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UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
URBANA. ILLINOIS 61801
CAC Document No. 237
AN ENTROPY MAXIMIZATION APPROACH
TO THE DESCRIPTION OF
URBAN SPATIAL ORGANIZATION
Robert M. Ray III
September, 1977
CAC Document No. 237
AN ENTROPY MAXIMIZATION APPROACH TO THE DESCRIPTION
OF URBAN SPATIAL ORGANIZATION
by
Robert M. Ray III
Center for Advanced Computation
University of Illinois at Urbana- Champaign
Urbana, Illinois 6l801
September 1977
A dissertation submitted to the faculty of the University of North Carolina,
Chapel Hill, in partial fulfillment of the requirements for the degree
of Doctor of Philosophy in the Department of City and Regional Planning,
June 1977.
Copyright by
Robert M. Ray III
1977
ABSTRACT
Within the modern city, spatial patterns of urban phenomena, e.g.,
areal distributions of differentiated populations, activities, and land
uses, represent the most immediate and tangible manifestations of all
social forces underlying the process of modern urbanization. Thus, it
would seem that rigorous methods for quantitative description and analysis
of specific characteristics of urban spatial organization would be con-
sidered fundamental to other more specialized studies of urban conditions.
However, despite the considerable attention paid by the various social
sciences to particular aspects of urban spatial organization, there appears
to be little tendency toward convergence on common analytic methods prac-
tical for treatment of the complex structure of real-world urban space.
This condition stems in large measure, we contend, from the inappropriate-
ness of conventional statistical data analysis techniques for quantifica-
tion of the degree of spatial co- organization, areal association, or con-
gruence between geographic distributions of urban phenomena.
Here, we develop an alternative method of urban spatial distribu-
tion analysis that is deoigned explicitly for quantitative characteriza-
tion of the structure of spatial associations existing between some set
of areally distributed urban variables. Our approach grows out of a par-
ticular combination, and in some instances generalization, of mathemati-
cal concepts developed previously within the areas of information theory,
urban trip distribution modeling, and the theory of multidimensional
scaling. From such a diversity of mathematical concepts there is con-
structed a pattern- information method of spatial distribution analysis
that appears applicable to the study of geographically distributed urban
phenomena in general.
The model developed unites in a unique manner quantitative measures
of the degree of spatial congruence existing between two areal distribu-
tions with information theoretic measures of the complexity of spatial
structure transmitted between them. The particular information theoretic
concepts developed lead directly to a cluster analysis procedure that is
shown to be applicable tc the analysis of structures of spatial associa-
tions determined among areally distributed urban variables.
Using data concerning the spacial distributions of thirty- two (32)
urban variables across a hypothetical urban area, we illustrate the method
proposed computing all measures of spatial association between all variables
and cluster analyzing the resulting structure of associations. As an
independent means of analyzing the structure of associations between
variables, a nonmetric multidimensional scaling analysis is also performed.
Close agreement between our intuitive notions of the interrelationships
between urban distributions and both cluster analysis and multidimensional
scaling results is observed.
TABLE OF CONTENTS
LIST OF ILLUSTRATIONS iv
LIST OF TABLES vii
ACKNOWLEDGEMENTS viii
Chapter
I. INTRODUCTION 1
The Organized Complexity of Urban Space 1
The Gap between Theory and Data ........ 6
The Deficiencies of Present Data Analysis Methods . . 8
The City as a Self -Organizing Spatial System .... 12
The Present Effort 14
II. INFORMATION THEORY, PHYSICAL DISTANCE, AND
URBAN SPATIAL ORGANIZATION 17
Introduction 17
Communication, Information, and Entropy 19
Information Transmission 24
The Entropy-Maximizing Model of Urban Trip
Distribution 27
III. SOME PRELIMINARY METHODS FOR ANALYSIS OF URBAN
SPATIAL DISTRIBUTIONS 42
Introduction 42
Characterization of Urban Patterns as Areal
Distributions 45
Basic Measures of Central Tendency and Dispersion
for Areal Distributions 49
An Alternative Method for Computing the Distance
Variance of a Distribution 52
Some Preliminary Measures of Spatial Association
Between and Within Areal Distributions 59
A Spatial Interaction Approach to Measurement
of Distribution Distance 61
Chapter
IV. NEW METHODS FOR ASSOCIATION MEASUREMENT AND CLUSTER
ANALYSIS OF SPATIAL DISTRIBUTIONS 67
A Unique Measure of Spatial Association Within and
Between Areal Distributions 67
An Information Theory Measure of Spatial Complexity
Conveyance Among Areal Distributions 73
A Procedure for Least Biased Grouping of Spatial
Distribution Elements 78
Cluster Analysis of Spatial Associations Between
Distributions 89
V. URBAN SPATIAL DISTRIBUTION ANALYSIS: A WORKED EXAMPLE . 92
The Hypothetical Urban Area 92
Urban Spatial Distributions Selected for Analysis . . 104
Example Analyses Performed 109
VI. SUMMARY AND CONCLUSIONS 123
Summary of Argument 123
Potential Applications of the Method 127
APPENDIX 1 130
APPENDIX 2 163
APPENDIX 3 168
BIBLIOGRAPHY 173
IV
LIST OF ILLUSTRATIONS
Figure
1. Schematic Diagram of a General Communication System
(after Shannon, 1949) 20
2. Mean Trip Length D and Spatial Information Transmission
T as Functions of 3 within the Constrained Entropy
Maximization Model of Urban Trip Distribution .... 32
3. A Hypothetical Region Containing Four Urban Areas ... 34
4. Mean Trip Length D and Spatial Information Transmission
T as Functions of 3 for Home-to-Work Trips within the
Hypothetical Region of Fig. 3 35
5. Spatial Distributions of Livelihood and Residential
Land Uses and Elementary Schools Within a Hypothetical
Urban Area 37
6. Mean Trip Length D and Spatial Information Transmission
T as Functions of 3 for Home-to-Work and Home-to-Shop
Trips Within the Hypothetical Urban Area of Fig. 5 . . 38
7. Mean Trip Length D and Spatial Information Transmission
T as Functions of 3 for Home-to-School Trips Within
the Hypothetical Urban Area of Fig. 5 38
8. First Example Cluster Analysis 84
9. Second Example Cluster Analysis 85
10. Third Example Cluster Analysis 86
11. Fourth Example Cluster Analysis 87
12. Generalized Land Use for the Hypothetical Urban Area . . 93
13. Zonal System Subdividing Urbanized Area into Areal
Units for Data Aggregation 95
14. Probability Distribution of Single-family Residential
Land Use 97
15. Probability Distribution of Two-family Residential
Land Use 97
16. Probability Distribution of Multi-family Residential
Land Use 98
17. Probability Distribution of Commercial Land Use ... 98
18. Probability Distribution of Public and Semi-public
Land Use 99
19. Probability Distribution of Parks and Playgrounds ... 99
20. Probability Distribution of Light Industry 100
21. Probability Distribution of Heavy Industry 100
22. Probability Distribution of Railroad Property .... 101
23. Probability Distribution of Vacant Land 101
V
Figure
24. Hierarchical Tree Showing Sequence of Cluster Mergers
within Cluster Analysis of [_ EDI Matrix of Areal
Distribution Dissociation Measures 110
25. Graph of Structural- Informat ion-Transmission-Loss
Function over Successive Stages of Cluster Analysis
of [. EDI2] Matrix Ill
f'g r 1
26. T0RSCA-9 Two- Dimensional Scaling Solution of [_ EDI*J
f »S
Matrix of Inter-distribution Distances 113
27. Hierarchical Tree Showing Sequence of Cluster Merges
within Cluster Analysis of [_ LDI J Matrix of Areal
Distribution Dissociation Measures 118
28. Graph of Structural-Information-Transmission-Loss
Function over Successive Stages of Cluster Analysis
of [. LDI2] Matrix 119
f'g r 1
29. T0RSCA-9 Two- Dimensional Scaling Solution of l_ LDI1 J
Matrix of Inter-distribution Distances 121
30. Pattern of Single-family Housing Units 131
31. Pattern of Two-family Housing Units 132
32. Pattern of Multi-family Housing Units 133
33. Pattern of Mobile-home Housing Units 134
34. Pattern of Transient Lodging Units 135
35. Pattern of Daycare Centers and Nursery Schools .... 136
36. Pattern of Elementary Schools (K-6) 137
37. Pattern of Junior High Schools (7-9) 138
38. Pattern of Senior High Schools (10-12) 139
39. Pattern of Colleges and Vocational Schools 140
40. Pattern of Neighborhood Parks and Playgrounds .... 141
41. Pattern of Regional Outdoor Recreation Areas 142
42. Pattern of Indoor Movie Theaters 143
43. Pattern of Churches 144
44. Pattern of Full-Line Department Stores 145
45. Pattern of Apparel Shops 146
46. Pattern of Furniture Stores (Not Department) 147
47. Pattern of Hardware Stores (Not Department) 148
48. Pattern of Food Supermarkets 149
49. Pattern of Quick-Shop Grocery Stores 150
50. Pattern of Specialty Food and Liquor Stores 151
51. Pattern of Pharmacies 152
52. Pattern of Auto Service Stations 153
VI
Figure
5 3. Pattern of Full- line Restaurants 154
54. Pattern of Fast-Food Drive-ins 155
55. Pattern of Hospitals 156
56. Pattern of Employment in Heavy Industry 157
57. Pattern of Employment in Light Industry 158
58. Pattern of Private Office Space 159
59. Pattern of Banking Activity . 160
60. Pattern of Major Arterial Street Frontage 161
61. Pattern of Railroad Property 162
VI 1
LIST OF TABLES
2 *
1. Values of GDV, -EDI , H(-Z), H(f fQ ), and f fC for
the Four Spatial Distributions of Figures 8, 9, 10,
and 11 89
2. Proportional Distributions of Land in Different Uses
for the Hypothetical Urban Area 94
3. Thirty-two Areal Distributions of Urban Phenomena for
Example Analysis 103
2
4. Values of _ EDI in Miles Squared for f = 1.....32
and g = 1,...,32 164
5. Values of ,, EDIf in Miles for f = 1.....32
and g = 1,...,32 166
2
6. Values of ,_ LDI in Miles Squared for f = 1.....32
f,g
and g = 1,...,32 169
7. Values of _ LDI' in Miles for f = 1.....32
f»g
and g = 1,...,32 m
ACKNOWLEDGEMENTS
The methodology of urban spatial analysis suggested in this paper
stems from a cross-fertilization of concepts explored in a number of
divergent research areas in which I have been involved over the past
several years both at the University of North Carolina at Chapel Hill
and at the University of Illinois at Urban a -Champaign. Thus, it is
with sincerity and regret that I note here the impossibility of expres-
sing specific appreciation to all of those who have directly or indir-
ectly contributed to the shaping of these ideas.
The completion of this work would have been impossible without
the assistance and patience of my dissertation committee in the Depart-
ment of City and Regional Planning of the University of North Carolina
at Chapel Hill. Special appreciation is given to Professor George C.
Hemmens, not only for his support as dissertation committee chairman,
but more for the constant inspiration and guidance that he gave to me
throughout the circuitous development of this thesis. Special acknow-
ledgements are also due Professor David H. Moreau for his thorough exam-
ination of the mathematical logic of the methodology presented and
numerous constructive criticisms. I also extend my appreciation to the
other members of my committee, Professors C. Gorman Gilbert, Edward J.
Kaiser, and Robert M. Moroney, for reading the thesis and discussing
with me its scope and format on several occasions.
IX
The University of Illinois at Urbana-Champaign, where much of
this work was done, provided extensive facilities both through the Center
for Advanced Study and through the Center for Advanced Computation. At
Illinois, Professor Daniel L. Slotnick was a valuable resource for numerous
discussions concerning the manner by which rapidly advancing computational
technologies might be most efficiently harnessed for social science data
analysis and modeling applications. Deep appreciation is also given to
Professor Hugh Folk for many discussion concerning the material presented
here and his detailed criticisms of earlier drafts.
On a more personal note, I extend my eternal gratitude to my dear
wife Alice, not only for the numerous early morning hours that she spent
proofreading, editing, and typing this manuscript, but more the constant
understanding, encouragement, and inspiration that she provided me through-
out this work. To my son Marsh, I am eternally indebted for the hours
that I have taken from him as a father in the course of this and other
related work too often brought home.
CHAPTER I
INTRODUCTION
The Organized Complexity of Urban Space
Summarizing a recent collection of essays focusing on better
definition of what a city is and how it can best be conceptualized to
serve the needs of urban and regional policy analysts, John Dyckman has
observed :
. . . the urban community is an extremely complex system, open to
change in many directions. In practice it may be difficult to
determine the number of significant variables which constitute the
environment of this system. Only by developing techniques compe-
tent to deal with "organized complexity," to use Warren Weaver's
term, can planning hope to deal with a changing city as a manageable
artifact. While many developments in data handling and data organ-
ization, the rise of computers, and great conceptual advances in
scientific methodology all promise some hope for this task, it
appears that little progress can be made until the existing under-
brush of poor and weak definitions is cleared and pruned.
(Dyckman, 1964, pp. 224-25)
In view of the state of the art of research methods within urban and
regional studies, especially as related to the description of urban
spatial organization, Dyckman 's use of Weaver's term organized complexity
to characterize present perceptions of our urban environments seems
particularly appropriate.
In his classic essay "Science and Complexity," Weaver (1947,
1948) proposed three general types of problems that modern science has
successively confronted. According to Weaver, the rise of modern
2
science throughout the nineteenth century could be attributed almost
exclusively to its treatment of problems of simplicity — problems for
which the workings of compound sets of variables might be adequately
described by sequential analysis and recombination of only first-order
causal relationships existing between pairs of variables, relationships
between all other variables at any one time held constant. The turn of
the century witnessed the refinement and application of specific con-
cepts of probability theory that enabled science to deal with certain
problems of disorganized complexity — problems involving very large num-
bers of variables for which, while the behavior of individual variables
might be essentially random, mean macroscopic properties might be pre-
dicted for the collection of variables as an aggregate, for example, the
prediction of macro properties of ensembles of gas molecules in accord-
ance with the laws of modern thermodynamics.
As an extension of nineteenth-century mechanics allowing scien-
tific analysis of simple deterministic systems and turn-of-the-century
statistical mechanics enabling quantitative treatment of disordered
probabilistic systems, Weaver argued that the true challenge of twentieth-
century science would be the development of new concepts sufficient for
analysis of problems of organized complexity. For Weaver, this category
included any scientific problem requiring simultaneous consideration of
large complexes of variables, all interacting in integrated fashion to
determine the behavior of the system as an organic whole. Examples here
are general problems associated with living organisms in biology as well
as basic problems concerning the organization of perception and behavior
in psychology, social organization in sociology, and the problem of
primary concern throughout this thesis — the problem of urban spatial
organization.
The overwhelming complexity of problems accompanying the acceler-
ating pace of urbanization occurring throughout the world requires that
we devote an increased share of our scientific resources to an under-
standing of the spatial dimensions of our urban environments. Many
problems of critical concern relate directly to the spatial pattern of
the city. Hence, there is an increasing need for methods for descrip-
tion of urban spatial organization that, while respecting the concepts
and theories of divergent academic disciplinary approaches, possess
sufficient generality and practicality to serve the needs of those
policy analysts required daily to advise public officials in making
decisions that will influence strongly the future complexion of urban
environments.
While the post-war era of rapidly developing transportation,
communication, and industrial automation technologies seemed to suggest
that the importance of physical distance as a determinant of spatial
patterns of urbanization would decline indefinitely into the future
(Webber, 196*4; 1968), today it seems clear that the "friction of dis-
tance," to recall Robert Haig's term (1926), will remain a viable con-
cept for urban and regional analysts for many years to come. All too
abruptly have we become aware of the finiteness of the supplies of
fossil fuels available for transportation of materials and persons with-
in and between our cities. Hence, transportation energy-efficiency
criteria should become increasingly important within the metropolitan
land use and transportation policy making of the future. As we as a
society become more aware of the inequities of opportunities for educa-
tion, employment, and housing experienced by different segments of our
urban populations due to patterns of residential segregation by socio-
economic classes, our need to comprehend the spatial organization of
the city and its relationship to such social inequities is heightened.
Viewing the city as a spatially organized physical entity, various forms
of environmental pollution become still another class of urban phenomena
that must be dealt with in the context of the total pattern of the city
if policy related to issues of environmental quality is to be both
efficient and equitable. (Berry, lQ?^)
That the social (economic, cultural, political) organization of
the city generally precedes and determines in large measure the complexion
of urban space is a proposition that we do not dispute. However, given
the complexity of economic and cultural forces at work determining the
organization of social and economic activities within urban areas, to
model with any precision urban spatial organization as the geographic
manifestation of social and economic forces represents, in our opinion,
an unmanageable task. Thus, the question arises: to what extent can we
work backward and, by improvement of our methods for analysis of the
spatial organization exhibited directly by urban areas, not only develop
the means for unambiguous description of the spatial patterns readily
observable within our cities, but perhaps also, by inference, enhance
our understanding of the social factors sustaining the spatial patterns
that we observe?
Thus we are suggesting that, for analysis purposes, the total
collection of issues associated with modern urbanization may be
5
subdivided into two broad component problems: (1) the problem of urban
spatial organization concerned with the analysis of phenomena that may
be considered, at least for a given period of time, as static spatial
patterns, e.g., geographic distributions of differentiated populations,
activities, and land uses; and (2) the problem of urban social organiza-
tion concerned with the analysis of phenomena that must be considered
as dynamic social processes, e.g., the actions, interactions, and trans-
actions of individuals and groups of individuals that inhabit the urban
environment and give to it all of the characteristics concomitant with
human life. Lacking such a partition between the issues of urban spatial
organization and those of urban social organization, we are left with
the more general problem of urban organization per se, encompassing the
totality of organized complexity with which any comprehensive theory of
urbanization must deal.
With considerable margin for error, it may be claimed that the
concept of urban organization represents the central concern of current
theory-construction efforts within urban and regional studies. While
other terms such as "urban structure" or "urban system" are often used
instead, through use of each of these phrases there is invariably an
attempt to establish some synoptic conceptualization of the total set
of social and spatial phenomena associated with the general notion of
urbanization. But rigorous definition of such concepts as organization,
structure, and system represents one of the most challenging intellec-
tual riddles of our day. (Boulding, 1956; von Bertalanffy, 1968;
Rapoport and Horvath, 1959; Meier, 1962) Hence, too often individual
attempts to provide comprehensive conceptual frameworks from which the
constituent elements of urban organization might be fruitfully analyzed
lead only to more terminological confusion and thus hinder the very task
for which urban and regional analysts have assumed responsibility.
The Gap between Theory and Data
Since the spatial pattern of our cities represents the most
visible manifestation of the social forces underlying modern urbaniza-
tion, it would seem that a rigorous scientific method for observation,
description, and quantitative analysis of the general characteristics
of urban spatial organization would be considered fundamental to any
more specialized studies of urban conditions. However, despite the
considerable attention paid by the various social sciences to specific
aspects of urban spatial organization, there appears to be little ten-
dency toward convergence on any common method practical for treatment
of the organized complexity of real-world urban space.
Sociological discussions of urban space, proceeding typically
in the tradition of human ecology (Park, Burgess, and McKenzie, 1925;
Hoyt, 1939; Harris and Ullman, 1945; Hawley, 1950; Duncan and Schnore,
1959; Theodorson, 1961), seem fundamentally correct in conceptualizing
urban space as a complex territorial arrangement of differentiated popu-
lation and socioeconomic activity patterns geographically structured
in accordance with the spatial dimensions of social organization. How-
ever, entangled in a complexity of concepts invoked for description of
social organization proper, such discussions have offered few method-
ological suggestions for quantitative analysis of the interdependence
between social organization and the organization of urban space.
7
Economic theories of urban space (Wingo, 1961; Alonso, 1965),
formulated in the fashion of the equilibrium-seeking deterministic (and
hence mechanistic) models of space-location theory (Losch, 1954; Isard,
1956), achieve admirable quantitative treatment of primary real-estate
market forces at work determining the overall "urban-suburban-rural"
distribution of land uses within metropolitan regions. However, con-
fronted with serious mathematical indeterminancies arising from intra-
regional location interdependencies among differentiated households,
firms, and institutions, the utility of such mechanistic models for ex-
plaining the richness of variety of population, activity, and land use
patterns observable in real-world urban landscapes is severely limited.
(Koopmans and Beckmann, 1957; Tiebout , 1961; Harris, 1961)
Geographers, such as Berry (1963, 1971), have sought a theoreti-
cal basis for explanation of intra-urban commercial activity structure
within the concepts and propositions of central place theory formulated
originally by Christ aller to explain the hierarchical pattern of cities,
towns, and villages within a region in terms of an efficient geographic
spacing of economic activities of varying degrees of specialization.
(Ullman, 1941; Vining, 1955; Berry and Garrison, 1958)
Given the discrete clustering of non-agricultural activities
into spatially separate urban centers, central place theory seems well
suited as a theoretical basis for spatial analysis at the regional
scale. In fact, Losch' s mathematical derivation of similar hierarch-
ical systems of regional settlement patterns and accompanying market
areas based on the scale economies of various economic activities demon-
strates that, within certain simplifying assumptions, the essential
8
characteristics of the macro- geographic phenomena conceived by Chris-
taller may be derived from micro-behavioral economic assumptions alone.
(Losch, 195«4)
However, upon entering the economic space of any single city,
the spatial clustering of economic activities becomes much more complex.
While scale economies and transportation costs continue to encourage
dispersion of similar retail and service activities over equi -populated
subareas of the city, other classes of similar activities often exist
side by side in Kotelling-competition fashion (Hotelling, 1929), and
thus the market areas of individual retail and service activities can
no longer as readily be assumed to be non-overlapping and disjoint.
Thus, while the concepts of central place theory and market-area analy-
sis often provide useful insights for organizing our perceptions of
certain aspects of the hierarchical structure of commercial activities
that we observe within urban space, the use of such theory remains very
much at the level of verbal conceptual frameworks aiding analysis, and
falls short of providing any meaningful theoretical basis for quanti-
tative analysis of urban spatial organization in general.
The Deficiencies of Present Data Analysis Methods
The search for viable quantitative methods for analysis of
spatial associations between geographically distributed patterns of
social phenomena has held the interest of statistically-oriented method-
ologists within the social sciences since the beginnings of urban and
regional studies.
Initial attempts to analyze relationships between urban spatial
patterns followed the ecological correlation approach using conventional
9
correlation techniques to quantify the extent of association among
sociological urban variables arrayed by geographic subareas of the
city. Such studies have provided summary descriptions of the mean
characteristics of individual subareas (census tracts, political wards,
transportation zones) as well as correlations between summary variables
across subareas. However, except where subarea characteristics have
been displayed graphically in map format, these studies have yielded
little information concerning the area-wide interdependence of spatial
patterns of urban phenomena.
Robinson (1950) has criticized the use of ecological correlations
as a basis for analysis of urban social phenomena by pointing out that
correlations of sociological variables over individuals within a study
group cannot be inferred from correlations computed between variables
representing mean characteristics of subgroups of the study population.
While as Menzel (1950) has suggested, ecological correlations may be
considered meaningful where the geographically delineated populations
themselves are clearly identified as the units of analysis, still it
must be remembered that ecological correlations are in no way dependent
upon proximity relationships between geographic subareas, and hence
spatial associations among urban patterns that extend across contiguous
subareas are in no way measured.
In similar fashion, more recent studies of specific cities
employing variants of the social area analysis technique of Shevky and
Bell (1955, 1961) focus on classification of prior delineated subareas
along a priori constructed sociological dimensions, independent of any
consideration of spatial relationships between geographic subareas.
10
Further, studies conducted using data analysis techniques in the tradi-
tion of ecological correlation methods do not in general yield results
that are appropriate as intermediate data for comparative analysis of
variations in urban patterns across urban areas. While exceptions to
this rule exist for specialized studies, for example, the study by
Taeuber and Taeuber (1965) of Negro residential segregation within U.S.
cities, data analysis methods for such studies tend to be selected with
respect to narrowly defined research objectives, and hence the applica-
bility of the methods chosen for more general problems of urban spatial
analysis is limited.
Summarizing and criticizing a wide variety of methods used for
measurement and analysis of geographically distributed social phenomena,
Duncan, Cuzzort , and Duncan (1961) refer to the collection of method-
ological problems involved as statistical geography. While they them-
selves propose no new solutions to the methodological issues that they
raise, their discussion is valuable in that it addresses in a compre-
hensive manner the variety of issues surrounding the dependence of
measures determined by most areal data analysis methods on the number
and size of the areal units chosen for data collection and tabulation.
In an effort to develop more general methods for quantifying
spatial associations between geographically distributed variables,
methods yielding measures of areal association less sensitive to the
specific number and size of areal units by which data are arrayed,
Warntz (1956, 1957, 1959) and others (see Neft, 1966) have approached
the problem of analyzing the interdependence of spatially distributed
phenomena in quite a different manner.
11
The approach taken by Warntz and followers requires initial
transformation of data arrayed by discrete areal units into potential
surfaces mathematically continuous across all areal units in the
geographic region of interest. Then, for any two areally distributed
variables (now represented as continuous mathematical surfaces), an
approximation to the true surface-to-surface correlation (the measure
that would be obtained by correlating the values of potentials for the
infinite set of points matched between the two surfaces) is obtained
by computing a measure of surface-to-surface correlation using only a
sample of points.
However, there are serious methodological questions surrounding
the method proposed by Warntz for analysis of the spatial interdependence
of geographically distributed social phenomena in that there exist an
infinite number of ways by which mathematically continuous surfaces may
be selected to fit a discrete set of spatially distributed observations.
Recognizing this condition, Warntz chooses to define his surfaces in
strict analogy to the concept of field potential as it is employed in
physics. To support intellectually this choice of a specific mathema-
tical function, Warntz allies himself with the arguments of the "social
physicist" John Q. Stewart (1947, 1948).
Stewart, like his contemporary Zipf (1949), held that there exist
general laws of nature governing the macro behavior of social systems
much in the same manner that the universal laws of physics govern the
behavior of complex physical systems. We acknowledge the wealth of
empirical evidence suggesting that mathematical equations fitting remark-
ably well data on macro distributions of social phenomena can be constructed
12
in the same form as the equations for the concepts of gravitational
force, energy, and potential in physics. Nevertheless, after at least
three decades of empirical research, there is little evidence for the
existence of any universal numerical constants for such mathematical
models of social phenomena analogous to the gravitational constant of
physics. (Isard, 1960) For example, given a new set of data on inter-
city travel within the U. S., the social scientist is forced to cali-
brate anew his gravity model determining empirically each time some set
of parameters best-fitting the data at hand. Thus, Warntz's decision
". . .to cling to the purely physical notions of Newton on gravity,
La Grange on potential and Stewart on social physics ..." (Warntz,
1957, p. 128), from the viewpoint of the statistically-oriented social
scientist, must be regarded as a rather arbitrary premise guiding the
selection of a specific mathematical function for characterizing discrete
geographic distributions as continuous surfaces.
The City as a Self -organizing Spatial System
Convinced of the need for more general methods for analysis of
the dimensions of urban space and feeling with others (Dyckman, 1964;
Rogers, 1967) that the problem of urban spatial organization is pri-
marily a problem of organized complexity as defined by Weaver, we are
compelled to seek an alternative approach to urban spatial analysis
that while consistent with the more general goal of urban studies, the
alignment of substantive theory and available data, will provide an
operational means for less ambiguous quantitative description of real-
world urban spatial organization. It will be a fundamental premise of
13
our approach that macroscopic patterns or areal distributions of urban
phenomena represent the most appropriate analysis units for description
of urban spatial organization. In a sense, we are simply aligning our-
selves with the view of the early urban ecologists that urban space is
most conveniently conceptualized and analyzed as a territorial arrange-
ment of differentiated population, social activity, and land use patterns.
Our primary task here, however, will be to explore alternative quanti-
tative methods better equipped to deal mathematically with areal distri-
butions and spatial associations between distributions as primary analy-
sis units within the study of urban spatial organization.
Focusing on the macroscopic phenomena of the urban landscape, we
view the urban process as a complex interacting system of patterns,
self-organizing in geographic space in accordance with the spatial dimen-
sions of the social organization that it seeks to accomodate. The
specific geographic outcome of this process of spatial self -organization
manifests itself at two levels of environmental complexity, that of
urban form and that of urban spatial structure. By urban form we mean
simply the external morphology, overall shape, or supra-pattern of the
city as it extends itself upward and outward in space as a physical
artifact. In contrast, by urban structure we mean the internal order
of physical integration, geographic association, or syntax of spatial
relationships exhibited between population, activity, and land use
patterns — internal spatial relationships resulting between patterns of
urban phenomena independent of whatever particular overall form might
be assumed by the city as a whole.
14
Of course, it is generally recognized that the specific forms
of individual urban areas, i.e., specific geographic arrangements of
population, activity, and land use patterns, vary widely from city to
city as a consequence of local variations of geophysical features of
the landscape and historical conditions. Nevertheless, while the varia-
tion of urban form across metropolitan areas is known to be great,
there exists a general consensus among urban analysts that the internal
spatial structures of cities, i.e., intra-urban spatial relationships
between patterns of urban phenomena, vary less widely across cities,
and in fact within specific regions, tend to conform to common struc-
tures determined almost entirely by cultural, social, and economic forces
at work within the region independent of local geophysical and histori-
cal conditions.
The Present Effort
Throughout the pages that follow, we investigate an alternative
method of urban spatial distribution analysis that is designed explicitly
for quantitative description of certain dimensions of urban spatial
structure. The method appears general to the analysis of a wide variety
of spatially distributed phenomena of interest to urban analysts, inclu-
ding the geographic patterning of differentiated socioeconomic popu-
lations, activities, and land uses.
Our approach grows out of a particular combination, and in some
instances generalization, of mathematical concepts developed previously
within the areas of information theory (Wiener, 1948; Shannon, 1948, 1949),
urban transportation trip distribution modeling (Creighton, 1970; Wilson,
15
1970; Potts and Oliver, 1972), and the theory of multidimensional scaling
(Torgerson, 1960; Green and Carmone, 1970). We shall see that out of
such a diversity of mathematical concepts there can be constructed a
pattern-in format ion method of spatial distribution analysis that is
applicable to the study of areally distributed urban phenomena in general.
In this chapter, we have presented our perception of the need for
such a method. Recognizing a fundamental gap between current concep-
tions of spatial organization and current theories of information pro-
cessing, in Chapter II we examine briefly the basic concepts of infor-
mation theory searching for some general mathematical basis for quanti-
tative characterization and analysis of spatially organized phenomena.
Here, a specific mathematical isomorphism is observed between the for-
mulas of information theory and certain concepts employed within entropy-
maximization models of urban spatial interaction. The relationship noted
seems particularly germane to our present problem in that it provides
an initial bridge between the concepts of information theory and current
behavioral models of urban spatial organization.
In Chapter III, we review certain basic measures commonly used
within the analysis of areal distributions. Following this investiga-
tion of existing methods, in Chapter IV we employ the fundamental
rationale of entropy-maximization in developing a new approach to the
quantitative characterization of spatial associations between areal
distributions. The model developed here unites in a unique manner
measures of the spatial congruence between areal distributions with
information theoretic measures of the complexity of structure transmit-
ted between them. In Chapter V we illustrate the utility of the method
16
developed by applying it directly to analysis of certain areally dis-
tributed phenomena of a hypothetical urban area. Possible applications
of the model for description of real-world urban spatial organization
are discussed briefly in a concluding chapter.
CHAPTER II
INFORMATION THEORY, PHYSICAL DISTANCE,
AND URBAN SPATIAL ORGANIZATION
Introduction
Immediately following the development of mathematical informa-
tion theory (communication theory) by Claude Shannon (1948, 1949) and
Norbert Wiener (1948), there existed much excitement throughout the
social and life sciences concerning application of the basic concepts
and formulas of Shannon-Wiener information theory to problems invol-
ving analysis of systems of organized complexity.
Such widespread enthusiasm resulted from the appearance in the
works of Shannon and Wiener, as a fundamental measure of information,
the mathematical expression of entropy — a concept employed in physics
to quantify the disorder of closed thermodynamic systems. Wiener him-
self had claimed that
the notion of the amount of information attaches itself very natu-
rally to a classical notion in statistical mechanics: that of
entropy. Just as the amount of information in a system is a mea-
sure of its degree of organization, so the entropy of a system is
a measure of its degree of disorganization; and the one is simply
the negative of the other. (1948, p. 11)
Thus , it was all too easy to relate directly the entropy of Shannon-
Wiener information theory with the entropy of physics that ever increas-
es according to the second law of thermodynamics — the law that accord-
ing to Eddington (1935), holds "... the supreme position among the
18
laws of Nature." (Weaver, 1949, p. 12) Likewise, it was all too easy
to relate the entropy of Shannon-Wiener information theory with the
semantic information of human thought and communication and, by casual
reference to Schrodinger's speculation (1945) that "life feeds on nega-
tive entropy," with the very concept of biological organization itself.
(Rapoport, 1956)
Following the excitement generated by the works of Shannon and
Wiener, there occurred considerable refinement, extension, and applica-
tion of the fundamental concepts of information theory toward solution
of complex problems in a wide variety of disciplines , including commu-
nications engineering (Goldman, 1953; Raisback , 1963; MacKay , 1969);
mathematics and mathematical statistics (Kullback, 1953; Khinchin,
1957); biology (Raymond, 1950; Quastler , 1953 ) psychology (Miller,
1953; Quastler, 1955; McGill , 1954; Attneave, 1959; Garner, 1962);
and urban and regional studies (Meier, 1962). In Miller's words, "the
reason for the fuss is that information theory provides a yardstick for
measuring organization." (1953, p. 3) Despite the attention devoted to
the applicability of information theory for solution of complex scien-
tific problems, to our knowledge, no one to date has demonstrated in
any practical manner the utility of information theory for descriptive
analysis of problems of organized complexity comparable to that of
urban spatial organization.
However, recently Wilson (1970) has shown the usefulness of the
entropy concept in a wide variety of urban and regional models , inclu-
ding models of trip distribution, residential location, and inter-regional
commodity flows. The fact that in all of these models the concept of
entropy is related directly to the spatial distribution of urban
19
activities and the distribution of flows between activities raises the
question of the extent to which the concept of entropy might be appro-
priated for general quantitative description of urban spatial organiza-
tion.
In this chapter we review the basic concepts and mathematical
formulas of information theory, attempting wherever possible to relate
the existing theory to issues associated with urban spatial structure.
Here, our purpose is two-fold. First, we wish to show how the concepts
of information theory may be applied directly to quantify certain
aspects of urban spatial organization related to the spatial distri-
butions of activity places and the circulation of persons between acti-
vities. Second, we wish to introduce into our discussion those concepts
and formulas that we will find useful throughout the following chapters
in developing our own alternative methodology for description of urban
space as a complex system of patterned phenomena.
Communication, Information, and Entropy
It is not surprising that the terms information theory and commu-
nication theory are often used interchangeably : wherever communication
occurs, information in some form is transmitted from one source to
another. Shannon formalized this proposition quite distinctly in stat-
ing that "the fundamental problem of communication is that of reprodu-
cing at one point either exactly or approximately a message selected
at another point." (1949, p. 31) Shannon conceived of any communica-
tion system as consisting of six essential components. An information
source selects for transmission a particular message from a finite set
of possible messages. A transmitter or encoder transforms the message
20
into a signal which is then actually transmitted over a communication
channel to a receiver or message decoder. Once the signal has been
received and decoded, it is then available for use at the information
destination. Communication problems arise from the fact that at any
stage of the communication process noise may be introduced, thus com-
plicating the task of accurate message transmission.
Information
Source
Noise
Channel
Information
Destination
Encoder
Decoder
Fig. 1. Schematic diagram of a general communication system
(after Shannon, 1949)
In Figure 1 we have revised Shannon's diagram of a general com-
munication system to emphasize the nature of the encoding and decoding
operations that occur at either end of a communication channel. It is
important to note that in Shannon's schema messages conveyed from
source to destination, however complex, are necessarily organized in
terms of a finite vocabulary of semantic elements or alphabet (e.g.,
the character set of a teletype) common to both encoding and decoding
operations alike. Note also that except for labels and schematic indi-
cations of information flow, the symmetry of the diagram reflects the
bi-directional nature of all communication processes.
21
The mathematical theory of communication proposed by Shannon
treats only the engineering problems associated with the transmission
of encoded messages or signals across channels in the presence of noise
Thus , while his broader conceptual framework recognizes the existence
of information sources, encoders, decoders, and information uses, de-
spite Weaver's speculations (1949a) concerning the more general appli-
cability of Shannon's theory to issues of meaning , Shannon himself
restricts the application of his theory to problems of signal storage
and transmission. In his own words, "the semantic aspects of communi-
cation are irrelevant to the engineering problem." (1949, p. 31) We
raise this issue here simply to express our opinion that the failure
of numerous attempts to generalize Shannon's mathematical theory to
treat problems of semantic information transmission is due to the in-
adequacy of the original mathematical concepts and formulas to treat
explicitly pattern information, i.e., information conveyed in the form
of spatial and/or temporal organizations of phenomena.
As noted above, Shannon's theory assumes that the message to be
transmitted from information source to destination must be selected
from a finite set of possible messages common to both encoding and de-
coding operations. We assume for the sake of generality that many mes-
sages are transmitted, some messages are transmitted more frequently
than others, and there is associated with any particular information
source a discrete probability distribution characterizing the relative
frequencies of messages emanating from the source.
Following earlier concepts of information used in communications
engineering (Hartley, 1928; Nyquist , 1924) and appealing to his intui-
tion, Shannon defined mathematically the amount of information that is
22
associated with any particular message transmitted over a specific
information channel as the log of the reciprocal of its probability of
occurrence. Since for any discrete probability p. we have 0<p.<l,
log (1/p.) = -log p.. Hence, -log p. is an equivalent measure of the
amount of information or "surprise" associated with a particular mes-
sage. Now if (x ,x ,...,x ) represents the discrete probabilities
associated with the n messages emanating from a particular information
source X , then
n
(2.1) H(X) = - I x.log x.
i 1 1
may be considered as the average quantity of information transmitted
from the particular source over a sequence of transmissions. Since
before a particular message is received from a source X, one would know
only the set of a priori probabilities (x1,x9,...,x ), the quantity
H(X) may also be considered a measure of the uncertainty associated
with source X.
Now the expression for entropy as defined in certain formulations
of statistical mechanics is
n
(2.2) H - - K Z p log p
i x -1
where p. is the probability of a system being in a specific state i
and where K is a positive constant that amounts merely to a choice of
a unit of measure.
Thus, Shannon's formula for the average amount of information asso-
ciated with a particular information source differs only from the entropy
concept of thermodynamics by the constant K. It can be shown that the
choice of a value for K is equivalent to the choice of a specific base
23
for the log functions of formulas (2.1) and (2.2). Intuition tells
us that the most elementary unit of information occurs in the form of
a binary or dichotomous outcome. Recognizing this condition and employ-
ing the base 2 for all log functions within his mathematical measures of
information, Shannon's formula (2.1) measures the number of binary units
(dichotomous messages) or bits equivalent to the expected information
from a source X. By analogy with formula (2.2), Shannon refers to this
quantity of a priori uncertainty or expected information as the entropy
of the information source X. For a particular information source X,
the maximum possible amount of information transmitted by the source
occurs when x =x =...=x and this quantity H(X) = log n bits.
Early arguments by Wiener, Weaver, and Miller that entropy repre-
sented a meaningful measure of the disorder of any probabilistic system
were based principally on certain mathematical properties satisfied
uniquely by the entropy concept. Here the notion of the disorganiza-
tion of a probabilistic system was equated with the randomness of a
discrete probability distribution characterizing the relative frequen-
cies of states of the system.
Let H(p ,p ,... ,p ) represent a measure of the randomness of any
discrete probability distribution (p1 ,p „,...,p ). Then it is reasonable
to require of such a function H the following properties.
a. H should be a continuous function of the p..
l
b. If all the p. are equal, p.=l/n, then H should be an
increasing function of n.
c. Suppose that the p. are grouped in various ways and let
wl = Pl+P2+*"+Pk
w2 = Vi+pkt2---+pe.
etc.
24
Then the following composition law should be satisfied:
(2.3) H(p1,p2,...pn) = H(w1,w2,. ..) + w1H(p1|w1,p2|w1,...)
It can be shown that the entropy function is unique in satisfying these
three conditions (Jaynes, 1957; Khinchin, 1957; Shannon and Weaver,
1949).
Shannon arrived at his choice of the entropy function of the
measure associated with an information source purely by means of prag-
matic reasoning and without need for the condition of its uniqueness
with respect to the above three properties. Others, however, recog-
nized the possibilities inherent in the uniqueness of entropy as a
measure of probabilistic disorder. By equating entropy with informa-
tion uncertainty, Shannon himself indirectly provided support for the
belief that entropy represented the most fruitful measure of order-
disorder relationships within complex systems.
Information Transmission
In this section we return to Shannon's engineering problem of
information transmission in the presence of noise and describe how
the concept of entropy is used within communication theory to measure
the rate of transmission between information sources and destinations.
Let X be an information source that encodes and transmits through
a particular communication channel messages drawn from a finite set of
m messages with associated probabilities (x ,x ,...,x ). At the other
end of tho communicul ion channel, let Z bo .in information sink that
receives and decodes sequences of the m messages transmitted by X,
25
and let (z ,z ,...,z ) be the probability distribution of messages re-
corded at Z. Now we may denote the average amount of information trans-
mitted by X and the average amount of information received at Z respec-
tively as
m
(2.iO H(X) = - I x log x
i x
m
(2.5) H(Z) = - E z.log z.
j J J
Now suppose the existence of an observer capable of recording for
each message transmitted from X the message as received at Z. Such an
observer would be capable of tabulating a joint probability distribu-
tion indicating the number of times that an i-th message encoded at X
was decoded as a j-th message at Z.
For the sake of simplicity, let us assume that the set of messages
at both X and Z are arranged in one-to-one correspondence and are both
rank ordered according to the values of their subscripts i and j. Thus,
whenever a message sent from X is received properly at Z, the value of
i equals the value of j ; otherwise, i/j.
Now let Q = [q. •] be the joint probability distribution observed
for a sequence of message encodings at X and message decodings at Z.
Then the joint entropy of X and Z, denoted H(X,Z) or H(Q) , is defined as
m m
(2.6)
H(Q) = - I I q, .log q .
Note that error- free transmission of messages from X to Z, i.e.,
the case of complete absence of noise , would result in a matrix Q where
q..=x.=z. wherever i=i and where q..=0 wherever i/j . Since Shannon
i] i ] iD
defines -x.log x. as 0 for x.=0, it should be obvious that for this
11 i
special case of noiseless transmission H(Q) =H( X)=H(Z) .
26
The introduction of noise into such a conmunication process im-
plies that, for some number of message transmissions, an i-th message
sent from X will be received and decoded improperly as a yth message
at Z. This means that q. .>0 for some i^j. Furthermore, it is shown
that H(Q) is greater than either H(X) or H(Z) and, in fact, H(Q)
approaches the limit H(X) + H(Z) as the level of noise within the
channel increases to the point of zero information transmission. This
represents the limiting case where the distribution of messages decoded
at Z exhibits complete statistical independence from the distribution
of messages sent from X.
Shannon defines the rate of transmission, or simply the trans-
mission of information from the source X to the destination Z through
a noisy channel as
(2.7) T(X,Z) = H(X) + H(Z) - H(X,Z) .
It may be shown (Goldman, 1953) that
(2.8) H(Q) = H(X,Z) < H(X) + H(Z)
with the equality holding only in the case of zero transmission. Since
H(Q)=H(X,Z)=H(X)=H(Z) in the case of error-free communication, via (2.7),
T(X,Z)=H(X)=H(Z) ; that is, all of the information produced at X is
received at Z. In the general case where noise is introduced at some
point within the communication channel H ( X ,Z) >H( X) and H (X,Z)>H(Z) ,
and thus the transmission will be imperfect between X and Z. Hence,
T(X,Z)<H(X) and T(X,Z)<H(Z). Note, however, that for all cases, the
transmission function is symmetric, i .e., T( X,Z)=T(Z ,X) .
27
While to this point we have restricted our discussion of Shannon-
Wiener information theory to the engineering problems of telecommunica-
tions, it should be noted that wherever there exists a joint probability
distribution recording the contingency of discrete probability distri-
butions the same theoretical concepts may be applied for quantification
of the statistical interdependence of the two distributions. In parti-
cular, information theoretic concepts have been used quite widely for
analysis of cross-tabulations of multivariate categorical observations
or contingency tables. For such applications information theory pro-
vides a means of non-parametric contingency analysis directly analogous
to methods based on the chi-square distribution. Furthermore, as
Attneave (1959) and Garner (1962) have demonstrated, the methodology
readily generalizes to the analysis of statistical interdependence
within three-way and higher-dimensional contingency tables.
The Entropy-Maximizing Model of Urban Trip Distribution
Trip distribution models are used as one component within the
metropolitan transportation-land use planning process. (Creighton,
1970; Wilson, 1970; Potts and Oliver, 1972) The purpose of such models
is to provide a meanc for simulating the travel behavior associated
with the socioeconomic behavior of inhabitants of the metropolitan
region.
Typically within the transportation-land use planning studies for
a metropolitan region, a large quantity of data is collected for a
random sample of households. For some 24-hour week day, data is record-
ed for each individual on certain socioeconomic variables and on every
28
trip away from home. For each trip, data concerning the geographic
location and land use for each trip origin and destination is recorded
along with the purpose for which the trip was made. From such data
our most comprehensive description of the interrelationships between
urban land use patterns and patterns of social behavior at the urban
scale are obtained.
Since the beginning of transportation studies it has been gene-
rally recognized that for any one particular trip purpose the number
of trips between any two locations varies inversely with some func-
tion of the distance separating the two locations. This simply means
that, all other things being equal, individuals have a propensity to
minimize distance travelled in the satisfaction of their activity
needs. Trip distribution models formalize in mathematical terms this
well-documented characteristic of urban travel behavior.
Regardless of the type of trip distribution model used (see Potts
and Oliver, 1972), the fundamental purpose of such models, e.g., gravity
models, intervening-opportunities models, is to simulate the distri-
bution of trips between spatial patterns of different land uses and
socioeconomic activities in a manner that best fits available data.
The entropy-maximizing model of trip distribution elaborated by
Wilson (1970) and Tomlin and Tomlin (1968) seems particularly attrac-
tive as a methodology for trip distribution modeling for a number of
reasons. First, as Wilson has shown, both the gravity model and the
intervening-opportunities model of travel behavior can be reformulated
with only minor alteration of certain parameters within the entropy-
maximizing framework. Second, the entropy-maximization methodology re-
lates directly the mathematical concept of entropy as used in statistical
29
mechanics and information theory to the probabilistic linkages between
spatial patterns of land use and activities within a metropolitan
region. Thus, the entropy-maximizing model would seem to provide an
appropriate method for measuring the degree of organization exhibited
by observed travel behavior.
Third, it is generally agreed that travel behavior patterns,
mediated by proximity relationships between urban locations, deter-
mine in large measure the spatial patterning of urban land uses and
activities. Since the entropy-maximizing approach provides a means
for unbiased simulation of urban travel patterns with respect to all
information available, the approach seems worthy of in-depth consi-
deration within the development of any methodology designed for mathe-
matical description of urban spatial organization in general.
The entropy-maximizing model of trip distribution can be formu-
lated mathematically in the following manner. The model assumes the
availability of survey data describing the spatial distributions of
social populations and economic activities over some set of analysis
zones subdividing an urbanized region, minimal travel distances (times,
costs) existing between all pairs of zones, and estimates of average
travel times for trips of specific purposes. To be specific, let D
represent the mean travel time for all home-work commuting trips, let
X be the probability distribution of workers over m residential zones,
let Z be the distribution of jobs over n employment zones, and let S
be a matrix of minimum network travel times between any residential
zone and any employment zone. The problem requires determination of
a most probable, mean, or maximum entropy joint probability distribu-
tion Q with marginals X and Z such that each element q. . represents
30
the forecasted proportion of all trips occurring between the i-th
residential zone and the j-th employment zone. Mathematically, the
problem is formulated
m n
(2.9) max H -- - II q. .log q. .
i j i.D ifj
subject to the constraints
m
(2.10) I q. . = z. j = 1,.. . ,n
i ^ »D D
n
(2.11) E q. . = x. i = l,...,m
j i»3 i
(2.12) q. . > 0 i = 1,.. . ,m
j = 1 , . . . ,n
and the additional mean travel time constraint
m n
(2.13) E E q. . s. . = D
i j i»: i.l
Note that constraint (2.13) may be taken as simply an a priori speci-
fication of overall network distribution efficiency or time expendi-
ture.
The solution to the problem is given by
(2.14) q. . = x.u.z.v. exp(-3s. .) i = 1 , . . . ,m
i,: i i i i i»:
j = 1 ,. .. ,n
where 3 represents the Lagrange multiplier associated with constraint (2.13)
31
and the u. and v. are functions of the Lagrange multipliers associated
with constraint sets (2.10) and (2.11). It is known (Evans, 1970)
that corresponding to any real 3 there exists a unique Q maximizing
(2.9) and satisfying (2.10), (2.11), and (2.12) given by (2. If) where para-
meters u. and v. may be determined by iterative solution of the equa-
tions
rn i-l
(2.15) u. = »Z z.v. exp(-Bs. .)]" i = 1 , . . . ,m
1 j ] D 1>D
(2.16) v. = [l x.u. exp(-$s. .)J ] - l,...»n
3 l. i l 1,1
Additionally, it has been shown that there exists a monotonic mapping
between all $ and all feasible D such that as 3 approaches -00, D
approaches D , and as 3 approaches +°°, D approaches D . , where D
max r rr mm max
and D • respectively denote the maximum and minimum values of D possible
min r j r
for given S, X, and Z. (A. W. Evans, 1971; S. P. Evans, 1973) Both
D . and D may be determined by solution of the Hitchcock or trans-
mm max J J
portation problem (Dantzig, 1963; Dorfman et al. , 1958) uniquely deter-
mined by S , X, and Z. Together these results yield theoretical justi-
fication for iterative determination of the unique Q maximizing (2.9)
and satisfying the network distribution efficiency constraint (2.13) as
well as constraints (2.10), (2.11), and (2.12). (See Eigure 2).
Now, the entropy-maximizing model of transportation flows is
based on the probabilistic spatial distributions of two activity classes,
and the simulated distribution of trips is represented within the model
as a joint probability matrix associating specific trip origins and
destinations. Hence, it is possible to apply directly Shannon-Wiener
information theory concepts to the model for quantifying the degree of
32
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33
randomness associated with the specific distribution of trips determined
by tne model. One such measure of the randomness or disorganization of a
particular trip distribution can be formulated simply as H(Q)/ [h(X)+H(Z)]
where Q is the joint probability distribution determined by the entropy-
maximizing model and X and Z represent the probability distributions
associated with the two activity classes between which trips are distri-
buted. Note here that the denominator of the measure above is simply
the maximum value that H(Q) can assume. This value of H(Q) would occur
only if all travel behavior occurred in a manner completely insensitive
to distances between analysis zones.
Note also that the concept of transmission of Shannon-Wiener
information theory can be usefully employed for quantification of the
level of organization exhibited by the simulated distribution of trips
between activities. In the above example, we may use the measure of
information transmission given by Shannon directly to measure the amount
of contingency existing between places of employment and places of
residence. Remember that the formulation of transmission between two
probability distributions given by Shannon is symmetric. Hence, given
a large value of transmission between X and Z , we cannot infer that
an individual's choice of a place of employment is highly dependent
upon the location of his residence; nor can we infer the converse, that
places of residence are chosen to a large extent with reference to
individual work locations. In fact, all that we can infer from a high
value of transmission is that, for individuals, work locations and
home locations are highly interdependent and that, knowing one's place
of residence gives us much information concerning his place of employ-
ment; likewise, knowing his place of work tells us much concerning
where he resides. (Again, see Figure 2).
-
-
' ' ' » " " « M M .5 || „ „ „
» n n a j4
LEGEND
BBSS LivELiHoon
liiiillD Residential
iij urban Vacant
AGP I CULTURAL
• I
t=k^M
ILES
Fig. 3 A hypothetical region containing four urb
an areas
35
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14.0
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12.0
11.0 .
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values of 8
3.0 3.5
4.0
Fig. 4 . Mean trip length D and spatial information transmission T
as functions of 3 for home-to-work trips within the hypothetical region of
Fig. 3.
36
To illustrate this point more dramatically, consider the regional
landscape depicted in Figure 3. Simply by visual inspection, most would
agree that there is apparent a high degree of spatial co- organization
between the geographic distributions of places of work and places of
residence. Furthermore, the relatively sharp curves in the graphs of
Figure 4 suggest that, if the inhabitants of our hypothetical region are
at all sensitive to commuting distances in their joint choices of places
of employment and places of residence, then most will live and work
within the same community.
In preparing Figures 3 and 4, we have assumed that: (1) jobs and
homes are distributed in uniform manner over all tracts of livelihood
and residential land uses, (2) that the ratio of jobs to residences is
constant over all four communities within the region, and (3) that all
home-to-work commuting patterns may be approximated well by the entropy-
maximizing model of trip distribution given above with all s. . *s expres-
sed in units of miles. Also for convenience, we have assumed that all
jobs and residences within each tract are concentrated at point loca-
tions representing the centroids of each tract.
To construct the graphs of Figure 4, ten different values of D
(mean trip length) and T (spatial information transmission) were computed
corresponding to ten different values of the parameter $. Notice how
quickly the mean commuting distance falls initially with increasing
values of £. Also, notice how quickly the information transmission func-
tion T rises as 8 increases.
Now, let us focus solely on the community of the northwest corner
of the hypothetical region and examine home-to-shop and home-to-school
37
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Legend
0 Livelihood land use
Q Elementary school
□ Residential land use
F=r
-\ y-
h h o
3
1 Mile
Fig. 5. Spatial distributions of livelihood and residential land
uses and elenientary schools within a hypothetical urban area.
38
3.0
home-to-work trips
4.0
6.0
_ 5.0
4.0
2.0
1.0
0.0
2.0 2.5
values of 8
Fig. 6. Mean trip length D and spatial information transmis
as functions of 8 for home-to-work and home-to-shop trips within the
thetical urban area of Fig. 5.
W
•H
s
•H
W
W
•H
e
w
3.0 u
c
o
•H
4->
c
•H
sion T
hypo-
to
•H
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o
c
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to
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II
Q
2.0 2.5 3.0
values of 8
3.5 4.0
to
•H
c
o
•H
CO
CO
•H
£
to
•H
g
•H
■M
to
O
C
•H
II
H
Fig. 7. Mean trip length D and spatial information transmission T
as functions of B for home-to-school trips within the hypothetical urban
area of Fig. 5
39
trips as well as the home-to-work trips considered before for the region
as a whole. Figure 5 depicts the northwest community in somewhat more
detail, this time showing the locations of all elementary schools as
well. The community consists of twenty square-mile sections with one
elementary school located in each section. To simplify our example, we
assume that shopping places and employment places are uniformly distri-
buted over all tracts graphically coded as livelihood land use, and,
again, all activities are assumed to be concentrated at the centroids
of sections.
Referring to Figure 6, it will be noticed immediately that home-
to-work mean trip lengths are not as sensitive to small values of 3 as
they were for the region as a whole. This is simply a consequence of
the fact that the variation of distances between all homes and jobs
within the single community is much smaller than that for the entire
region. Note also, with reference to Figures 3 and 5, that as we move
from the geographic scale of the region to that of the city it becomes
more difficult to think of residential and livelihood land uses as being
spatially co-organized.
Now for experimental purposes , let us make some behavioral assump-
tions concerning travel patterns within the hypothetical community of
Figure 5. From the results of numerous transportation studies, it is
widely recognized that the scale factor (or exponent) applied to travel
distances in fitting trip distribution models to observed data varies
in accordance with the characteristics of the trip maker and the specific
purpose of his trip. For example, most origin-destination survey data
suggest that the factor to be applied should be higher for most shopping
40
trips than it should be for work trips, and presumably much higher still
for trips between home and elementary schools. (Hoover, 1968)
Let us choose values of 3 of .6, 1.2, and 2.7 for home-to-work,
home-to-shop, and home-to-school trips respectively. Then, as shown in
Figures 6 and 7, mean trip lengths are, respectively, 1.75, 1.25, and
.25 miles for these trip purposes. More importantly, notice the rela-
tionship between the values of mean trip length D and the spatial infor-
mation transmission function T. As D decreases with successively higher
values of (3, T increases. This seems perfectly reasonable, since the
greater the sensitivity to distance for trips of different purposes, the
greater should be the spatial interdependence between trip origin and
destination locations. Thus, Figures 6 and 7 demonstrate the obvious
fact that, attempting to predict the location of a particular household
within our urban area, we should receive much more information from
knowledge concerning schools attended by the children of the household
than from knowledge concerning where the parents shop and work.
Here, one further observation is appropriate. Suppose that we
apply a similar form of analysis to a journey- to-work, origin-destination
contingency table, determined not by simulation, but rather taken
directly from actual survey data for an existing urban area. Suppose
further that the residences of blue-collar workers are clustered together
in downtown areas of the city and that all white-collar workers reside
in outlying suburban neighborhoods. Also, assume that the majority of
white-collar jobs are clustered in the central business district of the
city and existing industries are located at the intersections of major
roadway and rail transportation routes at the periphery of the city.
41
Then, while there may be quite large information transmission between
places of residence and places of work for all employed, inspection of
the particularities of the urban spatial structure exhibited would indi-
cate to us that this interdependence of places of home and work must be
due primarily to sociocultural forces at play organizing urban space
and that the friction of distance between residential and employment
centers is of little concern.
The question then arises: do there exist other areal distribu-
tions of landscape features or socioeconomic conditions that, acting as
other forces, bring about the apparent insensitivity of community mem-
bers to monetary and time costs associated with home-to-work commuting?
For example, are white-collar residences spatially co-organized with
respect to a particular set of elementary and secondary schools more
favored by that particular socioeconomic population, or are executive
residences aligned spatially along a scenic river front? Is the areal
distribution of low-cost and older housing such that blue-collar resi-
dences are clustered through economic segregation in downtown neighbor-
hoods?
These questions lead us directly to the problem of characterizing
urban spatial organization in terms of structures of spatial associa-
tions existing between general patterns of urban phenomena. To what
extent can the concepts of information theory assist us here? Before
presenting our specific answer to this question, let us turn in the next
chapter to an examination of certain basic issues confronting the analy-
sis of areally distributed data.
CHAPTER III
SOME PRELIMINARY METHODS FOR ANALYSIS OF URBAN SPATIAL DISTRIBUTIONS
Introduction
In this chapter we return to our main objective, namely, the
investigation of more general methods for quantitative description of
urban spatial organization as a complex system of differentiated popu-
lation, socioeconomic activity, and land use patterns. Again, our focus
is on the city as a system of geographically patterned phenomena. We
are concerned with social behavior only to the extent that macro behavior
patterns may be suggested by specific geographic configurations of popu-
lations and activity places. Our main objective is the development of
alternative quantitative methods better equipped for analysis of the
spatial interdependence exhibited among geographic patterns of urban
phenomena.
Four major problems confront us within this task. The first prob-
lem is that of representing specific urban patterns as discrete areal
distributions that characterize in economical fashion the essential pro-
perties of the phenomena of interest. Two fundamental issues involved
here concern the choice of a set of variables for point -by-point measure-
ment of all patterned phenomena and the choice of a frame of areal
subdivisions of the urban area for use as a common basis for aggrega-
tion of all measurements. A second problem concerns quantitative
43
characterization of overall distribution properties. Common measures
used here include the geographic coordinates of distribution centroids
as measures of central tendency and various statistical moments about
these centroids as measures of distribution dispersion. A third prob-
lem involves the measurement of spatial association between differen-
tiated urban distributions. It is by such measures that inferences
about the ecological interdependence of distributions can be made.
Finally, a fourth problem involves analysis of the structure of asso-
ciations among areal distributions. It is here that we hope to arrive
quantitatively at those syntactical regularities of urban spatial organ-
ization exhibited in comparable manner across urban areas.
It should be noted that these four problems confounding the anal-
ysis of geographic patterns of urban phenomena are highly interrelated.
Most importantly, the utility and validity of all analysis results will
depend on our choice of a specific set of quantifiable variables and
our selection of a particular spatial sampling frame for representation
of all patterns of interest. Of course, we should select that set of
variables most closely identified with the specific urban phenomena we
wish to analyze. Given that the discrete representations of patterns
will inevitably depend to some extent on the particular system of areal
units selected for aggregation of all data, we must expect our analysis
results to depend on the spatial sampling frame as well. Here, the best
we can do is to choose a system of areal units of sufficiently fine reso-
lution to capture the essential characteristics of all patterns of inter-
est, and to employ analysis methods that depend only incidentally on
the particular frame selected.
44
Most methodological issues confronting the quantitative charac-
terization and analysis of urban spatial patterns come sharply into
focus if we recall the distinction betv;een parametric and non-parametric
statistical distributions. A parametric distribution is a probability
series that may be completely specified with reference to some number
of numerical parameters quite small relative to the potentially infinite
set of data values associated with the distribution itself. For example,
if a univariate distribution is known to be normal, then the entire dis-
tribution is completely characterized by only two parameters, i.e., its
mean and its variance. On the other hand, if the distribution is known
to be non-parametric and not well approximated by any known parametric
distribution then, while we may compute any number of summary statistics
and moments based on discrete samplings of the distribution, these
measures may assist us little in characterizing the overall nature of
the distribution itself.
It is one of the fundamental premises of this thesis that geogra-
phic patterns of urban phenomena cannot in general be adequately approx-
imated in terms of bivariate parametric distributions. Thus, we contend
that the most appropriate characterization of any specific pattern is
given by the complete representation of the pattern itself, i.e. , its
representation as an areal distribution of some measurable variable
whose value is recorded across a complete frame of spatial sampling units.
This is not to argue that there exist no summary measures of overall
distribution properties of value. The issue is, rather, just what over-
all distribution properties, in addition to such properties as central
tendency and dispersion, should we attempt to quantify. For example,
it would seem desirable to have some measure of the overall spatial
45
complexity associated with a particular distribution. Here, with res-
pect to the problem of unambiguous definition of such a concept as spatial
complexity , the position we shall assume is that whatever concept we
employ, like the concept of distribution variance, will only be definable
in mathematical terms.
In the remainder of this chapter and in Chapter IV, we develop an
alternative approach to the analysis of urban spatial distributions that
addresses in unified mathematical format all of the methodological issues
discussed above. Based on a maximum-entropy formulation of spatial rela-
tionships among areal distributions, the model yields a variety of mea-
sures useful for quantitative characterization of certain aspects of
intra-distribution spatial complexity and organization and inter-distri-
bution spatial association. Surprisingly enough, the model also yields
a new technique for hierarchical cluster analysis of areal distributions
based on the structure of spatial associations determined among them.
Characterization of Urban Patterns as Areal Distributions
Like all other methods used for analysis of geographically distri-
buted socioeconomic data, the methods that we propose here depend in a
fundamental way on the manner by which we characterize urban patterns
as discrete areal distributions. Of course, we assume the existence of
measurable variables closely identified with all phenomena of interest.
In many instances, however, due to data collection costs, confidential-
ity restrictions, or qualitative judgements in codification, we are
forced to settle for only proxy variables.
A more ambiguous collection of methodological issues surrounds
our choice of a specific system or frame of areal subdivisions of an
U6
urban area for use as a common basis for aggregation of all data and
representation of all patterned phenomena as discrete areal distribu-
tions. The analysis methods that we will develop here require that we
select our spatial sampling frame with respect to three general sets of
conditions.
First, some a priori delineation of the outer boundaries of an
urban area is required. Then it is assumed that the subdivisions of
the area are non-overlapping and cover exhaustively the complete urban
area. Thus, each data measurement will fall within one and only one
geographic areal unit or tract. Further, the tabulation (or statisti-
cal estimation) of aggregate variable values across all tracts should
comprise sufficient information for representation of urban patterns as
area-wide probabilistic distributions.
Second, it is assumed that areal units are of sufficient number
and scale to capture the essential spatial properties of all patterns
of interest. This condition concerns the spatial resolution of the samp-
ling frame employed. At too coarse a level of resolution, spatial pat-
tern features of interest will be lost. For example, if we wish a
detailed characterization of the pattern of neighborhood commercial
establishments throughout an urban area, a sampling frame of relatively
fine resolution must be employed. On the other hand, if we are concern-
ed only with the pattern of major centers of commercial activity, then
a much coarser sampling frame will do.
Third, it is assumed that all areal units are compact in shape.
While we do not require a regular grid, no tract should be overly
elongated in any one direction or curvilinear. This condition arises
as a result of two basic requirements of our mathematical model. First,
i+7
it is important that the centroids of individual tracts represent good
approximations (relative to tract sizes) of the point locations of all
variable measurements taken within tracts. Second, we wish geographic
coordinate pairs for points within tracts to be uncorrelated and to
remain uncorrelated over rotational transformations of coordinates.
Where all of these conditions are met within the specification
of a frame of areal units, for the purposes of our modeling strategy,
the complete frame itself may be represented numerically in the follow-
ing manner. We first establish a planar geographic coordinate system
having x and y orthogonal axes and origin fixed relative to the geogra-
phy of the urban area. Any unit of length convenient for expression of
distances (miles, kilometers) may be selected for coordinate intervals.
Now let there be n tracts comprising the frame and let all tracts
be permanently numbered 1 through n. Associated with each tract i will
be four descriptive constants: Mx., My., Vx. , and Vy.. Mx. and My.
r l'-'l'l' J 1 1 J 1
represent the coordinates of the centroid of the i-th tract taken with
respect to the established x ,y coordinate system. Vx. and Vy. represent
x and y component variances associated with a uniform distribution of
points over the area defined by the i-th tract. Note now that our
numerical representation of the complete frame of areal units is simply
an array of summary measures describing the positions and sizes of all
n tracts. The x and y centroid coordinates of all tracts are taken as
measures of their relative positions, and, since we have assumed com-
pactness for all tracts, the x and y component variances of intra-tract
point di:;tril)u1 ion:; <ire r.lor.oly proport ionn] to tho squaren of x and y
tract dimensions.
48
Our requirement that all tracts be compact in shape will, of
course, imply that the values of Vx. and Vy. for each tract will not
differ by much. Thus, it will be convenient for many analyses to simply
assume that Vx.=Vy. for all i=l,...,n and reduce the number of descrip-
tive constants for each tract from four to three. It will facilitate
our mathematical discussion here, however, to maintain separate nota-
tions for Vx. and Vy..
1 1
Having described our method for selecting a specific spatial
sampling frame and representing it numerically, it remains only to be
said that all patterns of urban phenomena will be represented as dis-
crete probability distributions of specific variables across the set of
tracts comprising the frame. For maximum generality, we will assume
that data values for all geographic patterns to be analyzed have been
measured, either explicitly or implicitly, over all tracts. Thus, any
particular spatial distribution may be represented mathematically as a
vector J of n elements where n is the number of areal units, f denotes
the particular areal distribution, and the elements ^z.* i=l,...,n, are
probabilities proportional to the aggregated data values recorded for
each of the n areal units. Thus, ,-z.>0 for all i and for all f,
and
I . _z . = 1 for all f .
l fi
One further note concerning vocabulary is appropriate. We will
occasionally find it convenient to speak of the elements of an areal
distribution. By the term elements of a distribution, we intend gene-
rally to denote those areal units or tracts having non-zero quantities
of the variable measured in representing some pattern of phenomena as
a discrete areal distribution. For maximum mathematical generality,
however, we will preserve the option of characterizing all distributions
49
as consisting uniformly of n elements (n the total number of tracts)
where each particular probability vector JL may contain numerous zero
elements.
Basic Measures of Central Tendency and Dispersion
for Areal Distributions
For a given areal distribution f , let fMx and My denote the x
and y coordinates of the centroid or "center of gravity" of the distri-
bution considered as a whole. These distribution centroid coordinates
are defined by the formulas:
n
(3.1) Jx = I jz. Mx. ,
f j fi 1
(3.2) fMy = I fz. My. ,
where again the Mx.'s and My.'s are constants over all distributions
representing the x and y centroid coordinates of all n individual areal
units comprising the spatial sampling frame. Thus, fRx and My are
measures of distribution central tendency. As such, they represent the
average position or mean spatial coordinates for all point locations
of phenomena associated with the particular distribution f.
Now let fVx and ^Vy denote the two component variances associated
with the same areal distribution f measured with respect to the x and y
frame axes. We may then take as a generalized measure of overall spatial
distribution dispersion the quantity
(3.3) fDV = Vx + Vy
50
Following Neft (1966, p. 55), we will refer to this measure fDV as the
distance variance associated with the areal distribution f.
Let us consider in turn the two component variances fVx and _Vy
associated with f. From the definition of variance, we have
fVx = E(fCx2) - [E(fCx)]2
where Cx is a random variable denoting the x coordinate of any randomly
selected point of occurrence of phenomena contributing to the distribu-
r 1 2 - 2
tion f. Clearly, LE(fCx)J = Mx . This condition, together with cer-
tain additivity properties of expectation, allow us to write
(3.4) J/x = E _z. E(Xx2) - Jlx2
f i fi f i f
2
Considering the random variable ,.Cx . , note that
E(^Cx2) = E[(£Cx. - Mx.) + Mx.]2
f l L f l l iJ
0 0 i
= E[(rCx. - Mx.) + Mx. + 2Mx.(£Cx. - Mx.)J ,
Lfi l l lfi iJ
and since Ef2Mx.(^Cx. - Mx.)l =0,
L i f l i J
(3.5) E(rCx?) = Mx2 + Vx.
f l i l
where Vx. denotes the potential residual variance to be associated with
the random variable Cx to the extent that the randomly selected point
may be assumed to lie within the i-th tract. Clearly this potential
residual variance is exactly that same numerical constant of intra-tract
component variance aacribed to tract i above in our numerical represen-
tation of the complete spatial sampling frame. Then by substitution
51
9 n O
of (3.5) into (3.4) and noting that Mx = E. jl . Mx , we may write
n
..Vx = E _z. [Mx. + Vx. - -fix ]
f £ fi L i i f J '
n r2-2-n _
= E _z. [Mx. + _Mx - 2 _Mx E _z. Mx. + Vx.J ,
ifiLi f F j f j 3 iJ
n - ?
= E _z. [(Mx. - .Mx) + Vx.] ,
£ t 1 *" 1 f 1
n ? n
(3.6) = E _z. (Mx. - ^Mx) + E z. Vx.
j r l if £ f l l
In an identical manner, it may be shown that
n _ 9 n
(3.7) ^Vy = E _z. (My. - -My) + E ^z . Vy .
f J fl Jl f"7 £ f 1 ^ 1
Together (3.3), (3.6), and (3.7) imply
n j ~
(3.8) DV = E -Z. [(Mx. - Jlx) + (My. - J4y) 1
f ,• f l L l f ifJ
i
\ f2! IVxi + Vyil
n
+
This demonstrates that distance variance as a general measure of
overall distribution dispersion may always be decomposed into two dis-
tinctly different components, one determined by the probability vector
JZ in conjunction with the spatial coordinates of tract centroids and
the other determined by JL in conjunction with the residual variances
associated with intra-tract point distributions.
52
An Alternative Method for Computing the
Distance Variance of a Distribution
In this section we wish to demonstrate a method for computing
the distance variance of an areal distribution in a manner that is
independent of the centroid coordinates of the distribution. To do this,
we must first construct a symmetric matrix S (n x n) where any element
s. . represents the expected squared euclidean distance between any two
point locations within our urban area, one point being taken within the
i-th tract and the other taken within the j-th tract.
Let the random variable representing the expected squared dis-
tance between any pair of points of the i-th and j-th tracts be denoted
2
E(D. .). Given the additivity of squared distance components along
2 2 2
orthogonal axes, we may express E(D. .) alternatively as E(Dx. . + Dy . .)
2 2 .
where Dx. . and Dy. . are themselves random variables representing
1,3 i>:
squared distance components along the orthogonal x and y axes. Further-
more, given the fact that the expectation of a sum of random variables
is equal to the sum of the expectations of the random variables taken
2 2 2
individually, we may note that E(D. .) = E(Dx. .) + E(Dy. .).
1 »J 1»!) 1»3
2
Now consider simply the random variable E(Dx. .) which represents
the expected squared distance component along the x axis. Let Cx.
denote the x coordinate of the point taken within the i-th tract, and,
similarly, let Cx. denote the x coordinate of the point taken within
the i-th tract. As discussed above, Mx. and Mx. denote the mean x
coordinates of all points distributed uniformly thoughout the i-th and
j-th tracts respectively. Then it follows that
53
E(Dx2 .) = Ef(Cx. - Cx.MCx. - Cx.)]
i,3 L i 3 1 3 J
= E(Cx2) + E(Cx?) - 2E(Cx.Cx.) ,
1 3 13
and since the random variables Cx. and Cx. are assumed to be independent,
E(Dx2 .) = E(Cx?) + E(Cx2) - 2E(Cx.)E(Cx.) ,
1,3 1 3 1 ]
(3.9) = E(Cx2) + ECCx2.) - 2Mx.Mx.
1 3 1 :
Now with reference to (3.5) we know that
(3.10) E(Cx2) = Mx2 + Vx. ,
1 11
and similarly
(3.11) E(Cx2.) = Mx? + Vx.
3 D 3
Together, equations (3.9), (3.10), and(3.11) imply
E(Dx2 .) = (Mx2 + Mx2 - 2Mx.Mx.) + Vx . t Vx . ,
= (Mx. - Mx.)2 + Vx. + Vx.
1 : 13
In identical fashion, it may be shown that
E(Dy2 .) = (My. - My.)2 + Vy. + Vy .
i,J 1 D 1 ]
2 2 2
Now, from above, we know that E(D. . ) = E(Dx. . ) + E(Dy. .).
i,3 1,3 i,3
Also, it is clear that the squared distance between centroids of the
2 2
i-th and i-th tracts is exactly the sum (Mx. - Mx.) + (My. - My.) .
131J
54
Thus, it may be easily verified that the expected squared distance
between any two points in our city, one taken from the i-th tract and
the other from the j-th tract, is simply the squared distance between
the centroids of the two tracts augmented by the sum of four additional
terms: namely, the four component variances associated with the dis-
tributions of points within the two tracts relative to the x and y
axes.
Thus, the following representation of our S matrix is suggested.
Let , S denote an n x n symmetric matrix where each element , s. . repre-
b b 1,3
sents the squared euclidean distance between the centroids of the i-th
and i-th tracts. Here, of course, ,s. .>0 for all ij^j and ,s. .=0 for
' b 1,3 b i,]
all i=j according to:
(3.12) Ls. . = (Mx. - Mx.)2 + (My. - My.)2
b i,: i ] i ]
Also, let S denote an n x n symmetric matrix where each element s. .
w w 1 ,j
represents that additional sum of intra-tract component variances neces-
sary to account for the total expected squared distance between point
pairs of i and j due to our lack of knowledge concerning the exact loca-
tions of the two points within the two tracts. In this case, s. .>0
* w 1,3
for all i=j as well as all i^j according to:
(3.13) s. . = Vx. + Vx. + Vy. + Vy .
w 1,3 i ] Ji 3
Then , clearly
(3.14) s. . = s. . + s. . i,j = l,...,n
1,3 b 1,3 w 1,3
55
Now following Neft (1966), Bachi (1957), and others, for a given
areal distribution f with centroid coordinates _Mx and fMy, let us
define as an alternative measure of dispersion the generalized distance
variance :
n n
(3.15) J3DV =EI
L L _Z. _Z. S. .
i j r i f] !»3
Given that s. . = s. . t ,s. . , i ,j=l,. . . ,n, we may always decompose
1,3 w 1,3 b l,] J J
GDV into between-element and within-element components in accordance
with
fGDV = (w)fGDV + (b)fGDV .
n n n n
= Z Z jz . jz . s. . + Z Z _z . _z . ,s . .
. . fi f 3 w l , 3 . . f l r 3 b l , 3
i j J 'J l 3 J 'J
Considering first the expression for ..v-GDV, note that
(3.16) ,v\i=GDV = Z Z _z. jz. [(Mx. - Mx.)2 + (My. - My.)2]
(b)f • > fi 1 3 L i 3 J± J3
= I Z JZ. JZ. (Mx? + Mx? -2Mx.Mx.)
i j r l f ] i 3 13
n n 2 2
+ ZZ^z. ^z. (My. +My. -2My.My.)
i \ f i f : 'i J3 i J
This formulation demonstrates that the between-element component of
generalized distance variance itself may always be decomposed further
into additive x and y components in accordance with
(3.17) (b)fGDV = (b)fGDVx ♦ (b)fGDVy
For mathematical convenience, let us assume a translation of all
tract coordinates of the form M'x. = Mx. - ..Mx and My'. = My. - Jfy so
l if J l J± f J
56
that the centroid of the distribution f is now at the frame origin. Then,
Z? J.. M'x. = E? _z. (Mx. - _Mx) = 0, and f] J. M»y. = E1? z. (My. - -My)
l f l l l fi if ' ifiJi 1 f 1 'i T
- 0. Clearly, all elements s. ., s. ., and .s. . would be invariant
iij »i»j b 1,3
to such a translation of coordinates.
With reference to (3.16) and (3.17), note that ,, ,.fGDVx may now
be expressed as
n n 9 ?
,, ...GDVx = l I jz. jz. (M'xT + M'xf - 2M'x.Mx.) or,
(b)f . . t l f 3 l 3 13
(3.18) ,, . .GDVx = E E _z. _z. (Mx. - Jix)2
(b)f 1 j f 1 f 3 if
n n 9
+ I E _z. _z. (Mx. - .Mx)
i j fi f 3 ] f
n n
-211^.^. (Mx. - _Mx)(Mx. - iix)
i j r 1 f 3 if D f
The last term of (3.18) will always be 0 since, by manipulation of terms,
it may be written in the format -2[E. jz. . (Mx. - Mx)][E. _z. (Mx. - ,-Mx)]
and E. ~z . (Mx. - Mx) is clearly 0. Minor additional manipulation per-
mits us to write
n - 2 n - 2
,, ._GDVx = E _z. (Mx. - £Mx) + E _z. (Mx. - _Mx)
(b)f i fi if j f] 3 f
which, with reference to (3.6), yields
n n
,v,£GDVx = ^Vx - E _z. Vx. + .Vx - E _z . Vx.
(b)f f £ fi 1 f j f ] ]
In identical fashion, it may be shown that
(b)fGDVy = fVy - I fz. Vy. + fVy - Z fZ. Vy. .
57
Thus, via (3.17), we have
(3.19) /ux^GDV = 2(_Vx • E _z. Vx. + _Vy I jz . Vy.)
(b)f f ^ f i i f J t i Ji
n n
n n
= 2 £DV - 2(1 jz. Vx. + Z ^z. Vy.)
f i r l i j fi Ji
l * * * i
Now, let us consider the within-element component of our general-
ized distance variance measure and, with reference to (3.13), write
n n
GDV = Z Z
, vGDV = L L JZ. JZ . s. . ,
(w) £ • fi f] w 1,3
n n
Z Z _z. jz. (Vx. + Vx. + Vy. + Vy.) ,
i i r i f ] i 3 Ji Ji
n n
= Z jz. (Vx. + Vy.) + Z jz. (Vx. + Vy.)
i r l i Ji j f 3 3 J:
Since our summations here take place over the same set of terms, we may
rearrange the order of our summations and write simply
n n
, x.GDV = 2(Z jz. Vx. + Z jz. Vy.)
(w)f . fi i i r i Ji
But this is precisely the quantity by which , . GDV differs from 2 DV
in (3.19). Hence, given that ^GDV = ,1v.eGDV + , s^GDV, we have the
& f (b)f (w)f
major result :
(3.20) GDV = 2 DV
By its definition in (3.15), the generalized distance variance GDV for
for any distribution f may be computed solely in terms of the probability
vector JL associated with f and our matrix S of inter-point expected
squared distances which is determined solely by our choice of a specific
58
sampling frame. Additionally, from (3.20) above, we know that the dis-
tance variance of a distribution f is related to its generalized distance
variance by
DV = \ GDV
Thus, we have demonstrated that the distance variance of any specific
distribution f may also be computed directly from JZ, and the matrix S
in a manner independent of the coordinates of the distribution's centroid.
Specifically,
(3.21) J)V = h 1 Z jz. jz. s. .
• ^ fi f] 1,3
n n
Z Z
i J
Given that both DV and GDV are expressed in units of squared
distance, it will assist our thinking in practical applications to take
the square roots of both quantities as basic measures of overall distri-
bution dispersion. Then, the measures DV2 and fGDV2 will be expressed
directly in units of geographic distance (miles, kilometers). However,
names assigned to these measures differ among authors. Bachi (1957)
and Duncan, Cuzzort, and Duncan (1961), following Bachi, refer to JDV
h
as the standard distance of distribution dispersion and to J3DV as the
mean quadratic distance. We prefer the terminology given by Neft (1966),
however, and in keeping with our nomenclature for _DV and fGDV, will
refer to the measures DV2 and GDV2 respectively as the standard dis-
tance deviation and the generalized standard distance deviation of an
areal distribution.
It should be noted at this point, however, that our derivations
and expressions for both _DV and ^GDV differ from those of Bachi and
59
Neft in a basic manner. Both Bachi and Neft, following standard pro-
cedures for computing the variance of grouped data, neglect the contri-
bution to distance variance associated with intra-tract residual variances,
Thus, the numerical consistency of their measures over different sampling
frames would seem to depend strongly on the assumption that all areal
units are small relative to the size of the urban area and, thus,
potentially quite numerous. Bachi appears to acknowledge this condi-
tion in stating:
Other things being equal, that frame should be preferred
which . . . renders minimal the aggregate "within zone" squared
distance and which renders maximal the aggregate weighted squared
distance between the centers of the zones and the general center.
(Bachi, 1957)
The methods that we propose here, however, take full account of
the contributions to distance variance made by point distributions
within tracts. In essence, the methods proposed here are directly ana-
logous to procedures employed in physics for determination of moments
of inertia for irregular shapes. These procedures are based on the well-
known parallel-axis theorem of mechanics concerning the additivity of
component second moments. By analogy with such procedures, we have
chosen the above course in defining mathematically the distance variance
of areal distributions in an effort to obtain greater consistency of our
computations of DV and GDV over different spatial sampling frames.
Some Preliminary Measures of Spatial Association Between and Within
Areal Distributions
Using the same concepts employed above in our presentation of
general measures of areal distribution dispersion, we may define a
60
general measure of the spatial dissociation between two distributions
in the following manner. Let f and g be two areal distributions repre-
sented respectively by vectors JL and Z of n elements each. Again,
o
the elements of both JZ and Z will be discrete probabilities propor-
tional to aggregate data values recorded for each of the n areal units
of a common spatial sampling frame.
Then we may define the generalized squared distance of interaction
between the two distributions f and g as
o n n
(3.22) _ GDI = I Z jz. z. s. .
f,g i j r 1 g ] 1,3
where again the elements s. . represent expected squared distances
separating points paired randomly within and between tracts.
Now with simple but lengthy algebraic manipulation, it can be
demonstrated that
(3.23) _ GDI2 = ( Jbc - Mx)2 + ( Jfy - My)2 + £DV + DV ,
f »g r g fJgJfg
where Jlx, Jfy and Mx, My are the coordinates of the centroids of the
two distributions. Here, notice the similarity between our expression
2
for _ GDI and our formulation of the expected squared distance be-
2
tween points of different tracts, E(D. .) = s. . , as defined by (3.12),
1.3 !»J
(3.13), and (3.14). In both cases, our mean squared distance measure
may be considered as consisting of three distinct components: (1) the
mean squared distance from a randomly selected point of one distribution
(tract) to the centroid of that distribution (tract), (2) the squared
distance from the centroid of the one distribution (tract) to the cen-
troid of the other, and (3) the mean squared distance from the centroid
61
of the other distribution (tract) to some other point randomly selected
within it. Note also that where the distributions f and g are one and
2
the same, then _ GDI = _GDV = GDV.
The above conditions hold only because, in the formulation of
2
both -GDV and GDI , we assume complete spatial independence within
the pairing of points within and between distributions. In other words,
the present measures assume that the pairing of points within and between
distributions occurs in a manner that in no way depends on spatial
proximity relationships existing between distribution elements. The
probabilistic weighting of mean squared distance components is deter-
mined solely in terms of the cross-product elements of the probability
vectors -Z and Z which, taken by themselves, are completely aspatial.
Seeking more appropriate measures of spatial association between areal
distributions, in the next section we will explore an alternative
measure of mean squared distribution distance where spatial proximity
relationships between distribution elements determine in part the pro-
babilistic weighting of mean squared distance components.
A Spatial Interaction Approach to
Measurement of Distribution Distance
Seeking a more informative measure of spatial association between
areal distributions, by analogy with the intraurban trip distribution
models discussed in Chapter II, let us examine spatial interaction
models of the form:
(3.24) . MDI2 = Z Z . q. . s. .
f»g i j f,g i.l i.l
62
2
Here, _ MDI denotes the mean squared distance of interaction between
f»g a
two distributions f and g, s. . represents as before the expected squared
distance between points paired between the i-th and j-th tracts, and
q. . denotes a probabilistic weighting of s . . determined in part by
the value of s. . itself. Specifically, we will require that the matrix
_P Q(nxn)bea joint probability distribution with row marginals
^z., i = l,...,n and column marginals z., j = l,...,n where, again,
o J
JZ, and Z represent discrete probability vectors characterizing distri-
butions of the aggregate variables associated with f and g over the n
tracts comprising the spatial sampling frame.
Now let .. II denote the set of all ,_ Q joint probability matri-
ces having row marginals JZ and column marginals Z. Note then that any
Qe,. II may be considered as determining a probabilistic pairing of
points between the areal distributions f and g and thus an inter-distri-
bution pairing of points across all tracts as well.
One possible _ Q matrix occurs, of course, where _ q. . =
2
jz. . z. for all i,j = l,...,n. In this instance, our measure of - MDI
fig] f.g
2
is identically the same as our measure of ,. GDI defined in the preceding
* »g
section. This represents the case again where complete independence
exists within the pairing of points between the distributions f and g.
In general, however, it would seem desirable that our measure of
2 .....
r. MDI be a function of a r Q joint probability distribution exhibit-
f »g f ,g
ing some degree of stochastic interdependence or constraint attributable
to whatever spatial interdependence, association, or congruence that
may exist between the two areal distributions f and g. In other words,
we wish our f Q matrix, already constrained to be a joint probability
* »g
distribution with marginals ^Z and Z, additionally to be determined as
63
a function of spatial proximity relationships existing between the
elements of f and g. Just how this should be done represents a key
issue of our thesis.
Now suppose, by analogy, we appropriate directly the mathemati-
cal concepts of the entropy-maximization model of trip distribution in
an attempt to formulate an appropriate Q matrix. Our model would
then be:
(3.25) max - Z I _ q. . log(- q. .)
subject to the constraints,
n
(3.26) E _ q. . = z. j = l,...,n
i f.g i»: g j
n
(3.27) Z . q. . = _z. i = 1
j f,g i»D r i
(3.28) _ q. . > 0 i,j = l,...,n
f.gi,]
and the additional constraint,
n n o
(3.29) II£ q. . s. . = " MDI
i j f,gi»] i»: f»g
It should be immediately obvious that such a model is inappropriate for
our present task, since the very same variable that we wish to ultimately
2
determine, . MDI , appears in the constraint (3.29) as a numerical con-
f,g
st ant assumed to be known a priori.
In order to make several points, however, let us pursue further
the investigation of this entropy-maximization approach to our problem.
As we have noted above in Chapter II, the solution to the model (Wilson,
64
1970; Potts and Oliver, 1972) is given by
(3.30) q. . = u. jl. u. z. exp(-3s. .) i,j = l,...,n
where the vectors _U and U are determined by iterative solution of the
f g 3
equations
(3.31)
-1
_u. = [l u. z. exp (-3s. . )1 ,
fi lj g ] g ] i»3 J i = 1,... ,n
n _2
(3.32) u. = [E _u. _z. exp (-0s. .)]" j = l,...,n
g j Li f l fi i,] J
and where 3 is the Lagrange multiplier associated with constraint (3.29)
Now as discussed above in Chapter II, there is known to exist a
one-to-one mapping between all real values of 3 and all feasible values
2
of _ MDI . Further, we know that as 3 approaches + °°, the associated
2
value of _ MDI approaches its minimal feasible value. (A. W. Evans,
f,g
2
1971: S. P. Evans, 1973) This is the minimal value of - MDI that
f»g
would be obtained if we chose to solve the Hitchcock or transportation
minimization problem uniquely determined by equations (3.24), (3.26),
(3.27), and (3.28). (Dantzig, 1963; Dorfman et al. , 1958) Thus, one
possible way out of our dilemma concerning a choice for 3 would be
simply to assume theoretically a 3 equal to +00 and solve for the unique
. . 2
minimal _ MDI ,
ffg
(3.33) _ LDI2 = min Z Z _ q. . s. .
f>g ^ Q* n i j f'g ^ x-3
f,g f,g
65
This measure of minimal or least mean squared distance of inter-
action between distributions has some interesting properties. Elsewhere
(Ray, 1974), we have demonstrated its applicability to the solution of
certain pattern recognition problems. Among other desirable properties,
it has the advantage that it may be minimized, not only over all
, Qe_ II but over all scale, translational, and rotational trans-
f>g f>g
formations of the geometry of one spatial pattern relative to the geometry
of another as well.
It might appear that another logical solution to our problem con-
cerning a choice of a specific value for 8 might be simply to set 8=0.
Here, however, exp(-3s. .) = 1 for all s. . and thus the _ Q matrix
i.] i»3 f»g
obtained via (3.30), (3.31), and (3.32) will in no way depend on inter-
tract squared distances. In fact, it can easily be shown that, for this
2
case where 8=0, the value of r MDI will be identically equal to the
f.g
2
value of GDI given by (3.23).
* >g
Thus, the entropy-maximization model of trip distribution, applied
directly, seems to offer little toward the determination of a unique
- Q matrix reflecting spatial proximity relationships between distri-
* >g
bution elements. It leaves us with an arbitrary choice of a real value
for 8. Consequently, we must make an arbitrary selection of a single
_ Q matrix from an infinity of possible _ Q matrices.
f»g r,g
Throughout this discussion, we have assumed that all _. q. .fs
should be proportional to proximity relationships between distribution
elements and, hence, somehow inversely proportional to the s. .'s. By
the theory of the entropy-maximization model given in Chapter II, this
implies that any appropriate 8 must lie between 0 and + °°. At 8=0,
66
2 2 2 2
_ MDI reverts to , GDI . At 8=+°°, MDI becomes . LDI , a value
f.g f.g f,g f,g
that must be obtained by solution of a transportation programming pro-
blem. Adopting the transportation programming solution, we know that
only a small number of the _ q. .'s will be non-zero, i.e., a number
f»gl,]
on the order of n+n-1 if we assume all elements of JL and Z to be non-
r g
zero. Consequently, only a small number of proximity relationships
between distribution elements would contribute to the determination of
2
_ MDI . This condition seems highly undesirable. Thus we are left
f.g
with the conclusion that the entropy-maximization model of trip distri-
bution, applied directly, offers no satisfactory method for measurement
of spatial associations between areal distributions, and we must turn
in Chapter IV to the development of an alternative approach.
CHAPTER IV
NEW METHODS FOR ASSOCIATION MEASUREMENT AND
CLUSTER ANALYSIS OF SPATIAL DISTRIBUTIONS
A Unique Measure of Spatial Association Within
and Between Areal Distributions
In this section we shall develop a specific measure of distri-
2
bution distance of the form given for ,. MDI where the matrix _ Q is
determined in a unique manner relative to all spatial proximity rela-
tionships existing between distribution elements. Retaining the same
meanings as before for our notations f, g, JZ. Z, _ Q, _ II, and S,
f^' g f,g ' f,g
our model is derived as follows.
2
Note that our measure _ MDI given by (3.24) may be considered
r,g
simply as a weighted sum of squared distance components between all dis-
tribution elements paired between f and g. To demonstrate this condi-
tion clearly, let , r. . = _ q. . s. . for all i , j = l,...,n. Then
f,g 1,3 f,gMi,D 1,3
we may express (3.24) simply as
9 n n
. MDI = Z Z _ r. .
f>g i j f>g i»D
2
Thus, _ MDI is simply the sum of all elements of the new matrix * R
f ,g ^ J t»g
(n x n) and our problem is now to specify in an appropriate manner
the elements of ^ R«
68
Now, suppose we adopt the objective that the elements of R
should have values as evenly distributed as possible subject to the
conditions imposed on _ R given that . Qe_ IT. To formalize this
objective mathematically, scale _ R by the constant k =(£. E. _ r. .)
f»g i D f»g itD
so that the resulting matrix _ R' =[k , r. .]=[,. r! .] may be consi-
*»g i»g i»3 ^»g 1»D
dered as a joint probability distribution. Then our objective becomes
to determine that matrix _ R whose associated joint probability matrix
_ R* is maximally entropic subject to the constraint that ,. Qe,. II.
* >g ^»g * >g
In information-theoretic terms, the interpretation of this objective is
that we should select that _ R representing a least biased estimate,
i »g
i.e. , that _ R that is maximally noncommittal with regard to missing
* »g
information. (Jaynes, 1957)
Now considering R' as a joint probability matrix, let the
f »g
vectors U and V denote respectively its row and column marginal proba-
bilities such that
n
u. = Z _ r' . . i = 1, . . . ,n ,
i j f,g 1.3
n
v. = Z_ r * . . i = 1 .... ,n
1 i f.g i»3
Now necessarily H( R' ) < H(U) + H(V), and the upper bound of H( Rf )
t »g * >g
is obtained only if . R' has the form
f,g
r- r' . . = u.v. i , j = l,...,n
f»g i,J ii
Let us assume momentarily that H( _ Rf ) does indeed attain its upper
f,g
bound. Then, we must have
U2 -41
k _ q. .s. .=u.v. i,T=l,...,n
69
and consequently
-2 -1
(4.1) r _q.. . = u v k s. i,j = l,...,n.
■L»&-L»J * J J-jj
Now let u,i=k ui$ i=l ,...,n and v' .=k~ v., j=l ,...,n. Then (4.1)
may be expressed
■ i ... .-1
f
q. . = u* . v*. s. . i,j = l,...,n.
»g 1.3 i j 1,3 ,J * '
With reference to the constraint that _ Qe_ II, we have
f.g f .g
n
E£ q. .= z. j = 1,. . . ,n ,
i f.gHi.3 g 3
n -1
£ u'.v'.s. . = z. i=l,...,n ,
i i 3 1.3 g 3
n -1
v*. I u'.s. = z. j = l,...,n ,
3 i i 1,3 g 3
and thus ,
n _! _!
(4.2) v' . = z.(E u*.s. .) i = l,...,n,
3 g 3 i i 1,3
By an identical manner, it may be shown that
n _-, _-,
(4.3) u'. = jzAZ v'.s. .) i = l,...,n.
i r l j 3 i.3
Now (4.2) and (4.3) represent a set of 2n equations which, in
a manner identical to the determination of "balancing factors" within
trip distribution modeling, may be solved iteratively for the 2n
unknowns of the vectors U' and V. Solution may proceed in the following
70
manner. First, initialize the U* vector by setting u* . = 1 for all
i = l,...,n. Then with equations (4-. 2) , determine a first approxima-
tion of Vf. Use this V1 within the equations (4.3) to determine a
new U* , return to equations (4.2) 9 and so forth.
Such an iterative procedure will determine values for U' and V1
that are unique up to a positive scalar multiple; that is, given that
U' and Vf satisfy (**.2) and (4.3) , then U"=cU! and VM=%V« will also
satisfy (4.2) and (4.3) where c is any positive constant. For our
purpose here , we should periodically throughout the iterative solution
of (4.2) and (4.3) re-scale U' and V such that f! uf . = E? v' . . Then,
113:
_2
at convergence, we may determine the constant k ' of (4.1) from the
condition that E. u' . =k E. u.=k , or alternatively from the condition
1 1 1 1 ' J
E. v'.=k E. v.=k . These relationships between the vectors U, V, and
] 3 d :
U1 , V via k imply mathematical uniqueness for the values of U, V,
-2 -2
and k ; hence, by substitution of these unique U, V, and k into
equation (4.1) , the uniqueness of ,. Q itself is assured. Let us
*
denote the unique r Q so derived as r Q .
The fact that equation (4.1) has a solution satisfying all a
priori conditions insures that the entropy function defined for ,. R' ,
H( _ R') = - E. E. ^ r' . . log ,. r1 . . , does indeed attain its upper
f.g 1 : f,g i>D f,g ifD
bound, i.e., H( ,. R' ) = H(U)+H(V). Furthermore, we have seen that
f»g
this solution is unique. Thus, assuming only that ,. Qe_ II and that,
t»g t ,g
otherwise, all weighted component squared distances of interaction
between distribution elements should be allocated as evenly as possible
over all element pairs, i.e., their distribution should be maximally
entropic, we have arrived at the distance measure
71
o n n .
(4.*0 - EDI* = Z I . q*. . s. .
f»g i j f»g l.j ltD
We will refer to this measure of distribution distance as the entropic
squared distance of interaction between two distributions f and g. Taking
2
the square root of EDI , we have simply the entropic distance of
interaction between two distributions, r EDI.
f»g
2
The measure _ EDI appears to be unique with respect to the
* Ȥ
following five properties desirable for any measure of distribution dis-
tance.
2
1. As a weighted sum of squared euclidean distances, . EDI
is invariant with respect to all translations and rotations (orthogonal
transformations) of frame coordinates. This condition follows from the
translational invariance and the unique rotational invariance properties
of euclidean distance (Beckenbach and Bellman, 1961) together with the
fact that all weights themselves depend only upon their associated
squared distance components and the fixed vectors JZ and Z.
2
2. The square root of EDI , _ EDI, is homogeneous with res-
fȣ f >8
pect to scale transformations of frame coordinates. To illustrate,
suppose frame coordinates are converted from miles to kilometers. Then
-. EDI in kilometers, re-computed using the new frame, would be simply
* »8
the old _ EDI in miles times the conversion factor 1.6. (Note that
f.g
this property does not hold for the entropy-maximization model of trip
distribution because of the reliance of the model on the functional
exp(-6d. .). If the d. .'s are re-scaled, then the parameter 6 must
also be changed if the interzonal trip distribution matrix is to remain
unaltered. )
3. As an estimator of areal association between two distributions
72
2 .
f and g, _ EDI is numerically consistent with respect to the resolu-
tion of the spatial sampling frame. The smaller and more numerous the
areal subdivisions, the more accurate the measure obtained. Where f
2
and g are the same distribution, EDI approaches zero as the number
of areal subdivisions of the frame increases. At any intermediate level
2
of frame resolution, where either f=g or f^g, the value of _ EDI
* »S
depends only incidentally on the specific frame selected. Unlike tradi-
tional ecological correlation measures computed as a function of f and
2
g data values coincident within individual tracts, _ EDI is computed
*" >6
as a function of all data values associated within and between all tracts
in a manner proportional to spatial proximity relationships existing
among tracts.
4. As a weighted sum of squared distances between points of two
2
x,y bivariate distributions f and g, the value of EDI may be decom-
posed into a series of additive terms that includes measures of the x
and y component variances of the coordinates of points within both f
and g. Additionally, this series may be arranged to have terms expres-
sing the x and y component covariances of the coordinates of point pairs
spatially associated between f and g as a consequence of the probabilis-
*
tic matching of points between distributions that is implied by _ Q .
* »6
2
(Bachi, 1957) This decomposition property of _ EDI results uniquely
f >S
from its formulation as a sum of squared distances.
2
5. The measure ,_ EDI is formulated in a least biased manner.
As a weighted sum of squared distances between points of f and points
of g, the distribution of all component weighted squared distances is
made maximally entropic subject to the single constraint that the
73
weighting occur as a joint probability function having marginals JL
and Z .
g
While our measure of entropic squared distance of interaction
2 . . . . .
_ EDI is unique in satisfying the above five properties, we remain
* »S
faced with the condition that it, like all other distribution association
measures, depends at least to some extent on the choice of a particular
frame. Again, this is simply a logical consequence of the fact that
our choice of a specific frame determines directly the manner by which
a spatial pattern of phenomena is characterized numerically as a discrete
areal distribution. Recognizing the inevitability of this condition,
in the next section we turn our attention once more to information
theory in an effort to determine quantitatively the amount of informa-
tion captured by a particular sampling frame concerning the spatial
interdependence of urban patterns.
An Information Theory Measure of Spatial Complexity Conveyance
Among Areal Distributions
In the preceding section, we demonstrated how a unique measure
of entropic squared distance 'I EDI, characterizing the spatial associa-
tion (dissociation) between two areal distributions f and g may be formu-
lated and computed solely as a function of the probability vectors _Z
and Z and the matrix S of expected squared distances between all tracts
of a chosen frame. Our purpose here is to demonstrate a direct rela-
2
tionship between certain concepts of the _ EDI model and those concepts
of information theory discussed above in Chapter II. In a manner
mathematically isomorphic to the measurement of encoded information
transmission rates within telecommunications systems, we will find it
7U
possible to characterize the extent to which spatial structure, quanti-
fied in information theoretic terms, is conveyed between patterns of
urban phenomena. Additionally, we will find that this related method
of areal distribution analysis sheds some light on methodological issues
concerning the dependence of analysis results on the particular spatial
sampling frame selected for characterization of patterns.
Now given the assumptions of our model, the choice of a particular
frame determines directly the numerical characterization of a specific
pattern f as a discrete probability distribution JZ of the aggregate
data values of the variable associated with f across all n tracts of
the frame chosen. Thus, we may define immediately for f the entropy
function
(4.5) HCjZ) = -I fzi logfzi
which may be considered as a measure of the aspatial complexity of the
areal distribution f relative to the specific frame selected.
In the present case, it is important to note that our measure of
aspatial complexity for an areal distribution depends in a fundamental
way on the number and scale of areal subdivisions comprising the spatial
sampling frame. To illustrate this condition, consider the upper
bound of H(^) which, with reference to information theory, we know
to be log n. This is the value that would be obtained for some class
of urban phenomena (for example, raw population) for which aggregate
data values are distributed uniformly across all n tracts of the frame.
Now quadruple the number of areal units by subdividing all tracts of
the given frame into four new equi-area tracts. Assuming that aggregate
75
data values remain distributed in a uniform manner over all *+n tracts
of the resulting frame, the new value of H(^) would be log 4n. Thus,
in the extreme, we may expect our measure of the aspatial complexity of
an areal distribution to vary in a manner proportional to the logarithm
of the number of subdivisions of the frame.
This condition calls for no apologies. In fact, in a certain
manner it seems entirely reasonable, for as we increase the resolution
of our spatial sampling frame, we should expect to sift out an increasing
amount of information concerning the complexity of organization of urban
spatial patterns. It is important to note, however, that the informa-
tion recorded in the vector JZ, alone is completely aspatial. Any permu-
tation of the individual elements of ,-Z, z /•%» i = l,...,n, would yield
the same value of H( JZ) . Thus all information concerning the spatial
character of f depends directly on the one-to-one correspondence defined
between the probabilities z . , i = l,...,n, and the set of numerical
constants describing the spatial sampling frame: Mx. , My., Vx. , Vy.,
i = 1 , . . . ,n .
Now the matrix S of expected squared distances also contains all
information necessary for numerical description of a given frame up to
its specific geographic orientation. We may readily decompose S into
its two additive components S and S given in (3.12) and (3.13) by
w b
noting that S = S - S, and since the diagonals of S are known to be
,s. . = s. . - (^s. . + hs . .) i,j = l,...,n
b 1,3 1,3 1,1 3»3
We rely here on our assumption concerning the compactness of all tracts
76
to bring about the conditions that %s . . = Vx. + Vy., hs. . - Vx. + Vy.,
1,1 1 Ji» j,] 3 J2
Vx. = Vy., and Vx. = Vy., for all i,j = l,...,n.
Also, it is well known that the matrix .S may be factored to
yield a set of tract centroid coordinates Mxf . and My'., j = i ... n
differing only from the pre-specified tract centroid coordinates Mx
j
andMy,-i=l,...,n, by a rotational transformation. (Young and House-
3
holder, 1938; Gower, 1966; Green and Carmone, 1970, p. 102) Since our
mathematical model is completely invariant with respect to frame coordi-
nate rotations, the S matrix itself may be considered to represent a
complete and sufficient representation of its associated frame. Thus,
we may consider all information available concerning a specific spatial
distribution f to be represented sufficiently for the purposes of our
model jointly by the vector _Z and the matrix S.
Consider again two areal distributions f and g characterized by
the probability vectors JL and Z together with the frame expected
squared distances matrix S. In the preceding section, it was demon-
strated that corresponding to each JZ, Z, and S there exists a unique
.».
joint probability distribution Q characterizing in a least biased
manner the spatial interdependence between the elements of f and the
*♦*
*\
elements of g. Thus, given that Q is itself a discrete probability
distribution, we may define for any f and g the entropy function
(,.6) WftgQ*)--| figq*lfJ logf,/isj
which may be considered as a measure of the joint spatial complexity or
simply the joint complexity of the two areal distributions f and g,
again, relative to the specific frame associated with S.
77
Then, direct application of information theory leads to formula-
tion of the information transmission function
C+.7) C = H(_Z) + H( Z) - H(_ Q*)
f.g r g f,g
which will be taken as a measure of the spatial complexity conveyance
between f and g relative to the given frame. This measure - C may be
interpreted as a measure of the structural complexity shared between
f and g. Alternatively, _ C may be interpreted as the amount by which
the combined aspatial complexities of f and g are reduced by their joint
spatial complexity.
As in all other applications where the entropy function is used
to quantify order- disorder relationships exhibited by some complex of
variables, it is difficult to attach precise verbal meanings to the
mathematical concepts that we employ. For the present application, we
have chosen to associate directly the term complexity with the concept
of entropy to underscore the fact that our measures are here taken
relative to a specific level of spatial sampling, and hence relative to
some level of complexity of system description.
The measure H( J5) is aspatial in that it in no way depends on
distance relationships between distribution elements. Its value does,
however, depend in a fundamental way on the resolving power of the
spatial sampling frame employed; hence, H(fZ) is said to measure the
aspatial or raw complexity of an areal distribution captured by the
given frame. Since r Q is constrained to be a joint probability dis-
r,g
ft
tribution between JZ and Z, the joint entropy function H(,. Q ) always
exists. Further, since r Q is determined in part as a function of the
f.g
78
inverse elements of S, we refer to H( c Q ) as the joint spatial com-
plexity of f and g relative to the frame. The information transmission
function results immediately from the existence of H(^), H( Z), and
*
H(f Q ), and we may consider C as a measure of the amount of spatial
complexity conveyed or shared between f and g relative to the frame.
A Procedure for Least Biased Grouping
of Spatial Distribution Elements
Consider a specific areal distribution f sampled with respect to
a specific frame. Given the probability vector _Z and the matrix S
that together characterize the distribution, a unique joint probability
*
distribution Q may be computed using the method outlined above.
t » t
A
Here, of course, , J} will be symmetric since S is symmetric and both
row and column marginals of - _Q are Z. In this case, moreover, the
r ,r r
•*•
functional H( Q ) defined by (M-.6) will represent, not a measure of
the joint spatial complexity of two different areal distributions, but
rather a measure of the spatial complexity of f alone relative to the
selected frame. In a like manner, the information transmission function
(4.8) fC = HC^) + H(fZ) - H(f fQi{)
= 2H(fZ) - H(f fQ*'C)
may be considered as a measure of the intra-distribution spatial com-
plexity conveyance of f alone relative to the frame, i.e., a measure of
the structural complexity exhibited by f directly as a consequence of
its characterization with respect to the given frame.
79
Now assume that for some reason we wish to group together indi-
vidual elements of a particular areal distribution to simplify (or com-
press the data associated with) its numerical description. For example,
suppose that we have in block -by-block format aggregate measures of all
annual retail sales of goods and services within an urban area, and our
problem is to group individual blocks into commercial districts to
obtain a more efficient characterization of the pattern of retail acti-
vity throughout the city. One way that we might proceed is as follows.
Let us accept the block -by -block data concerning aggregate yearly
retail sales as our most complete description of the true pattern of
retail activity, g. Assume there are m blocks within our city and let
S (m x m), as before, represent all expected squared distances between
all m blocks. Again we will assume aggregate data values to be normal-
ized across blocks so that the distribution of aggregate data values
across blocks is represented as a discrete probability vector Z where
m
L. z. = 1.0 and z. > 0, i = l,...,m. Note that for this example most
11 g l ~ ' ' '
blocks within the city will contain no retail activity at all. Hence,
immediately we may simplify our numerical description of g by reducing
it to only those n blocks (n < m) in which commercial establishments are
located.
There will be absolutely no loss of information incurred in doing
this, since z. log z. = 0 for each block i holding no commercial estab-
g i g i
lishments and hence H(JZ) = H( Z) where Z represents the strictly posi-
r g » *"
tive probability vector of n elements associated with those n blocks
in which commercial establishments are located. Further, the unique
.'.
_. ,-Q determined by fZ and corresponding n x n elements of S will be
t » *
80
ft ft
such that H(- ,-Q ) = H( Q ) and thus _ -C = C. These conditions
f»f g>g f»f g,g
follow directly from the admissibility-of -null-events property of entropy
as employed in information theory. (Khinchin, 1957) Thus our problem
reduces immediately to the simpler problem of determining a more econo-
mical characterization of the areal distribution g by grouping only the
n elements of the areal distribution f characterized by the reduced vector
JL and the reduced matrix S (n x n). The question remains, however, of
how best to proceed to cluster the elements of f .
In answer to this question, we propose the following cluster
analysis procedure. For notational convenience here, we will denote
* 0 0
the unique f fQ as Q and JZ as Z or simply Z. Now consider the merger
of two elements of f such that its resulting characterization consists
of only n-1 elements. Further, define the structural information trans-
mission between f and its first-stage reduced characterization as
TCz/z) = H(Z) + H(1Z) - H(1Q)
1 . .
Here, the vector Z will have only n-1 non-zero elements, and, similarly,
1
the matrix Q will have only n-1 non-zero rows. Clearly if we merge two
elements together, we should add their associated probabilities that
are proportional to the aggregate data values recorded within them
1 _ 0 0
separately. Thus, here merging elements k and 1, let z,- z, + z and
1 1
to insure that Z remains a probability vector set z = 0. Also, if
\
T(Z, Z) is to be a legitimate information transmission function, then
Q must be a joint probability distribution with column marginals Z
1 10 0
and row marginals Z. Therefore, we must also set q, .- q . + q .
1 . 1 .
and q . = 0 for all j = l,...,n. The matrix Q will then contain one
81
row of all zeroes corresponding to the elements subsumed by the two-
element cluster k. Which two elements k and 1 should we merge? Clearly,
those two elements that render maximal the structural information trans-
1
mission T(Z, Z) between the original complete characterization of f
and its first-stage reduced characterization for merger of these two
elements will minimize the loss of structural information concerning f
over all possible pairwise element mergers.
This same reasoning may be employed to devise a general pairwise
cluster merging algorithm that moves progressively from an initial stage
of n clusters (n the number of given distribution elements) to a final
stage where all elements have been merged into a single cluster. At the
t-th stage of pairwise cluster merging, n-t clusters will remain dis-
tinct and unmerged. Let I denote the set of integers associated with
the n-t clusters remaining at the t-th stage. Also, I will denote the
set of subscripts of non-zero elements of the probability vector Z,
and hence the set of subscripts of non-zero rows of the joint proba-
bility matrix Q, corresponding to the n-t remaining clusters.
Now let k and 1, ke I and le I represent any two clusters consi-
dered for merger at the t-th stage. Then our pairwise cluster merging
rule states: merge clusters k and 1 such that, at stage t+1, the result-
ing structural information transmission between the reduced set of
n - (t + 1) clusters and the original full set of n clusters will be
maximal. Again, this condition is equivalent to the requirement that,
at each stage, that pair of clusters should be merged that involves
minimal loss of spatial complexity shared between the original descrip-
tion of an areal distribution and its reduced description. Formulating
82
this rule mathematically, we have
(4.9) max T(Z,t+1Z) = H(Z) + H(t+1Z) - H(t+1Q)
k,le I
where H(Z) is of course constant over all cluster mergers and where
H(t+1Z) = H(tZ)
t . t t , t
+ zklog zk + z^log z1
-( z. + z. ) log( z, + z.. )
k Ik 1
and
H(t+1Q) = H(tQ)
♦fW'Ac.j + f tql,jl0gt<1l,j
" ? (\,j + \,j) l0g(tqk,j + tqk,l} '
Immediately following any pairwise cluster merger, updating opera-
tions are necessary. If clusters k and 1 are merged at stage t, then
t t+1 t+1 t
the probability vector Z is updated to Z by letting Z = Z and
t+1 t t „ t+1 _ . . , v.,.^
resetting z, = z, + z and z = 0. Also, the ]oint probability
matrix Q is updated to Q by letting Q = Q and resetting q, . =
q + q and q = 0 for all j = l,...,n. Then, delete 1 from
k » D 1»3 1 » j
t t+1
the set of clusters I to obtain the reduced set I of n - (t+1) clus-
ters. A list structure should be maintained over all cluster mergers
recording the specific elements belonging to each of the n - (t+1)
clusters remaining at each stage t+1.
Note again that at the initial stage t=0, the set of clusters
ke I will be the full set of integers k = 1 , . . . ,n representing the
subscripts of the n non-zero rows and elements of Q and Z respectively.
83
At each stage t, the cardinality of the set I will be reduced by one,
so that at stage t = n-1 the set I will consist of only one cluster,
i.e. , all clusters will have been merged into a single cluster.
0 0 * 0
Also note that at stage t+0, Z = Z, Q = Q , and hence T(Z, Z)= £
t jt
Over successive stages of pairwise cluster merging, t=0,l ,. . . ,n-l , we
will have
H(°Z) > H(1Z) > ... > H(tZ) > ... > H(n"1Z) = 0
H(°Q) > H^Q) > ... > H(tQ) > ... > H(n_1Q) = H(Z) ,
T(Z,°Z) > Kz/z) > ... > KZ^Z) > ... > T(Z,n_1Z) = 0.
Thus, at each stage of pairwise cluster merging, there occurs necessarily
some loss of the structural information transmission between the original
characterization of an areal distribution and its reduced characteri-
zation. At each stage, our rule is to minimize this amount of struc-
tural information lost. Thus, we may consider the clustering technique
outlined as a minimum-pattern-information-loss cluster analysis procedure,
To illustrate the behavior of this cluster analysis procedure,
four small example problems are given in Figures 8, 9, 10, and 11. On
the left side of each of these four figures, a spatial distribution of
elements is shown together with a graphic description of the clustering
process. Alphabetic characters represent specific distribution elements,
numerals denote the specific order of pairwise cluster merging, and a
hierarchical outlining system is used to indicate the specific elements
grouped together at each stage. In all four cases all elements are
centered within unit cells of an 8 x 8 chessboard grid with centroid
84
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88
coordinates Mx. and My. taken from the set {(1,1) ,(l ,2) ,. . . ,(8,8)},
and spatial distribution elements are weighted equally. Further, each
element should be considered as square with unit x and y dimensions;
hence, Vx. = Vy. = 1/12 for all i = 1,...,64.
l J i
Figure 8 demonstrates the symmetric behavior of the cluster analy-
sis procedure given a symmetric spatial distribution. Element pairs
from each of the four obvious groups are merged in turn before merger
of the two double-element clusters of each group occurs. Notice here
the sharp elbow in the graph of the structural-information-transmission-
loss function accompanying the clustering display on the left. The
sharp break occurs at stage t=12, suggesting that the most appropriate
stopping point for clustering might leave the last four clusters unmerged.
Even here, however, some structural information transmission is lost in
moving from the original 16-element characterization of the spatial dis-
tribution to the more economical 4-element characterization. We have
simply destroyed information concerning the structure of the pattern by
simplifying its characterization.
While the spatial distribution of elements in Figure 9 lacks the
perfect symmetry of the pattern of Figure 8 , it too strongly suggests
four major clusters, and the route taken by the cluster analysis pro-
cedure to arrive at the obvious four clusters is similar in many respects
to the successive stages of clustering in Figure 8. Figure 10 depicts
a logical clustering of sixteen elements into three major clusters, but
here the elements of the three apparent clusters are more diffused and
hence the elbow in the graph of the structural- in format ion-transmission-
loss function is less sharp. Figure 11 demonstrates an extreme case in
89
which, while the clustering is reasonable given that we must cluster,
no elbow at all is apparent in the in format ion -loss function over all
stages and thus we may conclude that no simpler characterization of the
original distribution can be made without undue loss of information
concerning the pattern.
TABLE 1. Values of GDV, J^DI2, H(_Z), H( Q ), and C for the
four spatial distributions of Figures 8, 9, 10, and 11.
f fGDV f ^DI2 HC^) H(f fQ*) f fC
squared distances .... bits
37.33 3.07 4.0 5.98 2.03
29.29 3.24 4.0 5.97 2.03
20.29 3.04 4.0 6.22 1.78
21.33 1.98 3.0 4.34 1.66
Cluster Analysis of Spatial Associations Between Distributions
The cluster analysis procedure described and outlined above may
also be used for analysis of structure of spatial associations exist-
ing between areal distributions. While cluster analysis of a wide
variety of association matrices is possible using the technique, here
we will discuss only the application of the method to analysis of areal
2 2
distribution associations of the form ,. EDI and ,. LDI .
Note that distribution matrices of the form [ EDI and
[ LDI ] will be square and symmetric with all elements, including
diagonals, strictly positive. Diagonal elements will be strictly
90
positive for [_ LDI matrices, as well as for [_ EDI 1 matrices, due
lf,g J lf,g J *
to our inclusion of intra-tract residual variances within the definition
of the S matrix of expected squared distances and our definition of
. LDI2 via (3.33).
f »g
Assume we are given a set of n? urban spatial distributions F,
all characterized with respect to the same spatial sampling frame and
its associated S matrix. Then for all pairs of areal distributions, f
2
and g where feF and geF, we may compute ,. EDI using the method described
* »g
above. The result is a square symmetric matrix (n* x n') of strictly
2
positive elements where each element f EDI represents a measure of
the mean entropic squared distance between the two distributions f and
g-
2
Now let the nf x n* matrix of _ EDI measures be denoted simply
f »g
E. Also, assuming equal weights for all spatial distributions, f = l,...,nf,
define the maximally entropic probability vector W where w_ = 1/n' for
all f = l,...,n'. Thus, H(W) = log nf. Then consider the functional
o n ' n ' j.
EDI = £ E p" e^
f g f»g f ,g
li
where P (nf x n') is a joint probability matrix with row and column mar-
ginals equal to W such that EDI may be considered as a measure of the
grand mean entropic squared distance of interaction over all pairs of
distributions. Then, by reasoning identical to that given in the first
section of this chapter, we may determine a maximally entropic set of
weighted components of EDI , and in a manner identical to the formulation
* 2 . *
of _ Q for our _ EDI computations, determine here the unique P that
f,g f,g
makes the weighted components of EDI maximally entropic subject only
91
ft
to the condition that P have row and column marginals equal to the
maximally entropic W.
Again, EDI is determined in a least biased manner, i.e., it is
maximally noncommittal with respect to all missing information. It
represents a measure of the overall spatial dissociation existing among
all distributions. More importantly here, however, imbedded within its
formulation is the maximally entropic P matrix which, together with W,
allows us to use directly the cluster analysis method presented in the
second section of this chapter for analysis of the structure of associa-
tions existing between a set of spatial distributions. To illustrate
the utility of these methods for description of urban spatial organiza-
tion, let us now turn to an example application.
CHAPTER V
URBAN SPATIAL DISTRIBUTION ANALYSIS: A WORKED EXAMPLE
The Hypothetical Urban Area
To illustrate the application of the methods developed above in
Chapter IV for analysis of urban spatial distributions, a hypothetical
city was designed. We chose to work with a fictitious urban area rather
than an actual one, not only to avoid data collection problems, but also
to permit ourselves more freedom in the choice of specific spatial dis-
tributions to be included within the analysis.
Generally, two sets of concerns determined the nature of the
hypothetical community. On one hand, the objective was to illustrate
the application of analysis techniques developed with as little effort
as possible expended in data preparation and data processing tasks. At
the same time, however, we needed an example problem of sufficient rich-
ness of complexity to permit the full capabilities of the model to be
tested.
As a compromise between these two objectives, a fictitious Ameri-
can midwestern community of approximately 110,000 population was designed,
(Figure 12) Bartholomew (1955) was consulted to determine the average
land area and proportional distributions of specific land uses for a
sample of detached midwestern communities (Lincoln, Kansas City, and
93
LEGEND
Single-family Residential
H H \ L
1 MILE
I 1 I 1 I-
3
^§ [wc-Family RESIDE!*.! IAl
tffttffl MuLT I -FAMILY RtSITF.NVAL
Public and Semi -public
PAPKS AND PlAv'5i?-.JND'.'
Light Industpy
Heav, Ini>uc,tpy
§§i$§ Railroad Psopeptv
| ] Vacant
Fig. 12 Generalized land use for the hypothetical urban area
94
Wichita) having populations at survey dates of approximately 110,000.
Our hypothetical community occupied a land area of twenty square miles
or 12,800 acres. Proportional distributions of land uses for the commu-
nity are shown in Table 2.
TABLE 2. Proportional distributions of land in different uses for the
hypothetical urban area.
Land Use
of Total
Acres
30
3840
2
256
1
128
2
256
1
128
2
256
5
640
3
384
6
768
23
2944
25
3200
100
12,800
Single -family
Two- family
Mult i- family
Commercial
Light Industry
Heavy Industry
Railroad Property
Parks and Playgrounds
Public and Semi-public
Vacant
Streets
TOTAL
A frame for spatial aggregation of all land uses and other urban
phenomena was selected as shown in Figure 13. The frame was chosen
deliberately to have areal units of different sizes. Tracts containing
the central business district (CBD) and the four outlying commercial
centers were selected as quarter-quarter sections of a township-range
95
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96
93
. 4 6 ( 10 \l 14 16 1f ?0 ii i*- It ?K 30 3? 34 36 36 40
Fig. 13. Zonal system subdividing urbanized area into areal
units for data aggregation. (Coordinates in 1/8-miles) Also depicted
is network of major arterial streets.
96
land survey grid. All other tracts were taken as quarter sections with
the exception of four peripheral corner tracts which were taken as full
square-mile sections.
To simplify the layout of the hypothetical community, it was
decided to distribute the 25% of the total land area in streets uniformly
across all tracts. Approximately two-thirds of all vacant land was allo-
cated to tracts along the periphery of the urban area and one-third was
allocated in a random manner across interior tracts. The distribution
of industrial land use and railroad property was determined in large
measure by the placement of two major railroads, one running north-south
through the center of the city and the other cutting diagonally across
the southeast sector. The distributions of all land uses (except streets)
are depicted in the block diagrams of Figures 14-23 where the heights
of all tract blocks are scaled so that the sum of the volumes of all
blocks is constant over all diagrams. Thus, these diagrams may be viewed
as graphic presentations of the discrete bivariate probability distri-
butions characterizing the distributions of land uses across the city.
To facilitate further the design of the hypothetical city, all
104 tracts of the spatial sampling frame were subdivided into 2.5-acre
cells, and all land uses were allocated across all tracts in discrete
2.5-acre quantities. Thus the complete twenty square-mile area (12,800
acres) for design purposes could be considered as consisting of 5120
2.5-acre cells. (See Figure 12). The decision to allocate the 25% of
all land occupied by streets uniformly throughout the urban area simpli-
fied matters considerably. To account for land in streets, we had only
to multiply the total acreages of all land uses (except streets) given
97
Fig. 14. Probability distribution of single-family residential land use
Fig. 15. Probability distribution of two-family residential land use
98
Fig. 16. Probability distribution of multi-family residential land
use
Fig. 17. Probability distribution of commercial land u
se
99
Fig. 18. Probability distribution of public and semi-public land use
Fig. 19. Probability distribution of parks and playgrounds
100
Fig. 20. Probability distribution of light industry
Fig. 21. Probability distribution of heavy industry
101
Fig. 22. Probability distribution of railroad property
Fig. 23. Probability distribution of vacant land
102
in Table 2 by the factor 1.333 to obtain generalized land use acreages
in which associated street acreages were subsumed. These generalized
land use acreages were then divided by the factor 2.5 and truncated to
integer values to obtain a proportional distribution of the remaining
ten generalized land uses over the set of 5120 2.5-acre cells.
The specific allocation of land uses over tracts and cells depict-
ed in Figure 12 was made primarily in an intuitive manner with occasional
reference to land use survey and planning data given in Chapin (1965)
and Goodman and Freund (1968). To reflect more closely the spatial com-
plexity of an actual urban area, it was decided that the community should
be multinucleated with respect to centers of both industrial and commer-
cial activities. Two major industrial centers were located to the south
and to the east of the CBD along the two railway corridors, and both light
and heavy industrial land uses were interspersed within these two cen-
ters. Other light industrial land uses were located at the intersections
of major arterials with two interstate highways bypassing the community
on the north and west sides. (See Figures 20, 21, and 22).
In addition to the primary concentration of commercial land uses
within the CBD, four secondary concentrations of commercial activities
representing suburban shopping centers were located in each of the north,
east, south, and west sectors of the city. (Figure 17) Also, eight
smaller clusters of commercial land uses were scattered throughout the
community along major streets to represent ribbon commercial develop-
ments along arterials and small neighborhood shopping centers. (Figure
12)
103
The pattern of mult i- family residential land use followed closely
the distribution of commercial activity centers. (Figures 16 and 17)
Our rationale here was simply that both multi-family and commercial land
use centers would be expelled from low- density residential neighborhoods
and would tend to cluster together at locations along major arterials.
Duplex housing tended to lie close to the CBD and major industrial areas.
(Figure 15) Single-family residential land use was distributed in a
more uniform manner across the entire urban area. (Figure 14)
The pattern of public and semi-public land uses was determined
primarily by the placement of public and private schools. Our city includ-
ed a community college occupying the 160 acres of tract 47. Following
Bartholomew's land use classification system (1955), two golf courses of
160 acres each, one public and one private, were also included within
the distribution of public and semi-public land use. The private golf
course was located in tract 42 and determined in large manner the low-
density, high-rent character of the west side of town. The public golf
course was located in tract 50 in service to the newer suburban develop-
ment of the northeast sector. (Figures 12 and 18)
Parks and playgrounds were distributed fairly uniformly through-
out the urban area with the exception of one large central park of 240
acres, which was located across tracts 17 and 28 just to the northwest
of the CBD. A smaller municipal park of 40 acres was located in tract
55. All other parks and playgrounds were smaller (5 to 20 acres) and
assumed to be neighborhood-serving in character.
104
Urban Spatial Distributions Selected for Analysis
After delineating the general pattern of land uses for our hypo-
thetical community, it was then possible to focus on spatial distribu-
tions of specific urban variables. Our main objective was to select a
set of spatially distributed variables representative of a wide variety
of the socioeconomic activities of urban areas, including residential,
cultural, recreational, commercial, and industrial activities. Recog-
nizing the strong interdependence between the locations of certain urban
activities and transportation facilities, we wished also to include
variables related to the configuration of major arterial streets and
railroad facilities in the analysis. Within these broad objectives, our
selection of a specific set of spatially distributed urban variables was
somewhat arbitrary.
Table 3 lists 32 variables corresponding to 32 spatial distribu-
tions of urban phenomena selected for the example analysis. In each
case, aggregate data values for all variables, expressed in terms of the
units given in Table 3, were recorded for all 104 tracts of the sampling
frame. Figures 30-61 in Appendix 1 display the distributions of aggre-
gate data variables across all tracts for each of the variables selected.
It should be noted that the prior allocation of all 2.5-acre cells
of the city to specific land uses as shown in Figure 12 played a funda-
mental role in the subsequent estimation of aggregate data values across
tracts for all distributions. For example, given the specific allocation
across tracts of the 137 2.5-acre cells of two- family residential land use
implied by Table 2 (and accounting for the additional acreage included
lp
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106
for streets), the number of duplex housing units in each tract could be
determined immediately by assuming an average factor of 10 dwelling units
per acre for duplex development. In like manner, the 68 cells of multi-
family residential land use shown in Figure 12, using an average density
factor of 30 dwelling units per acre, determined the spatial distribu-
tion of apartment dwelling units depicted in Figure 32 of Appendix 1.
To bring about some variety of single-family residential densi-
ties, three density factors of 4, 6, and 8 dwelling units per acre were
applied respectively to 512, 992, and 496 cells of single-family land
use. Standard 1/6-acre lot development was distributed rather uniformly
throughout the urban area, 1/4-acre development was distributed mainly
in the western section of town, and the 1/8-acre lot development was
concentrated mainly in that area between the CBD and the industrial cen-
ters. In addition to these single-family housing densities, an average
density factor of 12 units per acre was employed for the number of mobile
home units of four trailer courts in tracts 36, 37, 93, 94, and 98.
(Figure 33, Appendix 1) Mobile home development is shown as single-
family land use in Figure 12.
All public and private schools (nursery, elementary, junior high,
and senior high) were distributed throughout the urban area in more or
less Loschian hierarchical manner. Here, 36 daycare centers and nursery
schools and 20 elementary schools were distributed rather uniformly
across all residential land. (Figure 36) Forming more stellated pat-
terns, 10 junior high and 5 senior high schools (both public and private)
were located at approximately equi-spaced points throughout the community.
107
Again, in our effort to reflect reality, junior and senior high schools
were occasionally placed side-by-side on a single parcel of public land.
(Figures 35-38) In addition to the community college occupying tract 47
a number of vocational or trade schools were located in tracts close to
the CBD. (Figure 39)
In defining the spatial distribution of outdoor recreation areas
it was decided that the central municipal park of 240 acres was of a
character sufficiently different from all other neighborhood parks that
it should not be included within the pattern of neighborhood parks and
playgrounds. (Figure 40) Since this single park comprised almost two-
thirds of the 384 acres of land devoted to all parks within the city,
to include it within the city-wide distribution of park and playground
acreage would have resulted in its complete dominance of the pattern
and destroyed the spatial association between local parks and neighbor-
hoods. Hence, this major central park was grouped with the two golf
courses of tracts 42 and 50 to define a separate pattern of regional
outdoor recreation areas. (Figure 41)
In addition to schools and outdoor recreation areas, two other
areal distributions of cultural and recreational activities, movie
theaters and churches, were defined for the hypothetical community.
Churches were distributed in a uniform manner over all non-industrial
land uses of the community. (Figure 43) Movie theaters were located in
major commercial centers where adequate parking facilities could be
assumed to be located.
Eleven different areal distributions of commercial establish-
ments ranging from full-line department stores to fast-food drive-ins
108
were defined for the community. (Patterns 15-25 of Table 3; Figures
44-54 of Appendix 1) Our attempt here was to select a variety of com-
mercial activities whose areal distributions would be representative
of activities typically associated with major shopping districts, strip
commercial developments along arterials, and local neighborhood retail
outlets. Thus, full-line department, furniture, and hardware stores
tended to cluster at the CBD and major shopping centers. Food, drug,
and liquor stores were more evenly distributed throughout the entire
community, and auto service stations and restaurants were distributed
along major arterials.
The distributions of major arterial street frontage and railroad
property (Figures 60 and 61), as well as the distributions of heavy and
light industry (Figures 56 and 57) patterned with respect to these trans-
portation facilities, were taken directly from the prior delineated land
use -transport at ion system of the community. Areal distributions of
private office space (Figure 58) and banking activity (Figure 59) were
defined with strong CBD orientations. As an additional item, four region-
serving hospitals were located at points close to the CBD.
It should be noted that a variety of measurement units were used
in quantifying the 32 areal distributions selected for analysis. Resi-
dential distributions were measured in terms of numbers of dwelling units,
school distributions in terms of enrollment figures, commercial estab-
lishments in terms of floor areas, and so forth. Since the areal dis-
tribution itself (characterized as a discrete probability function)
represents the unit of analysis, however, we should not be accused of
"mixing apples and oranges." Our method is explicitly designed to allow
109
analysis of spatially distributed urban phenomena quantified in terms
of whatever variables are convenient to observation and measurement and
highly correlated with the specific phenomena of interest. Of course,
there always remains the inevitable trade-off between the objectives
of precision of phenomena measurement and economy of data collection.
Example Analyses Performed
Having defined geographically the set of 32 areal distributions
for our hypothetical urban community, all distributions were character-
ized as discrete probability distributions across the tracts of our
2
sampling frame. Then, values of _ EDI were computed between all pairs
* »6
of distributions using the method described in Chapter IV.
2 2
Values of _ EDI and JCDI were computed independently for
each pair of distributions to evaluate numerical error effects within
2 2
computation. In theory, _ EDI should be identically equal to JjDI .
In practice, we found that, using single-precision arithmetic on an
2 2
IBM 360/91 computer, values of . EDI and _EDI differed almost
always after the fifth significant digit, and, where f and g both had
a large number of non-zero elements (both greater than 30), they differed
often after the third significant digit. Thus we conclude that any
future experiments or application of the method should employ double-
precision arithmetic within computations.
2
The result of these comDutations was the 32 x 32 matrix of _ EDI
f,g
values reproduced as Table 4 in Appendix 2. Then, weighting all distri-
butions equally, we applied the cluster analysis algorithm developed in
Chapter IV to investigate the structure of spatial associations existing
110
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112
among all 32 distributions. Figure 24 displays from top to bottom the
complete sequence of pairwise cluster mergers that occurred in moving
from the initial stage of 32 clusters (distributions) to the final stage
of a single cluster. Figure 25 graphs the structural-information-trans-
mission-loss function over the complete sequence of pairwise cluster
merges.
Up to about the 19th stage of cluster grouping, all results seem
reasonable. Particularly striking is the emergence of the cluster of
arterial street-oriented activities. The seed of this cluster is the
early merger of areal distributions corresponding to arterial frontage
(5), auto service stations (W), fast-food drive-ins (Y), and full-time
restaurants (X). Merging with this cluster soon after is the two-element
cluster of specialty food and liquor stores (U) and pharmacies (V).
Joining later is the two-element cluster composed of multi-family hous-
ing (C) and food supermarkets (S). With the exception of junior high
schools (H), which becomes part of this cluster much later, all of these
activities are typically strongly patterned with respect to the network
of major arterial streets. The weak (late) merging of junior highs with
this arterial-oriented set of activities is simply an artifact of our
specific placement of the 10 junior high schools within our fictitious
community .
Paralleling the sequence of cluster mergers resulting in the set
of arterial-oriented activities is the development of a cluster of
neighborhood-oriented activities. The seed for this cluster is the early
merger of spatial distributions corresponding to single-family housing
(A), churches (N), daycare centers and nursery schools (F), and quick-
113
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114
shop grocery stores (T). Joining this cluster soon thereafter is the
two-element cluster of elementary schools (G) and neighborhood parks
and playgrounds (K). Later, as a consequence of the specific spatial
layout of our hypothetical community, the two-element cluster of two-
family housing (B) and private office space (3) merges with the set of
neighborhood-oriented activities. Here again, the merger of private
office space (3) with other neighborhood-oriented activities must be
considered an accidental consequence of the specific layout of our com-
munity.
The merger of the distributions of heavy industrial employment
(1) and railroad property (6) seems of course entirely in order. This
two-element cluster remains distinct until the very last stages of
cluster merging when at last mobile homes (D), forced to join some clus-
ter, merges with it. Note that the distribution of light industry
employment (2) does not merge with rail-oriented industry due to the
location of considerable amounts of light industry at interstate high-
way interchanges.
The cluster of full-line department stores (0), apparel shops (P),
furniture stores (Q), and hardware stores ( R) may be considered as a
set of retail establishments representative of major commercial centers,
i.e., the CBD and the four suburban shopping centers. This cluster
remains intact until joined late in the clustering process by banking
activity (M-).
The remaining set of distributions all represent activities that,
in the given community, appear to lack co-organization with any other
activities. Mobile homes (D), colleges and vocational schools (J), high
115
schools (I), hospitals (Z), and outdoor recreation centers (golf courses
and major parks) (L) appear to be spatially distributed in a manner
independent of other activity distributions. For the most part, this
is due simply to the fact that each of these distributions consists of
so few elements that no complexity of pattern exists , and hence no co-
organization with other spatial distributions can possibly exist. Within
an urban area of the scale chosen, locations for such activities will
appear to be independent of the locations of other activities.
As an independent means of analyzing the structure of associations
between distributions, the methodology of nonmetric multidimensional
scaling seemed appropriate. Like all cluster analysis procedures, non-
metric multidimensional scaling procedures are heuristic data analysis
techniques designed explicitly to expose the structure of relationships
existing between elements of some data matrix. (Green and Carmone, 1970;
Shepard et al., 1972) In the words of one of the pioneers of multidi-
mensional scaling methods,
the unifying purpose that these techniques share, despite their
diversity, is the double one (a) of somehow getting hold of what-
ever pattern or structure may otherwise lie hidden in a matrix of
empirical data and (b) of representing that structure in a form
that is much more accessible to the human eye — namely, as a geome-
trical model or picture. (Shepard et al., 1972, p. 1)
Further, since nonmetric scaling techniques (unlike principal components
analysis and factor analysis methods) require no specific metric proper-
ties of data association measures to be analyzed, this mode of analysis
seemed particularly appropriate to our problem, since we know little
concerning the metric properties of our f EDI distance measure.
To obtain a matrix of inter-distribution distances appropriate
for multidimensional scaling, the symmetry of the [- EDI ] matrix was
* »g
116
forced by simply averaging corresponding off-diagonal elements. Then,
square roots of all elements of [f EDI ] were taken to obtain the matrix
of mean entropic distances of interaction [ EDl] . A matrix of pseudo-
metric, inter-distribution distances was then defined as [\. EDI'1 where
_ EDI' = EDI - \. -EDI - h EDI
f.g f»g ft^ g.g
for all f,g = 1,...,32. This matrix is given as Table 5 of Appendix 2.
The elements of this new matrix [f EDI*] are said to be pseudo-metric
inter-distribution distances, since, while they satisfy the conditions
that £ -EDI1 = 0 for all f = 1,...,32 and £ EDI1 = JJDI1 > 0 for all
f»r f,g g,r
f t g» ^»g = 1,...,32, there is no assurance that the triangular inequal-
ity metric property will hold for all triplets of distributions f, g,
and h, i.e., that
- EDI' > .EDI1 + . EDI1
ftg " ffh h,g
will be true for all f,g,h = 1,...,32.
The specific multidimensional scaling algorithm selected for
analysis of the [f EDI'] matrix was a procedure developed by Young called
TORSCA-9 (1967, 1968). Figure 26 displays the best-fitting two-dimensional
representation of the [_ EDI'] matrix determined by TORSCA-9. Here,
f»g
interpoint distances between individual symbols A-Z and 1-6 have been
made as proportional as possible to the original . EDI' measures subject
t >g
to the dual objective that all interpoint distances of the final solu-
tion have the same rank order as the original distance measures _ EDI'.
To display the agreement between our cluster analysis method and
Young's two-dimensional scaling solution of all - EDI' measures, in
* *g
Figure 26 we have indicated directly on the graphical output from TORSCA-9
117
the sequence of pairwise cluster mergers up to the 19th stage. We find
the agreement between cluster analysis results and the multidimensional
scaling solution rather striking. Clusters are clearly apparent for
neighborhood-oriented activities, strip commercial activities, major
commercial center activities, and so forth. Note here that three dis-
tributions, i.e., transient lodgings (E), mobile homes (D), and colleges
and vocational schools (J), were so spatially dissociated with all other
distributions that they fell outside the limits of TORSCA-9's display
and thus are not shown on Figure 26. Note also that these same three
distributions, (E), (D), and (J), were the last three individual distri-
butions to merge with other sets of distributions in our cluster analysis.
(See Figure 24).
2
As an additional exercise, we also analyzed the matrix of ,, LDI
spatial dissociation measures between all pairs of distributions. This
exercise was undertaken for two purposes. First, we wanted to see how
2 . 2
much our measures of ,. EDI would differ from corresponding ,. LDI
f»g f ,g
measures. Second, we wanted to examine the sensitivity of both cluster
analysis and multidimensional scaling procedures to at least one differ-
ent set of distribution dissociation measures.
2
For this experiment, we first computed values of _ LDI between
all pairs of distributions using an IBM-supplied computer program for
transportation programming problems. Solution of [32 (32 + 1)]/ 2
transportation problems resulted in the symmetric matrix [ ,. LDI of
t »g
2
Table 6, Appendix 3. Note that the values of . EDI of Table 4, Appen-
* *S
dix 2 are generally twice as large as the corresponding values of the
2
minimal measures of „ LDI .
ftg
118
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Fig. 27. Hierarchical tree showing seauence of cluster merges within
cluster analysis of L LDI2] matrix of areal distribution dissociation
measures.
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Using the same procedure as before, we then cluster analyzed the
matrix [f LDI ] and obtained the hierarchical cluster merging tree of
Figure 27 and the structural- information-transmission-loss function of
Figure 28. Note that, while some cluster merging sequences are similar
to those obtained before, e.g., arterial streets (5) still merge initially
with auto service stations (W) and fast-food drive-ins (Y), and heavy
industry (1) merges with railroad property (6), overall, the results of
the two cluster analyses are quite different. For example, churches
(N), nurseries (F), and quick-shop groceries (T) merge with duplex hous-
ing (B) and it is not until the 20th stage that these activities merge
with other obvious neighborhood-oriented activities such as single-
family housing (A), elementary schools (G), and parks (K). But by the
20th stage, single-family housing, elementary schools, and parks have
already been merged with arterial-oriented activities. Hence, we
2
evaluate this cluster analysis of . LDI measures inferior to our
f.g
2
prior analysis of - EDI measures.
3 fig
2
Continuing our experiment with the _ LDI measures, we performed
a multidimensional scaling analysis. As before, we took square roots
2
of all «. LDI measures to obtain linear _ LDI measures and defined a
f.g f»g
new set of pseudo-metric inter-distribution distances by computing
- LDI* = LDI - \. £LDI - h LDI
f.g ftg f»f g,g
for all f ,g=l,. .. ,32. This matrix [ LDI'] is given as Table 7 of
f ♦ £
Appendix 3. Comparing Table 7 and Table 5 (Appendix 2), we find that
values of _ LDI', in general, now are only slightly smaller than
r»g
corresponding values of _ EDI'.
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122
Applying TORSCA-9 to this new matrix of distribution dissociation
measures, we obtained the two-dimensional point representation of inter-
distribution distances depicted in Figure 29. Again, mobile homes (D)
and colleges and vocational schools (J) are so dissociated with all other
distributions that they fall outside the limits of TORSCA-9* s display
area. We will admit, however, that the multidimensional scaling repre-
sentation of the remaining distribution associations is more appealing
than the results of our cluster analysis. In fact, the two geometric
configurations of Figures 26 and 29 are quite similar despite the dif-
ferent inter-distribution distances scaled. This is not entirely unex-
pected, since the nonmetric scaling procedure considers primarily the
rank order of distances between distributions, and, while we know that
the elements . EDI* and . LDI* are different, their rank orderings
f,g f,g
should be not too dissimilar.
CHAPTER VI
SUMMARY AND CONCLUSIONS
Summary of Argument
In Chapter I, we presented our basic case concerning the need
for investigation of more general methods for quantitative description
of the organized complexity of real-world urban space. We argued that
existing social science theory concerning urban spatial organization
was much too limited in scope for explanation of the rich variety of
socioeconomic patterning that we observe across urban landscapes. We
pointed out unresolved methodological questions surrounding those methods
most commonly used for analysis of the ecological interdependence of
geographically patterned urban phenomena, and we called for the develop-
ment of alternative methods of urban spatial distribution analysis better
equipped for the task at hand.
In Chapter II, we reviewed the basic concepts of Shannon-Wiener
information theory seeking some more general mathematical basis for
quantitative description of the essential dimensions of urban spatial
organization. We examined the unique properties of the mathematical
concept of entropy specifically as a measure of informational uncertainty
within telecommunications theory and as a measure of the random complex-
ity of discrete probability distributions in general. With reference
to the entropy-maximization model of intraurban trip distribution, we
124
noted how information theory concepts might be used in conjunction
with origin-destination transportation study data to analyze the extent
of interdependence between the co-organization of various socioeconomic
activities in urban space and the ecology of sociocultural relation-
ships existing between activities. Here, however, the proposed para-
digm was essentially behavioral and thus dependent on extensive obser-
vation and analysis of social activity systems for operationalization.
In Chapter III, we returned to the principal research objective
of our thesis presented and defended in Chapter I, namely, the inves-
tigation of quantitative methods better equipped for analysis of urban
spatial organization as a complex system of differentiated population,
socioeconomic activities, and land use patterns. Translated into
methodological issues, our task became the exploration of more effect-
ive methods for analysis of spatial distributions as well as the
spatial interdependence exhibited between differentiated distribu-
tions. We began this exploration by first reviewing certain basic sta-
tistical concepts commonly employed for analysis of areal distributions
such as measures of distribution central tendency (p. 49), measures of
distribution dispersion such as distance variance (p. 50) and general-
ized distance variance (p. 55), and, as a measure of inter-distribu-
tion spatial dissociation, Bachi's square of quadratic averages of
distances (Bachi, 1957), which we chose to refer to as the generalized
squared distance of interaction between two areal distributions. Our
formulation and presentation of these basic measures of intra- and
125
inter-distribution properties differed from previous formulations for,
in every case, we considered not only distances between centroids of
distribution elements or tracts, but also intra-element residual dis-
tances resulting inevitably as a consequence of the spatial dispersion
of specific point locations within tracts. Despite our reformulations,
however, we remained dissatisfied with each of the above measures of
distribution dispersion and dissociation, for, while all might be trans-
lated mathematically into functions of probabilistic matchings of
elements within and between distributions, in every case the specific
probabilistic matchings implied were completely independent of any
consideration of proximity relationships existing between elements.
Facing the problem of characterizing in a more meaningful man-
ner the spatial organization of areal distributions and the spatial co-
organization exhibited between distributions, we then focused once more
on the entropy -maximization trip-distribution model of urban transpor-
tation systems modeling, this time seeking some unbiased means of gen-
eralizing previous measures of distribution dispersion and dissocia-
tion to depend more directly on proximity relationships between dis-
tribution elements. We found that the entropy-maximization trip-
distribution model (as any other type of trip distribution model would
do as well) left us with a completely arbitrary choice concerning the
specific distance deterrence function to be employed in determining
a spatially interdependent probabilistic matching of elements within
and between distributions.
126
To resolve this problem, in Chapter IV we appealed to information
theory, particularly as interpreted by Jaynes (1957). We pointed out
that each of our measures of distribution dispersion (distance variance)
and distribution dissociation (mean squared distance of interaction)
could be viewed as a sum of weighted squared distances between distribu-
tion elements. Further, the only information that we had concerning
the weights to be applied was that the matrix of weights should be a
joint probability distribution with marginal probabilities equal to
the probabilities associated with aggregate data values over areal dis-
tribution elements (tracts). Viewing our measures as sums of weighted
components, we then adopted the position that the distribution of
weighted components should be made maximally entropic subject to the
single constraint that the matrix of weights be a joint probability dis-
tribution with marginals equal to the given areal distribution proba-
bilities. This position leads to the formulation and solution of least
biased estimates for the weighted components of any of our distribu-
tion measures, and, hence, least biased estimates of the measures them-
selves. We say, following Jaynes, that the procedure is least biased,
since it results in a solution to our problem that is maximally non-
committal with respect to all missing information.
Continuing in Chapter IV, we demonstrated the direct applica-
bility of information theory as an instrument for characterizing the
spatial complexity conveyed by areal distributions. Here again our
information theoretic measures of distribution complexity conveyance
127
were formulated in terms of the unique set of component proximity
relationships determined by our entropy-maximization procedure. Further
it was shown that a minimum-structural-information-loss cluster analy-
sis procedure could be implemented in terms of the same information
theoretic concepts. The resulting procedure was shown to be applicable
for cluster analysis of elements of the same distribution to simplify
its characterization as well as for cluster analysis of sets of areal
distributions structured in accordance with the spatial dissociation
measures computed between them.
In Chapter V, using a hypothetical data set, we demonstrated
the application of the unique measure of distribution dissociation and
the closely associated cluster analysis procedure. As an independent
means of analyzing the structure of dissociations of all hypothetical
distributions, a nonmetric multidimensional scaling analysis was per-
formed. We found a close agreement between our intuitive notion of
how all distributions were spatially interrelated and both cluster
analysis and multidimensional scaling results.
Potential Applications of the Method
As pointed out above in Chapter IV, our unique measure of mean
entropic squared distance between distributions has the property that it
is numerically consistent with respect to the scale and number of areal
units of the spatial sampling frame employed. In other words, as the
resolution of the frame increases, the measure converges asymptotically
to its true value. On the other hand, because of data collection and
128
processing costs, we are typically forced to work with frames of varying
degrees of resolution. However, our methodology associates with each
measure computed to characterize some property of a distribution or the
extent of spatial co- organization existing between distributions informa-
tion-theoretic measures that quantify the amount of distribution complex-
ity with respect to which any particular distance measure has been com-
puted. Thus, while our intra- and inter-distribution dissociation mea-
sures will vary incidentally across different spatial sampling frames,
it is always possible to record for each measure the amount of informa-
tion processed.
This property of our method should make it well suited for analysis
of geographic distributions of a variety of socioeconomic phenomena.
For example, the problem of quantifying in unambiguous fashion the ex-
tent of residential segregation of socioeconomic and ethnic populations
would seem to be directly amenable to our approach. Furthermore, the
method proposed should permit quantitative measurement of the degree to
which certain ethnic populations are assimilated into the total social
fabric of the community as a function of such variables as educational
attainment or annual income.
In a manner similar to the hypothetical example presented in
Chapter V, it should also be possible to analyze the structure of asso-
ciations existing between distributions of any number of socioeconomic
activities within a city. While our model offers directly no predictive
capabilities concerning the spatial structure of any one particular
city, it most certainly can be used as an instrument for quantitative
129
description of urban space, and, hence, provides us with a tool by which
certain theories can be evaluated.
The method would be applicable to comparative analyses of spatial
structure across cities as well. Our model yields a set of distance
measures between various patterns of phenomena, and where the same
phenomena are measured and analyzed across a sample of urban areas, the
structure of pattern associations may be compared. Individual associa-
tion measures as output from our method may be taken as variables them-
selves and conventional multivariate analysis methods used for compari-
sons between cities.
In conclusion, it is our opinion that an understanding of the
total pattern of the city will always be instrumental to our efforts to
cope with the ever-increasing complexity of modern urbanization. To
understand the city as a complex set of patterned phenomena, it is
required that we further the development of methods for unambiguous
description of urban spatial structure. Our effort here has been con-
ducted toward this general goal.
APPENDIX 1
GRAPHICAL DISPLAYS OF THIRTY-TWO AREAL DISTRIBUTIONS
OF HYPOTHETICAL COMMUNITY SELECTED
FOR EXAMPLE ANALYSES
131
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parrcim «io. i
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SINtlf -(»"ll i housing U»MS (» 01 CO'S) S»NoOl • 4
Fig. 30. Pattern of single -family housing units
132
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<• OF OU'S) STFIPOl •
Fig. 31. Pattern of two- family housing units
133
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«iilll-l««H! HOIISI»0 UNITS C» Of 6U"1> SYNSOl « f
Fig. 32. Pattern of mult i- family housing units
131
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»0"llf-HO"r HOUSING UNITS (• Of OU*S> STNSOl > r
Fig. 33. Pattern of mobile-home housing units
135
■•----•- — ••
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P4TTC** tO. i 1MHMFNI LODGING UNITS <f Of TIU'S) STftBOl • f
Fig. 34. Pattern of transient lodging units
136
40
I
1
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C4TC4M CtMISS »ND SuHSfBT SCHOOLS (("HdUKNll STHBCU • I
Fig. 35. Pattern of daycare centers and nursery schools
137
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HHitlMT SCHOOLS <«-fc> (l»»0U«IHI) STdbOt • 6
Fig. 36. Pattern of elementary schools (K-6)
138
40
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JUNIOR NIGH SCH0OIS </-V) ( (NROLIMEND SVftBOl > H
Fig. 37. Pattern of junior high schools (7-9)
139
40
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SENIOR HIGH SCH0015 <10-1?> (l«OU«IKII STHHOl a I
Fig. 38. Pattern of senior high schools (10-12)
140
♦ 0
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rOHIGFS »*» VUC4T10N4L SCHOOLS (FNBOllHINT) STFflOl « J
Fig. 39. Pattern of colleges and vocational schools
141
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P»It(»n HO. 11
8 10 12 14 16 18 20 I? 24 26 28 30 32 34 36 38 40
N[ ItHPOftHOOO P4HCS AND PI » 1 GSOUN [> S (ACRES) S»«BOl ■ I
Fig. 40. Pattern of neighborhood parks and playgrounds
142
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AtklONAl OUTDOOR RECREATION AREAS (ACRtS) STABOl • I
Fig. 41. Pattern of regional outdoor recreation areas
143
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Fig. 42 . Pattern of indoor movie theaters
144
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CHU*C»CS IN0N-V4C4NT) <S»NC1U»R» SC01S) STHB01 « H
Fig. 43. Pattern of churches
145
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IUll-ll«U DtPAOTXtKI SIKH («»»» IN S«-M*100O> StPBOl • 0
Fig. 44. Pattern of full-line department stores
146
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Fig. 4 5 . Pattern of apparel shops
147
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I!:»«1IU»F STORFS (NUT DfP«H«fM) (»0F» IN Sa-FTMUOO) STHROl • 0
Fig. 46. Pattern of furniture stores (not department)
148
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18 m«*SW**I *fO«»S (xi f di P«s '«( xl ) llltl IN Sa-M*100L> SYHBOl • *
Fig. 47 . Pattern of hardware stores (not department)
149
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mi. u SUM »■«•«( TS <««(* IN S«-M«1000) stNMOl • s
Fig. 48 . Pattern of food supermarkets
150
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?U SuIH-ShOP GROCERY S10RIS <»»{» I h S8-MO000) SYMBOL ■ T
Fig. *+9. Pattern of quick-shop grocery stores
151
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ii SPiCIHTf »OO0 »N0 U0U0P. STOPIS (»»f» IN SS-fTOOOOl STHBOl • U
Fig. 50 . Pattern of specialty food and liquor stores
152
40
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»»M»t» no. <•< p>au(m costs s»ns so-*i) svmoi ■ v
Fig. 51. Pattern of pharmacies
153
40
18
36
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32
3C
28
26
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16
1 1
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MfTia* NO. 23 »UTO M8VICI S141I0NJ (LOTS S«-fT*1000> 5f«B0l • U
Fig. 52. Pattern of auto service stations
154
4U
1 H 1
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1
36
M
1/4
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1/4
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8 10 1/ 14 16 in /O // /4 /6 ?n 30 3/ 34 36 58 40
Mill -lint IIST*U**N1S (f Of SIA1S) STObOl • I
Fig. 53. Pattern of full-line restaurants
155
40
M
36
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H
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20
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23
1(1 12 14 16 1« 20 22 24 26 26 30 32 34 56 36 40
MST-FOOD OSIVf-INS (P»»«l«.& SPACES) STMBOl • T
Fig. 54 . Pattern of fast-food drive-ins
156
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i
2 4 6 8 10 M 1* 16 18 ?0 H ?* ?* ?« 30 5? S4 36 38 40
PATTftV NO. ?6 HOSPITALS Cf OP 8IDt) STMOL ■ 1
Fig. 55. Pattern of hospitals
157
40
58
56
M
32
JL
28
26
24
22
2D
1s
16
14
12
10
LL
!
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1 l 1
1
1
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l
1 1
Kan »8A
1 1
i
1 1
L.J..J
1
11*4
392
1
1
1
1
2270
192
2 96
188
2269
■
1
1
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1
1
4 6 8 10 12 14 16 18 2u ^^ 24 26 28 30 )2 34 36 M 40
2? I UPlOTftf NT IN M{»YT INOUStOT (f 0» fBPlOTUS) 31*801 • 1
Fig. 56 . Pattern of employment in heavy industry
158
40
M
»6
**
S?
JO
2*
?6
/4
il
10
14
16
1*
1?
10
J»»
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1021
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7?1
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P4TTIHK MO. ?H
6 1(1 1? 14 16 1* ?U « ?4 ?6 ?C 51) J? 54 56 50 40
f"PL0TME«1 IN IIOhT INDUS?** (* Uf KPluIlM) SYMBOL ■ ?
Fig. 57. Pattern of employment in light industry
159
16
if
Jil
1/
1
i
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1
i
1
i
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?*
2
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z « «
P*TTt»» NO. 19
• 10 1? 14 16 IK ?n <>? /4 ?6 28 50 32 34 36 3« 40
p»i»Mi office sp»ct (SQ-ft«iooo) sfneoi ■ $
Fig. 58. Pattern of private office space
160
40
it
36
34
J*
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►
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P4T11RK Mb. 3<
V 14 16 18 ?< ?? ?4 ?6 >8 50 J? 34 36 38 40
fKMKIDG 4CT1V1TT (( OF T F u I « S ) Sl'RIll •
Fig. 59. Pattern of banking activity
161
40
M
M
M
12
SO
28
26
24
22
20
18
16
14
12
U)
2640
2640
['"»
1320
2640
2640
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s;«o
S260
2640
2640
S28U
b?*0
3280
4280
264U
i3?n
1,20|
2640
S2HU
i2eO
4280
4280
2640
13?0
2640
15.0
132(1 26411
?640 13?0
1320
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1320 264C
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5200
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1520
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2 4 6
PAIltSK NO. 31
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
MAJOR AIKIIU STREET FRONTAGE (L1NE4L Mil) ST»«0l • 4
Fig. 60. Pattern of major arterial street frontage
162
1 1
,1
1
1 1 "I
<\J
>
i^
i-
ib
I 1
1 1
1
1 1 "'1
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30
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1
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l 4 6
PATT(*« NO. W
8 10 M 14 16 If i(j <V «?4 26 2* in 32 34 36 38 40
BA11ROA0 l»OPt«!l (ACRES) <t»HOl • 6
Fig. 61 . Pattern of railroad property
APPENDIX 2
MATRICES OF _ EDI2 and - EDI' MEASURES OF DISSOCIATION BETWEEN
ALL PAIRS OF THIRTY-TWO AREAL DISTRIBUTIONS
SELECTED FOR ANALYSIS
1
4
8
9
164
TABLE 4
2
Values of _ EDI in miles squared for f=l,...,16 and g=l,...,32
*S6
17 3 6 5 6 7 1!
9 1»! 11 1? 13 14 15 16
1/ 18 19 20 21 22 25
25 2< 2/ 2» 2V 30 31
15
1.55435* ?.'75'5* 2.159543 6. 554779 3.842176 1.8V5T73 1.999504 2.435476
2.484M42 3.23/019 2.13040/ 3.0*91(>/ 2.563.04 1.XK.5U9 2.1991*9 2.295635
7.38/336 /./4. .">/ 2.'./ '.'I 1.8655/0 1.91/ '-,6 1. 95032/ 1.81«48/ 1.9479/?
1. 802330 2.5..74J.6 5.//C61 2.9*9967 7.355408 2. i 2 2Vij 7 1.83713fc 3.517708
2
2.6/5911 0.7/6*19 2.7/»/4/ 3.515C/4 5.370548 1.631915 2.927*45 3.795604
3.001386 5./15559 2.8Q177* 5.56/(34 1.21/.M58 1.99SMR 2.669474 2.996513
7.463760 2.5' 6 .4 5 3.1/'. 4/5 /.Ot'562* 7.4*/ -y 2.1-/56 2.104219 2.7115-2
2.56109!' 1.4392V/ 1.619219 1.313537 1.711431 7.017284 2.265189 1.716375
3
2.13050.-, 2.7/6/34 ,). 366979 6.554321 J. 51V 55 2.161090 1.943850 1.950900
2.226/37 k.'HrSi 2.73/^44 J.'ilft(66 1.451. .7 2.156C15 0. 9819(19 1.267794
1.594297 l.:4 5/6 1.2f.96/4 2.252151 1.26-116 1.255045 1.607795 1.476696
1.666472 2.iV8c'62 5.5650/3 1.V,^7W 2.0/0313 1.855946 1.499695 3.2155/3
4
6.554794 3.515*6? 6.354305 0.2*4156 1 0. 86*r<69 4.906353 6.5»?891 7.757446
7.516092 12.VW7/2 6.59.461 t. 654617 7.5V5..U' 5.7255*6 6.282186 7.0781*2
6.4///2Z 5. 5248/ 7.125492 5.99131/ 6. 10*9/1 5. "31211 5.859993 6.769192
6.713314 4.666-01 1. 76/898 6.5911/34 5.104356 5.449333 6.013813 2.585400
5
3.8421(18 5.57'r,6r 3.511565 10.8,6B9Sfl f. 219655 4.192230 3.775140 3.432683
2.430939 3.CH9i11 4.551VU3 7.664248 2.555541 4.26802* 3.068567 2.382750
2.516141 S..,/S5f)4 3.36-/85 3.46b/-1 1.698/16 3.592990 3.318/48 2.8*9031
3.27280/ 5. '27/16 6. 8 1/661 4.0//3J9 5.012r23 2.777990 3.379224 6.5/7168
6
1.894984 1.631862 2.16U93 4.906345 4.192200 0.9o0396 2.134568 ?./?7B5?
2.33250* 5. •53591 2.14/556 ?.9f.89>5 7.54*2/1' 1.598324 2. 12908/ 2.316424
2.v>55354 2.M4/6/ 2.283*96 1. 5311:50 1.845553 1. 745792 1.624/40 1.935899
1.856647 1.393'.5? 2.535180 3.052062 1.675649 1.846372 1.743487 2.537313
/
1.9994*5 7.72/845 1.943*66 6.5B/9'I;6 5.775143 2.134(1? 0.884944 2.106397
2.353/63 5./1'5'.4 1.86 i/ft? 3.566895 i.">Jt,?72 2. 3^44". i 1.948152 2.1595*3
2.203496 2.151/5? 1.*96537 2. '81773 1.81/728 1.*258f3 1.911554 1.923750
1.919259 2.3*614/ 3.SL-C26 2.3*9595 2.463118 2.409183 1.779047 3.591447
8
2.43691/ 3.7^5^1? 1.95"939 7.757446 J.437'.80 2.727H7 2.106403 0.503724
2.353572 5.725 393 ?.3/./73 5.71435* 2.279701 2.85/C76 2.3023/2 2.285621
2./42J14 2.36/683 1. 92/75/ 2.529305 2.190o17 2.319129 2.379670 2.25C23/
2.2/391/ 2.V3C/9* 4./«i9116 2.52?'j31 /.99218« 2.813129 2.188369 4.462025
9
2.484835 3."MW ?. 226*33 7.5!6rv2 7.43:938 2.3525r</. 2.353764 2.553569
0.219855 4.541332 2. 549828 J.0'-3559 1.74/, 26 2.4*1579 2.260689 1.97694C
1.8862-4 2.2/1(51 e./<7tii 2.163//6 2.301/42 2.2e/*6* 2.244405 2. (1323/
2.132898 1.5/3731 4.521931 2.f>8/5«6 2. 34/526 2.2/6269 2.091218 4.237667
10 10
5.237202 5.7353/- 4.993240 12.998810 3.007511 5.253410 5.710339 5.725407
4.541328 •). 158531 5.91559? 2.80&72J 3.342'.21 4.9f-6132 4.33U89 3.600569
3.4(1514." 4.1/1"</8 5.221405 4.1726C4 5.035197 4.5364»'4 4.252464 4.(6/341
4.32156* 4.335499 8.:.3?.'>o4 6.872990 3.184922 3.196133 4.603783 7.933109
11 11
2.150414 2.8012*4 2.23/"66 6.5985(3 4.551l65 ?. 142612 1.860772 2.368/08
2.549*37 5.91S55.; 1. '6-514 3.779375 7.714597 2.237(47 2.25'652 2.541870
2.47888* 2.287*69 1.9762V? ?.2('66*.? 1.840403 ?. 00050/ 2.069973 2.124556
1.9/3759 2.5//990 3.998510 2.651129 2.656/09 2.597330 2.00*883 3.754651
12 12
3.0890*7 3.5670?7 3.I.8.-C65 ".654621 2.664248 ?. 96*981 3.566287 3.714352
3.06355* 2.-08 722 3.779391 0.4512.-7 2.359714 3.024161 2.985051 2.6T9479
2.631*26 2.7*65/2 3.403850 2.51/205 5.11/91 2.V77C62 2.5/2562 2.6134C7
2./822C4 2.4v5129 4.r?5233 4.8031/3 2.28V-0*. 2.258832 2.803015 4.932590
13 13
2.563536 3.216046 1.45V83 7.595*18 2.555541 ?.548?6r ?.5?4?77 2.77919*
1.747025 3.342'23 2.714403 2.357916 0.1/2705 2.5/63/2 1.4/9*44 1.199402
1.23/521 1.2130-T 1.5452/3 7.301M9 1. 81/3*5 1./u9*.*1 1.961064 1.548??C
1.8/0'52 2.15*'77 4.454591 2. 366698 1.832033 1.440761 1.896639 4.230904
14 14
1.8C64/9 1.995103 2.15602/ 5./255*6 4.268)1/ 1. '.98330 2.324430 2.8570*2
2.48l>83 4.97MC", 2.25/006 3.024163 2.5763/8 1.1 ()3i5 7.264891 2.3*1492
2.336>:/8 2.14736" ?.57.:/«5 1.7!. 1124 2.00<--24 1. "4681ft 1.790425 2. '.'55159
1.89662/ 1.911451 3.131552 3.14/M6 2.048/0/ 2.14t761 1.867023 2.989079
13 15
2.1997(7 2.66946C 0.9*1".')2 6.2821"5 3.068563 2.179C85 1.94*144 2.308331
2.2606»2 4.5i1i*.7 2.256*.35 2."<-br52 1.4/9-43 2.?'.4941 (1. 293565 0.*07316
0.915873 0./4C/.57 1.2-91/6 7.04293d 1.0*4-51 C>'-75n>' 1.300694 1.157156
1.414108 2.C2//88 3.505528 2.0065/6 1.4/(o54 1.200/30 1.362821 3.237142
16 16
2.295613 2.«76496 1.767/60 7.0/K182 7.382/48 7.31641/ 2.159558 2.285618
1.976934 5.600569 7.54V 5/ ?. 60 94 30 1.1994r>0 ?. 5-14*4 0.808444 0.764490
0.965367 0.9-m:1« 1.5lv>0() 2.C5*5*1 1.42553* 1.190520 1.442750 1.122618
1.4/5325 2.116253 4.037242 2.433181 1.439509 1.131663 1.5116/3 3. £04465
165
TABLE U (continued)
2
Values of ^ EDI in miles squared for f=17,...,32 and g=l,...,32
1 2 3 4 > 6 7 8
« It' 11 12 13 14 15 16
1/ 18 IV 20 21 22 21 24
25 26 22 28 2V 30 31 32
17 17
2.38/306 2.463945 1.59/.7.M 6.427/38 2.516139 2. 05514ft 2.203505 2.742009
1.*80283 3.41.514V 2.471"/? 2.ft«1>'-0 1.23/'45 ?.33ftOft5 0.V15997 0.965351
0.166984 i}.'»3iMV3 1./vru6" 1.6/V54 1.35'583 1.32919* 1.4o/994 1.78191?
1.SU8363 1.648/33 3.523680 2.5*1/11 1.014139 0 . V • 1 7 *• t f J 1.581, 327 3.343793
18 18
2.248241 2.58f.(.51 1.040522 5.«-9249(t 5.37R506 2.014766 2.131739 2.367684
2.27105* 4.1*1«'9> 2.28/562 2./.ft5/5 1.213'"T 2.149518 G. 740212 0.948017
fl.V3C'37V 1.124/5! 1.4/4396 2.045.563 1.107</> l.i. 3:1235 1.3UVM 1.254541
1.474040 1.735115 5. 742835 2.235453 1.30013/. 1.022590 1.397876 3.017156
19 19
2.0/8V26 3.1/«,47> 1.2H9f,29 7.125496 3.36*784 2.7<34P7 1.896533 1.9??758
2.227636 5.221 '7/ 1.9?ft«'J9 5.4!Ji:'53 1.545',36 7.5/8791 1.28V1M6 1.5195(6
1.796075 1.474402 (,.Wr-703 2.3245/2 1.17' 517 1.4407H7 1.707074 1.579967
1.S/30V? 2. 55^406 4.2COJ59 7.0169/1 2. 40(^49 2.1/1982 1.657116 3.936848
?0 20
1.865544 2.0G5625 7.252153 5. 99151? 3.46"/7< 1.531030 2.08.1713 2.529298
2.163774 4.177.96 7.20664/ 2.5177.3 2.301-.41 1./01119 2.042939 2.03858.1
I.6/V959 2.0«.3563 2.3206/ 0.833251 1.8ft5*2/ 1.66*147 1.394880 1.673907
1.51,5919 1.4/4225 3.145222 3.056055 1.473394 1./0305/ 1.586070 3.065136
21 21
1.912927 2.4n7"71 1.?6'-1P8 6.10i99O 3.A9K716 1.845336 1.816721 2.190609
2.301744 5.033180 1.840.592 3.11/*95 1.81/5*6 2.0(8825 1.084833 1.425259
1.536601 1.10739) 1.1/0312 1.865432 <).654?/.o 1.(_<.4364 1.255295 1.293162
1.28537* 2.rJ75?71 3.45114/ 2.1*0460 1.8*9/*? 1.632445 1.3271)05 3.188962
72 22
1.V50342 2.187359 1.255033 5.831254 3.592993 1.745/90 1.825/96 2.319126
2.287869 4. 5.564/6 2. 00^494 2.9// *5 1.7rt9>n5 1.84680/ 0.987504 1.190598
1.329194 1."3')74f) 1.44-'./a0 1.6t«l49 1.044353 0.4-9610 1.068155 1.211586
1.277M1 1.767/86 3.15/705 2. 3191-3 1.448280 1.221342 1.202175 2.897681
23 23
1. "19467 2.104203 1.6' 7^02 5.860075 3.318748 1.624744 1.911549 2.379617
2.244411 4.257'.6:> 2.U6WS4 7.5/2560 1. vol 69 1.79042/ 1.3(0702 1.442758
1.463.102 1.333V/0 1.7i7"66 1.3948.<2 1.255/95 1. '161110 0.7*4163 1.199492
1.113615 1. 691)198 3.05VJ1C 2.543*75 1.32(534 1.242851 1.119886 2.85786/
24 74
1.947909 2.701553 1.47f.678 6.769?, 7 7.88V '26 1.935894 1.923840 2.250237
2.013744 4.T47341 2.12453". 2.613413 1.54-2*7 2.r.'55133 1.157165 1.122627
1.28192' 1.<:54553 1.W942 1.6/39' 3 1.295148 1.211547 1.199480 0.838428.
1.11819/ 1.913/53 3.r..27/7 2.594552 1.5121/4 1.306655 1.320665 3.58/293
75 75
1.802306 7.56108.9 1./.6-480 6.713348 3.272^97 1.a56661 1.919?53 2.273913
2.152933 4. .'21568 1.V/5/33 ?./><}><> 1.87C53 1.*9ft63<* 1.414131 1.475323
1.5083// 1. 4/4,149 1.575.83 1.5 .5928 1.2853/8 1.72/824 1.113623 1.118226
0.954'Jlu 1.954;. 26 3.75254^ 2.6I.OVS7 1.68//00 1.546U15 1.317/10 3.499330
26 26
2.308638 1.439796 7.29825/ 4.666*05 5.827/15 1.393G51 2.386140 2.930792
1.5/392". 4.3355.13 ^.'>^/•<«^ 7.475126 2.15862ft 1.911449 2.02/28/ 2.116250
1.648/31 1.7J5"12 2.559<.„4 1.4/4225 2.0/3268 1.762781 1.690192 1.013729
1.954278 0.16"'362 2.397508 3.310118 1.177495 1.750132 1.815504 2.605093
27 27
3.776933 1.M9217 3.5650/3 1.76/96 6.ei7o61 7.535174 3.9P8g?? 4.769005
4.571927 8.037673 3.9V502 4.875276 4.45<.i*8 3.131334 3.5'i5327 4. 05925?
3. 52567ft 3.247'32 4.26(555 3.145223 3.451143 3.1<>2199 3.059001 3.802761
3.752538 2. 39/510 C.3v/2o5 4.C43488 2.5//'/3ft 2.838129 3.229137 1.113823
28 28
2.95997C 3.313534 1.HV6725 6.5"1C42 4.077797 3.052064 2.389591 2.521911
2.6S-/55S 6.-729c.3 2.651' 66 4.803127 2.366746 3.147813 2.C06528 2.433185
2.531/05 2.235<.4»- 2.1169/2 3.05ft056 2.18C462 2.319192 2.543818 2.594548
2.6035VU 3.51012( «..C43«.91 0.265031 3.24o635 3.042103 2.316958 3.428721
29 29
355315 1.7133^5 7.(7''260 5.164727 3.017-16 1.675632 2.463104 2.992168
I
..34857V 3.1'4950 2.6561/6 2.7-1*12 1.832C14 2.0486/1 1.4/C621 1.439487
1.01O10 1.3ui'1/1 2.4ui '26 1.4/35/* 1.859/44 1.4<,8247 1.320480 1.512106
1. 68.7647 1.U74V<, 7.5/7951, 3.746647 1.371753 0.641553 1.622880 2.690198
30 30
2.3/75/v 7.0172/4 1.855934 5.449314 2.77/V87 1.846366 2.409181 2.813127
2.226?'° 3.1''6153 2 . «• 9 / 5 1 •' 7.25^832 1.44; c55 7.148/55 1.200/23 1.131660
0.90/7/7 1. 275»-8 2.1/1«76 1./(5'S3 1.63743ft 1.271330 1.242^40 1.3(6635
1.545988 1.250130 2.33M32 3.0421 i3 f. 641558 0.115563 1.522354 2.849196
31 31
1.83/118 2.2*518.1 1.499/U9 6.0137/9 3.37°730 1.74350ft 1./7«»045 2.188352
2.0*1227 4.6 3/25 2. ft. "50 2.vi3077 1.89ftf47 1.86/07/ 1.367833 1.511587
1.580548 1. 397892 1.65/106 1.5/6'7 1.371V9 1.702182 1.119892 1. 37068b
1.31/68/ 1.815516 3.229161 2.31o9<(» 1.6229/5 1.522348 1.028159 2.948503
32 32
3.519V74 1.716383 3.71558,3 7.5»5442 <.577 99 7.53/32/ 3.591455 4.4619M
4.2376/5 7.953'22 3./54'55 4.932'>/5 4.230907 7.9.-i91(> 5 3.73/154 3.804475
3.343808 3.01/158 3.936-55 3.r;6514* 3.U8/65 2.V976V0 7.85/869 3.58/315
3.49933/ 2.605101 1. 115^29 3.4<nft5i; i.(,'i'.//e 2.>-4921'.l 2.948503 0.631344
166
8
TABLE 5
Values of _ EDI' in miles for f=l,...,16 and g=l,...,32
1 7 3 4 5 6 7 8
9 II 11 1? IS 14 15 16
1/ 18 19 20 21 ?2
75 26 if it 29 JO
I! 15
0.0 1.72.-'>57 1.05/65 ?.373''?1 1.71'' 41 '!. 798515 '». 8.83088 1.1*6735
1.264013 2.0"5591 0.905'2/ 1.44459/ 1.3U3/01 0.6o2694 1.129263 1.17/369
1.735576 1.1'f »6 1.C'.e./.l9 0.MVMJ5 ,:.hvV1i 0.963522 0.806560 C. 866919
0.741)361 1. /(.444ft 1.67J'.5'. 1.431776 1.U,).;6> 1.2420*7 0.738830 1.557534
2
1.72»-95/ 0.0 1.4*48/9 1.777*25 ? . ? 0 7 '■ 3 ? 0.8/365" 1.44*0*9 1.776327
i.i.-vitji i./<*r'?s \.st>:ui ^.nt^.^^ i.655o53 i.:.'.»ivs 1.460916 1.5/34*3
1. 4114(11' 1.3«?1«4> |.«,«>:6(i 1.(195714 1.53'"'// 1.246655 1.15'<530 1.376205
1.3021*0 f).>*5244 1.015'5« 1.671079 1.U67412 1.251433 1.167345 1.01)6129
1.085763 1.4i4«/9 0.1 7. 455355 1.79'v '■;■ 1. 223/94 1.142006 1.231093
1.391.51" 2.1///7' 1./32''77 1.636751 1.(18/2?" 1.175756 0. 807754 0.975739
1.152i99 i).'«145( 1.'7i;ft12'* 1.2.5375 n.M/i 3?4 G.9?9268 1.015997 0.934*;4
1.003995 1.4.M7) 1./84Gh9 1.252W2 1.3042V< 1.270705 0.895632 1.648163
4
2.373"71 1.72/25 7.455355 0.0 3.75*5*4 2.U9799 7.450172 2.713577
2.69334'.' 3.5/b<*53 /. 435566 2.8 78696 7.714273 2.2345*1 2.44M27 2.6C8470
2.49041*. 2.5*4959 ..5*-110 7.330/98 2.3/4.08 2.333310 2.307779 2.491567
2.468653 2.1U8/U9 1.19405H 2.513193 7.199224 7.29117b 2.314656 1.458653
5
1.719C43 7.2HMJ2 1.7939«.P 3.758384 n.Q 1.897943 1.795227 1.752423
1.487003 1.o*2524 1.976»40 1.5/GC32 1.535r*-p 1 . * * *920 1.676*88 1.463104
1.574 79 1.791 '-14 1./24541 1./12"8/ 1.806019 1.799544 1.67-344 1.536224
I.638VC7 1.907276 2.551313 1 .958728 1.648442 1.615666 1.659915 2.480248
6
0.798515 i).e/ii.t,r 1.723694 2. {,69/99 1.897943 0.0 1.100872 1.412719
1.31«i'>92 7.16' -.56 1.v.621f.1 1.5043/4 1.40//UG I). 776621 1.225603 1.305365
1.221334 1.21333*- 1.237314 0.746 371 1.018799 1.1, 1034(1 0.867446 1.U8U78
0.948394 3.912509 1.362477 1.561745 1.004-95 1.143*47 0.865575 1.319640
/
0.883' 8/ 1.44HKO 1.14*'U6 2.4S»172 1.795227 1.ir,(>«77 0.0 1.188303
1.34/14" 2.2. ,• /.: ',. •.4,17*6 1."/'257v 1.41256t 1.13651'/ 1.165715 1.258909
1.295197 1.2/540/ 1.!>11c4 1.105/21 1.'V5263 1.C67M5 1.037784 1.0305*6
0.999**9 1.365^96 1.. 0/683 1.346961 1.354i»26 1.3*1o39 0.906V15 1.683242
8
1.186235 1.776377 1.731(93 2.7135/7 1. 75,473 1.412719 1.188303 0.0
1.4113-4 2..S247 •'/ 1.75*.' 61 1.799124 1.393155 1.41952? 1.3*1921 1.378952
1.551340 1.4529*4 1.17/197 1.364116 1./r>94/4 1.3499"4 1.317449 1.256646
1.247999 1.612*63 2.C/MU3 1.461948 1. 59*34? 1.58/239 1.192651 1.973440
9 9
1.264J13 1.5*21?1 1.39DM* 2.693U1 1.467.03 1.319997 1.342148 1.411304
0.0 2.0885/3 1..'*G487 1.651661 1.245/49 1.334728 1.415619 1.317104
1.301100 1.448/T5 1.353737 1.279537 1.36S515 1.390371 1.319999 1.718235
1.243375 1.17655* 2.052649 1.563591 1.432-19 1.434768 1.211786 1.952452
10 10
2.095391 2.2<*7S22 2.17/270 3.5/5953 1.6«/324 2.168856 2.280040 2.324709
2.0*8575 O.i) 7.304(1' 1.5*55 :p 1./8519C 2.'.v2474 7.02^556 1.843653
1.80345- 1. 6975*6 2.706135 1. "/!"'/» 2.153308 2.''54*5( 1.94/(78 1.*917*8
1.943011 2.04S7S* 2.7.6534 2.5/3112 1.711/34 1.751**0 2.005095 2.74738.5
11 11
0.9C5027 1.3/0662 1.232977 2.423566 1.976.-4C 1.062181 0.940286 1.258061
1.I8J487 2. M. 4-10 •).'. 1.757/97 1.446*9} 1.054)53 1.255270 1.369474
1.344269 1.3(0319 1. ('75650 1.1706M f:.9r9473 1.1^5732 1.069450 1.0*2206
0.9-*1113 1.3*3546 1.80/C/4 1.40*592 1.39154* 1.416116 0.980117 1.704354
12 12
1.444392 1.71*422 1.636751 2.8 7e696 1.526032 1.504374 1.702579 1.799124
1.651661 1.5.-55C 1.7?7'>V7 0.0 1.431'. 20 1.4*6(5/ 1.616361 1.500513
1.524057 1.5r{)ft8C 1./ 1:0917. 1.3692*2 1.601582 1.5'322e 1.39*155 1 .403050
1.442. :66 1.4/9024 2. (.9/844 2.10S23/ 1.36/6/0 1.39835* 1.43641/ 2.095534
13 13
1.305/01 1.6556*3 1.1.8773'" 2.7147/3 1.535988 1. 407700 1.417568 1.393155
1.245249 1./>51V(; 1.446993 1.451020 G.O 1.3/8322 1.116515 0.990305
1.033538 1.(51625 l.i.fi.'v/ 1.541179 1.184/99 1.2i//35 1.21/592 1.02108,6
1.143456 1.4113/" 7.li41fS6 1.465419 1.2489/7 1.158649 1.13r468 1.956727
14 14
0.662494 1.00*188 1.175/56 2.234581 1.888.926 0.726620 1.136589 1.419529
1.334/28 2. 824/4 1..'i54"5 3 1.4-60!)/ 1.3/r522 0.1. 1.236115 1.288056
1.789351 1.72344* 1.22M.-.6 0 . 8.532/3 1.144/45 1.C{.5*r9 0.89*996 1.022633
0.910756 1.114154 1.53J541 1.55/191 1.12c233 1.225080 0.873381 1.443356
15 15
1.129263 1.46J916 0.807254 2.448127 1.67/<-8' 1.725603 1.165715 1.381971
1.415619 2.V/556 1.255/7'! 1.616561 1.116515 1.236115 0.0 0.727727
0.828046 ■J.77>"'.18 ('.7?5r'J3 1.216558 l'.781">8.: (.'.771958 0.877831 0.768*71
0.889J06 1.341761 I.///0I4 1.314120 1.0668// 0.9980/9 0.83/833 1.665/40
16 16
1.177349 1. 5754.-3 (.075739 2./'. 8470 1.4651(4 1.3P5365 1.75'««09 1.378952
1.31/104 1.843653 1.5694/4 1.500513 i.9V.';iS 1.7*&056 0. 727712 0.0
0.865*08 0.. '6/984 1. 049-35 1.220536 U.98,/87 0.90194/ 0.958346 0.755/54
0.93UO31 1.5/9/91 1.925/11 1.4/2454 1.059,6* ('. 9/03/9 C. 930203 1.832G89
167
TABLE 5 (continued)
Values of EDI' in miles for f=17,...,32 and g=l,...,32
1 2 3 4 5 6 7 8
9 1i 11 V 13 14 15 16
1/ 1* 1V 20 21 i? 23
25 26 7/ 28 2V 3d 31
a
w i?
1.735576 1.41Hi:r 1.1V V9 2.490414 1.5?'. - 7 <^ 1.271334 1.29519? 1.551340
1.301100 1.*0345* 1.364,«»v 1.52405/ 1.01i'l> 1.289351 C. 82^046 0.865*08
0.0 !.vtWV 1.1V4//.1 1 . . .J 6 ? < 3 1.0*1 ''1 1.1:0044/ 0.99:>692 0.882730
0.97358/ 1.21M.30 1.hUf431 1. 521643 '..86313? O.r/521/ 0.991345 1.715994
18 18
1.1X68*6 1.39154* 0 . * V 14 5 0 2.384V59 1./90614 1.21333* 1.275497 1.432984
1.44jj/':5 1. ,->r>»f, l.v»i?19 l.'.MK.-I, l.l-31'>/5 1.273443 U./7rV18 0.86/984
P. 885730 1.) 1.>61564 1.25074 3 0.V4/246 0.^5032/ 0.937*70 0.879179
C. 9667/9 1.261925 1. 726795 1.47a3/4 1.025731' 0.949964 0.906326 1.674533
19 19
1.00*?°9 1.5*2 60 0. 9. 6125 ?.5-o11w 1.724341 1.232314 1.081184 1.17/197
1.35373/ 2.2H6135 1.7565* 1. 7Kj912 1.'*'.59/ 1.2261.-6 0.925903 1.049835
1.1947/0 1. <■•».! 564 !>.• 1.2739' h 0.747 14 0. 954599 1.014*32 0.935755
0.900545 1.481256 1.943404 1.264536 1.3*9318 1.352440 0.926245 1.*26493
20 20
0.819605 1. '95714 1.2*5T25 2.330/Vf 1.712'**. 2 0.796370 1.105721 1.364116
1.279539 1.92'. '79 1.12.V.61 1.36V2fi2 1.34117V r.r',32/5 1.216358 1.220536
1.086203 1. 250/43 1.2/5C'0* 0.0 1.05<«65 1.003352 0./65620 0.915459
0./B2491 t).Vs.-.64 5 1.59U585 1.583221 ('.935545 1. 10844.$ 0.809549 1.52/365
21 21
0.899210 1.33r?/2 0.87°324 2.374*08 1.fcC6'!19 1.j1d799 1.023263 1.269474
1.365515 2.15330* 0.VV4/3 1.6015--2 1.1*4/99 1. 044743 0./81580 0.9*2/86
1.061091 ). 84/24/ 0.747 14 1.U59'65 '. . C 0.687292 0.732141 0.739431
0.693079 1.290699 1. 711.359 1.311663 1.1605K3 1.116V03 0.693353 1.595653
22 22
0.963522 1.246C55 O.V09268 2.3*3310 1./9''544 1.010340 1.06/015 1.349984
1.391)3/1 2.:j54-5i 1.1'. hci? 1.5M22* 1.2(7735 1.i'L59C9 0.771958 0.90194/
1.00044/ }..'50327 0.954!-*9 1.1.(3352 I .6* 7292 Co 0.656693 0.739964
0.711342 1.19V. '81 1.645*31 1.39540U 1 .0C.fcc.26 0.958514 0.665tC3 1.528792
23 23
0.806359 1.150530 1.015«9/ 2.30/777 1.67*344 0.867445 1.037/84 1.31/449
1.319999 1. "«.?;/* 1.'.. 694 5 1 1.39*155 1.21759? 0.*9*.995 0.872*31 0.95*346
0.993692 J. 957^7' 1.(14-3? 0. 76562' '.7 52141 0.656693 'i.O 0.623049
0.4V4501 1.103599 1.5/11/8 1.420*8/ G.*6l.-5r 0.850496 0.46230/ 1.46632/
24 24
0.866919 1.'/62)5 0.934*84 2.491567 1.53'274 1.01M7* 1. 03(586 1.756646
1. 21^235 1.i.V17>* 1.(*7?J6 1.4(3'.. 50 1.C21 *6 1.(22633 0.76%-.*71 0./55754
0.8*<2/31 0.8/91/9 0.9J5/56 0.915459 0./3V431 0./J9V64 0.623049 0.0
0.471161 1.1*V?58 1.7*4634 1.42916/ H.952522 0.910*51 0.622401 1.688910
25 25
0.740361 1.3T21SP 1.(03995 2.46*655 1.638907 0.94*394 0.999*90 1.742999
1.2433/5 1.°43'j11 G.'/*1113 1.442066 1.14J456 0.V10756 0.fP9CC6 0.930631
0.9/558/ J.r'66//<> f .9' .545 0./,-249l 0.695'./9 0.711 '4? 0.4945U1 0.471161
0.0 1.1M943 1.7541 111 1.411548 1.012443 1.r".5589 0.571501 1.645191
26 26
1.204446 0.9r5244 1.42639* 7.10*209 1.90727ft 0.912509 1.365096 1.612063
1.1/o35* 2.'.)459v4 1.3*3346 1.479629 1.4115/9 1.114(54 1.341761 1.379/91
1.218<30 1.?t1'*25 1.4*1,56 3.9*o645 1.29. '99 1.1990*1 1.105599 1.1*9258
1.181943 J.C 1.4555/3 1./59-64 (.92.^2/1 1.C54593 1.1C5101 1.486352
2/ 27
1.673658 1.01595* 1./84S9 1.194650 7.551313 1.3624/7 1.80/683 2.0781C3
2.057649 2.7*6534 1.*')/i/4 2.I97.44 2.T41-86 1.530541 1.777614 1.925711
1.800431 1.726795 1.945404 1.5935-5 1.7V-359 1.6'.5t31 1.5/10/8 1.7*4634
1. /54110 1.455573 C.C 1.926665 1.461108 1.606/71 1.586328 0.//4288
28 28
1.431776 1.771. 2C 1.25297? 2.513193 1.95*228 1.561745 1.346961 1.461948
1.563595 2.573112 1.40.-.5V2 2.1U„-\23? 1.465419 1.557191 1.314126 1.472454
1.521643 1.42*3/4 1.264536 1.5*3221 1.311(63 1.393400 1.420*8? 1.429167
1.41154*. 1.759.-64 1. 926655 O.tJ 1.71119/ 1.668640 1.292310 1.726324
29 29
1.180065 1.067417 1.V»t29'- 2.199224 1.64*442 1.'»(14S95 1.354626 1.598342
1.432"19 1.711/34 1.3V154H 1.5o/6/6 1.24*9/7 1.12*233 1.06687/ 1.059068
0.R63137 1.025/5G 1.5/31/- 0.933345 1.1ou5*3 1.0(88/6 0.861*58 0.952522
1.012443 0."2N271 1.4*11u>> 1.7111?/ U.C 0.6i0V*9 0.96CS44 1.479496
30 30
1.242C*/ 1.251433 1.2/0/05 2.2911/0 1.615o6o 1.14384/ 1.381639 1.5*2239
1.434768 1./51-8>. 1.41(,116 1.59635* 1.15-649 1.2250*0 0.99*079 0.9/0378
T.H/5217 J."4«»'»64 1.35244') 1.1lv,443 1.11ovl5 0.95.3514 0.89;, 495 0.91C850
1.005589 1.054593 1.606771 1.68&640 i.63l»V89 0.0 0.974930 1.573451
31 31
0.75*879 l.lr/545 li.8«5'-3? 2.314656 1.659"15 0.8655/4 0.906915 1.192651
1.211?"6 2.1(15, 95 u.9,':117 1.45/.417 1.13-46« 0.-733?1 0.837*33 0.'<30203
0.991345 0.91.6526 0.926245 ('. 809549 C. 673*53 0.665*05 D. 462307 0. 622401
0.5/1501 1.105101 1.5*6528 1.292310 O.Vf.L 44 0.V/4V31 0.0 1.455592
32 32
1.557534 1.0(6129 1.64-165 1.45F653 2.4"o?4r 1.519640 1.683242 1.973440
1.952452 2.747**5 1.7(4554 2. ..95554 1.95/727 1.443*56 1.665740 1.832089
1.715994 1.624533 1.K<64V3 1.527365 1.595653 1.528792 1.466327 1.688910
1.645191 1.4J-6552 0.//4C6* 1.726324 1.479496 1.573451 1.455592 O.C
APPENDIX 3
MATRICES OF _ LDI2 AND £ LDI f MEASURES OF DISSOCIATION BETWEEN
ALL PAIRS OF THIRTY-TWO AREAL DISTRIBUTIONS
SELECTED FOR ANALYSIS
169
TABLE 6
2
Values of _ LDI in miles squared for f=l,...,16 and g=l,...,32
* »6
1 ? 3 4 5 6 7 t
9 111 11 12 13 14 15 1ft
i/ in iv ^o i-\ a ?3 ?4
25 26 27 26. 29 30 31 32
1
0.221414 1.115(1/ 0. -51229 4.875216 2.0656C6 0.55/896 0.49/720 0.652383
i. 3/836/ k.ytitr* o.fti6?i)i) 2.05/0:5 1.3K/68 o. 497866 1.0006/8 1. 09/367
1.368/55 1.25. -Vf, IS. 68646? it. So/269 ".561/"' P. 643615 I). 525." 34 0.569657
0.436224 1.4150U 2. 462626 1.2V505 1.48,474? 1.71546V 0.455758 1.936773
2
1.115017 i). 164678 1.2/8.148 2.417/71 3.514<58 0.6120(1? 1.305758 2.006145
1.627744 5.591/V. 1.29jo7« ?.703<52 1.77.' 07 0. 7*3479 1.305974 1.572070
1.35K447 1.255524 1.57/15: «'.8?,j94S 1.C6«*56." P. 914094 Ce.45959 1.1691(4
1.0V3>43 O.U'VUt 0.940,05 1. 7.59563 1.049161 1.323822 0.886987 0.762825
3
0.831229 1.2/814* 0.07107" 4.766851 ?.14<\65 0.87538" 0.701305 1.009519
1.3/05r7 4.47 (2> 1. ;• '62 ) 1.89.SS44 l.M;v/1 0.,-79P.'>* 0.448815 0.759343
1.0508/8 0./i54?6 (1.58.2424 0.9104»,6 IJ.b/^U 0.55302'. 0.586V27 0.546647
0.63484H 1.353/4/ 2.58.159 0.95544*. 1.360159 1.5522u/ 0.496330 1.928784
4
4.675216 7.417771 4. 766.51 0.145853 9.059156 3.5.*?GP8 4.915008 5.796687
5.742764 12. .55/51 4.93.164.> ?.1555c5 f>. 101505 4.215152 5.03601V 5.786635
5.271496 4..14690C 5.389215 4.4/1l)«6 4.614112 4.5,'2/CP 4.425967 5.253695
5.092691 3.81145" 1.U27794 4.6lo52!) 4.5(7560 4.811080 4.425123 1.561431
5
2.065606 3.514338 2.140065 9.03V156 0.070C64 2.333890 2.095400 2.037948
1.318/44 2.095523 2.535M9 1.66032b 1.557518 2. 3*944/ 1.V/2166 1.4653/7
1.651907 2.5o5/?c 1.966954 1.""VM') 2.14'»115 2.112766 1.866962 1.586659
1.75M569 2.4808.45 5.263300 3.124556 2.259t62 2.2«67C4 1.911747 4.803767
6
0.557896 J.6179C? 0.875369 3.587P08 7.333890 0.19307(1 0.690541 1.151372
1.277123 4.^7651 0. '09121 ?. 1)84,041 1.295625 (.495956 0.911636 1.118087
1.03V7O9 0.9*99*4 0.90662* 0.490983 o.61557r 0.6'7071 0.5058.55 0.6464P0
0.555244 0.790515 1.592143 1.465959 1.004?c3 1.256058 0.555174 1.309538
7
0.497720 1.50575" J.7I.15U5 4.015(08 7.0954(0 0.690541 0.152350 0.755336
1.255832 5.231 '5? '.576557 2. 521). 49 1.315548 0.74741*. 0.791741 0.99335P
1.230648 1.109,61 0. 596902 0./1h8«,5 0.50v9?9 0.601690 0.567339 0.600H6
0.508326 1.539/19 ?.576/o4 1.u5?69il 1.586130 1.771628 0.452519 2.035115
8
0. 8573.-3 2.006145 1.^09519 5.796687 2.037948 1.15137? 0.755336 0.177317
1.320/68 5.468916 C. 9268*6 2.3114^0 1.535666 1.229889 1.25/996 1.394G16
1.91079/ 1.65V519 0.692 34 'J 0."*4613 0.8,33/05 1.1461"9 0.974500 1.(06700
0.854/34 1.99*205 3.405391 1.242460 2.197.5/ 7.37586/ 0.815875 2.900652
9
1.328367 1. 6/7/44 1.320567 5.742/64 1.318/44 1.272123 1.255"32 1.320766
0.109057 4.391(20 1.52/ 55 2.48U323 1.0*1406 1.3C9612 1.451295 .1.261291
1.330223 1.594/16 1.737993 1.158165 1.2>-5121 1.741469 1.224465 1.0°2222
1.1C6572 1.119452 3.22311 5 I.8.V06I? 1.791129 1.. 544375 1.097669 2.970997
10 10
4.865476 5.391/SO 4.47062* 12.853251 2.995523 4.887651 5.231059 5.468916
4.SV1O20 J.15.-.661 5.5350.)) 2.786'!51 3.208'.8.- 4.o4555«. 5.783621 3.172243
3.083429 3.6?''5'5 4.7*6340 5.964795 4.56?/?? 4.T6/C47 3.8*4473 3.661*-9t
3.939596 4.1/1851 7.^65135 6.402439 2.6993/0 3.C35316 4.237291 7.582731
11 11
0.646900 1.290679 1.P8»620 4.938645 2.535619 0.809121 0.576557 0.926866
1. 3271.55 S - b i 3 ' J i ! n.;'2546? 2.789255 1.539^25 P. 847937 1.208764 1.403927
1.534406 1.4o9;,65 0./"5152 0.8105jJ 0.685548. 0.848187 0.79828? 0.878014
0.660833 1.59/338 2.6464/4 1.22556/ 1.8180-0 2.U24915 0.649886 2.164656
12 12
2.037985 2.705*52 1.8.98.544 7.155503 1.660525 2.0*4041 2.37CP49 2.311480
2.4*0325 2.766051 2.789/35 0.309525 1.578163 2.058573 1.980954 1.606167
2.141r/>. 1.0/5/50 2.56.J569 1./9J528 2.105--4C 2.H55282 1.69C521 1.712915
1.887462 2. H55629 5.625157 3.535845 1.861807 1.8.56194 1.86^832 3.637758
13 13
1.316268 1.//70V/ 0.6022/1 6.101563 1.53251" 1.203623 1.305548 1.335666
1.081408 3.70*668 1.539525 1.5/81'. 5 0.056/13 1.325647 0.925379 C. 769897
0.878133 a.-7J4«*4 ".,»^/4?^ 1.155913 (.0«Q 45 C.O9K980 0.994579 0.74C159
0.95641e 1.2..2545 5.31 72j 7 1.622252 1.202417 1.19U594 0.6Vo274 2.976406
14 14
0.4V2-66 0./.-347O 0.879,08 4.215132 2.380447 P. 403936 0.74/418 1.22V889
1.3U9M2 4.645550 0.847057 2.!5;<573 1.575 47 0.215971 1.024764 1.144254
1.311592 1.15*422 (.962245 0.550017 ('. 741771- 0.6oV430 0.605528 0.6V6461
0.584072 1.11142b 2.035155 1.45/288 1.274190 1.462536 0.5/1660 1.609457
15 15
1.0OO67S 1.305O74 0.448815 5.056P19 1.977166 P. 911636 0.791741 1.757906
1.451795 3./8<6?1 1. 708764 1.9KUV54 ".975579 1. '24764 0.06O126 0.391776
0.556/06 0.462*84 0./49702 0.505' "O 0.516653 0.466362 0.513952 0.449273
0.635980 1.203264 2.485535 1.1959*8 0.84856? 0.9/9662 0.510971 2.063053
16 16
1.097367 1.577(70 0.759543 5.786535 1.4655/7 1.118087 0.993350 1.394016
1.261291 5.1/2245 1.40392/ 1.8ijo1o/ 0.76989/ 1.144254 0.391/28 0.065638
0.524751 0.5/8/6O (J.976V14 0.915O*,-> (.761280 0.609318 0.662064 0.4/0244
0.698390 1.220260 2.9/1669 1.5.uu/4.* 0. 606202 U. 923136 0.614733 2.520927
9
170
TABLE 6 (continued)
2
Values of - LDI in miles squared for f=17,...,32 and g=l,...,32
* »8
1 7 5 * 5 6 7 t
9 II' 11 12 13 1* 15 16
1/ 18 19 20 21 22 21 21
25 ?6 2f 28 if 311 31 12
I? 17
1.36823* 1.35*447 1.050P7*. 5.77149t> 1.65170/ 1.0J9/9V 1.730648 1.910797
1.33'i?23 3.'i<->34?9 1.534406 ?.U1V1 (.*?*133 1.31139? 0.556/06 0.524751
0.U53635 >.f/.15.'1 1.24V5VV 1. 923/6" f.9/1'511 0./96625 0. 739661 0.666982
C. 863757 O.Vtf.76? 2.618.';4S 1.821577 [>. 5 54 7*.' C. 66699(1 0.841351 2.2fch271
18 18
1.2509V6 1.255524 0. 735436 4.8469,0 2. 305/2:- 0.9809*4 1.109761 1.659519
1.594216 3.0*1''!"- 1.46V. 165 1.973/59 r.K/1'494 1.15*427 0.462*84 0.57*769
ti.641531 D..'4l6o7 1.1>;5?10 1. "76756 0. 67/039 0.619<-/-1 0.664134 0.607207
0.8399C? 1.0*//79 ?. 395344 1. 712209 0./<-6<4t G.u43626 0.6*8103 2.026364
19 19
0.68646/ 1.5/715' 0.5»?424 5.3; 9215 1.96/754 (.V(.66?< [». 596902 0.692340
1.237793 4./6<'.'' ::.7 515? ?.3-.o3'9 n.*/6?4?/ ".9<.??45 0.74970? 0.976914
1.24959V 1.08.52H C.1<5''13 0.>*6139 [).4475'.'j C.o?59>-/-. 0.6//G26 0.630290
0.544102 I.5516V4 2.r9n/3.3 1.047851 1.6 3Mi34 1.8484V4 0.519/90 2.483000
20 20
0.567269 !). 8?(;945 ').91'"-466 4.4/U36 I.KVMo 0.49C983 ".718865 0.984613
1.13Mo5 3.9t4/93 U.81U5U0 1./9052B 1.135'1! 0.550017 0.905690 0.91598*
C. 92366V 1.I/6/5*. U.*Hf.13" 0.1*2-46 0.6M441 0.6i59?4 0. 4*376? 0.5V9932
0.496532 0.9464 j7 2.ur.5/V7 1.423253 0.915654 1.173739 0.530016 1.694G60
21 21
0.561769 1.Go95o(" C. 522318 4.614112 2.149115 0.615370 0.509929 0.833705
1.2"3121 4.5(2*21 (..68554* 7.1(.5*4i (. 997. 45 0. 741770 0.516653 0.7612*9
0.921*30 'J. o77.39 0.4495U0 0.68,0441 0.132291 C. 3*4856 G. 413061 0.399035
0.399901 1.256197 2.3C5471 1.064669 1.167134 1.324144 C.357b65 1.657941
22 22
0.643613 D. 9U, 94 0.533026 4.S'27fO 2.112766 0.607071 0.601690 1.146199
1.291469 4.067! 47 (l . 8 4 *' 1 .5 7 2.0357-2 0.99*9*9 0.6o9430 0.466362 0.609318
0.796625 0.619*61 0.625vrr> .). '36924 r>. 3*4.-56 C.',9S2»!4 0.3613«3 0.384772
0.39/345 1.012622 2.112VV4 1.1924/3 G.*2120/ 0.9**143 0.3325*4 1.66036?
23 23
0.525H34 0.. 43959 0.5-6027 4. 475967 1.866>6? C. 505*55 0.567339 0.974300
1.224465 J..-<-4473 '.7V'?*? 1.690V1 r. 994379 C.6C352-) 0.513952 0.662064
0.739661 0.664134 0.67/. .26 0.4>.3/6/ 0.413061 0.3613*3 0.135656 0.3/1768
0.330**0 l.i, 193/9 1.o*>3M* 1.192342 li. 7394*4 0.89/443 0.254313 1.5984/4
24 24
0.569*57 1.1691C4 0.546547 5.?536«5 1.5*6659 r./484C0 0.6C0816 1.006700
1.097*22 3.ot1*Vt U.e78f;l4 1. 712915 0./40139 0.69*461 0.4492/3 0.4/0244
0.666V8? o.oi /*!,/ C.biCivn 0. 59W3? 0.399 35 0.3*4//7 0.3/1/68 0.12/509
0.307669 1.0/7956 *. 630439 1.3., 6659 0.*52954 0.777265 0.350915 2.140455
25 25
0. 430224 1.093.-43 C.63.'*4P 5.G92691 1.7585/9 0.555244 0.50*326 0.854?34
1.1065 7? 3.«39596 G.6*0.*33 1.88/46? I. 95641 •» 0.5F4O72 0.635980 0.698391;
0.863757 •).,'.'"o.>ir , 1.54410? 0.49653? 0.3V7-.01 0.377345 0.33':P8G 0.307669
0.159359 1.139360 2.53/055 1.2129:1(1 1.032561 1.?C4l,7«J C. 31/993 2.065068
26 26
1.415O«0 0.*29*4>- 1.35324/ 3.811458 2.48C-45 0.790315 1.539719 1.99P203
1.11945? 4.1.1 51 1.c9/33> 2.033(29 1.2'2545 1.11142. 1.203264 1 .220260
0.9'C/6? 1.'^7799 1.551674 0.9464-7 1.256197 1.012622 1.017379 1. ("77956
1.13936C O.QvltO/ 1.**40G6 2.32V926 0.769*29 0.*2*0/1 1.0*84/9 1.946658
27 27
2.467626 0. 940(15 ?.3B1*59 1.(<?7/94 5.763300 1.592143 ?. 576784 3.405391
I. 223113 7.-65135 2.646474 3.625'57 3.317207 7. "35155 2.4*5535 2.971669
2.618*48 2.395344 2.J-90733 2.0*5/9/ 2.3054/1 2.112994 1.9*3*88 2.630439
2. 53/. 55 1.8840G6 O.I06666 2.//546P 2.044532 2.319636 2.082168 0.5h9203
28 28
1.295'1S5 1.719563 P. 93544* 4.616320 3.124536 1.465959 1.057690 1.?4?460
1.*9C.619 6.402439 1.22558/ 3.535r45 1.622252 1.45/2/-8 1.1959J-8 1.5*0/4t
1.82152/ 1./122<9 1.04/^51 1.423253 1.(,64n69 1.192473 1.192342 1.3C6639
1.212VK0 2.329926 2.77546* 0.09292H 2.39-354 2.6359K 1.031013 2.223071
29 29
1.4*4742 1.P491o1 1.360159 4.5075o0 2.259>-62 1.004203 1.586130 2.197P57
1.791129 2.-V93/T 1.>1-'.,h0 1.S61?:./ 1.20241/ 1.2/4190 0.84K562 0.806202
0.5547.V" 1.72634/< 1.63*. 34 0.715654 1.167134 U.> 217^7 0.739424 0.*52954
1.032561 0.7/v?29 2.04453? ?. 39*354 0. 077666 0.799373 0.907903 1.9??414
30 30
1.713469 1..373-2? 1.55770/ 4.811080 ?.?*67C4 1.756058 1.771678 7.375R87
1.844375 3.'i3S31f ?.*>?4v15 1.*56lv4 1.17.594 1.467536 0.9796*.? 0.973136
0. 666990 ^.*43626 1.--4K494 1.173739 1.324144 0.92*143 0.897443 0.977265
1.204C/8 U.^28071 2.319636 2.6359*0 0.295373 0.048160 1.050516 2.234735
31 31
0.455751 [J.*1>6V*7 0.496330 4.425173 1.911747 0.535174 0.452519 G.P15P75
1.09/6.-.9 4.237291 f>.64VS.ift 1.*63*32 0.*9.'274 0.5716/0 f. 510971 0.614733
0.841351 0.6C8103 0.519/90 0.53o0l6 0.35/r65 0.3375*4 0.254313 0.350915
0.31/993 1.06o479 2.0*216* 1. 0311. 13 0.9L/9C3 1.050516 0.136261 1.646808
32 32
1.936773 0.76?>75 1.92*784 1.561411 4.S03767 1.30953A 2.035115 2.900832
2.9/099/ 7. 5*2/31 2.164656 3.o3//5K 2.9/6406 1./.0945/ 2.063053 2.520927
2.2*82/1 2.('<63a4 ?.4K3:<U« 1.694. 6j 1.85/941 1.66036P 1.59X474 2.140455
2.065068 1.V4665P 0.5o9203 2.27J./1 1.922414 2.234/35 1.646ri08 0.146444
171
1
7
TABLE 7
Values of r LDI* in miles for f=l,...16 and g=l,...32
1 ? 5 4 5 6 7 ft
* 10 11 1* 13 U IS 16
1/ 18 IV 20 21 22 73 24
2* 26 ?? ?« 29 50 51 12
1
0.0 0.9/019« 0.^2/Mf 2.16000ft 1.385592 0.592161 0.557529 0.823418
1.0/84/5 2.1CIVO 0.650/ J5 1.3M3V"- 1.f>*'»9» f). 523616 0.'9Z4Kfc3 0 .9 ^6!>9 ?
1.10«»57S 1. ..'58' 43 0.712' i»5 0.604." 7 0. 620416 0.695551 0.589321 0.628805
L.*91m1 » 1,1 .«■ I.MMf:? 1.066730 1.15'.59' 1.256455 C. 526232 1.323950
2
0.960193 II. 0 1.0//15/ 1.504166 1.?4.3 86 0.656808. 1.0,71095 1.363872
1.221014 7.7'6«42 1.'<46/12 1.5/05-9 1./6><55 1.7/0165 1. 090445 1.206985
1.11/716 1.1/'././' 1.1'M'ill 0.v14477 0.95772/ '..-,.• 4654 0.637942 1.011438
0.965310 0. i/tol 0.6/9962 1.269156 U.Vt4oU 1.103359 0.858206 0.779272
3
0.877637 1.77157 o.' 2.158351 1 . 4 58 5 7 », r •. "■ f 2 1 5 7 0.767648 0.954107
1.10927* 2.' 7' 45 (.'."6971'. 1.M6997 '.8 59*8 7 t.-'jTf.Vi 0.615396 0.831195
0.9V4244 0. <-4(.5J 0.(V,'W 0.66515/ 0. 64 8 '>»,2 0.66S56*, 0.695384 0.668845
0. 723o22 1.1*77/5 1.5(4322 0.923621 1.135 2/ 1./21/14 0.626626 1.349082
4
2.166l?h 1.5(4166 ?. 15*551 0.0 2.96851? 1.8.47310 2.183097 2.37909t
2.369665 3.565845 2.1'J136 2.632,0/5 2.44>-549 2.0('853c! 2.220031 2.383442
2.2/415:. 2.1M1/1 2.2'*tj«25 2.0/5257 2.115431 2.092998 2.0/iC77 2.262083
2.222bli 1.9<1<41 i;.V33565 2.120598 2.0V7215 2.171193 2.069600 1.169660
5
1.3855V2 1.843086 1 . * 3 *■ S / 3 2. 988,512 C . C 1.484022 1.408613 1.392571
1. 108685 1.69/398 1.545/62. 1.212654 1.212076 1. 491809 1.379336 1 . 1 8 2 1 2 7
1.2609/4 1.4991*3 1.565/71 1.5316/6 1.431 61 1.424286 1.528195 1.219783
1.262129 1.5491(17 2.26.-244 1.744431 1.4/9357 1.492512 1.344635 2.166913
6
0.592161 0.6581*0* 0.862157 1.847310 1.48*02? 0.0 0.?1°604 0.995579
1.05<MiO 2.1/0664 :). 774505 1.353/87 1. (.61078 0.537973 0.883480 0.994300
0.957312 1. 934159 C . - 6 1 4 7 5 0.55u4?7 .*72t22 0.679260 0.584373 0.6986*9
0.615653 0.VC4951 1.188391 1.150199 0.933453 1.0655/1 0.608694 1.067604
7
0.557529 1.0/1095 f!.76/>4S 2.183197 1.408613 0.7196C4 0.0 0.784539
1.060719 2.252/98 0./.27M6 1.4*5575 1.(95"*06 O.7505C5 0.82*926 0.940349
1.06l"11 1.0161U6 U. 6/2862 0. 2424/4 0.606507 0. 690 198, 0.650643 0.67888/
0.593693 1.1906/6 1.554/58 0. 9669*1! 1.21392/ 1.292815 0.555170 1.373214
e s
0.823418 1.563872 '). 954107 2.379098. 1.392571 0.995579 0.784539 0.0
1.096622 2.30//96 0.866300 1.446/40 1.115191 1.1)28/09 1.076927 1.139051
1.349195 1./550.0 ','.748-16 ().91'j/>6 '.8389*8 1. "16562 0.9160*9 0.93//O3
0.843146 1.5/43)3 1.*051l'3 1.064115 1.446400 1.512662 0.8270«»5 1.662513
9 9
1.0/648,5 1.221014 1.109/78 2.369665 1. 108685 1.058,800 1.060719 1.0*6622
0.0 2.-'l# 5/.K8 1. 076957 1.506995 '.999/61 1.0/1118 1.167133 1.0.'. 3**0
1.117531 1.232417 1. 5M75 0.V96i*99 1 . : 7^ 1 6 7 1. 119*60 1.0*9813 0.9r6883
0.98608,5 1.0LV500 1.757910 1.3*0/55 1.5(59*3 1.328820 0.98*900 1.666193
10 10
2.162275 2.2869*2 7.C8/:*5 3.563845 1.69/398 2.170664 2.252898 2.307796
2.063/-* 0.0 2.511 47 1.59/485 1.76'. 965 2.111455 1.915652 1.7*92'3
1.72548 1 1.-76203 ?. 155846 1.94/r?9 ?.1'.1->0«- 1.9/45-* 1.933215 1.675849
1.9443/2 1.9991/1 2.775532 2.505322 1.666444 1.712280 2.022332 2. 725835
11 11
G. 65073"* 1. 46717 0.969716 2.180156 1.545268 0.7745f3 0.622616 0.866300
1.076v57 2.mi47 O.C 1.58 79V5 1.1"-255* P. 791972 1.05275 1.121729
1.1B1L40 1.155:-38 0.///4/3 U.//8682 0. 711.08 0.l/8441 0./65954 0.83/573
0.69687/ 1.1994*8 1.56557/ 1.032661 1.291903 1.3/4(82 0.684854 1.406663
12 12
1.33135" 1.57nsrf9 1.306"97 2.632075 1.212654 1.353787 1.44«.375 1.446740
1.506995 1.597485 1. 56/998 0.0 1.1.M120 1.340383 1.33*516 1.272197
1.4M'0f8 1.54<*5c 1.4/2. /() 1.242/16 1.372"?* 1.3532"? 1.2115M 1.22/455
1.285698 l.353a9S 1.8403O9 1.826094 1.29/560 1.295126 1.280992 1.846557
15 15
1.084991 1.268953 0.85V2'/ 2.449549 1.212076 1.0810/8 1.095908 1.115191
0.999261 1.760965 1. lb/554 1.1M120 0.C 1.0«f:643 0.v28'.86 0.841797
C. 8/9161 ).90.'.259 0.93(52 1.1)06054 0.951074 C.9S9945 0.947731 0.8. 5102
0.921077 1.062240 1.790395 1.243958 1.C66643 1.J66844 0.895425 1.695531
14 14
0.523616 •). 7/0165 0.8.57603 2.008558. 1. 498609 0.5379/3 0.750505 1.028709
1.071118 2.111455 M.791«:/2 1 . 54iO>: 5 1.09(643 CO 0.938997 1.001/62
1.0847U6 1.1146"/ C. 8-6738 0.592122 0.753418 0.715753 0.653999 0.775756
0.629608 0.9/85/0 1. 35/879 1.14141a 1.062953 1.153460 0.628923 1.195093
15 15
0.924*8.3 1.ovn445 0..61S5V6 2.220051 1.379336 0.883*80 0.82*926 1.076927
1.167133 1.915652 1. 0 51 275 1.. 538516 n. «*?.', 686 0.95x997 0.0 0.569*26
0.703/93 0./585*/ 0.8U44/C, 0.865(08 0.644937 0.61859} 0.641530 0.592415
0.722314 1. .59653 1.538/15 1.055916 0.681*553 0 . 959/06 0.636966 1.396308
16 16
0.976597 1.2U6965 ".831195 2.3><344? 1.162127 0.794300 0.940349 1.139051
1.063440 1. 749283 1.1217/9 1.2/219/ 0.641797 1.C01662 0.569426 0.0
0.661555 'J.//4i«-c i;.998«-fc7 0.8-<9745 (V.815//1 0./7612* 0. 749211 0.611204
0.765370 1.1,68413 1.669/97 1.225302 (J.fc5r.451i 0.930664 0.716717 1.553958
172
TABLE 7 (continued)
Values of . LDI' in miles for f=17,...,32 and g=l,...,32
12 3 4 5 6 7 8
« 10 11 12 13 u 15 16
17 IB IV ?0 21 ?? ?3 2*
25 26 77 2* 29 3', 31 32
17 17
1.1CV3M 1.117/16 0. "94744 2.2/415.J 1.26("/4 0."S/312 1.061911 1.349193
1.117531 1 . / «' b *. /- 1 1.1."V:4„ 1. 401)08? 0.8/91M I.i>4/1* 0. 703/93 0.6(1553
0.0 .).//(. 65/ 1.I./462J- 0.89/45/ (.910421 0.-46"20 0. 803129 0.759217
P.8702J7 0.95/V4* 1.5*3566 1.322212 0.701163 0.7.*.4V16 0.863946 1.479267
18 18
1.05M143 1.'/34/6 H.??4'li( 7.1J-01/1 1.49"'V>3 (J. 934139 1.006106 1.255000
1.232417 1.-/6/03 1. 15565"- 1 . 34 j*»56 U. 90625* 1.(11469? 0.65(347 0.774581)
0.7/' 637 )..! O.V9i>2C-» 0.9./f;-9 r.76r15' f. 741542 0.75*599 0.722924
0.8598/7 1 .1:10510 1.51J662 1.2(2540 O.Altuit 0.-93/0/ 0.7/4041 1.3900/5
19 19
0.712'. '.3 1.19451" 0.'Q?'47 7.2" (925 1.365/71 0.*61473 0.67?»>2 0.748816
1.056175 2.15*. 46 (.77/475 1.462070 ".95(652 0.2(673* 0.(04476 0.90e(67
1.0/4'..?.- II. "«<«.•. .- 0.0 0.8536/4 I.561MI3 0./11365 0.735t.V1 0./C6102
0.62V655 1.199125 1.655126 0.966142 1.23(445 1.325314 0.619438 1.530300
20 20
0.60426/ 0.*044// C.l(515/ 2 . 075i "5 / ' 1 . 3 3 1 '. 76 0.55G477 0. 742474 0.91C786
0.996t,99 1.947-29 (,//*-.,::2 1.242/16 1.008-.'34 0.5921?? 0.883008 0.889745
0.89/45/ 1.9S-2lVn 0.853674 0.0 1. 723', 99 C. 704527 0.567663 0.666(99
C. 570464 D. £.9952(1 1. 38/403 1.133739 C. 887636 1.028706 0.608657 1.236694
21 21
0.62C416 0.959/?/ 0.648567 2.115431 1.431(61 C. 672872 0.606307 0.838988
1.0781*7 2.10T15 ;>.711 >o" 1.37292'- 0. 9511/74 0.75341* C. 644937 0.813771
0.91C471 J.7'M155 0.56161,3 0.773' "9 0.0 0.519200 0.528287 0.51&762
0.504U59 I.LcVhMJ 1.4685/9 0.9/5/35 1.031821 1.110(18 0.472852 1.310943
22 22
0.695551 U.V74654 !l.<69586 2.t'>29v*. 1.424286 0. 679260 0.690198 1.016562
1.085:60 1.9<s4584 0.(76441 1.3537'S 0. 959^43 0.715753 0.61*593 0.726124
0.846920 0. 7415'./ 0. 713565 0. 7(457/ C.519/00 0.0 0.494382 0.52141/
0.518193 ).957V39 1.4J/30B 1.04/314 0. 857/48 0.924619 0.464016 1.240162
25 23
0.589321 0. "32942 0./.953C4 2.07.K.77 1.32H95 0.5>4373 0.650643 0.918049
1.049.13 1. "33713 0./;-5V54 1.2115M C. 94/731 0.65399V 0.641530 0.749211
0.8U51?" J./5<i59v 0./356V1 0. 56966} i.5?*£8? 0.4V4382 0.0 0.490087
0.428221 0.951697 1.5537d? 1.03a7"1 U.79/U33 0. 1-97516 0.344027 1.207237
24 24
0.628805 1.01143" 0.66/>45 2.?620« 3 1. 21978-3 0.698649 0.6/8887 0.937703
0.9*6>«.3 1.e/5>49 0... 3/5/3 1.222455 ii.»05O02 G./25756 0.597415 0.611204
0.75<*717 'J./:<".24 0.7'.-6ll.2 0.66&.*v" 0.518/82 0.521417 0.4900C7 0.0
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