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UNIVERSITY 0F3 ILLINOIS LIBRARY AT UR3ANA-CHAMPAIGN ENGINEERING NOTICE: Return or renew all Librar each Lost Book is $50.00. The oerspn charging this material is responsible for its reffi the hbrarv >V6>Qf »fy» withdrawn on cfitjmctfliNii e,ow Theft, ntti JonTTn! SLUMP* *«J aMAsons for discipli- nary action and may result in dismissal from the Un.versrty. To renew call Telephone Center, 333-8400 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN L161— O-1096 Digitized by the Internet Archive in 2012 with funding from University of Illinois Urbana-Champaign http://archive.org/details/entropymaximizatOOrayr enter for Advanced Computation UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA. ILLINOIS 61801 CAC Document No. 237 AN ENTROPY MAXIMIZATION APPROACH TO THE DESCRIPTION OF URBAN SPATIAL ORGANIZATION Robert M. Ray III September, 1977 CAC Document No. 237 AN ENTROPY MAXIMIZATION APPROACH TO THE DESCRIPTION OF URBAN SPATIAL ORGANIZATION by Robert M. Ray III Center for Advanced Computation University of Illinois at Urbana- Champaign Urbana, Illinois 6l801 September 1977 A dissertation submitted to the faculty of the University of North Carolina, Chapel Hill, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of City and Regional Planning, June 1977. Copyright by Robert M. Ray III 1977 ABSTRACT Within the modern city, spatial patterns of urban phenomena, e.g., areal distributions of differentiated populations, activities, and land uses, represent the most immediate and tangible manifestations of all social forces underlying the process of modern urbanization. Thus, it would seem that rigorous methods for quantitative description and analysis of specific characteristics of urban spatial organization would be con- sidered fundamental to other more specialized studies of urban conditions. However, despite the considerable attention paid by the various social sciences to particular aspects of urban spatial organization, there appears to be little tendency toward convergence on common analytic methods prac- tical for treatment of the complex structure of real-world urban space. This condition stems in large measure, we contend, from the inappropriate- ness of conventional statistical data analysis techniques for quantifica- tion of the degree of spatial co- organization, areal association, or con- gruence between geographic distributions of urban phenomena. Here, we develop an alternative method of urban spatial distribu- tion analysis that is deoigned explicitly for quantitative characteriza- tion of the structure of spatial associations existing between some set of areally distributed urban variables. Our approach grows out of a par- ticular combination, and in some instances generalization, of mathemati- cal concepts developed previously within the areas of information theory, urban trip distribution modeling, and the theory of multidimensional scaling. From such a diversity of mathematical concepts there is con- structed a pattern- information method of spatial distribution analysis that appears applicable to the study of geographically distributed urban phenomena in general. The model developed unites in a unique manner quantitative measures of the degree of spatial congruence existing between two areal distribu- tions with information theoretic measures of the complexity of spatial structure transmitted between them. The particular information theoretic concepts developed lead directly to a cluster analysis procedure that is shown to be applicable tc the analysis of structures of spatial associa- tions determined among areally distributed urban variables. Using data concerning the spacial distributions of thirty- two (32) urban variables across a hypothetical urban area, we illustrate the method proposed computing all measures of spatial association between all variables and cluster analyzing the resulting structure of associations. As an independent means of analyzing the structure of associations between variables, a nonmetric multidimensional scaling analysis is also performed. Close agreement between our intuitive notions of the interrelationships between urban distributions and both cluster analysis and multidimensional scaling results is observed. TABLE OF CONTENTS LIST OF ILLUSTRATIONS iv LIST OF TABLES vii ACKNOWLEDGEMENTS viii Chapter I. INTRODUCTION 1 The Organized Complexity of Urban Space 1 The Gap between Theory and Data ........ 6 The Deficiencies of Present Data Analysis Methods . . 8 The City as a Self -Organizing Spatial System .... 12 The Present Effort 14 II. INFORMATION THEORY, PHYSICAL DISTANCE, AND URBAN SPATIAL ORGANIZATION 17 Introduction 17 Communication, Information, and Entropy 19 Information Transmission 24 The Entropy-Maximizing Model of Urban Trip Distribution 27 III. SOME PRELIMINARY METHODS FOR ANALYSIS OF URBAN SPATIAL DISTRIBUTIONS 42 Introduction 42 Characterization of Urban Patterns as Areal Distributions 45 Basic Measures of Central Tendency and Dispersion for Areal Distributions 49 An Alternative Method for Computing the Distance Variance of a Distribution 52 Some Preliminary Measures of Spatial Association Between and Within Areal Distributions 59 A Spatial Interaction Approach to Measurement of Distribution Distance 61 Chapter IV. NEW METHODS FOR ASSOCIATION MEASUREMENT AND CLUSTER ANALYSIS OF SPATIAL DISTRIBUTIONS 67 A Unique Measure of Spatial Association Within and Between Areal Distributions 67 An Information Theory Measure of Spatial Complexity Conveyance Among Areal Distributions 73 A Procedure for Least Biased Grouping of Spatial Distribution Elements 78 Cluster Analysis of Spatial Associations Between Distributions 89 V. URBAN SPATIAL DISTRIBUTION ANALYSIS: A WORKED EXAMPLE . 92 The Hypothetical Urban Area 92 Urban Spatial Distributions Selected for Analysis . . 104 Example Analyses Performed 109 VI. SUMMARY AND CONCLUSIONS 123 Summary of Argument 123 Potential Applications of the Method 127 APPENDIX 1 130 APPENDIX 2 163 APPENDIX 3 168 BIBLIOGRAPHY 173 IV LIST OF ILLUSTRATIONS Figure 1. Schematic Diagram of a General Communication System (after Shannon, 1949) 20 2. Mean Trip Length D and Spatial Information Transmission T as Functions of 3 within the Constrained Entropy Maximization Model of Urban Trip Distribution .... 32 3. A Hypothetical Region Containing Four Urban Areas ... 34 4. Mean Trip Length D and Spatial Information Transmission T as Functions of 3 for Home-to-Work Trips within the Hypothetical Region of Fig. 3 35 5. Spatial Distributions of Livelihood and Residential Land Uses and Elementary Schools Within a Hypothetical Urban Area 37 6. Mean Trip Length D and Spatial Information Transmission T as Functions of 3 for Home-to-Work and Home-to-Shop Trips Within the Hypothetical Urban Area of Fig. 5 . . 38 7. Mean Trip Length D and Spatial Information Transmission T as Functions of 3 for Home-to-School Trips Within the Hypothetical Urban Area of Fig. 5 38 8. First Example Cluster Analysis 84 9. Second Example Cluster Analysis 85 10. Third Example Cluster Analysis 86 11. Fourth Example Cluster Analysis 87 12. Generalized Land Use for the Hypothetical Urban Area . . 93 13. Zonal System Subdividing Urbanized Area into Areal Units for Data Aggregation 95 14. Probability Distribution of Single-family Residential Land Use 97 15. Probability Distribution of Two-family Residential Land Use 97 16. Probability Distribution of Multi-family Residential Land Use 98 17. Probability Distribution of Commercial Land Use ... 98 18. Probability Distribution of Public and Semi-public Land Use 99 19. Probability Distribution of Parks and Playgrounds ... 99 20. Probability Distribution of Light Industry 100 21. Probability Distribution of Heavy Industry 100 22. Probability Distribution of Railroad Property .... 101 23. Probability Distribution of Vacant Land 101 V Figure 24. Hierarchical Tree Showing Sequence of Cluster Mergers within Cluster Analysis of [_ EDI Matrix of Areal Distribution Dissociation Measures 110 25. Graph of Structural- Informat ion-Transmission-Loss Function over Successive Stages of Cluster Analysis of [. 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 +T information transmission units (ex. bits) 0.0 o 8 8 • o i + IT) (0 4-» CO CQ CQ CQ U U U O O «4-i O »M X X C •H OJ «0 e E 1 H ^ V, i / \ / \ / ^ \ / \ / \ / / N. / \^ y ^ **" S / / / / / / / / C f r r 8 / ° 8 1 1 / ° + to 1 ■M v <n (0 CQ CQ CQ u o fc U Mh O <4h O 1! C •H o 6 L (0 o E 8 t CQ o II CQ CQ 8 CO c O •H ■M O I c O (0 3 io ,o •H H k C CO O «H •H T3 CO CO O. •H «H E k CO -M G f0 c Pi 10 "•§ g 3 •H ^ ■M O (0 6 H U <D O X) Uh O C E •H C H O 10 «H •H -M 4-> «0 fO N CO E -O C 03 E 3 a ■f 3 P +-» u c c H Oh <U •H C H <-» (0 Pi C -P (0 CO a; as ♦D mean distance of travel units (ex. miles) 0.0 • 0) cm x: ti c •H -H +•» •H CQ O 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 $. Notice how quickly the mean commuting distance falls initially with increasing values of £. Also, notice how quickly the information transmission func- tion T rises as 8 increases. Now, let us focus solely on the community of the northwest corner of the hypothetical region and examine home-to-shop and home-to-school 37 1 1 , • i II ! r i • / \> [ X [ XX? ! ! 1 ! XX >& - Ml ' ! \ h ! I '1 ' i I 1 1 1 1 r — — . i ! L±i_ 1 II [ 1 1 i .J_ I i 1 1 i ! [ 1 1 - ! I 1 1 -. (. 1 i , ~i- | rri 1 1 W 1 r ij I | 1 ! T~ i ' ; 1 MXjX^JXlXiXX 1 . _ i i i U v. <15kjX x xX] 1 1 i i I i ! yboxj>dxTxix * 1 xS i ! ! ! y^ixixx^x; ■ I ' 1 >$X]X - 1 x>- x ;-ivN J xx^xiiXxXx i 1 1 XjXjXjXj , 9ffSi 1 x^xxxx --x ! 1 1 XiXiXix? 1 i XW;xjx>v.xiX 1 t 1 1 x$M&&6S< i i i I r; n ^ i 1 m ^! X ! | • i X 1 t M I 1 1 1 1 I 1 I 1 1 1 i - 1 1 1 Tr 1 i | 1 1 i i I 1 r r h ■ 1 ! J +1 T xx: x^< XXoX Xi/ V S/ X' xV X.X • X u Legend 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 to •H II Q 2.0 2.5 3.0 values of 8 3.5 4.0 to •H c o •H CO CO •H £ to •H g •H ■M to O C •H II H Fig. 7. Mean trip length D and spatial information transmission T as functions of B for home-to-school trips within the hypothetical urban area of Fig. 5 39 trips as well as the home-to-work trips considered before for the region as a whole. Figure 5 depicts the northwest community in somewhat more detail, this time showing the locations of all elementary schools as well. The community consists of twenty square-mile sections with one elementary school located in each section. To simplify our example, we assume that shopping places and employment places are uniformly distri- buted over all tracts graphically coded as livelihood land use, and, again, all activities are assumed to be concentrated at the centroids of sections. Referring to Figure 6, it will be noticed immediately that home- to-work mean trip lengths are not as sensitive to small values of 3 as they were for the region as a whole. This is simply a consequence of the fact that the variation of distances between all homes and jobs within the single community is much smaller than that for the entire region. Note also, with reference to Figures 3 and 5, that as we move from the geographic scale of the region to that of the city it becomes more difficult to think of residential and livelihood land uses as being spatially co-organized. Now for experimental purposes , let us make some behavioral assump- tions concerning travel patterns within the hypothetical community of Figure 5. From the results of numerous transportation studies, it is widely recognized that the scale factor (or exponent) applied to travel distances in fitting trip distribution models to observed data varies in accordance with the characteristics of the trip maker and the specific purpose of his trip. For example, most origin-destination survey data suggest that the factor to be applied should be higher for most shopping 40 trips than it should be for work trips, and presumably much higher still for trips between home and elementary schools. (Hoover, 1968) Let us choose values of 3 of .6, 1.2, and 2.7 for home-to-work, home-to-shop, and home-to-school trips respectively. Then, as shown in Figures 6 and 7, mean trip lengths are, respectively, 1.75, 1.25, and .25 miles for these trip purposes. More importantly, notice the rela- tionship between the values of mean trip length D and the spatial infor- mation transmission function T. As D decreases with successively higher values of (3, T increases. This seems perfectly reasonable, since the greater the sensitivity to distance for trips of different purposes, the greater should be the spatial interdependence between trip origin and destination locations. Thus, Figures 6 and 7 demonstrate the obvious fact that, attempting to predict the location of a particular household within our urban area, we should receive much more information from knowledge concerning schools attended by the children of the household than from knowledge concerning where the parents shop and work. Here, one further observation is appropriate. Suppose that we apply a similar form of analysis to a journey- to-work, origin-destination contingency table, determined not by simulation, but rather taken directly from actual survey data for an existing urban area. Suppose further that the residences of blue-collar workers are clustered together in downtown areas of the city and that all white-collar workers reside in outlying suburban neighborhoods. Also, assume that the majority of white-collar jobs are clustered in the central business district of the city and existing industries are located at the intersections of major roadway and rail transportation routes at the periphery of the city. 41 Then, while there may be quite large information transmission between places of residence and places of work for all employed, inspection of the particularities of the urban spatial structure exhibited would indi- cate to us that this interdependence of places of home and work must be due primarily to sociocultural forces at play organizing urban space and that the friction of distance between residential and employment centers is of little concern. The question then arises: do there exist other areal distribu- tions of landscape features or socioeconomic conditions that, acting as other forces, bring about the apparent insensitivity of community mem- bers to monetary and time costs associated with home-to-work commuting? For example, are white-collar residences spatially co-organized with respect to a particular set of elementary and secondary schools more favored by that particular socioeconomic population, or are executive residences aligned spatially along a scenic river front? Is the areal distribution of low-cost and older housing such that blue-collar resi- dences are clustered through economic segregation in downtown neighbor- hoods? These questions lead us directly to the problem of characterizing urban spatial organization in terms of structures of spatial associa- tions existing between general patterns of urban phenomena. To what extent can the concepts of information theory assist us here? Before presenting our specific answer to this question, let us turn in the next chapter to an examination of certain basic issues confronting the analy- sis of areally distributed data. CHAPTER III SOME PRELIMINARY METHODS FOR ANALYSIS OF URBAN SPATIAL DISTRIBUTIONS Introduction In this chapter we return to our main objective, namely, the investigation of more general methods for quantitative description of urban spatial organization as a complex system of differentiated popu- lation, socioeconomic activity, and land use patterns. Again, our focus is on the city as a system of geographically patterned phenomena. We are concerned with social behavior only to the extent that macro behavior patterns may be suggested by specific geographic configurations of popu- lations and activity places. Our main objective is the development of alternative quantitative methods better equipped for analysis of the spatial interdependence exhibited among geographic patterns of urban phenomena. Four major problems confront us within this task. The first prob- lem is that of representing specific urban patterns as discrete areal distributions that characterize in economical fashion the essential pro- perties of the phenomena of interest. Two fundamental issues involved here concern the choice of a set of variables for point -by-point measure- ment of all patterned phenomena and the choice of a frame of areal subdivisions of the urban area for use as a common basis for aggrega- tion of all measurements. A second problem concerns quantitative 43 characterization of overall distribution properties. Common measures used here include the geographic coordinates of distribution centroids as measures of central tendency and various statistical moments about these centroids as measures of distribution dispersion. A third prob- lem involves the measurement of spatial association between differen- tiated urban distributions. It is by such measures that inferences about the ecological interdependence of distributions can be made. Finally, a fourth problem involves analysis of the structure of asso- ciations among areal distributions. It is here that we hope to arrive quantitatively at those syntactical regularities of urban spatial organ- ization exhibited in comparable manner across urban areas. It should be noted that these four problems confounding the anal- ysis of geographic patterns of urban phenomena are highly interrelated. Most importantly, the utility and validity of all analysis results will depend on our choice of a specific set of quantifiable variables and our selection of a particular spatial sampling frame for representation of all patterns of interest. Of course, we should select that set of variables most closely identified with the specific urban phenomena we wish to analyze. Given that the discrete representations of patterns will inevitably depend to some extent on the particular system of areal units selected for aggregation of all data, we must expect our analysis results to depend on the spatial sampling frame as well. Here, the best we can do is to choose a system of areal units of sufficiently fine reso- lution to capture the essential characteristics of all patterns of inter- est, and to employ analysis methods that depend only incidentally on the particular frame selected. 44 Most methodological issues confronting the quantitative charac- terization and analysis of urban spatial patterns come sharply into focus if we recall the distinction betv;een parametric and non-parametric statistical distributions. A parametric distribution is a probability series that may be completely specified with reference to some number of numerical parameters quite small relative to the potentially infinite set of data values associated with the distribution itself. For example, if a univariate distribution is known to be normal, then the entire dis- tribution is completely characterized by only two parameters, i.e., its mean and its variance. On the other hand, if the distribution is known to be non-parametric and not well approximated by any known parametric distribution then, while we may compute any number of summary statistics and moments based on discrete samplings of the distribution, these measures may assist us little in characterizing the overall nature of the distribution itself. It is one of the fundamental premises of this thesis that geogra- phic patterns of urban phenomena cannot in general be adequately approx- imated in terms of bivariate parametric distributions. Thus, we contend that the most appropriate characterization of any specific pattern is given by the complete representation of the pattern itself, i.e. , its representation as an areal distribution of some measurable variable whose value is recorded across a complete frame of spatial sampling units. This is not to argue that there exist no summary measures of overall distribution properties of value. The issue is, rather, just what over- all distribution properties, in addition to such properties as central tendency and dispersion, should we attempt to quantify. For example, it would seem desirable to have some measure of the overall spatial 45 complexity associated with a particular distribution. Here, with res- pect to the problem of unambiguous definition of such a concept as spatial complexity , the position we shall assume is that whatever concept we employ, like the concept of distribution variance, will only be definable in mathematical terms. In the remainder of this chapter and in Chapter IV, we develop an alternative approach to the analysis of urban spatial distributions that addresses in unified mathematical format all of the methodological issues discussed above. Based on a maximum-entropy formulation of spatial rela- tionships among areal distributions, the model yields a variety of mea- sures useful for quantitative characterization of certain aspects of intra-distribution spatial complexity and organization and inter-distri- bution spatial association. Surprisingly enough, the model also yields a new technique for hierarchical cluster analysis of areal distributions based on the structure of spatial associations determined among them. Characterization of Urban Patterns as Areal Distributions Like all other methods used for analysis of geographically distri- buted socioeconomic data, the methods that we propose here depend in a fundamental way on the manner by which we characterize urban patterns as discrete areal distributions. Of course, we assume the existence of measurable variables closely identified with all phenomena of interest. In many instances, however, due to data collection costs, confidential- ity restrictions, or qualitative judgements in codification, we are forced to settle for only proxy variables. A more ambiguous collection of methodological issues surrounds our choice of a specific system or frame of areal subdivisions of an U6 urban area for use as a common basis for aggregation of all data and representation of all patterned phenomena as discrete areal distribu- tions. The analysis methods that we will develop here require that we select our spatial sampling frame with respect to three general sets of conditions. First, some a priori delineation of the outer boundaries of an urban area is required. Then it is assumed that the subdivisions of the area are non-overlapping and cover exhaustively the complete urban area. Thus, each data measurement will fall within one and only one geographic areal unit or tract. Further, the tabulation (or statisti- cal estimation) of aggregate variable values across all tracts should comprise sufficient information for representation of urban patterns as area-wide probabilistic distributions. Second, it is assumed that areal units are of sufficient number and scale to capture the essential spatial properties of all patterns of interest. This condition concerns the spatial resolution of the samp- ling frame employed. At too coarse a level of resolution, spatial pat- tern features of interest will be lost. For example, if we wish a detailed characterization of the pattern of neighborhood commercial establishments throughout an urban area, a sampling frame of relatively fine resolution must be employed. On the other hand, if we are concern- ed only with the pattern of major centers of commercial activity, then a much coarser sampling frame will do. Third, it is assumed that all areal units are compact in shape. While we do not require a regular grid, no tract should be overly elongated in any one direction or curvilinear. This condition arises as a result of two basic requirements of our mathematical model. First, i+7 it is important that the centroids of individual tracts represent good approximations (relative to tract sizes) of the point locations of all variable measurements taken within tracts. Second, we wish geographic coordinate pairs for points within tracts to be uncorrelated and to remain uncorrelated over rotational transformations of coordinates. Where all of these conditions are met within the specification of a frame of areal units, for the purposes of our modeling strategy, the complete frame itself may be represented numerically in the follow- ing manner. We first establish a planar geographic coordinate system having x and y orthogonal axes and origin fixed relative to the geogra- phy of the urban area. Any unit of length convenient for expression of distances (miles, kilometers) may be selected for coordinate intervals. Now let there be n tracts comprising the frame and let all tracts be permanently numbered 1 through n. Associated with each tract i will be four descriptive constants: Mx., My., Vx. , and Vy.. Mx. and My. r l'-'l'l' J 1 1 J 1 represent the coordinates of the centroid of the i-th tract taken with respect to the established x ,y coordinate system. Vx. and Vy. represent x and y component variances associated with a uniform distribution of points over the area defined by the i-th tract. Note now that our numerical representation of the complete frame of areal units is simply an array of summary measures describing the positions and sizes of all n tracts. The x and y centroid coordinates of all tracts are taken as measures of their relative positions, and, since we have assumed com- pactness for all tracts, the x and y component variances of intra-tract point di:;tril)u1 ion:; <ire r.lor.oly proport ionn] to tho squaren of x and y tract dimensions. 48 Our requirement that all tracts be compact in shape will, of course, imply that the values of Vx. and Vy. for each tract will not differ by much. Thus, it will be convenient for many analyses to simply assume that Vx.=Vy. for all i=l,...,n and reduce the number of descrip- tive constants for each tract from four to three. It will facilitate our mathematical discussion here, however, to maintain separate nota- tions for Vx. and Vy.. 1 1 Having described our method for selecting a specific spatial sampling frame and representing it numerically, it remains only to be said that all patterns of urban phenomena will be represented as dis- crete probability distributions of specific variables across the set of tracts comprising the frame. For maximum generality, we will assume that data values for all geographic patterns to be analyzed have been measured, either explicitly or implicitly, over all tracts. Thus, any particular spatial distribution may be represented mathematically as a vector J of n elements where n is the number of areal units, f denotes the particular areal distribution, and the elements ^z.* i=l,...,n, are probabilities proportional to the aggregated data values recorded for each of the n areal units. Thus, ,-z.>0 for all i and for all f, and I . _z . = 1 for all f . l fi One further note concerning vocabulary is appropriate. We will occasionally find it convenient to speak of the elements of an areal distribution. By the term elements of a distribution, we intend gene- rally to denote those areal units or tracts having non-zero quantities of the variable measured in representing some pattern of phenomena as a discrete areal distribution. For maximum mathematical generality, however, we will preserve the option of characterizing all distributions 49 as consisting uniformly of n elements (n the total number of tracts) where each particular probability vector JL may contain numerous zero elements. Basic Measures of Central Tendency and Dispersion for Areal Distributions For a given areal distribution f , let 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 * »§ 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- f»£ 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 o o o o O O o o o o (Nl patrre^aa uoxssxuisuea}. uot^pukiojut TPatuona^s ^.uaoaad r-Kl *-0 0) bo w bo 0rS *.H OO <y 9 o*r org Ov- oo o o w •H W >> U 0) 3 ( q r» x J o r-t fpl ^ J- CM CO * V o 0) 6 0) co •H *j tH -.)• cn l(< (D »). CO lb rH tH Hi ID 85 o o o o 00 ^3 o o pauxp^aa uoxssTuisueaq. uoiq-PuiaojUT iearuona:}.s ^-ueoaad «-0 OJ bO WW ^ ooo w bO o»- oo o w •H w (TJ W 3 <u § 3 cn W) •H Bh c 3 o X> o o o o o o pauxE^ea uoxssxuisuEaq. uox3.Eu1ao.1UT xEan^ona^s auaoaad r-O 4) oa * O00 W bO Or* c OKI Org Or oo o w •H W 4) O H 0) •H •H 87 o o o c oc o O rO O « So OO <T3 OU^ W 00 o*» c •H 3 O OO ■ ♦ o pauxp^aa uotsstuisuej} uot^puliojut TEan}orta:}.s ^-ueoaad w •H w H (TJ g a> ■p w H O 0) H 0) 3 o t>0 •H 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 ♦ H "| " 1) • • $T I 36 6t 46 6? 68 M 4* 4? 48 49 J4 4) to I? ?9 10 11 u u » 81 6* 44 21 1/ 11 IV 31 M 88 it H I ii, » 60 ., 4) 16 3 •1 ' - 34 3? 69 89 fl inj 79 It 4 •1 ' 70 90 71 in? '6 A.' 4/ ?6 u| , '1 ' *1 3* 71 91 .„.. M ..L»L«I i l» Hl| „ "\ " "1 *" u mo 61 41 73 ?4 ?1 ?? 36 34 93 u 1? 10 60 40 39 SI 37 33 » 3V it 37 36 4 1 76 73 74 "1 94 « 96 97 96 93 . 4 6 ( 10 \l 14 16 1f ?0 ii i*- It ?K 30 3? 34 36 36 40 Fig. 13. Zonal system subdividing urbanized area into areal units for data aggregation. (Coordinates in 1/8-miles) Also depicted is network of major arterial streets. 96 land survey grid. All other tracts were taken as quarter sections with the exception of four peripheral corner tracts which were taken as full square-mile sections. To simplify the layout of the hypothetical community, it was decided to distribute the 25% of the total land area in streets uniformly across all tracts. Approximately two-thirds of all vacant land was allo- cated to tracts along the periphery of the urban area and one-third was allocated in a random manner across interior tracts. The distribution of industrial land use and railroad property was determined in large measure by the placement of two major railroads, one running north-south through the center of the city and the other cutting diagonally across the southeast sector. The distributions of all land uses (except streets) are depicted in the block diagrams of Figures 14-23 where the heights of all tract blocks are scaled so that the sum of the volumes of all blocks is constant over all diagrams. Thus, these diagrams may be viewed as graphic presentations of the discrete bivariate probability distri- butions characterizing the distributions of land uses across the city. To facilitate further the design of the hypothetical city, all 104 tracts of the spatial sampling frame were subdivided into 2.5-acre cells, and all land uses were allocated across all tracts in discrete 2.5-acre quantities. Thus the complete twenty square-mile area (12,800 acres) for design purposes could be considered as consisting of 5120 2.5-acre cells. (See Figure 12). The decision to allocate the 25% of all land occupied by streets uniformly throughout the urban area simpli- fied matters considerably. To account for land in streets, we had only to multiply the total acreages of all land uses (except streets) given 97 Fig. 14. Probability distribution of single-family residential land use Fig. 15. Probability distribution of two-family residential land use 98 Fig. 16. Probability distribution of multi-family residential land use Fig. 17. Probability distribution of commercial land u se 99 Fig. 18. Probability distribution of public and semi-public land use Fig. 19. Probability distribution of parks and playgrounds 100 Fig. 20. Probability distribution of light industry Fig. 21. Probability distribution of heavy industry 101 Fig. 22. Probability distribution of railroad property Fig. 23. Probability distribution of vacant land 102 in Table 2 by the factor 1.333 to obtain generalized land use acreages in which associated street acreages were subsumed. These generalized land use acreages were then divided by the factor 2.5 and truncated to integer values to obtain a proportional distribution of the remaining ten generalized land uses over the set of 5120 2.5-acre cells. The specific allocation of land uses over tracts and cells depict- ed in Figure 12 was made primarily in an intuitive manner with occasional reference to land use survey and planning data given in Chapin (1965) and Goodman and Freund (1968). To reflect more closely the spatial com- plexity of an actual urban area, it was decided that the community should be multinucleated with respect to centers of both industrial and commer- cial activities. Two major industrial centers were located to the south and to the east of the CBD along the two railway corridors, and both light and heavy industrial land uses were interspersed within these two cen- ters. Other light industrial land uses were located at the intersections of major arterials with two interstate highways bypassing the community on the north and west sides. (See Figures 20, 21, and 22). In addition to the primary concentration of commercial land uses within the CBD, four secondary concentrations of commercial activities representing suburban shopping centers were located in each of the north, east, south, and west sectors of the city. (Figure 17) Also, eight smaller clusters of commercial land uses were scattered throughout the community along major streets to represent ribbon commercial develop- ments along arterials and small neighborhood shopping centers. (Figure 12) 103 The pattern of mult i- family residential land use followed closely the distribution of commercial activity centers. (Figures 16 and 17) Our rationale here was simply that both multi-family and commercial land use centers would be expelled from low- density residential neighborhoods and would tend to cluster together at locations along major arterials. Duplex housing tended to lie close to the CBD and major industrial areas. (Figure 15) Single-family residential land use was distributed in a more uniform manner across the entire urban area. (Figure 14) The pattern of public and semi-public land uses was determined primarily by the placement of public and private schools. Our city includ- ed a community college occupying the 160 acres of tract 47. Following Bartholomew's land use classification system (1955), two golf courses of 160 acres each, one public and one private, were also included within the distribution of public and semi-public land use. The private golf course was located in tract 42 and determined in large manner the low- density, high-rent character of the west side of town. The public golf course was located in tract 50 in service to the newer suburban develop- ment of the northeast sector. (Figures 12 and 18) Parks and playgrounds were distributed fairly uniformly through- out the urban area with the exception of one large central park of 240 acres, which was located across tracts 17 and 28 just to the northwest of the CBD. A smaller municipal park of 40 acres was located in tract 55. All other parks and playgrounds were smaller (5 to 20 acres) and assumed to be neighborhood-serving in character. 104 Urban Spatial Distributions Selected for Analysis After delineating the general pattern of land uses for our hypo- thetical community, it was then possible to focus on spatial distribu- tions of specific urban variables. Our main objective was to select a set of spatially distributed variables representative of a wide variety of the socioeconomic activities of urban areas, including residential, cultural, recreational, commercial, and industrial activities. Recog- nizing the strong interdependence between the locations of certain urban activities and transportation facilities, we wished also to include variables related to the configuration of major arterial streets and railroad facilities in the analysis. Within these broad objectives, our selection of a specific set of spatially distributed urban variables was somewhat arbitrary. Table 3 lists 32 variables corresponding to 32 spatial distribu- tions of urban phenomena selected for the example analysis. In each case, aggregate data values for all variables, expressed in terms of the units given in Table 3, were recorded for all 104 tracts of the sampling frame. Figures 30-61 in Appendix 1 display the distributions of aggre- gate data variables across all tracts for each of the variables selected. It should be noted that the prior allocation of all 2.5-acre cells of the city to specific land uses as shown in Figure 12 played a funda- mental role in the subsequent estimation of aggregate data values across tracts for all distributions. For example, given the specific allocation across tracts of the 137 2.5-acre cells of two- family residential land use implied by Table 2 (and accounting for the additional acreage included lp 105 3 co (TJ <v <P O CO P •H C ID o o o o o o o • O o o o o o o o o cr o o to o o o o o o O CO o o co co to co to +<rl rl rl H H T-l *H rH to vH +j V P P P <TJ q CU •H «H «H «H «H <y x x x x x X X'H X o X c c c c c V) (TJ -j -3 ^ £ p bo GO bo be bo a CO fc l^, Uh «4h (4h iJh v 4-» a> lp M-l fc « to a> CO p CU <P C C C C C •H «H «H »H «H >H rH M rH iH (TJ ITJ bO q •H CD >» O o cr CO CU P P P P P p CO *j r/ <r a* rr tr cr a 1 co O* CO rH rH rH rH rH c C c c c P o w co co co co co co cu CO P X CO rH •H CO H cu (D Q) Qj <D a> u a; 8 cu 8 «0 c rH id id- T> a. a H tp 5 5 3 2 S E e a> ro c c c q q q q <tj c <D V e e q CU TJ T3 T3 TD T3 rH rH rH rH rH w W «.H »H »H «H «H •H «H CO •H CO ax cu 0) «H P rH H H rH rH rH (0 (0 (TJ CO lp 14h lp ip cp O O O O <D 4) «P lp «TJ fO ffl IT) ITJ ro m bo CO lp ip ip »p >p id lp cu CD O O O O P p U u p u U o o a; a> a; o a) u u u u u =fc: < < < < < cu cu 3 P o o o O CD O q & C c q q c o o P p u < < Q o P. •H O -tfc ^rfc ^ffc -ft ^ffc w w w w w < < =»t 1-3 ^rfc -rfc : ?fc : rfc ^rfc <2 5»fc rj < q o •H P a. •H P o CO CU Q q o •H P X) •H h p CO ITJ •H P (TJ ex CO C0 rH CO 8 TJ CO /— s q (tj p "■» CO x: CO 3 cu q p <D o rH o P <u q P cu CO 8 P ITJ E CD O bO bO P E P && (TJ CO £ *-* jq >> q CO ITJ t, CO p P CO CN O (TJ O cu P P q •H V CO CD <^rl (/) rH »H P a, itj CO P CO CO o q to «h p CO »-» CT> 1 p. p o CD CU cu o 3 3 p 3 p q «h CO P CD 1 O rH (TJ p TJ CD P 3 TJ TJ ip •H 3 q p 3 1 r> tH (TJ TJ CU CO TJ O CT q q bO C 3 •H q ^ w >-» q q p CO <^«» P +J «H CO CO •H «H p q 3 bo q w o (TJ O p p p O P CO rH q p CD CU •h q bo 3 T3 W CO «H cu CD q q q o o § CO > x: id ID co bo«H q s CO rH rH +J CO P P (TJ CD v q CD (TJ •H q p 3 q co «h bO rH O O ITJ Ai . (TJ o E -— ' w p p •H (tj bu a p >, O'H 3 B q O O O O O 42 X O u u CU (TJ E CO P (TJ 3 1 CD «H CO & w p 1 £ M O 3 •H E (TJ O ,C > CD CO CU O p ITJ CU X -H 3 X O bO x: o o > cu o P | (TJ P cu rS<! O TJ CO P > CD •H rH CD > (tj a >i o .c TJ CD O CO CO TJ q CU to O P p bO O CO •H ceo •H x >, O P C0 TD TJ P CU o CD a p (TJ CD s P •H *H «H •H «H O •H rH CU rH q rC fl fi o a •H q TJ o to +J p ip O TJ lp P P. P e j^-h e CU >, bO bO (3 O O > >»• 4T CO pj CU CO •H P P IP o cu a (TJ rH E O p o P «H «H ,q O CD CO S cu cu O >> 0) > (U TJ CO q q o (TJ +J LP -H Id ,C q (TJ x: x: to P H E CO q ax: p •H u q 9 r CD CD P TJ 1 E tp I cu cu P CU O (TJ CU •H H 3 P 3 CO rH CJ CD •H O l> & -tJ bO (TJ (TJ CU (TJ 1 CU •H p q p p ben q rH jq rH B P (TJ w 1 (TJ id CO rH lp p .5 p 8 rH *P «H rH CO (TJ CD o o CU X O O p 1 •H 3 M «H E 1 1 •H O O (TJ bO 1 P -H q o E H H H bO-H o rH rO g*s T3 o o i o rH p a rH rH > X O H C OHfl (TJ >> cu q q <h •H b0T> P -H a O •H CU p rH CO CO a a«H q tth •h s a o p. (TJ rH P cu o CU CD q x: P a, 3 (TJ o 3 ax: 3 3 (TJ o E E P (TJ (TJ (d io h s; s E- Q W »-3 w c_> 2 « n o tw, < tM X Em O" en (Vi < U, U, PC W W Oh m s p^ <mOQuUOXM^^PJSZOCUO(^WH^)>^X>HNt-iCNCO^-ir)cD O cu p p (TJ a, H(Nfo^LOl0^coolOr((N( , )JLO^D^ooa)OHCN(^Jm^D^oocnoH(N rlrlHrlrlrlHHHrKNtNtNCNCNCNCNCNCNCNtOCOn 106 for streets), the number of duplex housing units in each tract could be determined immediately by assuming an average factor of 10 dwelling units per acre for duplex development. In like manner, the 68 cells of multi- family residential land use shown in Figure 12, using an average density factor of 30 dwelling units per acre, determined the spatial distribu- tion of apartment dwelling units depicted in Figure 32 of Appendix 1. To bring about some variety of single-family residential densi- ties, three density factors of 4, 6, and 8 dwelling units per acre were applied respectively to 512, 992, and 496 cells of single-family land use. Standard 1/6-acre lot development was distributed rather uniformly throughout the urban area, 1/4-acre development was distributed mainly in the western section of town, and the 1/8-acre lot development was concentrated mainly in that area between the CBD and the industrial cen- ters. In addition to these single-family housing densities, an average density factor of 12 units per acre was employed for the number of mobile home units of four trailer courts in tracts 36, 37, 93, 94, and 98. (Figure 33, Appendix 1) Mobile home development is shown as single- family land use in Figure 12. All public and private schools (nursery, elementary, junior high, and senior high) were distributed throughout the urban area in more or less Loschian hierarchical manner. Here, 36 daycare centers and nursery schools and 20 elementary schools were distributed rather uniformly across all residential land. (Figure 36) Forming more stellated pat- terns, 10 junior high and 5 senior high schools (both public and private) were located at approximately equi-spaced points throughout the community. 107 Again, in our effort to reflect reality, junior and senior high schools were occasionally placed side-by-side on a single parcel of public land. (Figures 35-38) In addition to the community college occupying tract 47 a number of vocational or trade schools were located in tracts close to the CBD. (Figure 39) In defining the spatial distribution of outdoor recreation areas it was decided that the central municipal park of 240 acres was of a character sufficiently different from all other neighborhood parks that it should not be included within the pattern of neighborhood parks and playgrounds. (Figure 40) Since this single park comprised almost two- thirds of the 384 acres of land devoted to all parks within the city, to include it within the city-wide distribution of park and playground acreage would have resulted in its complete dominance of the pattern and destroyed the spatial association between local parks and neighbor- hoods. Hence, this major central park was grouped with the two golf courses of tracts 42 and 50 to define a separate pattern of regional outdoor recreation areas. (Figure 41) In addition to schools and outdoor recreation areas, two other areal distributions of cultural and recreational activities, movie theaters and churches, were defined for the hypothetical community. Churches were distributed in a uniform manner over all non-industrial land uses of the community. (Figure 43) Movie theaters were located in major commercial centers where adequate parking facilities could be assumed to be located. Eleven different areal distributions of commercial establish- ments ranging from full-line department stores to fast-food drive-ins 108 were defined for the community. (Patterns 15-25 of Table 3; Figures 44-54 of Appendix 1) Our attempt here was to select a variety of com- mercial activities whose areal distributions would be representative of activities typically associated with major shopping districts, strip commercial developments along arterials, and local neighborhood retail outlets. Thus, full-line department, furniture, and hardware stores tended to cluster at the CBD and major shopping centers. Food, drug, and liquor stores were more evenly distributed throughout the entire community, and auto service stations and restaurants were distributed along major arterials. The distributions of major arterial street frontage and railroad property (Figures 60 and 61), as well as the distributions of heavy and light industry (Figures 56 and 57) patterned with respect to these trans- portation facilities, were taken directly from the prior delineated land use -transport at ion system of the community. Areal distributions of private office space (Figure 58) and banking activity (Figure 59) were defined with strong CBD orientations. As an additional item, four region- serving hospitals were located at points close to the CBD. It should be noted that a variety of measurement units were used in quantifying the 32 areal distributions selected for analysis. Resi- dential distributions were measured in terms of numbers of dwelling units, school distributions in terms of enrollment figures, commercial estab- lishments in terms of floor areas, and so forth. Since the areal dis- tribution itself (characterized as a discrete probability function) represents the unit of analysis, however, we should not be accused of "mixing apples and oranges." Our method is explicitly designed to allow 109 analysis of spatially distributed urban phenomena quantified in terms of whatever variables are convenient to observation and measurement and highly correlated with the specific phenomena of interest. Of course, there always remains the inevitable trade-off between the objectives of precision of phenomena measurement and economy of data collection. Example Analyses Performed Having defined geographically the set of 32 areal distributions for our hypothetical urban community, all distributions were character- ized as discrete probability distributions across the tracts of our 2 sampling frame. Then, values of _ EDI were computed between all pairs * »6 of distributions using the method described in Chapter IV. 2 2 Values of _ EDI and JCDI were computed independently for each pair of distributions to evaluate numerical error effects within 2 2 computation. In theory, _ EDI should be identically equal to JjDI . In practice, we found that, using single-precision arithmetic on an 2 2 IBM 360/91 computer, values of . EDI and _EDI differed almost always after the fifth significant digit, and, where f and g both had a large number of non-zero elements (both greater than 30), they differed often after the third significant digit. Thus we conclude that any future experiments or application of the method should employ double- precision arithmetic within computations. 2 The result of these comDutations was the 32 x 32 matrix of _ EDI f,g values reproduced as Table 4 in Appendix 2. Then, weighting all distri- butions equally, we applied the cluster analysis algorithm developed in Chapter IV to investigate the structure of spatial associations existing 110 ANrTGK'J3CSUVW5VXHOPOR4 21M2D16FJ TfHS UNIT 1 1 ? P 4 t> ? U 1 3 v 7 3 7 1 (i ? c 4 t l' 7 I n 4 1 n i i 4 6 i / 1 ? <; /. v IIHIHII 4O0/129.J9 4 S 1 '• \ J J 1 I 6 U 7 1 ..' 9 3 Hi 6 i. 7 -• 1 4 7 — 4 1 2 9 I 41 1 ? 9 I 1 J 1 ? 9 I 11 1 ? v * 1 J ? 9 ti 2 9 I! ? 9 41 ? 9 J$ ? 9 h ? 9 1 3 << 4 1 3 4 A I 3 9 1 1 3 9 • -♦ o 3 ? 2 ? 3 n ? 3 I •" ? ? ? 3 11 ? 3 M i 3 '* J 3 3 3 ? 1 b 3 ? 1 5 * 1 5 --♦ 2 4 >• ? 4 • 14 1 * 8 1 3 > r I 3 ! 3 ;- i; I I >• c I 3 >■ ( I i I 1 I i y o \ 3 H I 3 8 n I ! s n I 1 1 \ > 1 1 y v. \ 3 8 I s x \ 3 f I I \ 3 p r 4 1 o 3 D 3 2 1 6 ? r> 2 1 > C 2 I ? \ > 6 ? 4 I I v o ? ', II 9 6? J I i 9 6? 4 I I 9 6? J, I I 9 6 • A I \ 9 6 • III 9 6? Ill 9 6? 4 1 i 9 6? ill 9 6? 411 9 6? Ill 9 6? 4 1 I 9 6? 1 1 I 9 6? 4 1 1 9 6? 5 1 I V 6 ? 41 I 9 6? 41 I 9 6? Ill 9 6? ! 1 X 9 6? r I 9 2 4 I 9 2 4 ; 9 ? I I 2 12 2 3*4 7 1 ? n 2 3 f 4 7 1141 3 h 4 7 I ill 3 ;• 4 7 IW 3 s 4 7 I Jii 3X47 fin 3*47 I Hi 3 8 4 7 1141 3 b 4 7 111! 3 4 4 7 1141 3 X 4 7 1141 3 8 4 7 i m 3 y. 4 7 I i t V 3 if 4 7 (ill 3 u 4 7 1141 3 H 4 7 in 3 ft 4 7 1141 3 a 4 7 in 1 3 c 4 7 1141 3 y 4 7 1141 3 8 4/ 111! 3 8 4 7 •HI 8 4 7 ill 8 4 7 141 £4 7 — I 1 4 7 * -4 4 4 4 HI 3 n 1 ? 5 c Wl ? 5 c ill 2 S C in in 2 3 ill 2 5 i: III 2 5 ill ? 5 -0 14 1 2 5 ill m •n n 5 n II n n n 5 II n n s II n 5 C n n n n 5 (I «-♦ 4 I Fig. 24. Hierarchical tree showing sequence of cluster mergers within cluster analysis of [^ ^EDI ] matrix of areal distribution dissociation measures. f,g Ill fOO rgoo r\ir- rvnA INjfO <mo «-X «--o CL C- OX « 0"0 « uj c;<\i « £ 0«-« z oo« ♦ Or OOJ OlO o»» O-o ox oo «-o r— r- «-x »-:> INl«- r\Jf\J (Nt/>> (\i<3 f\i^ IV 00 r\0 roo O £ Z o ZCC I oz u- U* \A •-* »■« «->►- «£</»«C 0CH- <V)H IU< 0C — -UJ a. a. »- >- oc c o •H ■p o c • ,3 X 4-t •H fc W -P W 03 O £ 1 i — i (3 CM o M •H Q CO W W bC •H 01 E <U w i 1 s 4h fc O 4-> 1 CO c •H o CO •H >1 n3 tC B o IP fc c <u •H P 1 CO H 3 td H § U -M Mh O O 3 & to ■P (1) TO bfl rO Mh P o CO x: CD a, > <T> •H fc CO o CO CD O • O lO 3 CN CO • fc no <D •H > u. o 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 T3 Q) G 3 O CO u <v h o e 8 1 a •5 1(1 it w £ U n 00 LI & J-. >, C H ■ <N u io O *-< ■ « 11 —t •-> c ■ V) d 1 a. u 3 *- e VI t. o 1 O *-* t 2 r. <J a 1 t- ~-> ■n 'i •-J C J — c X. a u 11 t: J -J c o c u ;. 1^ c M 3 -O V) VI •-* fi 2 « % n CO r: r: V) •-( *-< *-< n c H •H t3 .3 ^ o o 1 o •J J y> c c o ij 'i. U a « £ 11 o -i ■tlE 5 .c x: n o > -J x: O u o > o. o *-» >N o ..- •O <D u V) .1 -o c ■c >, o ^ H T3 ♦> • »-< ■ ^ c x: x: C O 3 B Q >l o :-. t» &.- O > ~< E b *J o t. x: g l4- •r4 n c K x: x: U L, ~l 01 I f: <-, i 01 o t> O <3 •1 B Ifl i O "* L. r. t, U 'J x> C t, -C l«H •-I 1/1 <9 •: o o ,t O Q g *-» c o c: ■* O « 10 >, V c c •H wo -J :: a o i. o H 3 O c U <y c .C (fl t- ^ X H Q M T c/> o z a M o c w Si K o P C O "O O V> M <OUQUl>UXH>]X^XZ o •- o V) o I V t> o VI o o o o V) t. VI </l CO 01 3 J S f. 3 •o -o <E o o O !_- r. c o ♦J -H V) V) • H "H v o V) <H c i-< D 1) C V) rs O 01 f^» >,T3 V) U r. •H •a c: a E +^ (- 10 LA 3. M ■- « o o *-» o D to 3 I 11 .H u >i V) ti : o '. <J ^ IQ a x: *t ♦-< o 49 .. O T3 V) 4-* > a "H r* w . i. t. O n c c <j > ™ a. * rs tx o 0) u • H «^t •H ..H '.4 2 . U >H u C V ■tn p t. o o » o. ■H U t-> O 0) n. • O o o >* > i ■o V) c r: O '0 4J o ex: *-• Li !-- 8 --t f <D t •J ] w> 3 V) --i o 01 o «• CO to 10 * « i n 10 ■♦. ^ >> 4-» B s « e 1 ■H o o '0 •H f. c o -i u •• ;. o *-< n. 'H <-H > .yr O ^4 « o •<-! "J i-» •.; VI c- n. t: -r-> •-« 4 o o 3 r». 3 3 IB € L L. (U nj 5 • ^ u c to f- < U. U X W Ul a. PU K «KH3>*X>>NHwn*inio o o o o fj O O C r* r- ^ f\* flj Af M i O O O O O O C' Ci O O — O O O ■-• 'i O _3 c^ O f. i j ^- ." O - - . " w •H to >> CO P CT> CO •H x: ■M •H (T3 o o f= a a o i i • • ^ o o O 1 M o Q *■ W 3 o bO i a •i o 14h • • " < o o O 1 o 4h w o 1 o c o O •* •H C3 O ■P O 1 P a i/\ • O o o CO 1 o o •o bO t • c O 1 •H o H rw fO o o o 1 o CO o 00 .H o o CO O 1 o c o> o • •H o o 1 o o CO c a CD ^ p i •H T3 1 O ^ ■P CT> 4: o s o H • CD OJ • bO •H tM 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 « ( I S II V ) T M t Ml ( » n D l t M I ) 4 1f«S 11?? T r i » i s 11?? J 7 1 ? 1 3 J i I J }• UWll- 1 7 1 V 1 3 ni in if 1 V 1 3 HIM 17 19 13 r mm 1 9 1 3 4 Mr H J { ! I 3 ? ? ? 1 S 4 ? ? 6 3 ? ? ? n 1 5 4 ? ? 6 •il!U 5 4 ? ? 6 -! II 4 2 2 6 J J 4 :■ 4 2 i 6 4 ? ? 6 •— J ? ? 6 n ? ? 6 44 til J 4 n ? 6 44 i 6 • -♦ 1 ? ? 4 3 I * 9 1 ? ? 9 ♦ c I mm ( U ] M 9 I HI 3 8 8 « ? n j i 3 8 8V 44M 3 8 S> 9 In 3 I 8 9 i 3 8 R 9 I I 3 8 k V 44M In inn 44MI 3 8 f V 3 iill 4 Mill nn 3 8 8 9 3 4 4 1 M I I HI 4 i III I I III 3 • -- 8 9 I 9 44 8 I ! ? 1 1 6 5 6 ? 1 1 ft 5 6 2 \\ 6 -♦ 1 ? 3 8 9 1 ? 3 in in k 9 o I J I k 9 ill S 9 •) Ml 8 9 I 41 8 9 I 41 8 9 Ml 8 9 \ J J 8 9 'I I 41 8 9 I 41 8 9 I 4! K 9 C I H 8 9 (i M- 8 9 I 4 I! 8 9 M 8 9 I 4 8 9 I 4 n n 8 9 -I 4 I I I 1 6 f J ? 3 1 nn I ! ■in /MO Mil 7 2 5 4UI m nn 7 ? 5 MM i n n n n n n i n ! 1 I S • -4 5 5 4 4 7 -♦ Fig. 27. Hierarchical tree showing seauence of cluster merges within cluster analysis of L LDI 2 ] matrix of areal distribution dissociation measures. 119 X o a o O < Ik o at CD S 3 (N.O <VlOO OiN *<*•# *\ii*> cwoj ru»- <mo •-o •-•0 #-N »-•* • r-w « r-O* « r-»- • r-O • oo- « Oac« o^>« c-o « o%»« o-o« Or« oo« OfM o>r oo ON O00 «-o r-»» Ul T-Oi O < M X T-tA W» < o 2 T-OO <MO Z Mr- < K <N/<M ui ae M ui vt Oi<0 3 rvN> «-> (Mac u. o rgo at KIO Ui CD a o o o ao o a o o r\i o ►-2 2UI ZH I OS uj«r or — iui CvUKCOt c o •H 4-» O g • 3 X M-t •H fc (0 •P 0) rtJ o e r-\ 1 i ) C CM o M •H Q (0 ►J CO b •H e n- CO 1 * @ Mh fc O 4-* 1 CO C •H o CO •H >> s •s f; s O «4H u c a> •H ■m 1 CO n 3 % O 4-> <4-t O O g CO •M a) CO V «H ■H o CO x: 2> <x > <tj •H b CO O CO cu o • o CO 3 CM CO • U bO a> •H > CiH o 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 C 3 O w 0) w 8 ■83 m 8 3 5 r ?r c o — < ♦ • •» <• in v< M r *i *» " pi o. • * T S ■ »> 1 <i --* •J C, 3 i r» *} M c d — « r •» .- c 1. 11 r 9 tc c u k. —4 c tj 3 -J *> n -h g n 6.1 r: c: V) *> 9 r. n —4 W '0 •-< .) c n Jl. 3 b 3 VI -1 c |i 8 X .-. ^ V o O ? r. Q <-• jc: u u w X o .r S • o V) u -,) -i tr. >> ■ T3 O -i •-* | -t t : - --; r 3 t -t *• U t t ■3 o c > »« 10 r. c IT ii •- b i ■ >« i o f V <• •> V H 1 <v •-« I. r. '- (■ •.- r. h v.. ~* ■ m a (.■ V) t o kf! A w ■h r o a -4 r_ ■-« o E -4 r> O S o E — * r» -j •^ <•- ~l g E ■ 3 c ^ c V) i^ X t- t-i Id -i «/> o ~ u. K-4 <«oQw«-ox>-i->^Ji:5'- ^> 1 O *> o "■! r. o •~> c >>*o •»^ •> v. c t) «-■ « r3 rs no rv. o o a F VI t- o o (• 3 11 <0 t 13. O O €> O.V •com ■i -t y •< (i « « n i n« cj 2 ?. E "E "8 a s ^ u I I c ? ~4 4i —4 la >> C C O > <3 % "r« ^H «rt -^ ' ♦j v <fc> o o a. £^> o -> o> c> ij ti <; «■ »^ >- >> »j •-> o o m C»H ^ > f Cl. (1.-H C 1= ti fa lb v t. -o 60 n io ^ t. E j: ■-! C •€-,:■< OO-of- whJ> JX>.Nripin jma) o •- o o o c o o o o o o o s o O O hi CD m o c o o o o o o o c oo o o o o c o o o O I o 1 o o o = O I o o o I o o o o o • o o a • 3 o •o o o o o I c o (0 >> H CD +-• w H o •M W U •H ■P 1 ►J c o •H rH O w 130 c •H o w o o o t C o o 1 o o o 09 C E 0^<^0'» - >Or^"Oi3»>"-^0*»'00^'^'Ji>-'^.J O" O. X JI.NKK C ^ *L *r\ S .^ * *t |«ii^r.niiMft • •••••••••••••••••••••a >.^ 1M>10"ll-0«1-Ul f .l»OI1NJ«M-U*IN .MSOfll^O^^O . Cl C) « — '-J — • I I I I I I I I I I I < I I I I I I I • I I o 4-> I < o CM GO •H 122 Applying TORSCA-9 to this new matrix of distribution dissociation measures, we obtained the two-dimensional point representation of inter- distribution distances depicted in Figure 29. Again, mobile homes (D) and colleges and vocational schools (J) are so dissociated with all other distributions that they fall outside the limits of TORSCA-9* s display area. We will admit, however, that the multidimensional scaling repre- sentation of the remaining distribution associations is more appealing than the results of our cluster analysis. In fact, the two geometric configurations of Figures 26 and 29 are quite similar despite the dif- ferent inter-distribution distances scaled. This is not entirely unex- pected, since the nonmetric scaling procedure considers primarily the rank order of distances between distributions, and, while we know that the elements . EDI* and . LDI* are different, their rank orderings f,g f,g should be not too dissimilar. CHAPTER VI SUMMARY AND CONCLUSIONS Summary of Argument In Chapter I, we presented our basic case concerning the need for investigation of more general methods for quantitative description of the organized complexity of real-world urban space. We argued that existing social science theory concerning urban spatial organization was much too limited in scope for explanation of the rich variety of socioeconomic patterning that we observe across urban landscapes. We pointed out unresolved methodological questions surrounding those methods most commonly used for analysis of the ecological interdependence of geographically patterned urban phenomena, and we called for the develop- ment of alternative methods of urban spatial distribution analysis better equipped for the task at hand. In Chapter II, we reviewed the basic concepts of Shannon-Wiener information theory seeking some more general mathematical basis for quantitative description of the essential dimensions of urban spatial organization. We examined the unique properties of the mathematical concept of entropy specifically as a measure of informational uncertainty within telecommunications theory and as a measure of the random complex- ity of discrete probability distributions in general. With reference to the entropy-maximization model of intraurban trip distribution, we 124 noted how information theory concepts might be used in conjunction with origin-destination transportation study data to analyze the extent of interdependence between the co-organization of various socioeconomic activities in urban space and the ecology of sociocultural relation- ships existing between activities. Here, however, the proposed para- digm was essentially behavioral and thus dependent on extensive obser- vation and analysis of social activity systems for operationalization. In Chapter III, we returned to the principal research objective of our thesis presented and defended in Chapter I, namely, the inves- tigation of quantitative methods better equipped for analysis of urban spatial organization as a complex system of differentiated population, socioeconomic activities, and land use patterns. Translated into methodological issues, our task became the exploration of more effect- ive methods for analysis of spatial distributions as well as the spatial interdependence exhibited between differentiated distribu- tions. We began this exploration by first reviewing certain basic sta- tistical concepts commonly employed for analysis of areal distributions such as measures of distribution central tendency (p. 49), measures of distribution dispersion such as distance variance (p. 50) and general- ized distance variance (p. 55), and, as a measure of inter-distribu- tion spatial dissociation, Bachi's square of quadratic averages of distances (Bachi, 1957), which we chose to refer to as the generalized squared distance of interaction between two areal distributions. Our formulation and presentation of these basic measures of intra- and 125 inter-distribution properties differed from previous formulations for, in every case, we considered not only distances between centroids of distribution elements or tracts, but also intra-element residual dis- tances resulting inevitably as a consequence of the spatial dispersion of specific point locations within tracts. Despite our reformulations, however, we remained dissatisfied with each of the above measures of distribution dispersion and dissociation, for, while all might be trans- lated mathematically into functions of probabilistic matchings of elements within and between distributions, in every case the specific probabilistic matchings implied were completely independent of any consideration of proximity relationships existing between elements. Facing the problem of characterizing in a more meaningful man- ner the spatial organization of areal distributions and the spatial co- organization exhibited between distributions, we then focused once more on the entropy -maximization trip-distribution model of urban transpor- tation systems modeling, this time seeking some unbiased means of gen- eralizing previous measures of distribution dispersion and dissocia- tion to depend more directly on proximity relationships between dis- tribution elements. We found that the entropy-maximization trip- distribution model (as any other type of trip distribution model would do as well) left us with a completely arbitrary choice concerning the specific distance deterrence function to be employed in determining a spatially interdependent probabilistic matching of elements within and between distributions. 126 To resolve this problem, in Chapter IV we appealed to information theory, particularly as interpreted by Jaynes (1957). We pointed out that each of our measures of distribution dispersion (distance variance) and distribution dissociation (mean squared distance of interaction) could be viewed as a sum of weighted squared distances between distribu- tion elements. Further, the only information that we had concerning the weights to be applied was that the matrix of weights should be a joint probability distribution with marginal probabilities equal to the probabilities associated with aggregate data values over areal dis- tribution elements (tracts). Viewing our measures as sums of weighted components, we then adopted the position that the distribution of weighted components should be made maximally entropic subject to the single constraint that the matrix of weights be a joint probability dis- tribution with marginals equal to the given areal distribution proba- bilities. This position leads to the formulation and solution of least biased estimates for the weighted components of any of our distribu- tion measures, and, hence, least biased estimates of the measures them- selves. We say, following Jaynes, that the procedure is least biased, since it results in a solution to our problem that is maximally non- committal with respect to all missing information. Continuing in Chapter IV, we demonstrated the direct applica- bility of information theory as an instrument for characterizing the spatial complexity conveyed by areal distributions. Here again our information theoretic measures of distribution complexity conveyance 127 were formulated in terms of the unique set of component proximity relationships determined by our entropy-maximization procedure. Further it was shown that a minimum-structural-information-loss cluster analy- sis procedure could be implemented in terms of the same information theoretic concepts. The resulting procedure was shown to be applicable for cluster analysis of elements of the same distribution to simplify its characterization as well as for cluster analysis of sets of areal distributions structured in accordance with the spatial dissociation measures computed between them. In Chapter V, using a hypothetical data set, we demonstrated the application of the unique measure of distribution dissociation and the closely associated cluster analysis procedure. As an independent means of analyzing the structure of dissociations of all hypothetical distributions, a nonmetric multidimensional scaling analysis was per- formed. We found a close agreement between our intuitive notion of how all distributions were spatially interrelated and both cluster analysis and multidimensional scaling results. Potential Applications of the Method As pointed out above in Chapter IV, our unique measure of mean entropic squared distance between distributions has the property that it is numerically consistent with respect to the scale and number of areal units of the spatial sampling frame employed. In other words, as the resolution of the frame increases, the measure converges asymptotically to its true value. On the other hand, because of data collection and 128 processing costs, we are typically forced to work with frames of varying degrees of resolution. However, our methodology associates with each measure computed to characterize some property of a distribution or the extent of spatial co- organization existing between distributions informa- tion-theoretic measures that quantify the amount of distribution complex- ity with respect to which any particular distance measure has been com- puted. Thus, while our intra- and inter-distribution dissociation mea- sures will vary incidentally across different spatial sampling frames, it is always possible to record for each measure the amount of informa- tion processed. This property of our method should make it well suited for analysis of geographic distributions of a variety of socioeconomic phenomena. For example, the problem of quantifying in unambiguous fashion the ex- tent of residential segregation of socioeconomic and ethnic populations would seem to be directly amenable to our approach. Furthermore, the method proposed should permit quantitative measurement of the degree to which certain ethnic populations are assimilated into the total social fabric of the community as a function of such variables as educational attainment or annual income. In a manner similar to the hypothetical example presented in Chapter V, it should also be possible to analyze the structure of asso- ciations existing between distributions of any number of socioeconomic activities within a city. While our model offers directly no predictive capabilities concerning the spatial structure of any one particular city, it most certainly can be used as an instrument for quantitative 129 description of urban space, and, hence, provides us with a tool by which certain theories can be evaluated. The method would be applicable to comparative analyses of spatial structure across cities as well. Our model yields a set of distance measures between various patterns of phenomena, and where the same phenomena are measured and analyzed across a sample of urban areas, the structure of pattern associations may be compared. Individual associa- tion measures as output from our method may be taken as variables them- selves and conventional multivariate analysis methods used for compari- sons between cities. In conclusion, it is our opinion that an understanding of the total pattern of the city will always be instrumental to our efforts to cope with the ever-increasing complexity of modern urbanization. To understand the city as a complex set of patterned phenomena, it is required that we further the development of methods for unambiguous description of urban spatial structure. Our effort here has been con- ducted toward this general goal. APPENDIX 1 GRAPHICAL DISPLAYS OF THIRTY-TWO AREAL DISTRIBUTIONS OF HYPOTHETICAL COMMUNITY SELECTED FOR EXAMPLE ANALYSES 131 ira H ! I 1HU H |«» 361, */i» 31b «oi i IM 551 574 Ml 360 jn 54(. 180 6 3u /5I, 5/* 461 111 9U -c *v 540 IU 165 > M5 4 35 III 911 1 1 -| H 1 4Sl 5V6 wJ St |», > 5*5 315 90 mo 90 1«J 140 | „ mi ?/o 381 ??0 6*9 255 150 345 11*5 461 5*5 195 89 *■>(! 435 11 J H | 90 ...JH 135 t 4 6 parrcim «io. i 8 10 11 14 16 18 20 ii <■* 26 28 30 32 34 36 38 40 SINtlf -(»"ll i housing U»MS (» 01 CO'S) S»NoOl • 4 Fig. 30. Pattern of single -family housing units 132 it M it Vi if 11 u 1 1 1 1 1 1 1 » > n > fi 9* •>(, Jb 19 1 1 ,.| 1*1 10 113 ma I I -1 I I 1 1 1 1 <M „l „ 1 " ?04 94 I ,.j „ 1 » I /% 169 2»1 10 9* s» 7* 94 9* 1 /•> I 9* < • 1 1 ? « « FAIIfBK SO. / ID ^? 1* 16 1> ?0 ?/ KO-IHIi! HOIISIII. UNITS .'* *6 /* ill il 34 36 5C 40 <• OF OU'S) STFIPOl • Fig. 31. Pattern of two- family housing units 133 40 u M J* 1? ll> 78 76 ^L i? 10 » »00 I f% 110 71 110 71 1 1 < 1 100 „ i I H «10 600 1 6u(l 1 1 1 r\ 100 410 ► i 1 1 1 71 n < 75 1 ..!::: 1 i I 4 6 "!lll» NO. J ► 1U !<• It 16 1» 7(1 77 74 76 711 Id J7 14 36 5* 40 «iilll-l««H! HOIISI»0 UNITS C» Of 6U"1> SYNSOl « f Fig. 32. Pattern of mult i- family housing units 131 60 M M »« 3* 30 21 it /* 16 14 II If) I ■ 1 L i 1 ■ i 1 1 1 l mo MO 1*0 < 1 l l i 360 H 1 t 4 • MMIH NO. 4 8 1(J 1? 14 16 IB ?U ?? <•* ?6 ?H )r 3? 34 36 36 40 »0"llf-HO"r HOUSING UNITS (• Of OU*S> STNSOl > r Fig. 33. Pattern of mobile-home housing units 135 ■•----•- — •• III 1*0 1 1 i ?\y 1 I ?!>/ 1 1 ' 1 I I I | ,„ ,..| ^M its i i ' 1 » •"1 ► 1 I ........ | 1 i i 1 ! 1 1 I 2 4 6 I 10 %i 14 16 IK iCI It ?4 ?6 2* SO 5* J4 56 JS 40 P4TTC** tO. i 1MHMFNI LODGING UNITS <f Of TIU'S) STftBOl • f Fig. 34. Pattern of transient lodging units 136 40 I 1 44 4(1 $K *e 7* 6* 4« 1 1 1 Ml 1 1 Si 1 1 1 1 " 1 1 1 „ K/ 1 » 1 " „ .... 1 1 1 I 1 It l.t M W (8 W . /n 14 . 1 1 1 1 • 1 JJJJ 2 4 « rtium »U. 6 B 10 1? 14 16 IK ?0 ?i> 24 76 ?8 30 3? )4 \b 3D 40 C4TC4M CtMISS »ND SuHSfBT SCHOOLS (("HdUKNll STHBCU • I Fig. 35. Pattern of daycare centers and nursery schools 137 •0 >•• > M M M >4« M 10 »t» »*o it »M Ml) ill? ws Mil - 464 > i %»» ">S' i 1 .... •• H 1 16 1* •ts 440 4 7? 1? 10 Ma tor * 6 »n 4 1 1 2 "' 2 * 6 >IT!(>» «U. ' 10 1.' 14 16 Id ?0 ?? ?4 it, 2ft 30 32 34 36 38 40 HHitlMT SCHOOLS <«-fc> (l»»0U«IHI) STdbOt • 6 Fig. 36. Pattern of elementary schools (K-6) 138 40 M U M M So 7* 76 74 77 ?o if 16 14 1? > 7M »to PM 40? > 1 1 7*3 1 1 1 77 7 i i I I 60? *?6 « 61* 6*} 1 | 1 1 1 1 1 * * 6 PMTIIN NU. I> h 10 1? 14 16 If 70 7? 74 76 ?A SO 37 34 36 3D 40 JUNIOR NIGH SCH0OIS </-V) ( (NROLIMEND SVftBOl > H Fig. 37. Pattern of junior high schools (7-9) 139 40 ill M 1« U So ?0 1<! 10 8 1 I i i I ! 1 i*«* i fOO ' 1 I i L..U 1 1 1 I !"" 1 14*11 i I 1 1 I 1 | ! I5M0 > 1 1 1 I * • P«TIf«N NO. 9 8 10 11 14 16 IK 10 it 14 ?6 ^8 3U M 34 36 38 40 SENIOR HIGH SCH0015 <10-1?> (l«OU«IKII STHHOl a I Fig. 38. Pattern of senior high schools (10-12) 140 ♦ Id 16 M U 10 ?8 1? 1 1 SIO 1 1 J 1 ♦11 ' 1 - ! ■ I 1 1 1)0 10 to • 1 ' 1 1 1 14 6 PftMIIN Hm. 10 10 1? 14 16 1* ?0 ii ?4 26 IB 30 3/ 34 36 38 40 rOHIGFS »*» VUC4T10N4L SCHOOLS (FNBOllHINT) STFflOl « J Fig. 39. Pattern of colleges and vocational schools 141 " i MM • « > 6 20 10 n % « 3 1 6 > 1 '1 1 1 1 1 1 1 I : 1 " ! - i 6 I •1 i 3 -| 1 1 I • 1 I 20 * ' 40 6 1 1 -1 ! ■ 1 1 ► 2 4 6 P»It(»n HO. 11 8 10 12 14 16 18 20 I? 24 26 28 30 32 34 36 38 40 N[ ItHPOftHOOO P4HCS AND PI » 1 GSOUN [> S (ACRES) S»«BOl ■ I Fig. 40. Pattern of neighborhood parks and playgrounds 142 «0 M M 3* M SO ?« It la I • 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 BIBLIOGRAPHY Alexander, Christopher. "The City as a Mechanism for Sustaining Human Contact." In Environment for Man: The Next Fifty Years , pp. 60- 109. Edited by William R. Ewald , Jr. Bloomington, Ind.: Indiana University Press, 1967. . Notes on the Synthesis of Form . Cambridge: Harvard University Press, 1966. Allport , Floyd H. 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