
economic and political factors are not considered; nor are
the details for sex—age breakdown or the three components of
population (fertility, mortality and migration). The six
most common extrapolation models will be described in this

chapter.

10

11

Linear Extrapolation

The linear equation can be derived in two ways that are
graphically presented here in figure 1. One (a in Figure 1)
is by calculating the average population change in past time
intervals and another (b and c in Figure 1) is by regressing
past population data against time. The regression technique
has two variations, namely, line projection and point
projection which can also be applied to the exponential and
the double logarithmic regression models.

A line projection (c in Figure 1) is simply a method
that extends the regression line into the future. A point

projection (b in the figure 1) ia a parallel shift of the

population

 

time (year)

(Note) a = linear average model
b = linear regression model with
point projection
c = linear regression model with

line projection

FIGURE 1 : LINEAR EXTRAPOLATION MODELS

12

regression line of line projection, so that it passes
through the last observed population point (Isserman, 1977
250). In this study, point projection was used for linear,
exponential and double logarithmic regression. Isserman
also used this technique for his accuracy test of
extrapolation models and argued that it might correct for
cumulated errors in the regression line (Isserman 1977
250). This seems reasonable ; the last observed population
is the latest actual value, because it probably corrects the
mathematical error up to the last observation point.

The linear regression model, with point projection, can
be represented by the following mathematical equations

(Isserman, 1977 : 259).

P = P + bn ............ (Equation 1)
t+n t
where P = population
t = terminal year in historical data base
n = number of years from t to projection year
b = regression coefficient

The value of coefficient "b", that is drived from the
least square method, is the slop. The slope represents the
population growth increment per of unit of time. Therefore,
the larger value of coefficient "b" indicates the larger net
population growth per unit time as shown. The negative
value of coefficient "b" means that the population decreases
as the value of parameter "b" decreases per unit of time.

The linear regression model with point projection

assumes that a population will grow or decrease by the same

13

number of persons in each future time interval and the
projection line will include the last observed population

point.

Erpgnential Extraegletign

The exponential equation can be derived by caculating
the average population growth rate in past time intervals or
by regressing the logarithm of past population data against
time. The exponential regression with point technique was
used in this study for the same reason previously stated in
the linear regression model with point projection. The
exponential regression with point projection take following

form (Krueckeberg and Silvers, 1974 : 262).

n
P = P (1+r)
t+n t
where P = population
t = terminal year in historical data
r = the rate of population change
n = the number of year from t to projection year

calculated.

The (1+r) in the above equation can be replaced by "b",
Thus, the equation can be transformed to logarithmic form as
follows (Isserman, 1977 : 259)

In P = ln P + n ln b ......... (Equation 2)
t+n t
where population
terminal year in historical data
the rate of population change
the number of year from t to projection year
n b = regression coefficient

Hanna-J
II II II II

14

The exponential model equation is actually the same as
the linear regression model with addition of the logarithm
transformation of population. If the coefficient "ln b",
calculated by the least square method, is positive, the
population will grow at a constant rate per unit of time.
If the coefficient "ln b" is negative, the population will
decrease at a constantly decreasing rate over time. The
larger the absolute value of coefficient "ln b" shown, the
faster the growth or reduction rate.

The exponential regression model with point projection
assumes that a population will increase or decrease at a
constant rate over time, and the projected exponential curve

will pass the last observed population point.

Double logarithmic equations are derived by regressing
logarithms of past population data against logarithms of
time. The point projection technique was also used for this

model. The formula is as follows (Isserman, 1977 : 259)

ln P = 1n P + b ( ln (t+n) — ln t ) ... (Equation 3)
t+n t
where P = population
t = terminal year in historical data base
n = number of years from t to projection year
b = regression coefficient

The double logarithmic equation is very similar to the
exponential model with the addition of the logarithm

transformation of time.

15

The modified exponential curve is a curve of the
exponential family of mathematical functions, which shows a
continuously declining proportional population growth at
constant rate, but a continuously increasing net population
growth until the curve approaches an upper capacity limit.
The modified exponential function can be presented as the

following equation (Pittenger, 1976 : 67)

m
P = k + a b
i+m
where P = population
1 = initial year in historical data base
m = number of years from i to the projection year
k = the upper limit or maximum population
a and b = parameter

In this equation, the parameter "a" is the difference
between the value of "k" and the value of the population at
origin, so the parameter "a” should be negative. The
parameter "b" is the ratio of change, and the value of it
should be positive and less than one (Pittenger, 1976 : 67).
The parameters "k", “a" and "b" can be derived by the least
square method, but a selected points technique from Croxton
and Cowden (1945) was used to get two parameter in this
study as Pittenger (1976) did.

The following equations are a summary of the selected
points technique to get the values of parameters "k", "a"

and "b". If Y1, Y2 and Y3 are the partial population totals

16

—— three equal groups of population data from initial year
to terminal year -— and n is the number of elements

comprising the partial total, then ;

n

b = (Y3 — Y2) / (Y2 - Y3)
n

ln b = (ln b ) / n

n 2

a = (Y2 ~ Y1) (b-1) / (b - 1)

n
k = [Y1 — a (b - 1) / (b — 1)] / n

If parameter "b" is negative, then parameter "a" will
be negative through the selected points method. In that
case, the value of "bm" in the equation will be changed in
the negative and positive range according to the value of
"m". If the value of "m" is even, the value of "bm" will be
positive and if the value of "m" is odd, then it will be
negative. The projection curve with the negative value of
the parameter "b" will produce the curve of a zig—zag form.
It is not the modified exponential curve.

If parameter "b" is larger than one, then parameter "a"
will be positive through the selected points technique.
This means that the projected population is the upper limit
plus the value of "abm", which is positive in the case of
parameter "b" larger than one. Therefore, the projection
curve does not show any upper limit population while the
upper limit is the assumption of the modified exponential
model.

If parameter "b" is zero or one, then the value of

17

"abm" becomes zero. In that condition, the projected
population will be the same as the upper limit population
"k" through time.

Therefore, it is necessary to satisfy the indispensable
condition of the modified exponential model. The
indispensible condition is that the parameter "b" is
positive and less then one, and that the parameter "a" is
negative. Graphically the curve looks like that shown in
Figure 2.

The modified exponential curve assumes that there is
some upper limit to the population, which may be set by
zoning, subdivision control or any other public policy, and
the population growth rate will decline at a constant ratio

until it approaches the upper limit.

POP.

 

 

TIME

FIGURE 2 : MODIFIED EXPONENTIAL CURVE

18

§:§haee Curyg

There are tow common "S" shaped exponential formulas,
namely the Gompertz curve and the Logistic curve. The
S—shape curves have a lower limit and an upper limit. The
Gompertz curve was used for testing accuracy in this study.

By the S-shape curves, population growth starts from no
net change in the early stage, and then begin to increase in
rapid growth rate and net population and, at certain point,
starts to decrease growth rate while still increasing net
population, until there is finally no net change in
population again at the upper limit. The mathematical
formulas of the Gompertz curves are shown as following

equations (Pittenger, 1976 : 57)

If the equation is transformed to logarithm form

(Pitternger, 1976 : 58),

m
ln P = ln k + ( ln a ) b ..... (Equation 5)
i+m
where P = population
1 = initial year in historical data base
m = number of years from i to the projection year
k = the upper limit or maximum population
a and b = parameter
The parameter “k", "a" and ”b" was also derived in this

study by the selected points method shown by Pittenger

19

(1976). The equation of the Gompertz curve is basically the
same as the modified exponential model except for the
logarithmic transformation of population. Therefore, the
value of parameter "b" should also be positive and less than
one and the value of "ln a" should be negative for the same
reasons as described for the modified exponential model.
The Gompertz curve's graphic equivalent is pictured in
Figure 3.

The S—shape curve assumes that population growth begins
slowly and gains momentum until it reaches an inflection
point, after which it begins to decrease proportionately to

population growth.

POP.

 

 

inflection point

I
H

 

TIME

FIGURE 3 : GOMPERTZ CURVE

20

gglypgmial Curve

The polynomial curve gives possibilities for the
application of the various shape of curves. The curve
shapes are dependent on the degree of the polynomial and the
value of parameters. The general formula is represented as

following (Pittenger, 1976 : 53) ;

2 3 n
P = a + bm + cm + dm + .... + qm ... (Equation 6)
i+m
where P = population
m = number of years from initial year
to projection year
a, b, c, .. q = polynomial coefficients

In this study, the second and third degree of
polynomial were tested. The second degree polynomial model

was then selected to test the accuracy of the model, because

POP.

’ 2

    

\
\

2
(a - b /4c)———

*.

l
l
l
l
l
l
l
+
l

 

 

l
I (-b/2c) TIME

FIGURE 4 : THE SECOND DEGREE POLYNOMIAL CURVE

21

the second degree polynomial model showed less projection
error in the Michigan case.

The coefficient "a", "b" and "c" were derived from the
least square method. The second degree polynomial equation
can be expressed to as following

2 2
P = c (m + b/2c ) + (a — b /4c)
i+m
This formula is shown graphically in Figure 4.

The coefficient "c" decides the form of the curve. The
value of "b/2c" shows the distance of the horizontal
parallel shift of the equation cm and the value of (a +
b2/4c) shows the distance of the vertical parallel shift of
the equation cm

The polynomial model makes no assumption about growth
pattern. It merely fits the curve to past population data
and extends the curve into the future. But this model has
more various curve forms than other models to fit past

population data which are seldom linear through time.

CHAPTER 2

ACCURACY TEST

Assuragy £9: gysrall Elsies and Teens

Each of the six extrapolation models was used to
project the 1980 population with census data from 1940 to
1970 for 172 cities and towns in Michigan. The projected
population was then compared with the actual population in
1980 census data, and the percentage error for each of the
cities and towns was calculated. To compare the accuracy of
the six models, the mean and the standard deviation of the
absolute value of percentage error were calculated.

The results of the accuracy tests are presented in
Table 1. In the table, the positive value of the percentage
errors represents the over-projection and the negative value
represents the under-projection.

As shown in the Table 1, all six models tend to
over—estimate the population of 1980 in Michigan. For
instance, the 4.1 percent of the cities was under—projected
more than 10 percentage with the linear model, otherwise,
the 50.0 percent of the cities was over—projected more than
10 percent, although the linear model tend less to
over-estimatation than the exponential and double
logarithmic models. The three other models also show the

tendency to over—estimation.

22

23

The linear model had the smallest mean percentage error
and standard deviation. Conversely, the exponential
regression model had the largest mean percentage error and
standard deviation for the 172 tested cities.

The populations of 79 out of 172 cities in Michigan
were projected within 10 percent of their actual values with
the linear model, which was the highest percentage among
four models. The linear model had the smallest range
between the highest positive and the hightest negative
percentage error. The range with the linear model is from
—38.3 percent to 60.4 percentage error. Other models showed
a more extreme range from the highest positive to the

highest negative percentage errors. Through the analysis of

table 1
Frequency of Percentage Errors

for 172 Cities and Towns in Michigan

 

 

 

 

% ERROR LIN EXP DLOG POLY GOM MOD
25 + 36 57 57 4O 4 2
10 TO 25 50 51 51 35 24 24

-10 TO 10 79 59 59 71 45 44
-25 TO -10 6 4 4 21 8 6
< —25 1 1 1 5 0 0
MEAN ERROR 16.6 34.9 34.5 18.4 10.6 10.3
SD. 14.7 77.9 76.2 17.4 7.9 7.8

 

TOTAL CITY 172 172 172 172 81 76

 

24

mean percentage error and standard deviation, we find that
the linear model seems to be relatively the best model among
these four models for 172 cities.

The Gompertz and the modified exponential model were
not applicable to all cities because the two models require
a certain indispensible condition. The indispensible
condition is that the parameter "b" in the equations 4 and 5
should be positive and less than one, while the parameter
"a" in the equations should be negative. The accuracy of
the two models was tested separately for those types of
cities. The past population growth pattern of the 81 of the
172 cities satisfied the indispensable condition for the
Gompertz model, 76 of the 172 cities satisfied the modified
exponential model. The projected 1980 population for those
cities, using these two models, war relatively accurate.

The Gompertz model yielded 10.6 mean percentage error,
with 7.9 standard deviation of percentage errors. The
populations of 45 out of 85 cities were projected within a
10 percent error. The modified model had 10.3 mean
percentage error, with 7.8 standard deviation of percentage
error. The populations of 44 out of 81 cities were
projected within 10 percentage error. These models also
showed a relatively small range from the highest positive to
the highest negative percentage error. The error range with
the Gompertz model was from -21.5 to 33.5 percent and the

range with the modified exponential model was from —21.4 to

25

41.6 percent.

These two models showed more accurate population
projection than the linear model for specific cities, but
require a certain condition not met by all cities.

Overall, this accuracy test for all cities showed that
no model was best for all cities and towns, although the
linear model and the second degree polynomial model yield
comparatively better results than others. Some models are
better suited for particular types of cities, such as the
Gompertz and the modified exponential models, which produced
more accurate projections for a particular type of past

population growth pattern.

All of the extrapolation models project future
population dependant only on past population growth trend.
Therefore, the past population growth pattern inherently
affects the accuracy of an extrapolation projection model.

In this section, it is assumed that different
population trend extrapolation models can be applied to
different types of cities classified by past growth rate
changes. The hypothesis will be tested with six models.

To classify past population growth patterns for cities
and towns in Michigan, changes in the population growth rate

during two past time spans were used ; 1950 to 1960 and 1960

26

to 1970. Thus growth rate change implies the population
trends in these two time spans, although it does not give
any information about net population change or proportional
change during the last span period. The value of the growth
rate change is calculated by the growth percentage from 1960
to 1970, minus the growth percentage from 1950 to 1960. As
mentioned before, the value implies the population changes
over the past three observation points, while the growth
rate or net population change shows just the population
changes over the last two observation points. That means
the growth rate change characteristic can be used for
simplification of past growth patterns while it implies
deeper past population trends. The value of the growth rate
change indicates the degree of growth rate change. A
positive value indicates a degree of growth rate increase
and a negative value indicates a degree of decrease of the
proportional population growth.

In this study, seven types were clasified according to
past growth rate changes between the 19503 and 1960s as
follows : (1) Great increase type in which percentage growth
rate changes ranged more than 25 percent between the years
from 1960 to 1970 and the years from 1950 to 1960; (2)
Medium increase type in which the percentage growth rate
changes ranged from 10 to 25 percent; (3) Moderate increase
type in which the percentage growth rate changes ranged from

0 to 10 percent difference; (4) Moderate decrease type in

 

27

which the percentage growth rate changes ranged from —10 to
0 percent; (5) Medium decrease type in which the percentage
growth rate changes ranged from —25 to —10 percent; (6)
Great decrease type in which the percentage growth rate
changes ranged from —50 to -25 percent; (7) Extreme decrease
type which the percentage growth rate changes were less then
-50 percent.

If a local planner can roughly estimate the future
growth rate for a city, he may obtain better accuracy by
selecting a model according to type of future population
growth, because the future population changes actually
affect the accuracy of the population projections. To
improve the accuracy of projection, each of the seven types
was classified again by the growth rate change between 1960
to 1970 and 1970 to 1980. Growth rate changes for 1960 to
1970 and 1970 to 1980 were calculated in the same way as
previously used. The actual census population data of 1980
were used to get the growth rates from 1970 to 1980 for each
city, because accuracy information should be provided
through known rather than uncertain data. It is assumed
that the future growth rates can be perfectly anticipated.
Accordingly, three types of the future growth rate changes
were classified and the accuracy of the six models was
tested for each of the three types.

The three types of future growth rate changes are : (1)

a heavy increase in which the changes in percent growth rate

28

are larger than 10 percent. (2) a light change in which the
percentage growth rate changes ranged from -10 to 10
percent. (3) a heavy decrease that has the percentage
growth rate change of less than —10 percent.

From here on, changes of percentage growth rate from
1950 to 1960 and 1960 to 1970 will be termed as "the past
rate change", and "the future rate change" will designate
the growth rate changes from 1960 to 1970 and 1970 to 1980.
In following sections, accuracies are first tested with the
seven types of the past rate change. Then the accuracies
for the three types of future rate change in each of the
seven types of past rate change are tested to provide more
detailed information on each models' accuracy and to develop

composite methods.

figgag Increase ip Growth Rate

 

Cities were classified for this type by past rate
changes which were larger than 25 percent. For instance,
the population of Saline in Pashtenaw county grew at a 52
percent growth rate from 1950 to 1960 and grew at a 106
percent growth rate from 1960 to 1970. Thus growth rate
change, the past rate change, is 54 percent. Therefore,
Saline falls into this category.

Only seven towns were of the great increase type in
Michigan. While it is statistically difficult to test which

model is reliable for this type on the basis of only seven

29

cases, the fitness of a model to this pattern may be
logically described by combining the results of this
analysis and the assumption of each model.

All seven cities did not meet the indispensable
condition for the Gompertz or the modified exponential
models. The linear model showed a better mean percentage
error and standard deviation than other models (see Table
2). Just 2 cities had a percentage error within 10 percent.
That means the majority of cities‘ projected had large
errors. Therefore, the linear model should be used for this
type, with some care. For a more detailed classification,

this type can be classified again with the future rate

Table 2

Frequency of Percentage Errors for
Cities of Great Increase Type in Michigan

 

 

 

 

% ERROR LIN EXP DLOG POLY
25 + 2 3 3 4
10 TO 25 1 2 2 3

-10 TO 10 2 2 2 0
-25 TO -10 2 0 O O
< —25 O 0 0 O
MEAN ERROR 19.7 26.0 25.8 33.2
SD. 14.2 21.2 21.0 14.8

 

TOTAL CITY 7 7 7 7

 

30

change, but the population growth rate of all seven cities
was decreased more than 10 percent from 1960 to 1970 and
1970 to 1980. Therefore, the results would be the same as
the results for the great increase type of past rate
change.

The projection line and curves can be analyzed as an
example of this type. The accuracy of each model can then
be reviewed. City of Saline, a case in Pashtenaw county,can
serve as an illustration of this type. Saline showed 52
percent population growth from 1950 to 1960, 106 percent
growth from 1960 to 1970 and 54 percent growth from 1970 to
1980. With the pre-1970 data, all of the exponential family
models draw projection curves with fast growth rates in the
future, while the linear model draws a projection line with
the same net population per decade in the future.

Actually, the population of Saline grew more than the
average population, but the growth rate declined during the
19705. Therefore, the linear model yields a slight
under—estimation and other exponential family models yield
over—estimations for the Saline population of 1980, as
represented in Figure 5. If the rapid growth rate had
continued during 1970s in Saline, the exponential or the
double logarithmic models would probably have produced
better projections than the linear model.

As an other example, three cities of this type in

Michigan showed net population decrease during 1970s,

Population

12,0004

V

11,000..

10,000 p
9,000.
8,000.
7,000
6,000.

5,0000

4,000‘
3,000.
2,000“

1,000”

 

1940 '50 '60 '50 '00 '00 2000

 

----...O-gno-u

—-—~O-—.-

)k

31

 

L A

Year

Linear Regression.Mbdel _
Exponential & Double Logarithndc Mbdel

Second Degree Polynomial Model
Actal Pepulation

FIGURE 5 : EXample of the Great Increase Type

(City of Saline)

32

although rapid population growth had been shown earlier.
All extrapolation models yielded high over-estimation for
those cities. In this case, it is probably better to use
another method rather than any of the trend extrapolation
models.

The exponential and double logarithmic models seemed to
be more suitable for cities with continuous growth rate
increase in the future, and the linear model seemed to be
more suitable for cities in which net population will grow
at a declining rate in the future. All of the seven cities
in this type had declining growth rates during the 1970s. In
conclusion, the linear model seems to be the best model for
this type in Michigan cases.

To summarize, the linear model can generally be used
for cities with great increases in growth rates although it
did not yield satisfactory results in this accuracy test.
If a planner can roughly estimate future growth rate, he can
apply the exponential model to cities with future rate
changes of more than 10 percent, the double logarithmic
model to cities with future rate changes between —10 to 10
percent, and apply the linear model to cities with the
future rate change of less than —10 percent. However, a
planner could better project population on the basis of
local knowledge in cases where the population grew quickly
in the past, but net population is expected to decline in

the future.

33

Medium Increase In Grease Bats

This type of city has the past rate change that range
from 10 to 25 percent. For example, the population of Grand
Rapids increased about 0.5 percent from 1950 to 1960, and
increased again about 11.5 percent from 1960 to 1970. The
growth rate change between the 19503 and the 19603 was thus
11 percent (11.5 percent — 0.5 percent = 11 percent).
Therefore, Grand Rapids is a city of medium increase in
growth rate.

Cities of this type comprised nine cases in Michigan.
It is difficult to test which model is best for this type of
city with only nine cases, but an accuracy test with nine
cases can indicate some tendencies of each model.

With the nine cases of this type, the linear regression
model showed the minimum percentage error and the highest
percentage of cities within a 10 percent error. (See Table
3) The mean percentage error with the linear model is 11.2
percent and the populations of six out of the nine cities
were projected within a 10 percentage error. As a result,
it seems reasonable to use the linear model as a projection
model for this type. The exponential model and the double
logarithmic model are the next best models for this type.
The two models yielded more over—estimation of population
for 1980 than the linear model. None of cities met the
indispensible condition for the Gompertz and the modified

exponential model.

 

 

34

If the future rate changes were considered for this
type, the six cases in the nine cities showed negative
growth rate changes more than 10 percent. The mean
percentage error was 7.7 percent with the linear model.
Otherwise, the mean percentage error was 14.7 percentage
error with the exponential model, 14.5 percentage error with
the double logarithmic model and 22.3 percentage error with
the polynomial model. Just one city had a future rate
change from —10 to 10 percent. For that city, the
polynomial model yielded minimum error (29.4 percent
under-estimation), but the result is not significant with
one case. Two cities had future rate change more than 10

percent and the linear model yielded the minimum mean error

Table 3

Frequency of Percentage Errors for
Cities of Medium Increase Type in Michigan

% ERROR LIN EXP DLOG POLY
25 + 0 1 l 2
10 TO 25 l 3 3 4

-10 TO 10 6 3 3 1
-25 TO -10 1 1 1 1
< -25 1 1 1 1
MEAN ERROR 11 2 14 9 14 7 20 0
SD 10 8 10.1 10 1 9 8

35

for these cities. In conclusion, the linear model can be
considered as the best among the six models for this type of
future rate change, with less than 10 percent in the medium
increase type.

The city of Flushing in Genesee county is an
illustration of the medimum increase type. In the Flushing
case, the population grew 69 percent from 1950 to 1960, and
grew 91 percent from 1960 to 1970. The growth rate change
was thus 21 percent, which is the medimum increase type.

In the Flushing case, the exponential model and the
double logarithmic model extended projection curves with the
assumption that the growth rate would be kept continuous in
the future as past data showed. But the actual growth rate
from 1970 to 1980 did not increased, although the net
population increased. As a result, the exponential model
and the double logarithmic model produced over-estimations
for the Flushing population of 1980 as presented in Figure
6. Otherwise, the linear regression line assumes that the
same amount of the population will increase as much as the
number of people estimated by regression per unit of time.
The actual population increment from 1970 to 1980 was very
close to the predicted number by linear regression in the
Flushing case. If the population of Flushing increased
rapidly with the growth rate, as in the past, the
exponential model might be better for that case than the

linear model.

36

Population

24,000 ’ '/

22,000 r
20,000 i f/
18,000 r ;/

16,000 ” ,/

14,000 0
12,000 “

10,000

8,000
6,000

4,000

 

2,000

 

‘ A A A A A
V v-

1940 '50 '60 '70 '80 '90 2000 Year

 

Linear Regression.Model

“ ' ‘ " ‘ ' ' "' "" Exponential & Double logarithmic MOdel

Second Degree Polynomial Mbdel
X’ Actual Population

FIGURE 6 : Example of the Medium Increase Type
( City of Flushing )

37

In summary of the accuracy test for the medium
increase type, the linear model can be used with relatively
better accuracy for this type of city. If a planner can
predict rough growth rate into the future, he can apply the
exponential model for the type of the future rate change of
more than 10 percent, and the linear model can be still used
for the rest of the cities in this type.

But a planner has to execise caution in using this
extrapolation model for cities which showed a continuous
increase or decrease of population in the past, and the
population trends are expected to be reversed in the future,

when the net population decreases or the increases suddenly.

Moderate Increase ip Growth Rate

 

 

This type of city has the past rate change that ranged
from zero to 10 percent. For example, the population of Ann
Arbor showed a 40 percent increase from 1950 to 1960 and a
48 percent increase from 1960 to 1970. The growth rate
change between the 19503 and the 19603 is 8 percent.
Therefore, Ann Arbor is a city of this type.

27 cities in Michigan were categorized in this class.
Except the Gompertz and the modified exponential model, the
mean percentage errors with four models are ranged from 10.1
to 15.2 percent error. The mean errors seem to be
satisfiable as the degree of accuracy with the extrapolation

models.

38

The linear model yielded the minimum percentage error
and the minimum standard deviation of percent error for this
type (See Table 4). The mean percentage error is 10.1
percent and 17 out of 27 cities were projected within 10
percent of their actual census populations. The polynomial
model comes out as the next best model from the accuracy
test.

No city satisfied the indispensible condition of the
Gompertz model or the modified exponential model for this
type. The exponential model and the double logarithmic

model yielded extreme percentage errors in some cities,

Table 4

Frequency of Percentage Errors for
Cities of Moderate Increase Type in Michigan

% ERROR LIN EXP DLOG POLY
25 + 3 5 5 5
10 TO 25 7 7 7 6
-10 TO 10 17 14 14 13
~25 TO -10 0 1 1 2
< -25 O 0 O 1
MEAN ERROR 10 1 15 2 15 l 14 2
SD 9 4 18.2 18 0 13 7

39

compared with the linear model. The highest error with the
exponential model was 80.5 percent error,.while 39.1 percent
error was the highest with linear model. The two models
also have a higher mean percentage error than the linear
model.

It is useful to consider the accuracies for this type
with the future rate changes. There are two cases for the
type over 10 percent of the future rate changes (heavy
increase type of the future rate change), 14 cases for the
type with the future rate change ranged from —10 to 10
percent (light change type), and 11 cases for the type with
the future rate change less than -10 percent (heavy decrease
type).

For the heavy increase and the light change types of
the future rate change, the double logarithmic model and the
exponential model yielded the minimum percentage error. The
exponential model showed slightly better results in this
analysis.

The mean percentage errors are 4.3 percent with the two
models for the light change type of the future rate change,
while it is 4.9 percent with the linear model and 5.2
percent with the polynomial model. The populations of 13
out of 14 cities in the light change type were projected
within 10 percent error with the linear model, the
exponential model and the double logarithmic model. The

polynomial model projected population of 11 out of 14 cities

40

within a 10 percent error. All of the four models project
the population for this type relatively very well.

For the 11 cities of the heavy decrease type in the
future rate change, the linear model turned out as the best
model among four models. But the mean percentage error is a
little high at 16.5 percent, and the standard deviation of
percent error is not good, either. Only two out of 11
cities are projected within 10 percent error. Other models
produced a quite high mean percentage error, for example,
30.1 percent with the exponential model and 24.5 percent
with the polynomial model.

Each model can be discussed with a graphic
illustration. The Ann Arbor case was used as an example of
the moderate increase type in the past rate change. The
population of Ann Arbor grew with 40 percent growth rate
from 1950 to 1960, increased with 48 percent growth rate
from 1960 to 1970, and increased with 8 percent growth rate
from 1970 to 1980.

Because the similar curves were depicted by the
exponential model and the double logarithmic model, the
double logarithmic model only was represented in Figure 7.
As depicted in Figure 7, the double logarithmic model, the
exponential model and polynomial model keep the fast growth
rate into the future as like the past population trend until
1970. Those curves are very well fitted to the past

population points. However, the actual population growth

41

Population

240,000
220,000+
200,000*

180,000'

160,000»
140,000»

120,000”

100,000“
80,000»
60,000"

40,000”

 

20,0000;

 

A A
r v v v v v

Q Q -

1940 '50 '60 '70 '80 '90 2000 Year

 

Linear Regression Model
"" ‘~‘ " " ‘ ‘ Exponential & Double Logarithmic Model
“"""—' Second Degree Polynomial Model

)‘ Actual Population

FIGURE 7 : Example of the Mbderate Increase Type
( City of Ann.Arbor )

42

rate decreased during 19703. The future rate change for Ann
Arbor is -38 percent. Therefore, the three models yielded
high over—estimation for the population of 1980.

The linear model projected population with the same
population increment calculated by regression. With the
population increment about 22,903 persons per decade, the
linear model extends the line into future. But the 8,169
persons actually increased during the 19703 in Ann Arbor.
The linear model also produced over-estimation. The linear
model projected population more accurately for Ann Arbor
than other models did, however.

The city of Durand in Shiawassee county has a different
future rate change (9 percent) than the future rate change
(—38 percent) of Ann Arbor, though the past population
patterns of the two cities are similar. The population of
Durand grew 3.7 percent during the 19503, 11.0 percent
during the 19603 and 15.3 percent during the 19703. In this
case, the four models produced under-estimations because the
actual population- grew with a faster growth rate than
expected by the extrapolation models.

The polynomial model was best for Durand. It drew a
steeper curve than the exponential model because it
reflected the rapid growth rate increase from the 19503 to
the 19603. The exponential model and the double logarithmic
model were the next best models for the city of Durand.

Those models were better than the linear model in the Durand

43

case. The exponential model is better for the cities that
are expected to grow or decrease continuously while keeping
the past growth rate, because the exponential model projects
population with the same growth rate while the linear model
projects population with the same amount of population
growth. The linear model is better for cities that will
grow or decrease with approximately the same increments of
net population as estimated by the linear regression.
Because of these reasons, the linear model is better in Ann
Arbor and the exponential model is better in Durand.

In summary of the accuracy test for the moderate
increasing type, it can be concluded that the linear model
is generally best for this type. If a local planner can
roughly estimate growth rate in future, he can use the
exponential model for the heavy increase type and the light
change type. If the future rate changes are expected to be
less than —10 percent, then he can still use the linear
model. He should be careful to use those models for the
cities that are expected to decrease or increase suddenly as
the opposite direction with the past trends during the next

time span, as illustrated with Ann Arbor.

Moderate Decrease ip Growth Rate

 

 

The cities of this type were categorized by the past
rate changes which are ranged from -10 to zero percent. The

past rate changes were calculated by the percent growth rate

44

from 1960 to 1970 minus the percent growth rate from 1950 to
1960.

There are 54 cities of this type in Michigan. That is
the largest number among seven types of past rate changes.
The 54 cases may be statistically enough to generalize the
results of this accuracy test.

The linear model and the polynomial model have minimum
mean percentage error. The mean percentage error with the
polynomial model is slightly lower than with the linear
model. The mean percentage error is 9.097 percent with the
polynomial model, and 9.128 percent with the linear model.
Therefore, it can be regarded that the two models actually
yield the same result in the aspect of mean error. As
another aspect of the accuracy measure, the standard
deviation of the percentage error can be considered to
compare the accuracy of two models. The standard deviation
of the percent errors with the linear model is 6.5, and 7.5
with the polynomial model (See Table 5). The linear model
shows a better result in the aspect of error distribution.

The highest error with the two models is considered the
accuracy measure. The highest error with the linear model
was -18.0 precent error in the aspect of under-estimation
and 30.2 percent error in the aspect of over—estimation,
while there was a -27.7 percent error of under-estimation
and 29.6 percent error of over—estimation with the

polynomial model. The linear model is slightly better in

45

the viewpoint of error range, too. Therefore, the linear
model was considered the best model for this type in this
study.
Other models also show good accuracy for this type.

The exponential model produces 10.1 mean percentage error
and the double logarithmic model shows 10.0 mean percentage
error. 31 cities for the Gompertz model and 30 cities for
the modified exponential model out of 54, are satisfied with
the indispensible condition of the two models. The Gompertz
model shows 7.5 mean percentage error and the modified
exponential model shows 7.0 mean percentage error. The

populations of 24 in 31 cities were projected within 10

Table 5

Frequency of Percentage Errors for
Cities of Moderate Decrease Type in Michigan

X ERROR LIN EXP DLOG POLY GOM MOD
25 + 2 3 3 4 0 O
10 TO 25 14 20 20 7 5 4
~10 TO 10 36 29 29 35 24 24
~25 TO ~10 2 2 2 6 2 2
< ~25 0 O 0 2 0 0
MEAN ERROR 9.1 10 1 10 0 9 1 7 5 7 0
SD. 6 5 7 5 7.4 7 5 6 7 5 9

46

percent error with the Gompertz model and 24 in 30 cities
with the modified exponential model. It seems best to use
the modified exponential model or the Gompertz model for
these types of cities if those cities are proper to the two
models. If the past population trend of a city can not
satisfy the indispensible condition, the linear model is the
optimum model for the moderate decrease type.

This type can be further classified according to the
future rate change. The first type is the heavy increase
type of the future rate change that is larger than 10
percent. Three cities out of 54 cities are categorized for
this type. The range of the mean percentage without the
Gompertz and modified exponential models is from 11.2 to
15.9 percentage error. The models tend to under-estimate
the populations. Though the exponential model yields
minimum mean percentage error and standard deviation of
percent error, populations of two out of three cities were
under—estimated with more than 10 percent error. Only one
in the three cities satisfy the indispensible condition for
the Gompertz and modified exponential models, but the
percentage errors are very high.

The second type is the light change type of the future
rate changes which are ranged from -10 to 10 percent. 41
out of 52 cities fall into this type. The linear model has
a minimum mean percentage error which is 6.9 percent error.

Other models also show fairly good results. The mean

47

percentage error is 7.2 percent with the polynomial model,
7.3 percent with the double logarithmic model, and 7.4
percent with the exponential model. The populations of 33
out of 42 cities were projected within 10 percent error with
the linear model and 31 cities with the polynomial model.

The Gompertz model and the modified exponential model
show very accurate projection for the cities which satisfy
the indispensible condition for two models. There are 22
cities for each of two models in 41 cities. The mean
percentage error for those cities is 4.6 percent with the
Gompertz model and with the modified exponential model. The
populations of 21 out of 22 cities were projected within 10
percent error with the two models. These two model can be
used as the best model for this type of the cities if those
cities meet the indispensible condition of two models. If
not, then, the linear model can be used for this type.

Third type is the heavy decrease type of the future
rate changes which are less than ~10 percent. There are 10
cities for this type. All models show a little high mean
percentage error. The polynomial model represents the best
results for the 10 cities. The mean percentage error is
14.8 percent with the polynomial model and 17.3 percent with
the linear model. All of six models tends to over-estimate
the populations. The polynomial model also over-projected
the populations for 7 out of 10 cities with more than 10

percent error, but it is still best for this type among four

 

 

48

models except the Gompertz model and the modified
exponential model.

The 8 cities for the Gompertz model and the 7 cities
for the modified exponential model are satisfied with the
indispensible condition of two models. The mean percentage
error for those cities is 13.8 percent with the Gompertz
model and 12.4 percent with the modified exponential model.
This results with two models are slightly better than the
polynomial model.

The graphical example will explain the tendency of each
models for the moderate decrease type of the past rate
change. Pontiac are illustrated for the type. The
population of Pontiac showed 11.6 percent increase from 1950
to 1960 and 3.7 percent increase from 1960 to 1970.

The past growth pattern of Pontiac meet the
indispensible condition for the gompertz model and the
modified exponential model. The projection curve with the
Gompertz model showes the rapidly increasing growth rate in
early stages and the rapidly decreasing growth rate after
inflection point, and then the projection curve approaches
the upper limit. The modified exponential model drew the
projection curve with the same decreasing growth rate
continuously until approaching the upper limit. The figure
8 depicts how models are fitting the past observed
population points. And it also represent why the projection

errors with six models are small or high.

 

 

49

 

 

 

Year

Population
d

110,000 1
100,000 “

90,000 1

80,000 0

70,000 "

60,000 0

50,000 “

40,000"

30,000;

20,000" :

10,000 ‘ ,l

I G v ¢ v 6
1940 '50 '60 '70 '80 '90 2000
' Linear Regression Model
- ------------- Exponential & Double Logarithmic Model
//”,,.~a—--~ Gompertz Model
.-/’7'-.‘. Mbdified Exponential Model

/

—————— Second Degree Polynomial Model

X Actual Population
FIGURE 8 : Example of the Moderate Decrease Type

( City of Pontiac )

50

The Pontiac showed the net population decreased from
1970 to 1980 though the population had increased from 1940
to 1970. All of models drew the projection line or curves
reflecting the past population trend, but the population
change during 19703 in Pontiac does not follow the past
trend. Therefore, all of model over—estimate the population
for 1980. However, the Gompertz model showes relatively
better accurate projection for Pontiac, because it drew
projection curve with the assumption of rapid decreasing
growth rate until approaching upper limit. If the
population of Pontiac were stable during 19703 which may be
classified as the light change type, the Gompertz or
modified exponential model may yield a better accurate
projection.

In summary of this section, the linear model yields the
minimum mean error for cities in this type overall. But, if
cities in this type satisfy the indispensible condition of
two models, the Gompertz model and the modified exponential
model can be used as the best model for the moderate
decrease type in growth rate. If not, the linear model is
the optimum model for the type.

If a local planner can roughly estimate future growth
rate, he can use different models according to the future
rate change. If the future rate change is larger than 10
percent, he can use the double logarithmic model. If the

changes are ranged from -10 to 10 percent, the linear model

51

can be used for this type. It is recommended to, however,
use the Gompertz model when the cities satisfy the
indispensible condition of that model. If the future rate
change is less than —10 percent, the modified exponential
model can be used when the cities meet the indispensible
condition of it, otherwise, the polynomial will be the best

model for this type.

Medimum Decrease ip Growth Rate

 

 

These cities have past rate changes that range from -25
to ~10 percent. For example, Adrian, in Lenawee county,
grew at 10.6 percent growth rate from 1950 to 1960 and at
0.2 percent growth rate from 1960 to 1970. The change in the
growth rate between the 19503 and the 19603 is -10.4
percent. Therefore, Adrian falls into this type.

There are 32 cities of this type in Michigan. With
these 32 cities, the polynomial model generates a minimum
percentage error of 14.0 percent (See Table 6). The linear
model yields results very close to the polynomial model.
The polynomial model and the linear model projected the
populations of 14 out of 32 cities with less than 10 percent
error. But the linear model projected the populations of 4
out of 32 cities with an error greater than 25 percent of
actual population, while it was only 3 out of 32 cities with
the polynomial model. In the aspect of standard deviation

of percent error, the linear model is slightly better than

52

the polynomial model. Since the two models yielded very
similar results, either can probably be used for this type.
In this study, the model that yielded the minimum mean error
was selected as the best model. If the mean error was same,
then the standard deviation of percent error was used as the
measure of accuracy. Therefore, the polynomial model is
selected as the best model for the medium decrease type in
this study.

The exponential and the double logarithmic models
tended to over—estimate the population more than the linear
model. When the accuracy of the Gompertz model and the

modified exponential model were tested with the cities that

Table 6

Frequency of Percentage Errors for
Cities of Medimum Decrease Type in Michigan

% ERROR LIN EXP DLOG POLY GOM MOD
25 + 4 7 7 1 1 0
10 TO 25 14 14 14 6 2 2
~10 TO 10 14 11 11 14 10 10
~25 TO ~10 0 0 O 9 3 2
< ~25 0 0 O 2 0 0
MEAN ERROR 14.3 16 8 16 7 14 0 9 0 7 8
SD 9 5 11 1 11.0 10 5 8 O 6 1

53

satisfied their indispensible condition, the accuracy of
these model was better than any other models. The condition
was satisfied with 16 cities for the Gompertz model and with
14 cities for the modified exponential model.

With those cities satisfing the condition, the Gompertz
model had 9.0 mean percentage error and the modified
exponential model has 7.8 mean percentage error. In the
aspect of standard deviation of percent error, the 10 out of
the 16 cities had the projection errors within 10 percent
with the Gompertz model and 10 out of the 14 cities with the
modified exponential model.

Therefore, if the cities in the medium decrease type
satisfy the indispensible condition of these two models,
their populations can be projected better by these two
models than by the polynomial model. If a city does not
meet the condition, the polynomial model can be used.

The accuracy of the models also depends on the type of
future population changes. Therefore, the medium decrease
type was classified into three types by the future rate
change. The first type was the heavy increase type in which
the future rate changes are greater than 10 percent. Only
three out of 32 cities fell into this category in Michigan.
The exponential model, the double logarithmic model and
linear model projected population very accurately for these
three cities with less than 10 percent error. The

exponential model and the double logarithmic model had 5.3

54

mean percentage error for the three cities and the linear
model had 5.9 mean percentage error.

The second type is the light change type in which
future rate changes range from —10 to 10 percent. This type
comprised 20 out of 32 cities. The polynomial model showed
the best result for these cities. The mean percentage error
was 9.5 percent with the polynomial model, while it was 11.5
percent with the linear model. The Gompertz model and the
modified exponential model yield a lower mean percentage
error if the models are applied to cities which satisfy the
indispensible condition of the two models. With these
cities, the Gompertz model had 4.6 mean percentage error
with 9 cities, and the modified exponential model had 4.3
mean percentage error with 8 cities. Therefore, the
modified exponential model can best be used for this type if
a city satisfies its indispensible condition. If it does
not satisfy this condition but satisfies the condition of
the Gompertz model, the Gompertz model can best be used for
the city. If the city is not appropriate for either of
those, then the polynomial model can be applied.

The third type of the future rate change is the heavy
decrease type in which future rate changes are less than -10
percent. There were 9 cases of this type in Michigan. Four
models, except the Gompertz model and the modified
exponential model, had high mean error for this type. The

modified exponential model and the Gompertz model yielded

 

55

better results for specific cities. Among 9 cities, 6
cities satisfied the indispesible condition of the Gompertz
model and 5 cities for the modified exponential model. The
modified exponential model had 12.0 mean percentage error
and projected the population of 3 out of 5 cities within 10
percent error. The Gompertz model had 14.7 mean percentage
error. The other four models tend to over—estimate the
populations. The mean percentage error was 18.4 percent
with the polynomial model, 23.3 percent error with the
linear model, 28.4 percent error with the double logarithmic
model and 28.5 percent error with the exponential model.
Therefore, if a city satisfied the indispensible condition
of the modified model, its population can be projected by
the modified exponential model. If it did not satisfy the
condition for the modified model but satisfied the condition
for the Gompertz model, the Gompertz model could be used for
the city. If it did not satisfy the conditions of either of
these, the polynomial model could be used, with care.

Adrian was graphically illustrated as one of the medium
decrease cities of the past rate change in Figure 9. Adrian
grew quickly from 1940 to 1960, but the population changed
little from 1960 to 1980. The exponential model and double
logarithmic model drew projection curves into the future at
a certain population growth rate. The polynomial model drew
a projection curve which showed the rapid population

increase at a decreasing growth rate until the mid—19603 and

 

56

 

 

 

 

 

 

 

Population
24,000 ‘
22,000 0
20,000 1
18,000 “
16,000 0
14,000 1
12,000 ”
lo I 000 1'
8,000 “
6,000 “
4,0004
2,000 0
1940 '50 '60 '70 '80 '90 2000 Year
Linear Regression Model
"“"‘“““"‘ Exponential & Double Logarithmic Model
“““““ Second Degree Polynomial Model
’//,,»~””'_-“ Gompertz Mbdel
”’,»~“”’“' .Mbdified Exponential Model
/‘
)t Actual Population

FIGURE 9 : Example of the Medium Decrease Type
( City of Adrian )

57

showed a rapid population decrease after the mid—19603.

Adrian's past population trends satisfied the
indispensible condition for the Gompertz and the modified
exponential models. The two models drew projection curves
that indicated extreme population growth in the 19403 and
then a rapidly decreasing growth rate in the 19503 and
little change in population after 1960. Actually, the
population of Adrian increased by only 804 persons during
the 19703. Therefore, the Gompertz model and the modified
exponential model projected populations very close to the
actual populations of 1980. The exponential model, the
double logarithmic model and the linear model yielded
over-estimations, and the polynomial model under-estimated
the population of Adrian for 1980.

As a summary of this section, the accuracy test of the
polynomial model showed the best overall results for cities
in the medimum decrease type, although the accuracy of the
linear model was very close to that of the polynomial
model. But, if the cities satisfy the indispensible
condition of the Gompertz model or the modified exponential
model, one of those models can best be used for this type.

If a planner can roughly estimate future growth rate,
he is advised to use the exponential model when the future
rate changes are greater than 10 percent. When the future
rate changes are less than 10 percent, the modified

exponential model yields the minimum mean error if the

58

cities satisfy the indispensible condition of the model. If
the modified model or the Gompertz model cannot be applied,
the polynomial model seems the best alternative for these

cities.

Great Decrease 12 Growth Rate

 

 

This type of city has past rate changes that ranged
from -50 to -25 percent. For example, the population of
Holland grew at a 56 percent growth rate from 1950 to 1960,
and at a 7 percent growth rate from 1960 to 1970. The growth
rate change was ~49 percent.

Fifteen cities of this type were found in Michigan.
Four models, expecting the Gompertz model and the modified
exponential model, had comparatively high mean percentage
errors for this type. The Gompertz model and the modified
exponential model yielded better accuracy the 9 cities that
satisfied the indispensible condition of the Gompertz model
and 8 cities that met the condition of the modified
exponential model (See Table 7). The mean percentage error
was 11.5 with the modified exponential model and 13.3 with
the Gompertz model.

The polynomial model was the next best method for the
15 cities of this type. It had 18.5 mean percentage error.
The polynomial model have greater frequency in the category
of "lower than 10 percentage error" than the Gompertz model

and the modified exponential model, but the polynomial model

59

produced more extreme error than the two models. The
modified exponential model or the Gompertz model can be used
if the cities meet their indispensible condition.
Otherwise, the polynomial model can best be used for the
cities which can not meet the condition. The other models
showed highly over-estimations for this type of city. The
linear, exponential and double logarithmic models yielded
more than 10 percent over-estimations for all 15 cities.

If future rate changes are considered, there are three
types within the great decrease cities in growth rate. The
first type is characterized by heavy increase in future rate

changes, but there were no cases of that type in Michigan.

Table 7

Frequency of Percentage Errors for
Cities of Great Decrease Type in Michigan

% ERROR LIN EXP DLOG POLY GOM MOD
25 + 8 11 11 4 1 0
10 TO 25 7 4 4 3 5 5
~10 TO 10 0 0 0 6 3 3
~25 TO ~10 O 0 0 1 0 0
< ~25 0 0 O 1 0 0
MEAN ERROR 26.4 40 6 4O 2 18 5 13 3 11 5
SD 8 8 18 9 18.7 14 6 8 8 5 2

60

The second type is the light change type in which future
rate changes are between -10 and 10 percent. There are
seven cities of this type in Michigan. Although that is a
statistically small number of cases, the polynomial model
seemed to be the best method for the type. The mean
percentage error with the model was 11.0 percent error. The
populations of four out of seven cities were projected
within 10 percent of actual population and all cities had
errors less than 25 percent. Only the city of Holland met
the indispensible condition of the Gompertz and modified
exponential models. These two models project the population
of Holland very accurately with about 2 percent error. But
one case is not sufficient to generalize the accuracy of the
two models for the type. Three other models: the linear
model, the exponential model and the double logarithmic
model, projected all cities with over—estimations of more
than 10 percent.

The third type is the heavy decrease type, in which the
future rate changes are less than —10 percent. There are
eight cities of this type in Michigan and all eight cities
satisfied the indispensible condition of the Gompertz model,
and seven cities satisfied the modified exponential model.
The modified exponential model yielded the minimum mean
error for the seven cities. The mean percentage with the
model was 12.8. The Gompertz model model was the next best

model for the type and had 14.7 mean percentage error.

61

However, only two cities were projected with less than 10
percent error. The other four models had high mean
percentage errors. The polynomial model was slightly better
than other three models. The exponential model and the
double logarithmic model projected populations for all eight
cities with higher than 25 percent error.

A graphical example can explain how each of models
projects population for a city of the great decrease type.
In the case of the Holland, rapid growth in population
between 1950 and 1960 was reflected in equations.
Therefore, the four models, excepting the Gompertz model and
the modified exponential model, drew projection line or
curves extending the rapid growth into the future (See
Figure 10). The Gompertz model and the modified exponential
model are applicable to the city and drew projection curves
to show rapid growth in population until 1960, with little
growth in population after 1960. The actual net population
of Holland was decreased by very few, about 200, persons.
Therefore, the Gompertz model and the modified exponential
model projected the 1980 population relatively well. The
other four models highly over—estimated the population. If
the population of 1980 had been increased at a rapid growth
rate, one of the four models might have projected the
population more accurately than the Gompertz model and the
modified exponential model.

To summarize this section, the modified exponential

62

Population

33,000

30,000

27,000 4

A

24,000

21,000 '

v

18,000 ‘

15,000 «

12,000 “

9,000 -

6,000 .

V

3,000 “

 

 

‘ ‘ A A A
‘v w v v v
- ‘

1940 '50 '60 '70 '80 '90 2000 Year

 

 

Linear Regression Model
------------- Exponential 5. Double Logarithmic Model
————— Second.Degree Polynomia1.Model

/— Gompertz Model
.. Modified Exponential Nbdel

X Actual Population

FIGURE 10 : Example of Great Decrease Type
( City of HOlland )

63

model is the best method for the great decrease type, if the
cities meet the indispensible condition of this model. If a
city cannot do so, but is appropriate for the Gompertz
model, the Gompertz model can be applied. If the city is
not appropriate for either of these models, the polynomial
model can be used.

When a planner can roughtly predict a future growth
rate for a city, he can use different models for different
types according to future rate change. The polynomial model
turned out to be the best model for the cities which had
future rate changes from -10 to 10 percent. There were not
enough cases of cities which satisfied the indispensible
conditions of the Gompertz and modified exponential models.
those models may have better accuracy for cities that meet
their indispensible conditions. When the future rate
changes are less than ~10 percent, the Gompertz model is the
best method among the six models. For cities which cannot
meet the indispensible condition of the Gompertz model, the
polynomial model can be used. A planner should also keep in
mind that any of these models has the potential to produce

high projection errors for this type of city.

Extreme Decrease ip Growth Rate

 

 

Cities classified as this type are those which have
past rate changes less than —50 percent. For instance, the

population of Midland, in Midland county, increased at a 94

64

percentage growth rate from 1950 to 1960, and at a 27
percent growth rate from 1960 to 1970. The growth rate
change from the 19503 to the 19603 was thus -67 percent.
Therefore, Midland is a city of the extreme decrease type.
There are 28 cities for this type in Michigan. Of these
28 cities, 25 satisfied the indispensible condition of the
Gompertz model and 24 satisfied the modified exponential
model. The Gompertz model yielded the best accuracy for
cities that satisfied its indispensible condition. The mean
percentage error was 14.4 for 25 cities with the Gompertz
model. The next best model is the modified exponential
model if cities satisfy its indispensible condition. The

mean percentage error with the modified exponential model

Table 8

Frequency of Percentage Errors for
Cities of Extreme Decrease Type in Michigan

% ERROR LIN EXP DLOG POLY GOM MOD
25 + 17 27 27 21 1 1
10 TO 25 6 1 1 5 13 15
~10 TO 10 5 0 0 2 8 6
~25 TO ~10 0 O 0 0 3 2
> ~25 0 0 0 0 0 O

MEAN ERROR 35 3 127 6 125 5 39 8 14 4 15 5
SD 19 5 162 7 158 7 22 9 7 5 8 9

65

was 15.5 for 24 cities. The other four models had extremely
high mean percentage errors. The mean percentage error was
35.3 percent error with the linear model and 39.8 percent
with the polynomial model. The exponential and double
logarithmic models had higher than 100 mean percentage
error. All six models tended to over—estimate the
populations for this type. Although the Gompertz model was
the best method for this type, only 8 out of 28 cities were
projected within 10 percent error (See Table 8).

The linear model was slightly better than the other
four models, but its mean percentage error was 35.3
percent. Only 5 of 28 cities were projected within 10
percent of their actual population. The exponential model
and the double logarithmic model had extremely high error
rate (i.e., 904 percent error with the exponential model).

Therefore, the Gompertz model can best be applied to
those cities that satisfy the indispensible condition of the
model. For the others, the linear model may be used with
the realization that, in this type of city, there is the
potintial for severe over-estimation with the linear model.

If the growth rate can be roughly estimated, accuracy
might be improved through using different models to project
different types of future rate changes. The first type of
future rate change is a heavy increase with future rate
change more than 10 percent. There is no case of this type

in Michigan. The second type is a light change which affects

66

the group of cities with future rate changes from —10 to 10
percent. There are five such cases in Michigan. The
Gompertz model and the modified exponential model yielded
relatively good projections for the four of the five cities
that satisfied their conditions. The modified exponential
model yielded 9.6 mean percentage error and the Gompertz
model had 10.8 mean percentage error.

The polynomial model is the next best model for this
type. The mean percentage error was 14.2 percent and two
out of five cities were projected with less than 10 percent
error. Other three models tended to highly over-estimate
the population and while the linear model was slightly
better than the others, it still had a high (18.0) mean
percentage error.

The third type is a heavy decrease type in which future
rate changes are less than -10 percent. There are 23 cases
of this type in Michigan. All six models showed high mean
percentage errors, and tended to over—estimate populations
of this type. The linear model and the polynomial model had
39.1 and 45.4 mean percentage errors, respectively, and the
exponential model and the double logarithmic model had
extremely high errors (145.3 and 143.0 mean percentage
error).

In the 23 cities, 21 satisfied the indispensible
condition of the Gompertz model and 20 met the condition of

the modified exponential model. With those specific cities,

Population

60,000(
55,000”
50,000“

45,0000

40,000“

35,000“

30,000L

25,000

I

20,000”
15,000'

10,000 .

5,000?

 

1940 '50 '60 '70 '80 '90 2000

 

_—~—_

FIGURE 11 z

67

L A— ; A L

___A

v v—
V V —v—

Year

Linear Regression Model

Exponential & Double Logarithmic Model
Second Degree Polynomial Model
Gompertz Model

Modified Exponential Model

Actual Population

Example of the Extreme Decrease Type

( City of Midland )

68

the Gompertz model had 15.0 mean percentage error and the
modified exponential model had 16.7 mean percentage error.

Midland, in Midland county, is illustrated as an
example of the extreme increase type of past rate change in
Figure 11. The population of Midland increased at a rapidly
increasing growth rate in the 19503, but the growth rate
rapidly declined through the 19603 and 19703 although the
net population continuously increased. The growth pattern
of Midland is consistent with the assumptions of the
Gompertz model. As shown in Figure 11, the Gompertz model
projects Midland's population very accurately. The modified
exponential model yielded more over-estimation than the
Gompertz model.

The exponential model and the double logarithmic model
drew the projection curves with the tendency of the high
growth rates in 19503 and 19603 and thus greatly
overestimated the population. The linear model drew a
projection line with an excessively larg increment per unit
of time, because it also reflected the fast growth in an
earilier time. The polynomial model also showed a rapid
growth projection curve. If the population of the Midland
had increased from 1970 to 1980 at the same growth rate as
it had the previous decade, the linear model might have
produced minimum error for the 1980 population of the city.

To summarize this section, none of the six models

yielded good accuracy for this type. The Gompertz model was

69

the best method for this type and the modified exponential
model was next. If cities cannot meet the indispensible
condition of these two models, the linear model may be used,
with caution, because of the potential for high error with
the model for this type.

With specified types of future rate changes, the
accuracy was slightly improved for cases of light change in
which the growth rate change was between ~10 and 10 percent
from the 19503 to the 19603. The modified exponential model
was the best method for this type and the Gompertz was
next. If a city cannot meet the indispensible condition of
those two models, the polynomial model can be used. For
cities in which future rate changes are less than -10
percent, all six models showed high mean percentage errors.
The Gompertz model yielded the best accuracy for this type
among the six models and the modified exponential model was
next. When a city cannot meet the indispensible conditions

of those models, the linear model may used with great care.

Cities Classified according :9 1970 Population

 

 

Number of population at a given census data is another
characteristic that can be used to classify cities. Four
population categories, according to 1970 population figures,
were used to test the accuracy of the six models (See Table
9). The linear model yielded relatively good results for

cities in all four population categories. For cities

70

between 10,000 and 50,000 population, however, the
polynomial model showed slightly better results. Among the
four models tested (excepting the Gompertz and modified
exponential models), the linear model was dominant in
accuracy.

The modified exponential model is more accurate for
smaller cities than the Gompertz model, if the cities
satisfy its indispensible condition. The Gompertz model
produced the lowest error for larger cities, if the cities
are appropriate for this model.

All six models showed similar projection accuracy

tendencies for each population size category. Each yielded

Table 9

Mean Percentage Error of Each Model
for City Types Based on 1970 Population

———~————-—————_———————_—----fl——-—-—-—~----———~————-——--———--

Model | ————————————————————————————————————————————————————
| 50 + 10 to 50 5 to 10 2 5 to 5
LIN | 32 6 (14) 23.3 (54) 9.3 (41) 11.8 (63)
EXP | 122.3 (14) 50.1 (54) 15.0 (41) 15.2 (63)
DLOG | 120 1 (14) 49.6 (54) 14.8 (41) 15.1 (63)
POLY | 35.4 (14) 21.3 (54) 12.9 (41) 15.1 (63)
GOM | 17.9 (7) 11.5 (24) 7.4 (19) 9.9 (31)
MOD | 20.1 (6) 12.0 (24) 6.6 (17) 8.7 (29)

(Note) number in parentheses indeicates number of cities
of that type in Michigan.

71

its best result for the cities between 5,000 and 10,000
population, and next for the cities between 2,500 and 5,000
population. Each of the six models yielded its highest mean
percentage error for cities over 50,000 population.

The results of this study are not consistent with those
obtained by Isserman (1977) who found relatively high
projection errors for small townships and relatively small
errors for the large townships (Isserman, 1977 : 256). In
contrast, the results of this study showed that the
populations of smaller cities were projected with greater
accuracy than those of larger cities. This difference,
however, was likely due, in part, to changes in population
trends in large and small cities.

However, past population size was not used as a
criterion to classify the cities in this study for several
reasons. First, population size does little reflect the
peculiar nature of each model. The extrapolation models
originally extrapolated cities' historical population growth
trends into the future, whatever the size of their
populations. If there is a correlation between a city's
population size and its population growth pattern, it may be
possible to judge the accuracy of an extrapolation model by
the population size in some cases. but I believe its past
population growth pattern is a better criterion than
population size, because the origin of extrapolation models

is based in the past population trends.

72

The second reason, for not using population size as a
classifier, was to avoid complexity in classification. If a
combination of seven types of growth rate change and four
categories of population size were used, the combination
would result in 28 types of cities. Considering future
growth rate change, 84 city types would be produced from all
possible combinations of each characteristic. When criteria
become so complex, the point of clear and simple
classification is lost.

The third reason is apparent the results of the
accuracy test for cities of varied population sizes. The
linear model dominates for cities of almost all population
sizes. The Gompertz model or the modified exponential model
will be used for all cities that satisfy the models'
indispensible condition. There is no point to
classification of city types if a model that is applied to
all fails to discriminate between the categories.
Therefore, I think the growth rate change is a better basis
for classifying cities than population size to apply

extrapolation model according to certain city types.

§EEE§EY

Overall, the linear model was, generally, the most
accurate method for projecting the 1980 populations of
cities in Michigan, but it is preferable that specific

extrapolation models be applied to those types of cities for

73

which they aremost accurate, rather than relying on one
method. This is especially true when the Gompertz and
modified exponential models are appropriate for those cities
that satisfy their indispensible conditions, because these
two models yield excellent quality population projections
for those cities.

To classify cities, growth rate change was used. The
differential pattern of errors for types of growth rate
change made it a more useful classification criterion than
population size. Since growth rate change during 19603 was
found to be related to population size in 1970, it might
also reflect this characteristic some extent.

Past and future growth rate changes were considered in
classifying cities by type for this study. The results of
the accuracy tests for growth rate change are presented in
Table 10. Given past growth rate changes, the linear model
is generally well suited to the increasing or moderate type
of change in growth rate.

The polynomial model is relatively accurate for the
rapidly decreasing type of growth rate, but the Gompertz or
the modified exponential model are more accurate for cities
of this type if the cities meet the indispensible conditions
of one of the two models.

For cities with a dissimilar combination of past and
future growth rate changes, the exponential model and the

double logarithmic model seem to be appropriate to cities

74

with rapidly increasing future growth rates, but where the
growth rate decreased more than 25 percent in past. The
exponential model tends to work well with rapidly increasing
growth rates. These two models generally produce very

similar population projection in most cases.

Table 10

Population Projection Model
for Cities Based on Growth Rate Change

Past Growth | | Future Growth Rate Change
Rate Change | All | ------------------------------
| | 10 + —10 to 10 < -10
25 + ( Lin l — - Lin
l l
10 to 25 | Lin | Exp Lin Lin
l I
0 to 10 | Lin | Exp Exp Lin
l l
—10 to O | Lin | Dlog Lin Poly
|* Mod | * Gom * Mod
l l
-25 to —10 | Poly | Dlog Poly Poly
|* Mod | * Mod * Mod
l l
-50 to -25 | Poly | — Poly Poly
|* Com | * Gom * Mod
I I
< —50 | Lin | - Poly L1n
|* Gom | * Mod * Gom

(Note) * indicates that the Gompertz model or the modified
exponential model is the best method for that type
if the cities satisfy the indispensible conditions
of one of them.

75

The linear model yields the accurate projections for
city populations that have shown rapid growth rate increases
in the past and rapid growth rate decrease for the future or
those that have shown moderate growth rate change in the
past and for the future. The polynomial model seems to be
the best model for cities with growth rate decreases in the
past and moderate or decreasing growth rate change in the
future. However, if cities satisfy the indispensible
condition of the Gompertz or the modified exponential model,

those models are the best methods for those types of cities.

CHAPTER 4

DEVELOPMENT OF COMPUTER PROGRAM

Criteria Basis :9 Select Best Model

 

 

A composite method which applies

extrapolation models to different types of

different

cities has

resulted from the accuracy test used with Michigan cases.

TABLE 11

Composite Method

| Composite II

|
Past Growth | Composite | Future Growth Rate Change
Rate Changes| | ————————————————————————————
I I I 10 + -10 to 10 < -10
25 + | Lin | Exp Dlog Lin
I l
10 to 25 | Lin | Exp L1n Lin
1 l
0 to 10 | Lin | Exp Exp Lin
l l
-10 to 0 | Mod | Dlog Gom Mod
| * Lin | * Lin Poly
I I
—25 to -10 | Mod | Exp Mod Mod
| * Poly l * Poly Poly
l |
-50 to ~25 | Gom | L1n Gom Mod
| * Poly | * Poly Poly
| l
< -50 | Gom I L1n Mod Gom
I * Lin l * Poly Lin
(Note) * indicates that, if a city does not satisfy

the indispensible condition of the Gompertz

model or the modified exponential model,

the

designated model will be used for that type.

76

77

In order to develop a composite method, cities were
classified according to their growth rate changes in the
same manner used for accuracy testing in the previous
chapter.

Following this, a two—part composite method was
developed. Composite I uses the most accurate method
employing past growth rate changes only. This can be
applied to cases in which a planner cannot anticipate a
future growth rate. Composite II uses the most accurate
method employing the appropriate combination of past and
future growth rate changes.

Seven types of past growth rate change and three types
of future growth rate change were used, resulting in a total
of twenty—one types for the composite II. The composite
method selects a model exactly as shown in Table 11. The
composite method was derived from the results of this study
as presented in Table 10. For some of the types, there were
few or no cases. To develop the composite method for those
types, the tendencies and basic assumptions of the six

models were comsidered.

Micro-computer Program with BASIC Language

 

 

The computer program provides alternatives. Users can
chose the model they prefer to use or can allow the
selection to be made by the program. If a user decide to

use a composite method and cannot anticipate future growth

78

rate, a model will be selected through the Composite I. If a
user can make a rough estimate of a future growth rate,
Composite II will select an appropriate model for the city.

If a user runs the population projection program, it
will ask the user to enter data (i.e., number of cities,
number of data points, time of the data points, city names
and population data). Once the user enters the data, the
program will ask for a projection year, and then display the
alternatives (seven methods and the composite method).

If a user chooses one of the seven models, the program
will immediately project the population to the desired time
for those cities. If a user chooses the composite method,
the program asks whether or not a rough future growth rate
can be given.

When a user chooses the composite method, the program
will decide to use whether Composite I or Composite II,
according to the user's answer. Then, the program will
calculate the growth rate over the past two time spans and
the growth rate change for each city. If a user answers
that he cannot anticipate the future growth rate, the
program will select a model with a past growth rate change
according to the Composite I, as shown in table 11. Then,
the program will immediately project the population of the
cities for the desired projection time with the model
selected. If a user answers that he can give rough future

growth rate, the program will call up future growth rates,

79

such as ~10, 0, 10, 15, etc. Once a user enters a rough
growth rate, the program will calculate growth rate change
between the growth rate in last time span and the rough
growth rate entered.

Then the program will select a model with the proper
combination of past growth rate changes and future growth
rate changes for each city according to Composite II. And
the program will immediately project the populations of the
cities for the projection time with the selected model.
After finishing the projections, the program will ask
whether or not a user wants to project the population of the
cities again, with another model.

There are other facilities in the program. A user can
save his data base in his disk through the program and he
can use the data base again and again later for projecting
the populations of the cities in the data set. He can also
save the results of projections into his disk.

The program will reject requests to project population
if a user does not enter the same time intervals as the
times of past population observation points. The program
may also recommend population projections for the
short—term, so that it automatically suggests the year of
the next span from the last population observation time as
the projection year. If a user still wants to project
populations over a longer term than one more year span, the

program will project the population for the long—term using

80

the projection year that a user types in.

In choosing the Gompertz model or the modified
exponential model, the program will test the appropriateness
of the city to that model. If a city cannot meet the
indispensible condition of the modified exponential model,
it will test for appropriateness with the Gompertz model.
In some cases, it will meet the indispensible condition of
the Gompertz model even though it could not satisfy the
indispensible condition of the modified exponential model.
If the city is not appropriate for either of these models,
the program will select the next best model according to the

composite method.

CHAPTER 5
CONCLUSION

This study was done in an effort to meet the needs for
the information about the accuracy of extrapolation models
which are the popular techniques used to project population
in local planning offices. Any given population
extrapolation model may be inappropriate or inapplicable to
the cities for which a planner seeks information, because
each of models projects populations with different
assumptions and there are various population growth patterns
in cities. Therefore, specific models should be used for
different type of cities.

To develop the composite method, the accuracies of six
extrapolation models were tested for various types of cities
with varied growth rate changes. The composite method was
based on the results of this accuracy testing and employs
the most accurate among the six models for particular types
of cities, according to the growth rate changes.

The exponential model and the double logarithmic model
are generally the best method for cities of which the growth
rate increased or changed moderately in the past and the
growth rate in future is expected to rapidly increase.

The linear model is generally the best technique for
the cities of which growth rate have changed moderately or
increased in the past, or will moderately change or decrease
with an earlier pattern of increasing growth rate.

81

82

The Gompertz model, the modified exponential model and
the second degree polynomial model are generally the best
methods for the type of cities which have had decreasing
growth rates in past, or which will moderately change or
decrease in growth rates, following a past growth pattern of
decreasing growth rate. But the Gompertz model and the
modified exponential model require certain conditions. The
parameter "b" in the equations should be positive and less
than one and the parameter "a" in the equations should be
negative. Once cities satisfy the condition, the two models
provide good quality projection for those cities.

It is remarkable that the Gompertz model and the
modified exponential model played an important role in
projecting 1980 populations of cities in Michigan. Isserman
(1977) found that one of three methods (the linear model,
the exponential model and the double logarithmic model) was
the most accurate method for certain types of townships
classified by growth rate. His findings can still be useful
except in the case of the double logarithmic model for
cities losing more than 25 percent of their population. For
instance, he recommends using the exponential model for
townships increasing more than 25 percent, or decreasing
less than 25 percent in populaton during the last decade.
This is also recommended by this study, but this study added
another condition. The added condition is that cities are

expected to increase more than 10 percent in growth rate,

83

along with the previous conditions, in order to use the
exponential model.

Because of the changes in population trends during the
19703 in Michigan, the recommendations of Isserman (1977)
are not sufficient for cities which more recently show
decreasing growth rate. This study found models for those
types of cities. It is recommended that the Gompertz model
or the modified exponential model be used if cities satisfy
the indispensible condition of those two models. It was
also found that the polynomial model yields good projections
for that type of cities, if cities are not appropriate for
the Gompertz model or the modified exponential model.

However, a planner has to use extrapolation models with
some care. All six models yielded relatively high errors
for the cities which had rapid growth rate changes. For
example, the great increase type or the extreme decrease
type. All of extrapolation models have especially extreme
errors with cities which have had sudden reversals of
population trends for the future. For instance, when the
population of a city shows highly increasing population
through an earlier time but suddenly decreases for the
future, all of models will have high errors. Therefore, in
such a case, the local planner should use a population
extrapolation model with caution. Although the
extrapolation models yield high errors in some case, they

are still useful when data, time and funds are limited.

APPENDICES

 

 

APPENDIX 1

RESULTS OF ACCURACY TEST FOR EACH TYPE

BASED ON PAST GROWHT RATE CHANGE

APPENDIX

1 :

84

RESULTS OF ACCURACY TEST FOR EACH TYPE
BASED ON PAST GROWTH.RATE CHANGE

 

 

 

 

 

 

 

 

Past Growth.Rate Change Lin. Exp Dlog Poly' Gom Mbd
mean % error 19.7 26.0 25.8 33.2 — —

25 + 3d. 14.2 21.2 21.0 14.8 - -
# of cases 7 7 7 7 O 0

mean % error 11.2 14.9 14.7 20.0 — —

10 to 25 3d. 10.8 10.1 10.1 9.8 — —
# of cases 9 9 9 9 O 0

mean % error 10.1 15.2 15.1 14.2 - —

O to 10 3d. 9.4 18.2 18.0 13.7 — -
# of cases 27 27 27 27 0 0

mean.% error 9.1 10.1 10.0 9.1 7.5 7.0

—10 to 0 3d. 6.5 7.5 7.4 7.5 6.7 5.9
# of cases 54 54 54 54 31 30

mean % error 14.3 16.8 16.7 14.0 9.0 7.8

-25 to —10 3d. 9.5 11.1 11.0 10.5 8.0 6.1
# of cases 32 32 32 32 16 14

mean % error 26.4 40.6 40.2 18.5 13.3 11.5

~50 to —25 3d. 8.8 18.9 18.7 14.6 8.8 5.2
# of cases 15 15 15 15 9 8

mean % error 35.3 127.6 125.5 39.8 14.4 15.5

< —50 3d. 19.5 162.7 158.7 22.9 7.5 8.9

# of case 28 28 28 28 25 24

 

APPENDIX 2

RESULTS OF ACCURACY TEST FOR EACH TYPE BASED ON

THE COMBINATION OF PAST GROWTH RATE AND FUTURE GROWTH RATE

 

 

85

APPENDIX 2 : RESULTS OF ACCURACY TEST FOR EACH TYPE BASED ON
THE COMBINATION OF PAST GROWTH RATE CHANGE AND
FUTURE GROWTH RATE CHANGE

GREAT INCREASE TYPE

 

 

 

 

 

 

 

 

 

Future Growth Rate Change Lin Exp Dlog Poly
mean 96 error — - - ~
10 + 3d. - — — -
# of cases 0 O O 0
mean 96 error - — — —
—10 to 10 3d. - - — -
# of cases 0 O 0 0
mean 96 error 19.7 26.0 25.8 33.2
< -10 3d. 14.2 21.2 21.0 14.8
# of cases 7 7 7 7
MEDIUM INCREASE TYPE
Future Growth Rate Change Lin Exp Dlog Poly
mean 96 error 38.3 29.4 29.6 26.6
10 + sd. - - — —
# of cases 1 1 1 1
mean 96 error 7.9 7.9 7.9 9.8
-10 to 10 3d. 5.1 5.1 5.1 2.1
# of cases 2 2 2 2
mean % error 7.7 14.7 14.5 22.3
< —10 3d. 4.1 9.8 9.7 9 8
# of cases 6 6 6 6

 

86

APPENDIX 2 Continued

MODERATE INCREASE TYPE

 

 

 

 

 

 

 

 

 

 

Future Growth.Rate Change Lin. Exp Dlog Poly
mean X error 11.3 10.0 10.1 20.8
10 + sd. 2.2 1.3 1.3 7.5
# of cases 2 2 2 2
mean.X error 4.9 4.3 4.3 5.2
—10 to 10 sd. 3.1 3.2 3.1 5.4
# of cases 14 14 14 14
mean X error 16.5 30.1 29.8 24.5
< -10 sd. 11.5 20.9 20.6 14.2
# of cases 11 11 11 11
MODERATE DECREASE TYPE
Future Growth Rate Change Lin. Exp Dlog Poly Gom Med
mean X error 12.4 11.2 11.2 15.9 21.5 21.4
10 + sd. 5.0 4.8 4.9 7.1 - —
# of cases 3 3 3 3 1 1
mean X error 6.9 7.4 7.3 7.2 4.6 4.6
~10 to 10 sd. 4.4 4.9 4.9 6.1 2.4 2.4
# of cases 41 41 41 41 2 2
mean X error 17.3 20.9 20.8 14.8 13.8 12.4
< —10 sd. 7.7 7.3 7.3 9.0 8.6 7.8
# of cases 10 10 1O 10 8 7

 

APPENDIX 2

Continued

MEDIUM DECREASE TYPE

 

 

 

 

 

 

 

 

 

Future Growth Rate Change Lin Ebcp Dlog Poly Gom Mod
mean X error 5.9 5.3 5.3 30.4 14.5 14.5
10 + sd. 1.9 1.3 1.4 14.4 — -
# of cases 3 3 3 3 1 1
mean X error 11.5 13.2 13.1 9.5 4.6 4.3
—10 to 10 sd. 7.8 8.5 8.4 6.3 3.9 3.9
# of cases 20 20 20 20 9 8
mean X error 23.3 28.5 28.4 18.4 14.7 12.0
< —10 sd. 8.5 8.4 8.3 10.4 9.3 5.9
# of cases 9 9 9 9 6 5
GREAT DECREASE TYPE
Future Growth.Rate Change Lin Exp Dlog Poly Gom Mod
mean X error — — — - — -
10 + sd. — - - — - -
# of cases 0 0 O 0 0 0
mean X error 23.7 31.0 30.8 11.0 1.8 2.0
-10 to 10 sd. 10.2 19.8 19.6 9.1 — -
# of cases 7 7 7 7 1 1
mean X error 28.8 49.0 48.5 25.1 14.7 12.8
< ~10 sd. 7.3 14.4 14.2 15.9 8.2 3.7
# of cases 8 8 8 8 8 7

 

APPENDIX 2 : Continued

88

EXTREME DECREASE TYPE

 

 

 

 

Future Growth Rate Change Lin. Exp Dlog Poly Gom Mbd
mean X error ~ - - -

10 + sd. — — — — - -
# of cases 0 0 0 0 O 0

mean X error 18.0 45.8 45.1 14.2 0.8 9.6

-10 to 10 sd. 15.1 30.3 30.1 6.6 8.0 6.6
# of cases 5 5 5 5 4 4

mean X error 39.1 145.3 143.0 45.4 5.0 16.7

< -10 sd. 18.5 174.6 170.2 21.3 7.5 9.0

# of cases 23 23 23 23 1 0

 

 

APPENDIX 3

COMPUTER PROGRAM FLOWCHART

APPENDIX 3 :

89

COMPUTER PROGRAM FLOWCHART

 

Z input data (city name, year 8: population) 7

 

 

   
 

4}
@
no

yes‘

 

 

 

 

calculate the growth rate
& rate changes in past

 

 

     

 

vailability of future

 

 

 

 

 

 

 

growth rate no
yes.
input data select optimum
(future growth rate) model by
Composite I

 

 

 

 

 

 

calculate the growth rate
6': rate change in future

 

 

 

11
r select a optimum model by Composite II ]

 

I - selected mode
no Gom. or Mod. model

   

 

 

 

 

 

 

 

 

 

 

 

 

yes
1 yes Of two model
no
r reselecting optimum model j
fir ijectireLponnation jg————J
1: writing tHe results j
yes

 

try another model

no

APPENDIX 4

COMPUTER PROGRAM WITH BASIC

 

 

90

APPENDIX 4 : COMPUTER PROGRAM WITH BASIC

10 REM POPMODEL

2O REM PROJECTING POPULATION WITH 'I'HE POPULATION TREND
3O REM EXTRAPOLATION ADDELS

4O REM PROGRAIVMED BY IKKI KIM. AUGUST 5, 1985

50 REM VARIABLE (ALPHABETICALLY)

 

 

 

60 REM A() ~ MATRIX OF PARAMETERS FOR POLYNQdIAL EQUATION
7O REM A$.A2$.A3$ - USER RESPONSES

80 RE“! A,B,NB - COEFFICIENT IN EQUATION

9O REM AVl ,AV2 - AVERAGE

100 REM C() ~ CLASSIFYING USER'S CHOICE

110 REM C$() - CITY NAME

120 REM CL() ~ CLASSIFICATION OF CITY TYPE

130 REM COV ~ COVARIANCE

140 REM D ~ DEGREE OF POLYNOMIAL EQUATION

150 REM DIF(),DIF2() - DIFFERENCE OF GRWI‘H RATE

160 RBI! E - EXPONEINT IN 'I'HE GOMPERTZ & 'I'HE MDDIFIED MODEL
170 REM F1$,F2$,F3$ - FILE NAME

180 REM G ~ NIMBER OF POINTS IN GROUP- PERIOD FOR USING
190 REM THE SELECTED POINTS TECHINIQUE

200 REM H,I,J,L - FOR/NEXT VARIABLES

210 REM K ~ UPPER LIMIT POPULATION

220 REM M - NUMBER OF CITIES

230 REM N - NUMBER OF DATA POINTS

240 REM P ~ PROJECTION YEAR

250 REM P$,P2$ ~ USER RESPONSES RELATED TO PROJENCTION YEAR
260 REM Q,R,T - INITIALIZING NUMBER

270 REM Ql$.Q2$:Q3$ ~ USER RESPONSES

280 REM R1(),YR1(),LP ~ TRANSFORMED POPULATION OR YEAR
290 REM S ~ PERIOD SPAN

300 REM VAR - VARIANCE

310 REM X( ) ,Y() ~ MATRIX FOR SOLVING POLYNCMIAL EQUATION
320 REM *******************************************************1”:
330 REM [] [1

340 REM [] ENTERING DATA []

350 REM [] []

360 PRINT "NUMBER OF CITIES AND TONNS FOR PROJECTION":

370 INPUT M

380 PRINT

390 PRINT "NUMBER OF DATA POINTS";

400 IINPU'I' N

410 PRINT

420 PRINT "DO YOU HAVE A DATA-FILE (Y/N)";

430 INPUT A3

440 PRINT

450 IF A$="N" OR A$="n" THEN 510
460 PRINT "NAME OF INPUT DATA-FILE";
470 INPU'I' F13

 

91

APPENDIX 4 : Continued

480
490
500
510
520
530
540
550
560
570
580
590
600
610
620
630
640
650
660
670
680
690
700
710
720
730
740
750
760
770
780
790
800
810
820
830
840
850
860
870
880
890
900
910
920
930
940

PRINT
OPEN "I", #1, F13
INPUT #1, M,N
DIM YR(N) :P0P(M.N) .C$(M) .CL(M) .P1(N) :YR1(N) .C(M)
DIM X(6,6) ,Y(6) ,A(6) ,DIF(M) ,DIF2(M)
IF A$="Y" OR A$="Y" THEN 1130
REM *******#*******akakakakak****#**************************1H:
REM ENTERING DATA DRING RUNNING PROGRAM
PRINT "DO YOU WANT TO CREAT A DATA-FILE (Y/N)";
INPUT A23
PRINT
IF A2$="N" OR A2$="n" THEN 640
PRINT "NAME FOR THE DATA-FILE";
INPUT F2$ : PRINT
OPEN "O",#2,F2$
PRINT #2, M,N
FOR I=1 TO M
PRINT "NAME OF CITY OR TONN : NO.";I;
INPUT C$(I)
IF A2$="N" OR A2$="n" THEN 690
PRINT #2: CHR$(34) 30$“) :CHR$(34):
NEXT I
PRINT : PRINT
FOR J=1 TO N
PRINT "TIME (YEAR) FOR DATA POINT #";3:
INPUT YR(J)
IF A2$="N" OR A2$="n" THEN 760
PRINT #2, YR(J);
NEXT J
PRINT : PRINT
FOR I=1 TO M
PRINT II II
PRINT "ENTER POPULATION DATA OF ";C$(L)
PRINT II II
FOR J=1 TO N
PRINT "POPULATION OF ";YR(J);
INPUT POP(I,J)
NEXT J
PRINT II II
PRINTzPRINT
NEXT I
IF M<3 TIEN 1040
PRINT "DO YOU WISH TO REVIEW TIE POPULATION DATA (Y/N)";
INPUT AS
PRINT
IF A$="N" OR A$="n" TIEN 1040
FOR I=1 TO M

 

 

 

 

APPENDIX 4

950
960

970 PRINT "FOR ":YR(J);" ---

980

990

1000
1010
1020
1030
1040
1050
1060
1070
1080
1090
1100
1110
1120
1130
1140
1150
1160
1170
1180
1190
1200
1210
1220
1230
1240
1250
1260
1270
1280
1290
1300
1310
1320
1330
1340
1350
1360
1370
1380
1390
1400
1410

92

: Continued

PRINT " CITY NAME :
FOR J=1 TO N

";C$(I)
";" POPULATION =";POP(I,J);" OK (Y/N)";
INPUT AS : PRINT
IF A$="Y" 0R.A$="Y" THEN 1020
PRINT "CORRECTED POPULATION : ";
INPUT POP(I,J)
NEXT J
NEXT I
IF A2$="N" 0R.A2$="n" THEN 1260
FOR I=1 TO M
FOR J=1 TO N
PRINT #2, POP(I,J);
NEXT J
NEXT I
GOTO 1260
REM *************************************************************
REM LOADING DATA FROM DATA-FILE
FOR I=1 TO M
INPUT #1, C$(I)
NEXT I
FOR J=1 TO N
INPUT #1, YR(J)
NEXT J
FOR I=1 TO M
FOR J=1 TO N
INPUT #1, POP(I,J)
NEXT J
NEXT I
REM *****x*******************************************************
REM JUSTIFICATION 0F PROJECTING YEAR
3=YR(2)4YR(1)
PfiYR(N)+S
PRINT "PROJECTION YEAR IS ":P;"
INPUT P$ : PRINT
IF P$="Y" 0R.P$="y" THEN 1390
PRINT "PROJECTING POPULATION FOR.THE SHORT-TERM.IS"
PRINT "RECOMMANDED SUCH AS FOR NEXT SPAN POINT(YEAR)."
PRINT:PRINT:
PRINT "D0 YOU‘WANT TO PROJECT POP. FOR ":P;" (Y/N)";
INPUT P2$ : PRINT
IF P2$="Y" OR.P2$="y“ THEN 1390
PRINT "TIME (YEAR) FOR PROJECTING POPULATION";
INPUT P : PRINT
FOR J=2 TO N-l
IF 8=YR(J+1)4YR(J) THEN 1420
PRINT "YOU NEED THE CONTINUOUS EQUAL PERIOD SPAN":STOP

(Y/N)";

 

93

 

 

APPENDIX 4 : Continued

1420 NEXT J

1430 REM *******************#*******************#***************
1440 REM [1 [1

1450 REM [J SELECTING A MOEDL [1

1460 REM [J [1

1470 PRINT "DO YOU WANT TO SAVE THE RESULTS (Y/N)";

1480 INPUT Q$ : PRINT

1490 IF Q$="N" OR Q ="n" THEN 1530

1500
1510
1520
1530
1540
1550
1560
1570
1580
1590
1600
1610
1620
1630
1640
1650
1660
1670
1680
1690
1700
1710
1720
1730
1740
1750
1760
1770
1780
1790
1800
1810
1820
1830
1840
1850
1860
1870
1880

PRINT "FILE NAME FOR SAVING THE RESULTS":
INPUT F3$ : PRINT
OPEN "O",#3,F3$
FOR I=1 TO M
IF I=1 THEN 1570
IF A3$="N“ OR.A3$="n” THEN 1570

 

 

CL(I)=CL(I*1) : GOTO 1740

PRINT II II
PRINT " WHICH MODEL DO YOU'WENT TO USE FOR PROJECTION ?"
PRINT : PRINT " 1. LINEAR.REGRESSION MODEL"

PRINT " 2. EXPONENTIAL REGRESSION MOD "

PRINT " 3. DOUBLE LOGARITHMEC REGRESSION MOD "
PRINT " 4. GOMPERTZ CURVE"

PRINT " 5. MODIFIED EXPONENTIAL CURVE"

PRINT " 6. SECOND DEGREE POLYNOMIAL CURVE"

PRINT " 7. THIRD, FORTH OR.FIFTH DEGREE POLYNOMIAL"
PRINT " 8. COMPOSITE METHOD (TESTED IN '85, MICH.)"
PRINT II N
PRINT

PRINT "ENTER THE NUMBER.OF SPECIFIC METHOD FOR ";C$(I);
INPUT CL(I) : PRINT : PRINT

IF M=1 THEN 1740
PRINT "USE SAME METHOD FOR REST OF OTHER CITIES (Y/N)";
INPUT A33 : PRINT
IF CL(I)=8 THEN C(I)=1
IF CL(I)<8 THEN C(I)=2
NEXT I
REM ************************************************************
REM SELECTING A MODEL BY PROGRAM
FOR I=1 TO M
IF CL(I)<8 THEN 2230
R1=(POP(I,N)-POP(I,N-1))/POP(I,N-1)*100
R2=(POP(I,N—1)-POP(I,N-2))/POP(I,N-2)*1OO
DIF(I)=R1-R2
IF I>1 THEN 1880
PRINT "CAN YOU GIVE THE ROUGH 9s GROWTH RATE FOR SOME OR ALL"
PRINT "OF CITIES FROM ";YR(N);" TO ";YR(N)+S;" (Y/N)";
INPUT 02$ : PRINT
IF 02$="N" OR Q2$="n" THEN 2200

 

94

APPENDIX 4 : Continued

1890 PRINT "ROUGH % GROWTH RATE FOR ";C$(I);" (..-10..0..10..)";
1900 INPUT R3 : PRINT

1910 DIF2(I)=R3—R1

1920 IF DIF(I)>10 AND DIF2(I)>10 THEN CL(I)=2
1930 IF DIF(I)>25 AND DIF2(I)>=—10 AND DIF2(I)<=10 THEN CL(I)=3
1940 IF DIF(I)<=25 AND DIF(I)>10 AND DIF2(I)<=10 THEN CL(I)=1
1950 IF DIF(I)>25 AND DIF2(I)<-10 THEN CL(I)=1
1960 IF DIF(I)>0 AND DIF(I)<=10 THEN 1980

1970 GOTO 2000

1980 IF DIF2(I)>=-10 THEN CL(I)=2

1990 IF DIF2(I)<—10 THEN CL(I)=1 : GOTO 2230

2000 IF DIF(I)>=-10 AND DIF(I)<=0 THEN 2020

2010 GOTO 2050

2020 IF DIF2(I)>10 THEN CL(I)=3

2030 IF DIF2(I)>=—10 AND DIF2(I)<=10 THEN CL(I)=4
2040 IF DIF2(I)<-10 THEN CL(I)=5 : GOTO 2230

2050 IF DIF(I)>=-25 AND DIF(I)<-10 THEN 2070

2060 GOTO 2090

2070 IF DIF2(I)>10 THEN CL(I)=2

2080 IF DIF2(I)<=10 THEN CL(I)=5 : GOTO 2230

2090 IF DIF(I)>=—50 AND DIF(I)<=—25 THEN 2110
2100 GOTO 2140

2110 IF DIF2(I)>10 THEN CL(I)=1

2120 IF DIF2(I)<=10 AND DIF2(I)>=-10 THEN CL(I)=4
2130 IF DIF2(I)<-10 THEN CL(I)=5 : GOTO 2230

2140 IF DIF(I)<—50 THEN 2160

2150 GOTO 2230

2160 IF DIF2(I)>10 THEN CL(I)=1

2170 IF DIF2(I)<=10 AND DIF2(I)>=—10 THEN CL(I)=5
2180 IF DIF2(I)<—10 THEN CL(I)=4

2190 GOTO 2230

2200 IF DIF(I)>=0 THEN CL(I)=1

2210 IF DIF(I)<0 AND DIF(I)>=-50 THEN CL(I)=5
2220 IF DIF(I)<—50 THEN CL(I)=4

2230 NEXT I

2240 Rm *************##3##*Shluk3|!*Jk*ilflk*1:*IIUIUR**************************

2250 REM SEND A TYPE OF CITY TO A SPECIFIC NDDEL CALCULATION
2260 FOR I=1 TO M

 

2270 IF CL(I)<=3 THEN 2430

2280 IF CL(I)=4 0R CL(I)=5 THEN 3090

2290 IF CL(I)=6 0R CL(I)=7 THEN 3880

2300 NEXT I

2310 PRINT " "
2320 PRINT : PRINT

2330 PRINT "PROJECT AGAIN WITH OTHER ALTERNATIVE NDDEL (Y/N)":
2340 INPUT 03$
2350 PRINT

 

APPENDIX 4

2360
2370
2380
2390
2400
2410
2420
2430
2440
2450
2460
2470
2480
2490
2500
2510
2520
2530
2540
2550
2560
2570
2580
2590
2600
2610
2620
2630
2640
2650
2660
2670
2680
2690
2700
2710
2720
2730
2740
2750
2760
2770
2780
2790
2800
2810
2820

95

: Continued

IF Q3$="N“ OR Q3$="nP THEN END
GOTO 1530
REM ********************************$******************#********
RPM [1 [1
RM [1 [1
RPM [1 [1
REM CALCULATING REGRESSION AND PROJECTING POPULATION
FOR J=1 TO N
IF CL(I)=1 THEN 2490
P1(J)=LOG(POP(I,J))
IF CL(I)=2 THEN 2500
YR1(J)=LOG(YR(J))
GOTO 2510
P1(J)=P0P(I,J)
YR1(J):YR(J)
NEXT J
IF CL(I)=3 THEN 2550
LP=P
GOTO 2560
LP=LOG(P)
T20
FOR J=1 TO N
TfiT4P1(J)
NEXT J
AV1=T/N
R=0
FOR J=1 TO N
R=R+YR1(J)
NEXT J
AV2=R/N
VARFO :COV20
FOR J=1 TO N
VAR#VAR+((YR1(J)RAV2)“2)
COVECOV+(P1(J)-AV1)*(YR1(J)-AV2)
NEXT J
B=COV/VAR
Y:P1(N)+B*(LP4YR1(N))
REM *****************************************************
REM PRINTING RESULTS
PRINT
PRINT II II
IF CL(I)=1 THEN 2960
Y=EXP(Y)
IF CL(I)=3 THEN 2880
PRINT "PROJECTED POP. OF ";C$(I);" FOR ":P;" ..... ":Y
PRINT " BASED ON THE EXPONENTIAL MODEL."
PRINT

 

LINEAR,EXPONENTIAL AND DOUBLE LOG. MODELS

 

 

 

96

 

 

APPENDIX 4 : Continued

2830 IF Q$="N" 0R Q$="n" TIEN 2300

2840 PRINT #3, "PROJECTED POP. OF ";C$(I);" FOR ";P;" ..... ";Y
2850 PRINT #3," BASED ON TIE EXPONENTIAL mDEL."

2860 PRINT #3,

2870 GOTO 2300

2880 PRINT "PROJECTED POP. 0F ";C$(I);" FOR ";P;" ..... ";Y
2890 PRINT " BASED ON TIE DOUBLE LOG. MODEL."

2900 PRINT

2910 IF Q$="N" 0R Q$="n" TIEN 2300

2920 PRINT #3,"PROJECTED POP. 0F ";C$(I);" FOR ";P;" ..... ";Y
2930 PRINT #3," BASED ON TIE DOUBLE LOG. NDDEL."

2940 PRINT #3,

2950 GOTO 2300

2960 PRINT "PROJECTED POP. 0F ";C$(I);" FOR ";P;" ..... ";Y
2970 PRINT " BASED ON TIE LINEAR MODEL."

2980 PRINT

2990 IF Q$="N" 0R Q ="n" TIEN 2300

3000 PRINT #3,"PRO.JECTED POP. 0F ";C$(I);" FOR ";P;" ..... ";Y
3010 PRINT #3," BASED ON 'TIE LINEAR MODEL."

3020 PRINT #3,

3030 GOTO 2300

3040 REM *********************************************************
3050 REM [] [J
3060 REM [] GOMPERTZ AND MODIFIED EXPONENTIAL NDDEL []
3070 REM [] [J
3080 REM GROUPING POINTS FOR USING TIE SELECTED POINT TECHNIQUE
3090 G=N/3

3100 G=INT(G)

3110 N2=(N—G+1) :N3=(N—G) :N4=(N-G—G+1) :N5=(N—G—G) :N6=(N-G—G-G+1)
3120 FOR J=1 TO N

3130 IF CL(I)=5 TIEN 3160

3140 P1(J)=LOG(POP(I,J))

3150 GOTO 3170

3160 P1(J)=POP(I,J)

3170 NEXT J

3180 T=O:R=0:Q=0

3190 FOR J=N2 TO N

3200 T=‘I‘+P1(J)

3210 NEXT J

3220 FOR J=N4 T0 N3

3230 R=R+P1(J)

3240 NEXT J

3250 FOR J=N6 T0 N5

3260 Q=Q+P1(J)

3270 NEXT J

3280 REM *****************************************************

3290

REM TEST FEASIBILITY FOR USING TN) MODEL

APPENDIX 4

3300
3310
3320
3330
3340
3350
3360
3370
3380
3390
3400
3410
3420
3430
3440
3450
3460
3470
3480
3490
3500
3510
3520
3530
3540
3550
3560
3570
3580
3590
3600
3610
3620
3630
3640
3650
3660
3670
3680
3690
3700
3710
3720
3730
3740
3750
3760

97

: Continued

NB=(T-R)/(R-Q)

IF NB<O TIEN 3350

B=EXP(LOG(NB)/G)
A=(R-Q)*(B-1)/((NB-1)‘2)

IF NB>O AND NB<1 AND A<O THEN 3590
IF C(I)=2 AND CL(I)=5 THEN 3390

IF C(I)=2 AND CL(I)=4 THEN 3440

IF CL(I)=4 THEN 3500

CL(I)=4 : GOTO 3120

PRINT " ***********#*******************************************"

PRINT " THE PAST POPULATION TREND 0F ";C$(I)
PRINT " CAN NOT BE FITTED TO THE MODIFIED MODEL"

PRINT " *******************************************************N

GOTO 2300
PRINT n ******************************#************************H

PRINT " THE PAST POPULATION TREND 0F ";C$(I)
PRINT " CAN NOT BE FITTED TO THE GOMPERTZ MOD "
PRINT n *******************************************************H

GOTO 2300

REM RESELECTING OPTIMUM MODEL

IF 02$="N" 0R 02$="nP THEN 3550

IF DIF(I)>=—1O AND DIF(I)<=O AND DIF2(I)>=—1O THEN 3540
IF DIF(I)<—50 AND DIF2(I)<-10 THEN 3540

GOTO 3880

CL(I)=1 : GOTO 2430

IF DIF(I)>=-50 AND DIF(I)<-10 THEN 3880

GOTO 3540

REM *****************************************************
REM PROJECTING POPULATION

K=(Q-( (NB~1)*A/(B*1) ) )/G

E=G+G+G+(P-YR(N)-S)/S

Y=K+(A*(B“E))

REM *******************************************#*********

REM PRINTING RESULTS

 

PRINT

PRINr II II
IF CL(I)=4 THEN 3740

PRINT "PROJECTED POP. OF ";C$(I);" FOR ";P;" ..... ";Y
PRINT " BASED ON THE MODIFIED MOD ." : PRINT

IF Q$="N" OR Q$="n" THEN 2300

PRINT #3,"PROJECTED POP. OF ";C$(I);" FOR ";P;" ..... ";Y
PRINT #3," BASED ON THE MODIFIED MODEL."

PRINT #3,

GOTO 2300

Y=EXP(Y)

PRINT "PROJECTED POP. OF ";C$(I);" FOR ";P;" ..... ";Y

PRINT " BASED ON THE GOMPERTZ MOD ."

 

APPENDIX. 4

3770
3780
3790
3800
3810
3820
3830
3840
3850
3860
3870
3880
3890
3900
3910
3920
3930
3940
3950
3960
3970
3980
3990
4000
4010
4020
4030
4040
4050
4060
4070
4080
4090
4100
4110
4120
4130
4140
4150
4160
4170
4180
4190
4200
4210
4220
4230

98

: Cont inued

PRINT
IF Q$="N“ OR Q$="n" THEN 2300
PRINT #3,“PROJECTED POP. OF ";C$(I);" FOR ";P;" ..... ";Y
PRINT #3," BASED ON THE GOMPERTZIMODEL."
PRINT #3,
GOTO 2300
RB]! *3“:akalcakakslnlc*akakakakakaltall*3”:alt*******#**********************##altal:
REM [1 [1
REM [1 [1
REM [I [1
REM GERNERATING MATRIX FOR.USING THE LEAST SQUARE,METHOD
IF CL(I)=6 0R C(I)=1 THEN 3910
PRINT "DEGREE OF POLYNOMIAL FOR ";C$(I);"
INPUT D : GOTO 3920
D=2
FOR H=1 T0 D+1
FOR L:1 TO D+1
T=0
IF (HHL)=2 THEN 4010
FOR J=1 TO N
T=T+((J—1)“(H+L-2))
NEXT J
X(H,L)='I'
GOTO 4020
X(H,L)=N
NEXT L
NEXT H
FOR H=1 T0 D+1
Q=0
FOR J=1 TO N
IF H:1 THEN 4100
Q=Q+POP(I.J)*((J-1)‘(H-1))
GOTO 4110
Q==Q+POP(I.J)
NEXT J
Y(H)=Q
NEXT H
Rm *************************akakaluknkak*Ilnkakakakakakak****************
REM PROJECTING POPULATION
GOSUB 4350
R=0
FOR H:1 T0 D+1
R=R+A(H)*(((P-YR(1))/S)‘(H-1))
NEXT H
Rm *********¥******#*******************#******************
REM PRINTING RESULTS
PRINT

 

EKIATKWEAL.MODEL

 

(TYPE 3, 4 OR 5)";

99

APPENDIX 4 : Continued

4240
4250
4260
4270
4280
4290
4300
4310
4320
4330
4340
4350
4360
4370
4380
4390
4400
4410
4420
4430
4440
4450
4460
4470
4480
4490
4500
4510
4520
4530
4540
4550
4560
4570

 

PRINT H II
PRINT "PROJECTED POP. OF ";C$(I);" FOR ";P;" ..... ";R

PRINT "BASED ON THE ";D;"-DEGREElflflfiflflrflllaMODEL."

PRINT

IF Q$="N" OR Q$="n" THEN 2300

PRINT #3,"PROJECTED POP. OF ";C$(I);" FOR ";P;" ..... ";R
PRINT #3,"BASED ON THE ";D;"-DEGREE POLYNOMEAL MODEL."
EFENT #3,

GOTO 2300

RE]! *SI‘******************************************************

REM MATRIX INVERSION AND MULTIPLACATION
FOR L=1 TO D+1
B=X(L.L)
IF B=O THEN STOP
X(L,L)=1
FOR J=1 TO D+1
X(L.J)=X(L.J)/B
NEXT J
FOR H=1 TO D+1
IF H:L THEN 4490
B=X(H.L)
X(H,L)=O
FOR J=1 TO D+1
X(H.J)=X(H,J)—X(L.J) *3
NEXT J
NEXT H
NEXT L
FOR,H:1 TO D+1
A(H)=O : REM INITIALIZE EACH ELEMENT OF B
FOR J=1 TO D+1
.A(H)=A(H)+X(H,J)*Y(J)
NEXT J
NEXT H
RETURN

 

APPENDIX 5

EXAMPLE OF PROJECTING POPULATION

WITH COMPUTER PROGRAM

 

 

100

APPENDIX 5 : EXAMPLE OF PROJECTING POPULATION
WITH COMPUTER PROGRAM

RUN

NUMBER OF CITIES AND TOWNS FOR PROJECTION? 1
NUMBER OF DATA POINTS? 4

DO YOU HAVE A DATA-FILE (Y/N)? N

DO YOU WANT TO GREAT A DATA-FILE (Y/N)? N

NAME OF CITY OR TOWN : NO. 1 ? SAGINAW

TIME (YEAR) FOR DATA POINT # 1 9 1940
TIME (YEAR) FOR DATA POINT # 2 ? 1950
TIME (YEAR) FOR DATA POINT # 3 ? 1960
TIME (YEAR) FOR DATA POINT # 4 9 1970

POPULATION OF 1940 9
POPULATION OF 1950 ?
POPULATION OF 1960 ? 98265
POPULATION OF 1970 9

—~---———----*_-_——--~—-~--*--*—_-——‘-——~

YOUR PROJECTION YEAR IS 1980 (Y/N)? Y

DO YOU WANT TO SAVE THE RESULTS (Y/N)? N

WHICH MODEL DO YOU WANT TO USE FOR PROJECTION?

LINEAR REGRESSION MODEL

EXPONENTIAL REGRESSION MODEL

DOULE LOGARITHMIC REGRESSION MODEL
GOMPERTZ MODEL

MODIFIED EXPONENTIAL MODEL

SECOND DEGREE POLYNOMIAL MODEL

THIRD, FORTH OR FIFTH DEGREE POLYNOMIAL
COMPOSITE METHOD (TESTED IN '85, MICH.)

comment-comp

 

101

APPENDIX 5 : Continued

ENTER THE NUMBER OF SPECIFIC METHOD FOR SAGINAW? 8

CAN YOU GIVE THE ROUGH % GROWTH RATE FOR SOME OR ALL
OF CITIES FROM 1970 TO 1980 (Y/N)? Y

ROUGH % GROWTH RATE FOR SAGINAW? -15

——--—-—--—I—————“————-—fi—w———————-—~—————————————-flfl_—-”_“

PROJECTED POP. OF SAGINAW FOR 1980 ........ 78909.72
BASED ON THE POLYNOMIAL MODEL

PROJECT AGAIN WITH OTHER ALTERNATIVE MODEL (Y/N)? N

 

OK

 

APPENDIX 6

POPULATION DATA

102

APPENDIX 6 POPULATION DATA

CITY 1940 1950 1960 1970 1980

Adrian 14230 18393 20347 20382 21186
Albion 8345 10406 12749 12112 11059
Algona 1931 2639 3190 3684 4412
Allega 4526 4801 4822 4516 4576
AllenP 3487 12329 37494 40747 34196
Alma 7202 8341 8978 9611 9652
Alpena 12808 13135 14682 13805 12214
AnnArb 29815 48251 67340 99797 107966
BadAxe 2624 2973 2998 2999 3184
Battle 43453 48666 44168 38931 35724
BayCit 47956 52523 53604 49449 41593
Beldin 4089 4436 4887 5121 5634
Benton 16668 18769 19136 16481 14707
Berkle 6406 17931 23275 21879 18637
Bessem 4080 3509 3304 2805 2553
BigRap 4987 6736 8686 11995 14361
Birmin 11196 15467 25525 26170 21689
Blissf 2144 2365 2653 2753 3107
Boyne 2904 3028 2797 2967 3348
Buchan 4056 5224 5341 4643 5142
Cadill 9855 10425 10112 9990 10199
Caro 3070 3464 3534 3701 4317
Center 3198 7659 10164 10379 9293
Charle 2299 2695 2751 3519 3296
Charlo 5544 6606 7657 8244 8251
Cheboy 5673 5687 5859 5553 5106
Chelse 2246 2580 3355 3858 3816
Chesan 1807 2264 2770 2876 2656
Clare 1844 2440 2442 2639 3300
Clawso 4006 5196 14795 17617 15103
Coldwa 7343 8594 8880 9155 9461
Corunn 2017 2358 2704 2829 3206
Daviso 1397 1745 3761 5259 6087
Dearbo 63587 94994 112007 104199 90660
Dowagi 5007 6542 7208 6583 6307

103

APPENDIX 6 : Continued

 

 

CITY 1940 1950 1960 1970 1980
Durand 3127 3194 3312 3678 4241
EDetro 8584 21461 45756 45756 38280
EGrand 4899 6403 10924 12565 10914
ELansi 5839 20325 30198 47540 51392
EatonR 3060 3509 4052 4494 4510
Ecorse 13209 17948 17328 17515 14447
Escana 14830 15170 15391 15368 14355
Essexv 2390 3167 4590 4990 4378
Farmin 1510 2325 6881 10329 11022
Fenton 3377 4226 6142 8284 8098
Fernda 22523 29675 31347 30850 26227
FlatRo 1467 1931 4696 5643 6853
Flint 151543 163143 196940 193317 159611
Flushi 1806 2226 3761 7190 8624
Franke 1100 1208 1728 2834 3753
Fraser 747 1379 7027 11868 14560
Fremon 2520 3056 3384 3465 3672
Garden 4096 9012 38017 41864 35640
Gaylor 2055 2271 2568 3012 3011
Gd.Blan 1012 998 1565 5132 6848
Gladst 4972 4831 5267 5237 4533
GrandH 8799 9536 11066 11844 11763
GrandL 3899 4506 5165 6032 6920
Gr.Po.Sh 801 1032 2301 3042 3122
GrandR 164292 176515 177313 197649 181843
Grandv 1566 2022 7975 10764 12412
Greenv 5321 6668 7440 7493 8019
Grosse 6197 6283 6631 6637 5901
GPFar 7217 9410 12172 11701 10551
GPPark 12646 13075 15457 15641 13639
GPWood 2805 10381 18580 21878 18886
Hamtra 49839 43355 34147 27245 21300
Hancoc 5554 5223 5022 4820 5122
Hartford 1694 1838 2305 2508 2493

Hastin 5175 6096 6375 6501 6418

 

APPENDIX

6

Continued

104

Highla
Hillsd
Hollan
Holly

Hought

Howell
Hudson
Hudsvi
Huntin
Inkste

Ionia

IronMo
Ironwo
Ishpem
Ithaca

Jackso
Kalama
Kingsf
LakeOr
L'Anse

Lansin
Lapeer
Lauriu
Lineol
Livoni

Lowell
Luding
Manist
Maniqu
Marine

Marque
Marsha
Marysv
Mason

Melvin

107807
6160
3058

53933
66702

2545
9421
8324
4875
4404

19824
6736
4065
4522

13089

131403
6314
2868

52984

110109

3068
9021
7723
4324
4567

21967
7253
5610
5468

13862

130414
6198
2678

45105

104814

3707
8937
7566
3962
4414

23288
7201
7345
6019

12322

 

 

APPENDIX

6

Continued

105

Menomi
Midlan
Milan

Milfor
Monroe

MoClem
MoMorr
MoPlea
Munisi
Muskeg

MuskHe
Negaun
NewBal

New Buff

Niles

NorthM
Northv
Norway
OakPar
Otsego

Owosso
Oxford
PawPaw
Petosk
Plainw

Pleasa
Plymou
Pontia
PoartH
Portla

Richmo
RivRou
Riverv
Roches

Rockwood

1147

 

 

APPENDIX

6

Continued

106

Rogers
Romeo

Rosevi
RoyalO
Sagina

StClai
StClSh
StIgna
StJohn
StJose

StLoui
Saline
SaultS
SouthH

8. Lyon

Sparta
Spring
Sturgi
Tecums
ThreeR

Traver
Trento
Utica

Vassar
Wakefi

Warren
Wayne
Whiteh

LO

Willians

Wyando

Ypsila
Zeelan

14455
5284
1022
2154
3591

582
4223
1407
1704

30618

12121
3007

89246
16034
2590
2214
43519

20957
3702

179260
21054
3017
2600
41061

29538
4734

161134
21159
2856
2981
34006

24031
4764

 

 

 

LI ST OF REFERENCES

 

 

107

LIST OF REFERENCES

Chapin, Francis Stuart, Jr., and Kaiser, Edward J., (1979).
Urban Land Use Planning. Third Edition. University
of Illinois Press.

 

Goodman, Willian I. and Freund Eric C., (1968). Pginpiples
apd Ppaggice 9; Urban Planning. International City
Managers' Association.

Greenberg, Michael R. and Kruckeberg, Donald A. and Mautner,
Richard. (1978). ngal Population and Employment
Projection Techniques. The Center for Urban Policy
Recearch. Rutgers University.

 

Isard, Walter,. (1960). Methods 9: Regional Analysis A an

Introduction 39 Regional Science. Massachusetts
Institute of Technology.

 

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