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Full text of "An entropy maximization approach to the description of urban spatial organization"

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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 [. EDI 2 ] 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 [. LDI 2 ] Matrix 119 

f ' g r 1 

29. T0RSCA-9 Two- Dimensional Scaling Solution of l_ LDI 1 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 f Q ), and f f C 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 ,, EDI f 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 (x 1 ,x 9 ,...,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 (p 1 ,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 

w l = P l +P 2 + *" +P k 

w 2 = Vi +p kt2--- +p e. 

etc. 



24 
Then the following composition law should be satisfied: 

(2.3) H(p 1 ,p 2 ,...p n ) = H(w 1 ,w 2 ,. ..) + w 1 H(p 1 |w 1 ,p 2 |w 1 ,...) 

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 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. . > 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 

r n 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 



15.0 



14.0 



13.0 . 



12.0 



11.0 . 



W 

v 

r-\ 
•H 

E 
(1) 

& 

V) 
•H 

c 

3 



I 

(0 

u 

<M 
O 

tt) 
O 

§ 

W 
•H 

c 

i 
it- 
O 



10.0 . 




0.0 0.5 1.0 1.5 2.0 2.5 

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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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 



0) 

o 

c 

fO 

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•H 



II 

Q 




2.0 2.5 3.0 
values of 8 



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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 f Mx 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) f My = I f z. 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, f Rx 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 f Vx 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) f DV = Vx + Vy 



50 
Following Neft (1966, p. 55), we will refer to this measure f DV as the 

distance variance associated with the areal distribution f. 

Let us consider in turn the two component variances f Vx and _Vy 

associated with f. From the definition of variance, we have 

f Vx = E( f Cx 2 ) - [E( f Cx)] 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( f Cx)J = Mx . This condition, together with cer- 
tain additivity properties of expectation, allow us to write 

(3.4) J/x = E _z. E(Xx 2 ) - Jlx 2 
f i fi f i f 

2 
Considering the random variable ,.Cx . , note that 

E(^Cx 2 ) = E[( £ Cx. - Mx.) + Mx.] 2 
f l L f l l i J 

i 

= E[( r Cx. - Mx.) + Mx. + 2Mx.( £ Cx. - Mx.)J , 

L fi l l lfi i J 

and since Ef2Mx.(^Cx. - Mx.)l =0, 
L i f l i J 

(3.5) E( r Cx?) = Mx 2 + 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 r 2-2-n _ 

= E _z. [Mx. + _Mx - 2 _Mx E _z. Mx. + Vx.J , 

ifi L i f F j f j 3 i J 

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 J l 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 if J 



i 

\ f 2 ! I Vx i + Vy il 



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(Dx 2 .) = Ef(Cx. - Cx.MCx. - Cx.)] 

i,3 L i 3 1 3 J 



= E(Cx 2 ) + E(Cx?) - 2E(Cx.Cx.) , 

1 3 13 



and since the random variables Cx. and Cx. are assumed to be independent, 
E(Dx 2 .) = E(Cx?) + E(Cx 2 ) - 2E(Cx.)E(Cx.) , 

1,3 1 3 1 ] 

(3.9) = E(Cx 2 ) + ECCx 2 .) - 2Mx.Mx. 

1 3 1 : 



Now with reference to (3.5) we know that 



(3.10) E(Cx 2 ) = Mx 2 + Vx. , 

1 11 



and similarly 



(3.11) E(Cx 2 .) = Mx? + Vx. 

3 D 3 



Together, equations (3.9), (3.10), and(3.11) imply 



E(Dx 2 .) = (Mx 2 + Mx 2 - 2Mx.Mx.) + Vx . t Vx . , 



= (Mx. - Mx.) 2 + Vx. + Vx. 

1 : 13 



In identical fashion, it may be shown that 

E(Dy 2 .) = (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) L s. . = (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 ] J i 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 f My, 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 

f GDV = (w)f GDV + (b)f GDV . 

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 ± J 3 

= 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 J 3 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)f GDV = (b)f GDVx ♦ (b)f GDVy 



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. = E 1 ? z. (My. - -My) 

l f l l l fi if ' ifi J i 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 ,, ,. f GDVx 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 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)f GDVy = f Vy - I f z. Vy. + f Vy - 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 J i 



n n 



n n 

= 2 £ DV - 2(1 jz. Vx. + Z ^z. Vy.) 
f i r l i j fi J i 



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 J i J i 



n n 

= Z jz. (Vx. + Vy.) + Z jz. (Vx. + Vy.) 
i r l i J i 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 J i 

But this is precisely the quantity by which , . GDV differs from 2 DV 

in (3.19). Hence, given that ^GDV = , 1 v. e GDV + , 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 DV 2 and f GDV 2 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 f GDV, will 
refer to the measures DV 2 and GDV 2 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) _ GDI 2 = ( Jbc - Mx) 2 + ( Jfy - My) 2 + £ DV + DV , 
f »g r g f J g J fg 

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) . MDI 2 = 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. . > 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 l j g ] g ] i»3 J i = 1,... ,n 

n _2 

(3.32) u. = [E _u. _z. exp (-0s. .)]" j = l,...,n 

g j L i 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) _ LDI 2 = 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. . f s 
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 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,g M i,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( R f ) 

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( _ R f ) does indeed attain its upper 

f,g 

bound. Then, we must have 

U 2 -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 u i$ 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.g H i.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 V f . Use this V 1 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 V 1 
that are unique up to a positive scalar multiple; that is, given that 
U' and V f satisfy (**.2) and (4.3) , then U"=cU ! and V M =%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! u f . = 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 : 

U 1 , 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 ,. r 1 . . , 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 l t D 

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 f z i log f z i 



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 J i» j,] 3 J 2 

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 Mx f . 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) W ftg Q*)--| figq * lfJ log f ,/ 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( f Z) 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) f C = HC^) + H( f Z) - H( f f Q i{ ) 

= 2H( f Z) - H( f f Q*' 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. = 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 f Z 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 

* 

the unique f f Q 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( 1 Z) - H( 1 Q) 

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 _ 

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 

and row marginals Z. Therefore, we must also set q, .- q . + q . 

1 . 1 . 

and q . = 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+1 Z) = H(Z) + H( t+1 Z) - H( t+1 Q) 
k,le I 

where H(Z) is of course constant over all cluster mergers and where 

H( t+1 Z) = H( t Z) 

t . t t , t 

+ z k log z k + z^log z 1 



-( z. + z. ) log( z, + z.. ) 
k Ik 1 



and 



H( t+1 Q) = H( t Q) 



♦fW'Ac.j + f tq l,j l0gt<1 l,j 

" ? ( \,j + \,j ) l0g(tq k,j + tq k,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 = 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. 

* 

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( 1 Z) > ... > H( t Z) > ... > H( n " 1 Z) = 

H(°Q) > H^Q) > ... > H( t Q) > ... > H( n_1 Q) = H(Z) , 

T(Z,°Z) > Kz/z) > ... > KZ^Z) > ... > T(Z, n_1 Z) = 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^DI 2 , H(_Z), H( Q ), and C for the 
four spatial distributions of Figures 8, 9, 10, and 11. 

f f GDV f ^DI 2 HC^) H( f f Q*) f f C 

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 
l f,g J l f,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 

. LDI 2 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 n f x n* matrix of _ EDI measures be denoted simply 

f »g 

E. Also, assuming equal weights for all spatial distributions, f = l,...,n f , 
define the maximally entropic probability vector W where w_ = 1/n' for 
all f = l,...,n'. Thus, H(W) = log n f . Then consider the functional 

o n ' n ' j. 

EDI = £ E p" e^ 

f g f»g f ,g 

li 

where P (n f 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 PlA v '5 i ?-.JND'.' 

Light Industpy 

Heav, Ini>u c ,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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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 



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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 



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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 £ -EDI 1 = for all f = 1,...,32 and £ EDI 1 = JJDI 1 > 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' > .EDI 1 + . EDI 1 
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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cluster analysis of L LDI 2 ] matrix of areal distribution dissociation 
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119 



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120 
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'. 



121 



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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 



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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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Fig. 31. Pattern of two- family housing units 



133 



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Fig. 32. Pattern of mult i- family housing units 



131 



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Fig. 33. Pattern of mobile-home housing units 



135 



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Fig. 34. Pattern of transient lodging units 



136 



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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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Fig. 36. Pattern of elementary schools (K-6) 



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Fig. 37. Pattern of junior high schools (7-9) 



139 



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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 



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Fig. 39. Pattern of colleges and vocational schools 



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Fig. 40. Pattern of neighborhood parks and playgrounds 



142 



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la 







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• 






























160 






















160 


80 












1 

1 1 








. 






























1 






161 




1 


1 










II 




1 


1 


























. 


< 














1 














4 

1 


| 












1 


1 1 





2 4 
PA1MRN NO. M 



1 1(1 H 14 16 IB ?0 2i ?4 ?6 ?8 JO J? 54 J6 38 40 

AtklONAl OUTDOOR RECREATION AREAS (ACRtS) STABOl • I 



Fig. 41. Pattern of regional outdoor recreation areas 



143 







L...L 


1 


» 


1 






Hull 












































Wfi 












1 1 
1* 






JSL 




too 


1 


1 I 


1?UU SOO 


1 1 
1 1 






1 


1 1 1 


t 




' 


H 1 

i" i 


1 1 1 
























' 






























• 




1 


► 






■ 


1 


1 1 





? * 6 8 10 1? 1* 16 ID i"J li ?* It 78 30 3? 3* 36 38 *0 

?•??(«•) no. is iNOoot ncvii imtiiti <» of sc»ts> sthbol » « 



Fig. 42 . Pattern of indoor movie theaters 



144 



40 

M 
M 
M 
M 

10 
?t 



fti 



1? 



















1 






































IIHl 




iti 






400 














I Ml 


14)0 


100 


600 














/on 






600 


1*00 


600 




400 




1 




,„ 






1 Ml 


JJ 


l„„ 


>•>(> 


16) 








tfS 










/no ?nu 


1 












**c 






»0 • 


410 ISO 

! 


1 


woo 






300 






I — « — , 






SOU 


ISO 


16»0 






ICO 




ISO 








l.i(j 




»S0 








1100 






< 






»l>0 






40U 












i 




1 1 
H 











2 4 4 

r«niim wo. 1* 



C 1(. 1/ 1* 16 1« ?P « ?* *6 ?t> 50 5? 5* 56 J8 40 

CHU*C»CS IN0N-V4C4NT) <S»NC1U»R» SC01S) STHB01 « H 



Fig. 43. Pattern of churches 



145 



i 




i 


1 


> 


► 






I/O 


1 


















■ 
























i?n 










/Ju 






in 


1 


H 


13C 




1 1 
1 1 


nn 5n(il i*n 


1 1 
1 1 








1- 


100 








11(1 

1 1 


1 


1 












1/(1 














■ 










i 
























WO 


'"1 


i 






1 






1 


► 



2 « 6 

P«Mf«» NO. T> 



a 11 1.' 1* 16 II* ?C ii ?* ?6 ?R SU 52 5* 5ft 3d 40 

IUll-ll«U DtPAOTXtKI SIKH («»»» IN S«-M*100O> StPBOl • 



Fig. 44. Pattern of full-line department stores 



146 



M 
U 
SI 

3d 
?a 

»6 



1? 








1 


I 




1 






1" 


1 








■ 
















Ml 


SI 
















it 












1 




1 






\ 


-1 


i 




i 




SO 100 


H 


1 1 


no 


1 




1ft 




90 


"1 








56 









1 


























' 


































»l 








< 




1 1 1 





? 4 6 

PtTTMW *U. 16 



6 10 1/ 1* 1« 16 ? ii ?* ?6 ?8 30 3? 3* 36 38 *(1 

»pp«»ll SHOPS <««i« is SS-fT*100> S»«B0l • P 



Fig. 4 5 . Pattern of apparel shops 



147 



3A 
56 
5» 

v 



16 









1 




t 






in 




















> 






































t(i 


' 






jn 


35 


■ 






1 1 
1 I 


3 : 3S 


ill 










■ 


4 | 


AO 


"1 


■ 




' 








1 
























' 








*•*> 






















' 


| 








1 


! ! ! 







2 4 6 

PATTERN NO. 1/ 



10 1? 1* 16 IP ?0 ^? ?* 26 28 JO 3? 3* 36 38 60 

I!:»«1IU»F STORFS (NUT DfP«H«fM) (»0F» IN Sa-FTMUOO) STHROl • 



Fig. 46. Pattern of furniture stores (not department) 



148 



M 

34 

Ju 

/8 



1? 

10 







1 






i 


i 






1 " 






















i 




































1 
















2* 


















11) 


20 








1 






1 » 








- 


" 


1 


1 




1 


1 


1 ! 


































i 




















1 






i 














1 





PATTCIN No. 



4 6 8 10 12 1* 16 1" 20 ii 2* 26 28 30 32 3* 36 38 40 
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 



M 
U 
it 

SO 

t» 
?» 

M 

n 

?0 



1? 
lb 



1 






i 






1 


i 








»0 






■ 
































N 




70 




18 








1 


[ 


1 




70 


1 


"1 


1 ,. 






! 


_LU 












16 








' 






30 






















?4 




















?4 






> IS 




















» 

1 


_LUJ 










_JJJJ 





I « • 

PAIIttN NO. 1v 



8 1U 1/ It 16 1« ?U ?? ?4 ?6 ?c 50 ]? 34 36 36 40 

mi. u SUM »■«•«( TS <««(* IN S«-M«1000) stNMOl • s 



Fig. 48 . Pattern of food supermarkets 



150 



•o 

M 

M 
M 
M 

50 
?* 
2* 
?4 
It 
ib 
1* 
u 
1* 
1? 
If. 







1 


1 








1 






1 












M 
















so 


?» 




2> 














J? 




II 










' 


?» 






20 






1 


?* 










I 
















,.l 


•i 1 




i% 














I ! 












so 




It 


SO 




21 










/o 






s» 






















i 


! 1 












1 1 


1 1 





»*TlttN NU. 



4 6 A 10 1? 14 16 1h ?u ?? ?<• It ?* 30 5? J4 56 It 40 

?U SuIH-ShOP GROCERY S10RIS <»»{» I h S8-MO000) SYMBOL ■ T 



Fig. *+9. Pattern of quick-shop grocery stores 



151 





! 


1 




► 










" 


1 




















' 








s 












10 






i 


10 






! 


ii 




10 


< 


1 1 "1 


1 


IS 


1 




1 
1 




1 •■ 


1 










10 


12 




'1 







, 


» 


I 

1 


1 








1 








10 


r 


•> 












■ 










10 






















1 » 


1 










I....I..J 


1 


■ 



PATtltB H\i. 



4 6 P 1(1 1? 14 16 IP ?0 l? ?4 ?6 ?» 30 3? 34 36 3* 40 
ii SPiCIHTf »OO0 »N0 U0U0P. STOPIS (»»f» IN SS-fTOOOOl STHBOl • U 



Fig. 50 . Pattern of specialty food and liquor stores 



152 



40 

M 

56 

M 

M 

30 
?8 
M 
M 

?? 

?0 
IS 
16 

u 
» 

10 













1 


■ 






i 














1 

moo 


...J..... 

1 






i 






000 


1000 






















«oo 
















lino 


»00 


1 














•00 






1000 


600 


eoo 




















• 


loon 

| 


too 

| 




900 






1100 










1000 


»no 
















1 








1100 






i 






|""° 


1 


















1 


1 


1 













? 4 6 II 10 M 1* 16 IK ?0 !? ?4 ?6 ?8 SO 3? 14 36 38 40 

»»M»t» no. <•< p>au(m costs s»ns so-*i) svmoi ■ v 



Fig. 51. Pattern of pharmacies 



153 



40 
18 
36 
J* 
32 
3C 
28 
26 
M 
II 
20 
10 
16 







1 1 


• 


i 






1 "1 
















i 






20 


75 




22 








o 




n 


/s 


80 


50 








6* 




ft 


f> 


1 » 


-1 


65 


80 




1 1 
1 1 


"1 "• 


"1 










,„. 


"1 '° 


"1 


65 






"1 1 
1 1 


1 


1 






2t 


CO 


TO 


80 




25 








i 


23 


7b 




55 
























1" 


45 










' 













2 4 6 8 10 12 14 16 18 2 li ?l 24 26 28 )C 52 54 56 38 40 
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 




» 






S> 








| ,00 


1 










36 






















M 




1/4 


















1/ 




























1/4 


44(1 




140 








50 






















/* 








m 




140 


HO 


//4 


1/% 






/6 






















- 


1 






100 


- 


[.„ 


'"1 


300 




! 














,„ 




" 


)0U 










/UO /OO 


H 








1 


* 


2UU /CO 










100 SOU 


1*0 






300 | 












17% 






1/4 








- 


1 










| 




|_L 


16 








1/4 


SCO 


1/4 


1/4 




1/4 






1* 
























1? 








in 


1/4 




1/4 








10 


















40 




a 




















6 






















4 








| 100 


100 










i 








II 


I 











/ * 6 

»<n(iN 40. /4 



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 
M 
II 

SO 

2a 

M 

H 

12 

20 
IK 
16 

u 
1? 
Ill 











L...L... 






1 


> 






i 




60 


60 


•0 




94 












5* 




56 16 60 


40 












36 


. 


i 


»l 


36 


38 












- 




"1 


H 




















36 


SO 


?6 


JJ 
1 


32 








SB 












!« 


32 


SA 


in 




32 












■ 


32 


*0 




36 




36 














«| 1 






















1 


J- 


I 1 















l«TTIBN NO. 



4 6 

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 



< 












i 


► 




















1 










■ 








































1 






400 


1 


> 








1 


;oo 






1 














> 
























m 










?*) 
















































■ 
















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 


! 


' 


» 






1 l 1 
















1 






























1 

! 










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 








i 


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»» 


ri> 




i 




























i 






.••*.—•. - 






























1021 








' 


\ 


I 


► 




' 






\ 


1 
















< 


1 


1 


► 


7?1 


7?1 


u 

1 


1 


1- 
























■ 






10/1 
























■ 


1 


! 










1 


-I 





? 4 t 



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 












!.... 

l„, 


1 


i 






1 




i 


% 


?* 




2 










« 




IS 


s 


in 


t 






1 I 






?r 




» 


. "I ! 


5 




1 


! 1 H 




li 


?SL 


n:o| 




1 


LU 

i i 




.. 


» 


•- 


ioo| 

1 


is 






H 
1 








1U 


IS 


*o 




i 








■ 


s 


s 




R 




5 




i 




1 




s 


•i 

| 






















1 











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* 

S'i 















: 


► 








5 












' 
















































' 






I 


1 


1 


i 




< 


J 


1 


! 


1 " 


-1 


! 








| 

• 




6 


"I 


► 






s 




1 






1 








4 














< 




































► 






4 


l 







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 




































2640 


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 


?6*ll 


1320 








1320 264C 


?640|13?a 








1320 


13?0 






2640 


i28d 


S2KC < 






4280 


42*0 


2640 ' 




















1- 


H 
















2640 


3280 


5>280 


4280 


4280 


2640 












?64P 


5200 


4280 


2640 














2640 


2640 


















1520 


,320| 














« 




H 


,320| 











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 


■ 




J 


•I 


1 1 -1 


1 


1 
1 1 


► 




■ 


1 1 H 


' 


vo 


>£ L 


1 1 ,5 I 




1 












65 


30 


20 














1*0 


5P 










to 


DO 










l» 


'1 


^o io 










1 


-1 





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 _ EDI 2 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.'i l ft(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.iV8 c '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 4.9 f -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 4.551 l 65 ?. 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./ u 9*.*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 . 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. r J75?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')74 f ) 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. v i3077 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 ? . ? 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 1.535 r *- 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 1.41256 t 1.1365 1 '/ 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 1.5'322 e 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 . 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?5 r '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 !. v tWV 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* . * V 14 5 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*5 T 25 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 1.2/5 C '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 1.33 r ?/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.'.)459 v 4 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 1.59 6 35* 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 _ LDI 2 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.61557 r 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. u u/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.9/1'51 1 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. u 43626 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. c 9/33> 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 ?.3 B 1*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 .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. 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. v1 4477 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. 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 . 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./.953 C 4 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< i 59v 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 

0.405759 0.Vt4'.57 1.5/5"64 1. 093(09 0.6676/9 0.V43096 0.466007 1.415442 

25 25 

0.495*21 O.Vfc531( 0.//3t2? l.iilty/ 1.282129 0.615653 0.593693 0.843146 

0.986iX5 1.944577 Q.t,1**72 1.2*569- C. 971; 77 11.629608 0.722314 0.76537T 

C.87U707 0.1-5**77 •'.^2'"i5 5 0.570464 <.5"4'59 0.518.193 0.42*221 0.405259 

0.0 1.006900 1.540/V2 1.042514 0.95/366 1.04896U 0.412533 1.382811 

76 26 

1.121*46 J. -3/661 1.127/73 1.971641 1.549187 0.8'4951 1.190676 1.374303 

1.0C95'"J 1.999171 1. 1994a>- 1. 353*95 1.06724" 0.97 c 571 1.05«653 1.06(413 

0.957948 1.010510 1.199125 0.-W57* 1.06Vt.*:. 0.95/939 0.95169? 0.9(4057 

1.006900 ').(i 1.324702 1.4958/1 P. 628*92 C.fc/0/22 0.987175 1.351888 

27 27 

1.50618? 0.y79v,s? 1.5^432? 0.933565 ?.76(?44 1.1 l 8391 1.554758 1.805103 

1. 75/910 2.7/553? 1.565577 1.84o36.9 1.79(395 1.357879 1.53>713 1.6J-9/97 

1.583*t6 1.51366? 1.655176 1.3>-24o5 1.4o.*5?9 1.4('73 n ^ 1.353782 1.5/5864 

1.54i)792 1.52470? 0.'.. 1.626551 1.387395 1.4(7353 1.3S9497 0.657760 

78 78 

1.066/30 1.769156 0.923'21 2.170598 1.744431 1.150199 0.966980 1.064113 

1. 340/55 2.5(5572 1.' 37161 1.H?oi74 1.243 r5 5J ; 1.141418 1.055C16 1.225302 

1.32221? 1.2-2S40 :). 966142 1.133739 0.975735 1... 47314 1.03F791 1.093809 

1.042514 1.495871 1.6/6551 rj.11 1.521695 1.6.169/ 0.957296 1.450305 

29 29 

1.156590 0."<4M(. 1.135. 27 7.097215 1.47935/ 0.933453 1.213927 1.448400 

1.303^43 1.6',8444 1.?91«.i.3 1.?"?560 1.066643 1.062953 0.881*53 0.^58458 

0.7C116J 0. '■in 34 1.?5-'.43 (l.r87636 1.031821 ?.rS774r 0.797^33 0.(67679 

0.95/366 Ci-t'-BV? 1.387395 1.521695 0.0 0.4»-6(35 0.896348 1.346424 

30 30 
1.256455 1.10M59 1.721714 2.1711«3 1.49?M? 1. '65571 1.292"15 1.517662 
1.32- v 2t. 1./177/-0 1.3741.5? 1.2»51?6 1.0/>6'44 1.1S3460 0.959708 0.930664 
0./I-4V16 0./V3/0/ 1.575314 1.C78/l6 1.110-18 0.v,r46l9 0.89/516 0.943096 
1.04*960 J. 870/22 1.48/553 1.601697 0.4*>>35 0.0 0.978931 1.461996 

31 31 

0.52673? 3. '58206 .f,?6'?6 2.0698"0 1.344-35 0.60h694 0.555170 0.^27095 

0.9K49O1 ?.(u?53? 0.6>4>54 1.?»099? 0.(95425 0.6/M973 0.638966 0.716/1? 

0.863946 Q. 7/4il41 0.619438 0.6;'8t57 n. 4/7*5? 0.464016 0.344027 0.468C07 

0.412555 11.987175 1.389497 5.95/298 0.896348 0.97(931 0.0 1.226970 

32 32 

1.323950 0.//97/2 1.349^2 1.189660 2.1fc6«.1j 1.06/604 1.373214 1.662513 

1.686lv\ 2.7?5(t.*5 1.4066O3 1. 84655/ 1.6'/5531 1.195(93 1.39*308 1.553958 

1.47926/ 1 . 5« 75 1.53 $00 1.75o6V4 1.51 '45 1.240167 1.7'.!/737 1.415442 

1.382.-. 11 1.3511 jK 0.657760 1.45u305 1.346424 1.461996 1.??6970 0.0 



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