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From structure to agency: Essays on the spatial analysis of residential segregation

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FROM STRUCTURE TO AGENCY:
ESSAYS ON THE SPATIAL ANALYSIS
OF RESIDENTIAL SEGREGATION

by

Yiming Wang

_________________________________________________________
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(POLICY, PLANNING AND DEVELOPMENT)

May 2011

Copyright 2011 Yiming Wang
ii

Dedication

To My Children

iii

Acknowledgements
In Mandarin, “father” and “mentor” are two closely related words. My father
taught me how to be a man. My mentor, Professor Heikkila, further turned me into
a scholarly man. As a Chinese saying goes, “respect your mentor as if he were
your father”, which is exactly how I do.

My dissertation committee is a highly capable and instrumental team. Apart from
Professor Heikkila, Professor Philip Ethington supported my research with the Los
Angeles County Union Census Data Series. Professor Peter Gordon is one of the
most prestigious regional scientists in America. He spent a lot of time talking with
me about my research ideas, especially at the initial stages of my dissertation. I
am not sure if I will be as successful as Professor Gordon, but he is my role model.

I am grateful to Professor Chris Webster at Cardiff University’s School of City and
Regional Planning, where I have been lecturing since 2009 whilst finishing this
dissertation. Professor Webster has given me crucial advice both on teaching and
on academic research, especially when I was composing the third essay included in
this dissertation.

iv

I would like to thank my family and friends for their constant love and caring. I
feel particularly indebted to my parents and parents-in-law, because I have been
unable to accompany and take regular care of them for most of the last few years.
Special compliments belong to my wife, who is always understanding and
supportive and has taken over much of my familial duty when I am pursuing my
PhD.

Last I acknowledge my great pleasure to have received two awards from the
Western Regional Science Association (WRSA) respectively for two essays
included in this dissertation. I was presented the 23
rd
Charles Tiebout prize in 2009
for authoring “White Flight in Los Angeles County, 1960-1990: A Model of Fuzzy
Tipping”. In 2010 I won the 14
th
Springer award for producing another paper,
titled “Decomposing the Entropy Index of Racial Diversity: In Search of Two
Types of Variance”. The both papers have been published in the Annals of
Regional Science, the official journal of WRSA.

v

Table of Contents
Dedication ii
Acknowledgements iii
List of Tables vii
List of Figures ix
Abstract x
Chapter 1: Prologue 1

Chapter 2: Decomposing the Entropy Index of Racial Diversity 6
2.1: Introduction 6
2.2: Racial Residential Segregation and Entropy 10
2.2.1: Two Types of Variance 10
2.2.2: The H Index 12
2.2.3: The E Index 17
2.3: A Decomposition of the H index 21
2.3.1: A Region at Two Levels of Granularity 21
2.3.2: A Spatial Adjustment of Racial Entropy 22
2.3.3: A Racial Adjustment of Spatial Entropy 24
2.3.4: Three Factors in the Entropy Index of Racial Diversity 26
2.4: The Case of Los Angeles County 30
2.4.1: Basic Application 30
2.4.2: A Longitudinal Comparison 34
2.4.3: Policy Implications 36
2.5: Conclusions 38

Chapter 3: White Flight in Los Angeles County, 1960-1990 40
3.1: Introduction 40
3.2: Segregation as Neighborhood Preference 43
3.3: A Model of Fuzzy Tipping 45
3.4: Data and an Ad Hoc Gravity Model 51
3.5: Results and Discussions 56
3.5.1: General Results 56
3.5.2: Spatial Autocorrelation 59
3.5.3: A Longitudinal Comparison of White Flight 63
3.5.4: Comparing the Results of Fuzzy and Crisp Tipping Model 65
3.6: Conclusions 66
vi

Chapter 4: Modeling Segregation under Regulation 70
4.1: Introduction 70
4.2: Schelling and Pigou 73
4.2.1: Schelling’s Spatial Proximity Model 73
4.2.2: The Pigovian Housing Regulations 77
4.2.3: Toward an Integrated Modeling Approach 79
4.3: A Cullar Automata Approach 80
4.3.1: Defining Neighborhood 81
4.3.2: The Moving Algorithm 81
4.3.3: The Rule of Transition without Regulation 83
4.3.4: The Pigovian Regulation 84
4.3.5: The Rule of Transition under Regulation 87
4.3.6: The Stop Conditions 87
4.3.7: The Initial Segregation Pattern 88
4.3.8: Assessing the Segregation Pattern 88
4.4: The Results of Cellular Automation 90
4.4.1: A Case Study of Simulation Results 91
4.4.2: Comparing General Impacts of Regulation versus Non-regulation 94
4.5: Discussion 98
4.5.1: The Inefficiency of Pigovian Regulation 99
4.5.2: Racial Integration under Pigovian Regulation 100
4.5.3: Recounciling Preference and Institution 102
4.6: Conclusions 103

References 105

vii

List of Tables
Table 2.1: Two Types of Variance in Racial Residential Segregation 12
Table 2.2: Regional Racial Composition 21
Table 2.3: Racial Composition by Local Spatial Units 21
Table 2.4: Spatial Distribution of Regional Population 25
Table 2.5: Spatial Distribution of Regional Population by Racial Groups 25
Table 2.6: A Decomposition of County Level H index, 1960-1990 34
Table 2.7: A Decomposition of Change in County Level H Index, 1960-1990 35
Table 2.8: Change in Two Types of Variance, 1960-1990 36
Table 3.1: Descriptive Statistics (Non-spatial Data) 52
Table 3.2: Descriptive Statistics of Location of Tract Centroids
and Pair-wise Distance 53
Table 3.3: R Square and Coefficient Values 57
Table 3.4: Correlation of Coefficients and Values of
Variance Inflation Factor (VIF) 58
Table 3.5: Spatial Autocorrelation Tested with Geary’s C 61
Table 4.1: Comparing the Outcome Average Utility
(50 Whites and 50 Blacks) 96
viii

Table 4.2: Difference in the Mean of Average Utility
(50 Whites and 50 Blacks) 96
Table 4.3: Comparing the Outcome Degree of Segregation
(50 Whites and 50 Blacks) 96
Table 4.4: Difference in the Mean Degree of Segregation
(50 Whites and 50 Blacks) 96
Table 4.5: Comparing the Outcome Average Utility
(Random Black/White Ratio) 97
Table 4.6: Difference in the Mean of Average Utility
(Random Black/White) 97
Table 4.7: Comparing the Outcome Degree of Segregation
(Random Black/White) 98
Table 4.8: Difference in the Mean Degree of Segregation
(Random Black/White) 98

ix

List of Figures
Figure 2.1: Entropy in Spatial Analysis and Segregation Research 20
Figure 2.2: Mapping H Index in the Census Tracts of LA County, 1990 31
Figure 2.3: Mapping Population Distribution in Los Angeles County, 1990 33
Figure 3.1: White Flight in Schelling’s Tipping Model 45
Figure 3.2: Two Versions of White Flight 47
Figure 3.3: Different Curve Shapes based on Equation (3.3), given β and λ 49
Figure 3.4: White Population in Los Angeles County, 1940-1990 50
Figure 3.5: White Flight in Los Angeles County, 1960-1990 (L-M method) 63
Figure 4.1: Private Racial Preference in Schelling’s Spatial Proximity Model 74
Figure 4.2: An Illustration of Neighborhood in Schelling (1971) 76
Figure 4.3: A Sample Comparison between Two Simulations, N = 10 92
Figure 4.4: Comparing the Trends of ψ
t
between Two Simulations 93
Figure 4.5: Comparing the Trends of | I
t
| Between Two Simulations 93

x

Abstract
Contemporary urban spatial analysis arguably features an emphasis on spatial
structure, whereas understudying the background agent actions. This dissertation is
intended to address such oversight, by presenting three independent essays on the
spatial analysis of residential segregation by race
1
in urban America. Combining
the three essays, this dissertation demonstrates a series of methodological
considerations and innovations to rediscover agency in modern quantitative urban
spatial analysis. Aside from its methodological orientation, this dissertation
implicates a number of issues related to planning theory as well as public policy.

The first essay sets out by recognizing two types of variance generally associated
with the phenomenon of racial residential segregation. One type involves differing
racial compositions between spatial locations. The other is concerned with varying
residential spatial distributions between ethnic groups. The essay then presents a
spatial analytical approach to identify and measure the two types of variance
through a decomposition of an entropy index regarding racial diversity. A region’s
racial diversity entropy is found to comprise three factors: a) the overall spatial
distribution of regional population; b) the ratio between the number of ethnic

1
While 'race' as a conception per se is admittedly socially constructed (Lopez 1994), the first two
essays in this dissertation adopt the racial categorization as per the Los Angeles County Union
Census Data Series (Ethington et al. 2000). One caveat about the dataset is that the exact racial
definition might vary slightly between the different census years, even though this has been
accounted for by Ethington et al. (2000). Another disclaimer is that the data are collected at the
census tract rather than individual level. Thus the results of analysis should not be generalized to
individuals, lest the rise of ecological fallacy (Robinson, W 1950).
xi

groups and that of spatial areal units within the region, and, c) the differential
between two types of variance aforementioned. For demonstrative purpose census
data from Los Angeles County are studied using this approach. The results suggest
the second type of variance to be the primary contributor to the increasing racial
diversity in Los Angeles. Implications regarding affordable homeownership and
inclusionary housing policies are discussed accordingly.

The second essay delivers a nonlinear econometric model about White flight in Los
Angeles, alluding to Thomas Schelling’s (1971) classic neighborhood tipping
model.
2
In this essay, the Schelling original is translated into a fuzzy set version
and tested against demographic census data in Los Angeles County from 1960 to
1990. Results of nonlinear least squares regressions indicate that the tipping point
has shifted from around 0.36 between 1960 and 1970 to 0.78 between 1980 and
1990. Regression results also suggest a constantly decreasing extent of White flight
in the census tract level. These findings confirm the existence of the fuzzy tipping
mechanism. They also reflect steady progression toward racially integrated urban
residential pattern in Los Angles County from 1960 to 1990.

2 Schelling (1971) actually contains two segregation models. The first is a spatial proximity model
or the so called checkerboard model of segregation between neighborhoods. The other is a bounded
neighborhood tipping model. The both models are revisited in this dissertation, with the tipping
model addressed in the second essay and the checkerboard model examined in the third essay.
xii

The third essay posits that segregation models often focus on private racial
preference but overlook the institutional context. The essay thus represents an
effort to move beyond such preference centricity. In the paper, an ideal Pigovian
regulatory intervention is emulated and added into Schelling’s (1971) spatial
proximity model of racial segregation, with an aim to preserve collective housing
welfare against the negative externalities induced by the moving of individual
agents. A large number of cellular automations generate some intriguing results.
A key discovery is that the Pigovian regulation tends to render less efficient
whereas more ethnically integrated residential patterns than laissez faire. This
finding informs some current policy debates about the racial impacts of such
housing regulation as zoning, which, albeit complex in practice, is arguably
Pigovian by economic nature. On top of its policy implications, this paper
demonstrates a micro simulation approach to reconcile the preference-based and
institution-orientated intellectual perspectives regarding racial residential
segregation.
1

Chapter 1: Prologue
This dissertation contains three independent essays on the spatial analysis of
residential segregation by race within an American context. Notwithstanding the
essays’ different contents they address one common question: How to reconcile
structure and agency in urban spatial analysis? This methodological question, first
of all, arises from the social theories regarding the dialectical relation between
structure and agency:
... it is in fact a problem which is the concern of all forms of social thought
ranging from sociology to theology. 'Structure and agency' essentially deals
with the relationship between two opposing tendencies: 'structure' can be
thought of in a rough hand way as a system of rules, 'agency' as the extent to
which people are, or are not, able to act within a given structure. (Burke et al.
2000, p. 13)

Much of the above quote can be traced back to the works by Anthony Giddens
(1979, 1984, 1990). Giddens refuted both the structural-functionalist view of
individual as passive entity and the phenomenological understanding that personal
free will can always surpass societal constraints. Instead, Giddens argued that
there is a recursive “structuration” process, which involves the repeated personal
practice of reflexive action, whereby an individual negotiates between his or her
intention and the restraining social structures.

In the literature of urban theory, similar points of view can be found, for instance,
in Storper’s (1997) characterization of city as the locus of reflexive actions or
2

Schön’s (1983) depiction of planners as “reflective practitioners”. Furthermore
there is an added interest in the spatial manifestation of urban social structure and
its dialectic connection with human agency. A good example could be the seminal
writings by Jane Jacobs (1961, 1979), who called for proactive agent interventions
in urban spatial structure to achieve more desirable local social outcomes.

Prevailing as the theoretical notion regarding structure and agency, the spatial
analytical methods that most urban planners employ are arguably structure-centric;
much less attention has been paid to agency. For example, there are a variety of
descriptive spatial indices widely used for spatial analysis, such as spatial entropy
on geographic dispersion (Batty 1976), Moran’s I on spatial autocorrelation
(Odland 1988), or LISA as the local indicator of spatial association (Anselin 1995).
However a common limitation of these indices is an utter focus on spatial structure:
They only measure prima facie spatial patterns but say little about the background
agent actions, even though the two factors actually interact with each other as per
the aforementioned social and urban theories.

In contrast to descriptive spatial indices, spatial econometrics poses as a more
sophisticated inferential statistical approach. By translating each observation’s
relative location into a spatial weight matrix, spatial econometricians are able to
account for the effects of spatial interdependence and heterogeneity which are
otherwise almost ignored in regular regression (Anselin 1988). However, the
3

major pitfall with spatial econometrics is its often reduced functional form. This
entails a potential risk of misrepresenting the underlying interaction between
human agency and spatial structure. Indeed, with spatial econometrics, any kinds
of spatial relation (e.g., first or second order, queen or rook contiguity) can be
tested and used to explain data, even if a regression equation contains no variable
specifically capturing the location behavior of individual agents!

Compared with the aforementioned data-based approaches, new urban economics
is less empirical and yet better informed by the mainstream modern economic
theories. New urban economics is “new”, because it is more focused on the spatial
dimension than classic urban economics (Richardson 1977; Richardson et al.
1996). A new urban economic model often builds upon an assumed private utility
function, which is later solved through constrained optimization or similar calculus
protocols, and finally ends up with a one shot identification of spatial equilibrium
conditions. However, like its successor, new urban economics is overly drawn to
the equilibrium spatial structure but understudies the dynamic spatial transition
process wherein agency and structure mutually interact under disequilibrium
conditions. While a single utility function suffices to derive the eventual
equilibrium spatial structure, reflexive agent is largely missing in new urban
economics.

4

The recent paradigm of agent-based micro simulation seems to finally bring agency
to the fore (Batty and Torrens 2001). Agent-based modelers now can simulate
emergent agent behaviors such as learning and bargaining in the face of perceived
spatial structures, which in turn are transformable by agents (e.g., Webster and Wu
2001). Agent-based as its name suggests, this type of spatial models does involve
the specification of spatial as well as social structures that constrain the behavior of
agents, while leaving room for reflexive individual actions. However, quite many
agent-based models lack systematic measure of simulation results (e.g., using
descriptive spatial indicator) which are thus barely comparable with empirical data.
Even more agent-based models fail to test whether the simulation results are
statistically generalizable, in which respect spatial econometrics clearly has a
comparative advantage. Perhaps a most obvious weakness with many agent-based
models is the often arbitrary assumptions about agent behavior, posing a stark
comparison with the rigorousness of new urban economics (Heikkila and Wang
2009).

The three essays included in this dissertation are intended to address the
aforementioned methodological limitations in contemporary urban spatial analysis.
For this purpose the phenomenon of racial residential segregation is studied from
three distinct spatial analytical perspectives. The following chapter 2 presents an
essay which illustrates a decomposition of a descriptive entropy index regarding
racial diversity. The decomposition identifies two types of measurable variance,
5

one regarding the differing racial compositions between locations and the other on
the varying residential spatial patterns between ethnicities. This essay
demonstrates that spatial structure and agent behavior (corresponding to racial
identity in this case) are mutually involved, albeit analytically decomposable.

The essay in chapter 3 delivers a nonlinear econometric model about White flight
in Los Angeles County, based on a set of local census data spanning three decades
from 1960 to 1990. The regression equation is made up of two parts. The first
part centers on the tipping point beyond which a White resident would decide to
leave the local neighborhood. The second part emulates the former’s choice of
destiny location using a spatial gravity model. This essay explains how White
flight arises from a certain mode of agent behavior and results in the observed
spatial structure of racial segregation in the reality.

Chapter 4 contains the third and last essay, which builds upon the results of an
agent-based cellular automation. The paper revisits Thomas Schelling’s (1971)
classic spatial proximity model of racial segregation and examines the consequence
of adding an ideal Pigovian regulatory intervention into the Schelling original. Not
only is Moran’s I applied to measure the spatial outcomes of simulation, but an
inferential statistical test is also conducted upon the automation results. This essay
showcases an agent-based simulation informed by some classic microeconomic
concepts and validated within a standard inferential statistical framework.
6

Chapter 2:
Decomposing the Entropy Index of Racial Diversity:
In Search of Two Types of Variance

In effect, there is no place without self and no self without place.
--Edward Casey (2001, p. 684 in "Between Geography and Philosophy")

2.1 Introduction
Space is the physical facet of place while race is the ethnic aspect of self identity.
As identity and place are essentially commingled, race and space are also two
mutually involved spheres of urban life. For example, racial composition usually
varies between spatial locations. So does the spatial distribution of people between
racial groups. These two types of variance, which may be denoted respectively as
type I and type II variance, are not exactly the same thing, albeit related. The
conceptual difference is perceivable in the seminal work by Massey and Denton
(1988, 1989), on the five dimensions of racial residential segregation. Arguably
their definitions of “evenness” and “exposure” are more related to the type I
variance, while “concentration”, “centralization” and “clustering” are more
associated with type II.

7

Massey and Denton (1988, 1989) restricted their inquiry within the
“majority/minority” dichotomy. In contrast the actual US urban demography has
increasingly featured a multiethnic pattern, especially in such large metropolitan
areas as New York and Los Angeles (Clark 1992; Logan et al. 2002; Myers 2007;
Rearden et al. 2009; Wilkes and Iceland 2004). For its unique mathematical
properties,
3
the entropy index of racial diversity, or H index, has become
particularly popular among researchers who want to address the contemporary
multiethnic scenario (Rearden and Firebaugh 2002; Reardon and O´Sullivan 2004;
Wong 2003a, b).

One noteworthy methodological piece is by Wong (2003b), which involves a
decomposition of a regional H index. Wong (2003b) suggests that a regional H
index can always be divided into a weighted sum of local H indices plus a residual
which captures the “additional regional diversity… incurred onto the local units
through the spatial aggregation or smoothing process”. Wong’s (2003b) approach
is inspiring, for it sheds light on the important yet implicit spatial dimension of
racial entropy. However Wong (2003) is definitely not an endpoint; it marks a
starting point from where we can further spell out the spatial-racial interaction
underpinning the observed pattern of racial residential segregation. This paper
represents an effort in this direction, mainly alluding to the recent publication by

3 Unlike the more classic segregation measures such as the dissimilarity or isolation index, the
entropy index, in terms of its functional form (as per equation (2.1) below), imposes no limit on the
number of ethnicities involved (see Reardon and Firebaugh, 2002; Reardon and O’Sullivan, 2004).
8

Heikkila and Hu (2006), regarding an adjustment of spatial entropy. To
distinguish, by notation, between the racial and spatial entropy, E index stands
hereafter for the latter while H denotes the former.

In a nutshell this paper delivers a spatial analytical approach to decompose H, the
entropy index of racial diversity. The general idea is to associate a regional H
index (denoted as H
r
) with a regional E index (denoted as E
r
) by recognizing that
both, after corresponding adjustment, can describe the exactly same space-by-race
or race-by-space pattern of demographic distribution. This readily suggests a
possibility of decomposing H
r
based on the information carried in E
r
and thereby
identifying the magnitude of two types of variance mentioned above. In this
paper, after according mathematical manipulations, H
r
proves to comprise three
factors: a) the overall spatial distribution of regional population; b) the ratio
between the number of ethnic groups and that of spatial areal units, and, c) the
differential between the type II and type I variance.

Specific operations go through the following steps. The first step is to open up the
internal geography of an urban region and thereby obtain a number of local areal
units, such as census tracts. The next step is to calculate two regional entropy
indices, H
r
and E
r
, with the former adjusted by the number of spatial units while
the latter by the number of racial groups. The two adjustments produce two
distinctive indicators, one measuring type I variance and the other gauging type II.
9

A decomposition of H
r
is exercised in the final step, revealing that the overall racial
diversity of an urban region critically depends on the spatial distribution of regional
population and on the differential between type II and type I variance. For
demonstrative purpose, a set of historical census data regarding Los Angeles
County are analyzed in this study. The results implicate the persistence of
localized segregation within the region.

Apart from its theoretical and methodological significance, the approach presented
here may also improve our understanding regarding policy issues related to racial
residential segregation. For example, the Community Reinvestment Act and the
associated legislations (collectively referred to as CRA hereafter) are claimed to
curb racial residential segregation by improving the minorities’ access to the
mortgage market (Squires and O'Connor 2001). Viewed in the lens of this paper,
CRA is however found to affect the type II variance only, but not type I, given the
policy design.

The remainder of this paper is structured as follows. The next section 2.2 reviews
literature on the different dimensions of racial residential segregation in the US
urban context, followed by a scrutiny over Wong’s (2003b) decomposition of the H
index. A proposal of improving Wong’s (2003b) method, by linking it to the
analysis of spatial entropy, is brought up accordingly. Also included in the review
section is a canonical mapping of the use of entropy in both the field of spatial
10

analysis and that of segregation studies. The method section 2.3 elaborates an
analytical decomposition of the H index and thus comprises the core substance of
this paper. The penultimate section 2.4 reports and discusses the results of
applying the above method upon data from Los Angeles. The last section 2.5 sums
up conclusions and speculates on future research.

2.2 Racial Residential Segregation and Entropy
2.2.1 Two Types of Variance

Massey and Denton (1993), in their seminal book, “American Apartheid”,
suggested that racial segregation in urban US tended to exhibit a general residential
spatial pattern, with African Americans in the central city and Whites in the
suburbs. Massy and Denton (1989, 1993) termed this kind of spatialized
segregation pattern “hypersegregation”. They further pointed out five dimensions
of measuring hypersegregation: evenness, exposure, concentration, centralization
and clustering:
…groups may live apart from one another and be ‘segregated’ in a variety of
ways. Minority members may be distributed so that they are overrepresented
in some areas and underrepresented in others, varying on the characteristic of
evenness. They may be distributed so that their exposure to majority members
is limited by virtue of rarely sharing a neighborhood with them. They may be
spatially concentrated within a very small area, occupying less physical space
than majority members. They may be spatially centralized, congregating
around the urban core, and occupying a more central location than the
majority. Finally, areas of minority settlement may be tightly clustered to
form one large contiguous enclave, or be scattered widely around the urban
area [all emphases in original]. (Massey and Denton 1988, p. 283)

11

Though claiming the conceptual difference, Massey and Denton (1988) did
acknowledge the empirical correlations between evenness and exposure (p.287),
and, among concentration, centralization and clustering (p.293). Both the degrees
of evenness and exposure essentially depend on the variance in local racial
compositions between different locations or spatial units. This kind of variance is
thus race-by-space wise. Another way to understand this is to envision n spatial
observations of local racial composition. Large racial variance across the n spatial
observations signal unevenness and tends to result in the minority’s low level of
exposure to the majority.
4

In contrast, concentration, centralization and clustering are all about variance in the
spatial distributions of residents between different racial groups. This type of
variance is thus space-by-race wise. One may also conceive of m observations of
residential spatial pattern, each per racial group, given m racial groups. Large
deviation across the m observations points to the fact that minority may follow a
very different residential pattern than majority, for example, more or less
concentrated, clustered or centralized. The table 2.1 below accordingly
summarizes the two distinct types of variance aforementioned.

4
Unevenness is however not always simultaneous with low exposure. For example, if an ethnic
minority happens to have a relatively large population size in an urban region, its exposure to the
ethnic majority is alleviated correspondingly, regardless of the spatial pattern of minority residents
(Blau, 1977, cited in Massey and Denton, 1988, p287).
12

Table 2.1: Two Types of Variance in Racial Residential Segregation
Type of variance
Character of
variance
Units of observation
Concerned
dimensions of racial
residential
segregation
I:
in racial
composition between
spatial units

race-by-
space

areal unit (e.g.,
census tract)

evenness;
exposure
II:
in spatial
distribution between
racial groups

space-by-
race

ethnic group (e.g.,
Asian/Black/Latino/Whi
te)

concentration;
centralization;
clustering

2.2.2 The H Index

A comprehensive summary and a now-classic piece, Massey and Denton (1988)
had exclusively focused on a dichotomous majority-minority scenario, as in many
other publications on similar topics at that time (Krieger 1971, 1991; Schelling
1971). This constraint can be conceivably attributed to the empirical White-Black
dichotomy which had characterized the US urban demography until the large scale
influx of Latino and Asian immigrants since the late 1970s. The contemporary US
urban demography features a promising trend towards the diversification of ethnic
groups, especially in such large metropolitan areas as New York and Los Angeles
(Clark 1992; Logan et al. 2002; Myers 2007; Rearden et al. 2009; Wilkes and
Iceland 2004). This actual transition has accordingly called for metrics that can be
used to assess racial residential segregation in a multiethnic context. One of the
indices that have earned popularity is the entropy index of racial diversity, or
13

namely, H index (Rearden and Firebaugh 2002; Reardon and O´Sullivan 2004;
Wong 2003a, b).

Entropy as a mathematical construct was firstly introduced into social science by
Theil (1967, 1972). In the field of segregation studies entropy has been found
particularly suitable to measure racial diversity in a multiethnic context (Rearden
and Firebaugh 2002; White 1986). The basic mathematical form of racial diversity
entropy, or the H index, can be expressed as follows:

k k
p p H
m
k
log *
1
∑
=
- =
(2.1)

where p
k
(0 ≤ p
k
≤ 1) stands for the share of aggregate population that belongs to the
k
th
of m racial groups, or formally,

t k
m
k
k k k
p Ρ Ρ = Ρ Ρ =
∑
=
/ /
1
(2.2)

where P
k
is the total count of people within the k
th
racial group and P
t
is the
aggregate regional population. H in equation (2.1) reaches its maximum when H =
log (m), if p
1
= p
2
= … = p
m
, suggesting a purely even population distribution by
racial groups. In reverse, H reaches its minimum, when H = 0, if p
k
= 1 (for a k of
certain value), corresponding to an extreme racial homogeneity.
14

Equation (2.1) can be applied at multiple spatial scales, for instance, at both the
regional and local levels. Massey and Denton (1988, p. 285) suggested a way of
comparing a regional H index against a local H index. Wong (2003b, p. 186) later
extends their approach into a multiracial context. Specifically, let ∆H stand for the
difference between a regional H index, denoted as H
r
, and a local H index, denoted
as H
i
, and thus Wong (2003b) submits,

] ) ( * [ *
1 1
∑ ∑
= =
- = - = Δ
n
i
n
i
i r i i i r
H H q H q H H
(2.3)

where q
i
(0 ≤ q
i
≤ 1) represents the share of regional population in the i
th
of n local
areal units, or formally,

t i
n
i
i i
i
q Ρ Ρ = Ρ Ρ =
∑
=
/ /
1
(2.4)

where P
i
is the total count of people within the i
th
local areal unit and P
t
is still the
aggregate regional population as in equation (2.2). Given equation (2.3), Wong
(2003) suggests a way of decomposing a region’s racial diversity:

∑
+ Δ =
n
i
i i r
H q H H *
(2.5)

15

∆H in equation (2.5), as per Wong (2003b), captures a “spatial aggregation effect”
whereby “local differences are smoothed when data of smaller areal units are
spatially aggregated to larger units, and the aggregated data can only report the
more evenly distributed population mixes at the regional level” (p. 186).

Equations (2.3) to (2.5) contain a couple of potential issues. First of all, the
inclusion of q
i
in equation (2.3) is ad hoc, only justifiable from a certain intuitive
perspective. According to Wong (2003b, p. 185), “…averaging the local diversity
values will not take into account the differences in population sizes among local
areal units…A more appropriate diversity value for the entire region is based upon
population weighted local diversity values”. However, equation (2.3) may also be
written in the following manner, given equation
(2.1):

∑∑
= =
- = Δ
n
i
m
k
k k
k i k i i
p p p p q H
1 1
)] log * log * ( * [
, ,
(2.6)

where p
i,k
(0 ≤ p
i,k
≤ 1) stands for the k
th
racial category’s share of local population
within the i
th
areal unit, or formally,

i k i
m
k
k i k i k i
p Ρ Ρ = Ρ Ρ =
∑
=
/ /
,
1
, , ,
(2.7)

16

where P
i,k
is the total count of people within the i
th
local areal unit that can be
grouped into the k
th
racial category and P
i
follows the definition in the equation
(2.4) above. Note that slightly adapting equation (2.7) gives further the i
th
areal
unit’s share of regional population within the k
th
ethnic group, or formally,

k k i k i k i k i
n
i
S Ρ Ρ = Ρ Ρ =
∑
=
/ /
, , , ,
1
(2.8)

Mathematically identical with equation (2.3), equation (2.6) now elicits a
comparison between racial groups, while equation (2.3) centers on the difference
between spatial scales. Note that the existence of q
i
in equation (2.6) becomes
barely justifiable at this time. For the same logic Wong (2003b, p. 185) relies on, if
a weight is to be placed in equation (2.6), it should be the local share of regional
population that belongs to a certain racial group, q
i,k
, rather than the local share of
aggregate regional population, q
i
. The former differs from the latter, because the
equation (2.9) below is clearly not the same as the equation (2.4) above.

t k i
n
i
m
k
k i k i
k i
q Ρ Ρ = Ρ Ρ =
∑∑
= =
/ /
,
1 1
, ,
,
(2.9)

where P
i,k
follows the definition in the equation (2.7) above and P
t
is the aggregate
regional population. Given the difference between equation (2.4) and (2.9),

17

] ) log * log * ( * [ )] log * log * ( * [
, , , , , ∑∑ ∑∑
- ≠ - n
i
m
k
n
i
m
k
k k k i k i k i k k k i k i i
p p p p q p p p p q
(2.10)

unless under two rare conditions, that either p
i,1
= p
i,2
=…= p
i,m
= 1/m, or, q
i
= 0
) ,..., 1 ( n i ∈ ∀ . The general non-equality shown in the equation (2.10) above hence
suggests that q
i
as a choice of weight in equation (2.3) is only justifiable from a
certain perspective. In other words q
i
is an ad hoc parameter.

Second, because q
i
is ad hoc, the magnitude of spatial aggregation effect, measured
with ∆H in equation (2.3) and (2.5), also becomes unstable. Moreover, ∆H is very
much a black box. One can hardly tell how much of the variance in ∆H is
contributed to by q
i
and how much by (H
r
- H
i
), when the two terms actually
interact with each other in equation (2.3). In this vein, while Wong (2003b)
correctly recognizes that “local differences are smoothed [in the aggregate scale]”,
the equation (2.3) above does not help us further clarify the mechanism behind the
spatial aggregation effect.

2.2.3 The E Index

While entropy is applied more and more frequently as an important indicator of
racial diversity, it has claimed its presence in spatial analysis for several decades.
Batty (1976) and Batten(1983) were those of the earliest suggesting that Theil’s
18

(1967, 1972) entropy can also be employed to measure spatial patterns. For
instance, in assessing the spatial distribution of population,

i
n
i
i
q q E log *
1
∑
=
- =
(2.11)

where q
i
follows its definition in equation (2.4) and E stands for spatial entropy,
which reflects the overall degree of population’s geographic dispersion within a
region. Analogous to H in equation (2.1), E in equation (2.11) is maximized when
E = log (n), if there is a purely even spatial distribution of population. On the other
hand, E reaches its minimum when E = 0, if all of the population agglomerates
within a single local areal unit.

Batty (1976) further pointed out that, if q
i
is always evenly distributed within the i
th

areal unit for all i, the following equation holds:

) / log( *
1
i i
n
i
i
x q q B Δ - =
∑
=
(2.12)

where ∆x
i
stands for the size or area of the i
th
unit, and, where B , according to
Batty (1976) is a spatial entropy index. Recently, Heikkila and Hu (2006, pp. 852-
853), based on Theil’s (1967, 1972) classic entropy decomposition algorithm,
suggest B to be a special case of E. In fact, E can always be transformed
19

accordingly to B, as far as the internal distribution of q
i
is specified. In making this
point, Heikkila and Hu (2006, p. 852) argue that E per se is spatial entropy.

Heikkila and Hu (2006) moved on to demonstrate an adjustment of spatial entropy.
An important concept arising through their paper is “extensive replication”. It
describes a scenario, wherein a pattern of spatial distribution across n spatial units
is exactly replicated for v

times
5
and the number of areal categories accordingly
increases to v*n. Under the condition of extensive replication, Heikkila and Hu
(2006, p. 851) indicate,

) log(
) ( ) * (
v E E
n n v
+ = (2.13)

where E
(n)
is equivalent to E in equation (2.11) and E
(v*n)
is the entropy of a new
urban region that is a v-tuple replica of the original. As to be illustrated in the next
section, this idea of “extensive replication” is readily transferable to the case of
racial residential segregation. However, for now, the figure 2.1 below perhaps best
summarizes the research output that has been reviewed above. With a conceivable
likelihood of missing important literature, figure 1 delivers a speculation that

5
Whereas Heikkila and Hu (2006) use k
1
, v is adopted here to avoid notational confusion, since k is
already used beforehand for another meaning. Detailed proof of equation (2.13) can be found in the
appendix to Heikkila and Hu (2006) and thus not reiterated here.
20

progress in the field of spatial analysis may continue to inform ongoing research
regarding the use of entropy in measuring racial residential segregation.
6

Figure 2.1: Entropy in Spatial Analysis and Segregation Research

Note:
: field of spatial analysis
: field of segregation research

6
Apparicio et al. (2008) presents a detailed review of spatial analysis’ intellectual contributions,
including but not limited to the application of entropy index, to the study of racial segregation.
21

2.3 A Decomposition of the H Index
2.3.1 A Region at Two Levels of Granularity

The first step of this method is to partition an urban region into a number of local
areal units, for instance, census tracts. This very first step is illustrated with the
table 2.2 and 2.3 below. A major difference between table 2.2 and 2.3 is that the
latter contains a shaded column which enumerates all of the local areal units within
the region. However, note that the both tables are describing a same region, simply
at two levels of granularity.

Table 2.2: Regional Racial Composition (Racial Groups in Column)
race
space
1
st

group
2
nd
group
… (m-1)
th
group
m
th

group
region p
1
p
2
… p
(m-1)
p
m

Table 2.3: Racial Composition by Local Spatial Units
race
space
1
st
group 2
nd
group … (m-1)
th

group
m
th
group
1
st
unit q
1,1
q
1,2
… q
1,(m-1)
q
1,m

2
nd
unit q
2,1
q
2,2
… q
2,(m-1)
q
2,m

.
.
.
.
.
.
.
.
.
…
…
…
.
.
.
.
.
.
(n-1)
th
unit q
(n-1),1
q
(n-1),2
… q
(n-1),(m-1)
q
(n-1),m

n
th
unit q
n,1
q
n,2
… q
n,(m-1)
q
n,m

22

Substantial caution should be exercised regarding the entries into table 2.3. While
the aforementioned p
k
can be accordingly filled into table 2, because 1
1
=
∑
=
m
k
k
p as
per equation (2.2), it is incorrect to enter p
i,k
into table 3 for 1
1
,
1
≠
∑∑
= =
n
i
k i
m
k
p . Table
2.3 now describes the allocation of aggregate regional population into n*m
categories. Because q
i,k
in equation (2.9) denotes the share of regional population
in the i
th
areal unit and meanwhile also in the k
th
racial group, only q
i,k
can be filled
into table 2.3 in order to make the aggregate of all entries equal to one, or
mathematically, 1
1
,
1
=
∑∑
= =
n
i
k i
m
k
q .

2.3.2 A Spatial Adjustment of Racial Entropy

Since H
r
in the review section measures the regional racial diversity, let H
r (m)

denote regional racial diversity across m ethnic groups. Another way to understand
this is by observing that table 2.2 has m cells in total for its 1-by-m dimension.
Thus H
r (m)
= H
r (1* m).
Now assume that racial composition is identical across the n
local areal units, or put in another way, type I variance is zero. In that case, racial
diversity within any local areal unit must be exactly the same as the overall
regional counterpart. Table 2.3 thus contains n extensive replications of local
racial composition, each of which equals exactly to H
r (m)
. Notice that now the
number of cells in table 2.3 has increased to n*m. The pattern of distribution
23

within table 2.3 can thus be characterized using H
r (n*m)
, by directly adjusting H
r (m)

as per equation (2.13) in the previous section:

) log(
) ( ) * (
n H H
m r m n r
+ = , if ) ,..., 1 (
,
n i p p
k k i
∈ ∀ = (2.14)

Although equation (2.13) in the review section adjusts spatial entropy while
equation (2.14) here deals with racial entropy, there is no de facto difference.
Indeed, as Heikkila and Hu (2006, p. 852) have deliberately emphasized, “there is
in fact nothing intrinsically spatial about [spatial entropy]”. Thus if spatial entropy
can be adjusted for the extensive replication effect, there is no reason why racial
entropy cannot.

Because n in equation (2.14) denotes the number of spatial units within the region,
equation (2.14) may be considered a spatial scale adjustment of racial entropy, H
r
(m)
. Note that p
i,k
, rather than q
i,k
, should appear in equation (2.14), since this time
our interest lies in the racial proportions of local population, which must add up to
unity within each local areal unit. This point may be further clarified with the
following mathematical expression:

∑ ∑
≠ =
m
k
k i
m
k
k i
q p
, ,
1 (2.15)

24

Nevertheless, equation (2.14) is based on a null assumption of type I variance.
When however type I variance does exist, equation (2.14) no longer holds and must
be rewritten as in the following form:

φ + + = ) log(
) ( ) * (
n H H
m r m n r
(2.16)

where φ may be seen as a residual due to the existence of type I variance and thus φ
simply equals to zero in the equation (2.14) above. In this sense equation (2.16) is
actually a general form of equation (2.14). Also note that the absolute value of φ,
or |φ|, now directly measures the overall magnitude of type I variance, given the
possibility that φ might be negative.

2.3.3 A Racial Adjustment of Spatial Entropy

There is an analogous way to adjust the spatial entropy index. Notice that just as
table 2.2 can be extended to table 2.3, table 2.4 below may also follow suit, ending
up with simply a transposed version of table 2.3, as per table 2.5, which illustrates
the spatial distribution of regional population by racial groups.
However, while table 2.2 is extended to a higher level of geographic granularity,
table 2.4 is now adjusted along the racial dimension.

25

Table 2.4: Spatial Distribution of Regional Population (Areal Units in Column)
space
race
1
st
unit 2
nd
unit … (n-1)
th
unit n
th
unit
people q
1
q
2
… q
(n-1)
q
n

Table 2.5: Spatial Distribution of Regional Population by Racial Groups
space
race
1
st
unit 2
nd
unit … (n-1)
th

unit
n
th
unit
1
st
group q
1,1
q
2,1
… q
(n-1),1
q
n,1

2
nd
group q
1,2
q
2,2
… q
(n-1),2
q
n,2

.
.
.
.
.
.
.
.
.
…
…
…
.
.
.
.
.
.
(m-1)
th
group q
1,(m-1)
q
2,(m-1)
… q
(n-1),(m-1)
q
n,(m-1)

m
th
group q
1,m
q
2,m
… q(
n-1),m
q
n,m

Now assume type II variance to be zero and thus the population’s spatial
distribution is exactly identical between all of the racial groups. In that
case, ) ,..., 1 (
) (
m k E E
n r k
∈ ∀ = and there arises m extensive replicas of the spatial
pattern indicated in table 4. Analogous to the spatial adjustment of racial entropy
in equation (2.14), a racial adjustment of spatial entropy may also be operated as
follows:

) log(
) ( ) * (
m E E
n r n m r
+ = , if ) ,..., 1 (
,
m k q s
i k i
∈ ∀ = (2.17)

26

A noteworthy element in equation (2.17) is s
i,k
, which is aforementioned in
equation (2.8) and which differs both from p
i,k
and q
i,k
by definition. As indicated
above, s
i,k
stands for the i
th
areal unit’s share of regional population that belongs to
the k
th
racial category, and,

∑ ∑ ∑
≠ ≠ =
n
i
k i
n
i
k i
n
i
k i
p q s
, , ,
1 (2.18)

Because E
r (n)
in equation (2.17) is adjusted by m, the number of racial categories,
equation (2.17) can be considered a racial adjustment of spatial entropy, E
r (n)
. As
before, it is possible to write a general form of equation (2.17), lest type II variance
is actually not zero. In that case,

η + + = ) log(
) ( ) * (
m E E
n r m n r
(2.19)

Analogous to φ, η serves as a residual if there is a certain degree of type II
variance. And for the same logic as in the case of |φ|, |η| measures the overall
magnitude of type II variance.

2.3.4 Three Factors in the Entropy Index of Racial Diversity

There is no actual difference between H
r (n*m)
and E
r (m*n)
, as both measure the
distribution of aggregate regional population across n*m cells. Because H
r (n*m)
=
27

E
r (m*n)
, equation (2.16) and (2.19) may be directly associated in the following
manner:

η φ + + = + + ) log( ) log(
) ( ) (
m E n H
n r m r
(2.20)

Equation (2.20) immediately allows expressing H
r(m)
in an additive fashion:

) ( ) / log(
) ( ) (
φ η - + + = n m E H
n r m r
(2.21)

Equation (2.21) is the key point made in this study, showing a decomposition of
regional H index. Equation (2.21) has several important properties. First, it
attributes a region’s racial diversity to three factors. One is the overall spatial
distribution of regional population. This is a non-racial factor. Another is the ratio
between the number of ethnic groups and that of spatial units. This factor is clearly
as per external data standard and should be fixed given a single dataset. The last
factor is the differential between type II and type I variance. This factor has a
general pertinence to racial residential segregation, as shown in table 2.1 in the
review section.

Second, there is an alternative way to interpret the meaning of equation (2.21).
While in this study racial entropy is decomposed into spatial entropy, one may well
28

do the reverse by easily adapting the same equation. Indeed, this reversibility is
because race and space are inseparable as illustrated with the two types of
variances in table 2.1 and as implied with the very quote at the paper’s outset.

Third, even though seemingly so, equation (2.21) does not serve as a fair basis for
comparing racial diversity between regions that differ in n or m. Note that log (n)
is endogenous to φ, given equation (2.16). So is log (m) to η, as per equation
(2.19). Therefore, even if one adjusts two regional H indices by respectively
deducting log (m/n), the two H indices are still incomparable. n’s interference with
measurement has been well recognized as partly
7
a manifestation of “modifiable
areal unit problem” (MAUP) in assessing racial residential segregation (Rearden et
al. 2009; Reardon and O´Sullivan 2004; Wong 2003a, b). In contrast, m’s
analogous impact, namely a “modifiable racial unit problem”, is less frequently
mentioned in the literature other than in Voas and Williamson (2000) and Mateo et
al. (2009). Analysis above however suggests that the both are important aspects of
measuring racial segregation.

7
According to Openshaw (1984), MAUP contains two subordinate problems. One is the problem
of scale and resolution, concerning inconsistent measurement due to data acquired upon the same
geographic space but at location units of different sizes. The other is the problem of zoning,
regarding the dependence of measurement upon the geometric shape of observation’s location unit,
even if the total number of location units is constant upon the study space.. The interference from n
is more related to the first sub-problem.
29

Last, equation (2.21) differs from equation (2.5) (Wong 2003b) both in implication
and in style. In terms of implication, equation (2.5) attributes regional racial
diversity to local racial diversity plus a residual which captures the “additional
regional diversity… incurred onto the local units through the spatial aggregation or
smoothing process” (Wong, 2003b, p.186). Therefore Wong (2003b) claims
equation (2.5) to be a spatial decomposition of H index. However, equation (2.21)
points out that race also plays an important role in the game. Notwithstanding this
race-space interaction, type I and type II variance are clearly detached in equation
(2.21), while remaining confounded in equation (2.5). In terms of style, equation
(2.5) is an ad hoc decomposition as aforementioned. In contrast, equation (2.21)
follows a purely analytical approach which allows for general applications. To
demonstrate these distinctions, the next section applies equation (2.21) upon a set
of historical census data from Los Angeles County.

30

2.4 The Case of Los Angeles County, 1960-1990
For demonstrative purpose, Los Angeles County Union Census Data Series,
version 1.01 (Ethington et al. 2000) are studied here, using the method presented
above. This is a historical-spatial panel dataset that contains basic demographic
information in 1,641
8
census tracts between the census years of 1940 and 1990.
The dataset classifies population into four ethnic groups: White non-Hispanic,
Black non-Hispanic, other non-Hispanic, and Hispanic. Therefore, m = 4 and n =
1,641 in the context of this study. In terms of time span, this inquiry is specifically
focused on the post civil rights movement era between 1960 and 1990.

2.4.1 Basic Application

Digital mapping in ArcGIS allows an initial investigation over type I and type II
variance. For example, figure 2.2 below maps the spatial distribution of local H
indices measured at the census tract level in 1990. With an average of 0.36, Los
Angeles County appears to have a modest overall level of racial diversity in 1990,
since the maximum of H is log (4) or about 0.6 in this case. There is also a
standard deviation of approximately 0.12, suggesting a considerable extent of
variation in these local H indices, or, type I variance.

8
Actually the data set contains information regarding 1,656 census tracts, 15 of which however
involve unusable demographic data for this study.
31

Figure 2.2: Mapping H Index in the Census Tracts of LA County, 1990

32

Based on the same data in 1990, it is also possible to map the population
distribution respectively for each of the four racial groups. Figure 2.3 below tends
to suggest a presence of type II variance, with Blacks densely congregating in the
south central LA County, posing a contrast to the dispersion of Whites across the
peripheral areas. The remaining two racial groups generally cluster in the eastern
side of LA County, mainly along the San Gabriel valley.

Figure 2.2 and 2.3 are both visually informative. However, it is hard to directly
associate them heuristically. Yet a statistical approach is readily available as per
equation (2.21). The table 2.6 further below shows the results of decomposing
regional H index for Los Angeles County in each census year between 1960 and
1990. Note that the fourth column in table (2.6) directly illustrates the differential
between η and φ, whereas their respective values are independently suggested in
the fifth and sixth column. For a purpose of comparison, the last column gives the
measure of spatial aggregation effect, or, ∆H, as indicated by Wong (2003b).
33

Figure 2.3: Mapping Population Distribution in Los Angeles County, 1990

34

Table 2.6: A Decomposition of County Level H index, 1960-1990
Year H
r(m)
E
r(n)
η - φ η φ log
(m/n)
∆H
(Wong,
2003b)
1960 0.29 3.16 -0.25 -0.44 -0.19 -2.62 0.13
1970 0.40 3.18 -0.17 -0.37 -0.20 -2.62 0.17
1980 0.49 3.18 -0.07 -0.28 -0.21 -2.62 0.15
1990 0.53 3.17 -0.03 -0.24 -0.21 -2.62 0.17

2.4.2 A Longitudinal Comparison

Table 2.6 above delivers several important messages. First, the regional level
racial diversity, H
r(m)
, has been steadily increasing in Los Angeles County since
1960. This is consistent with the conventional perceptions and with findings
claimed in the mainstream literature (Myers 2007). Second, the overall spatial
distribution of population is almost constant in Los Angeles over the decades.
Third, while |η| has been constantly decreasing, there is however a slight but steady
increasing trend in |φ|, though the former changes much faster. In other words,
type II variance (i.e., in the residential spatial distributions between races) has been
quickly shrinking while type I variance (i.e., in the racial compositions between
spatial units) persists and has even expanded a little. Finally, it is difficult to relate
fluctuations in ∆H, as shown in the last column of table (6), with any of these
trends.

Based on table 2.6, table 2.7 below is focused on the longitudinal difference in the
regional H index of Los Angeles County. An historical shift in the county level H
index is now attributed to the according change in regional E index and in type II
35

and type I variance, since log (m/n) is controlled. Percentage in the parenthesis
indicates the contribution of respective factor. Notice that a decrease in the value
of φ however raises the value of H
r(m)
, as per equation (2.21). According to table
2.7, η is the main pushing force behind the mounting racial diversity of Los
Angeles County. Growth in η contributed to 70% and 80% of the increase in H
r(m),
respectively for the two decades between 1960 and 1980. The very number further
rose to an astounding 133% during 1980 and 1990, with 33% offset by decrease in
E
r(n)
.

Table 2.7: A Decomposition of Change in County Level H Index, 1960-1990
Decade H
r(m)
E
r(n)
η - φ η φ log (m/n)
1960-
1970
+0.10 +0.02
(+20%)
+0.08
(+80%)
+0.07
(+70%)
-0.01
(+10%)
0
(0%)
1970-
1980
+0.10 +0.01
(+10%)
+0.09
(+90%)
+0.08
(+80%)
-0.01
(+10%)
0
(0%)
1980-
1990
+0.03 -0.01
(-33%)
+0.04
(133%)
+0.04
(+133%)
0.00
(0%)
0
(0%)

Because spatial entropy is not conceptually related to race, it might be worthwhile
to specifically focus on the change in type I and type II variance within Los
Angeles County. Table 2.8 below shows shift in the value |η| and |φ|, which
respectively gauges the magnitude of type II and type I variance. Clearly type II
variance has been steadily and quickly shrinking since 1960, while type I variance
largely sustains, with a slight expansion during the first two decades.

36

Table 2.8: Change in Two Types of Variance, 1960-1990
Decade |η| ~ type II variance |φ| ~ type I variance
1960-1970
(1960 = 100%)
-0.07
(-16%)
-0.01
(+7%)
1970-1980
(1970 = 100%)
-0.08
(-22%)
-0.01
(+4%)
1980-1990
(1980 = 100%)
-0.04
(-15%)
0.00
(0%)

2.4.3 Policy Implications

The above case study of Los Angeles is mainly for the purpose of demonstrating a
method. However, the application also generates some relevant policy
implications. As the table 8 above reveals, the increasing racial diversity in Los
Angeles County is primarily because of convergence in residential spatial
distributions between the four racial groups; racial compositions between local
neighborhoods have however remained relatively divergent. In terms of intuition,
this may tell the following story: On one hand, ethnic minorities, who used to
concentrate and cluster in urban centers, have now relocated to low density
neighborhoods in the urban periphery and thus result in a more dispersed
residential spatial pattern than before. This hence discounts the magnitude of type
II variance. On the other hand, the new communities that minorities move into
indeed exhibit a similar degree of racial homogeneity to its counterpart in the
central city. That thus makes little difference in type I variance.

Numerous policy efforts have been made to address racial residential segregation.
However it is not always clear what kind of segregation or which dimension of
37

segregation a policy specifically aims to address. If the potential benefits of
implementing a policy cannot be clearly identified, there is no reason to expect the
cost of intervention to be accordingly offset. For example, the current subprime
mortgage crisis has been arguably attributed to the enforcement of Community
Reinvestment Act and the related legislations (collectively referred to as CRA
hereafter) (Bernanke 2007). At the cost of intervening in the housing market, CRA
impels federally insured banks to provide home purchasing loans to individuals
from low-income minority neighborhoods (Squires and O'Connor 2001). CRA is
claimed to curb racial residential segregation by improving the disadvantaged
minorities’ access to the mortgage market. However, in the sprit of this paper, one
may well ask which dimension of segregation CRA specially targets?

Providing easy credit clearly gives people more freedom to move, or in the
language of this paper, to reduce type II variance and to induce more even
residential spatial distributions between ethnic groups. However, there is no
logical reason to expect CRA to equalize racial compositions between local
neighborhoods. An African American homebuyer, for instance, may move from a
high density Black community to a low density Black neighborhood, simply
because of personal preference for racial homogeneity. While the moving clearly
causes change in type II variance, it may have no impact at all on type I.
Therefore, to judge whether it is worthwhile to enforce CRA critically depends on
38

which kind of variance CRA aims to reduce. If type II is the objective, perhaps
CRA has worked. Otherwise, CRA is a wrong choice.

2.5 Conclusions
A spatial analytical approach is delivered in this paper for a decomposition of racial
entropy index into three factors: a) the overall spatial distribution of regional
population; b) the ratio between the number of ethnic groups and that of spatial
areal units, and, c) the differential between two types of variances. The last factor
is most important, because these two types of variance are generally associated
with the phenomenon of racial residential segregation. Type I variance can be
found in the differing racial compositions between local areal units. Type II
variance is concerned with the uneven residential spatial distributions between
ethnic groups. These two types of variances are differentiated conceptually,
analytically and empirically in this paper, generating relevant policy implications,
based on a case study regarding Los Angeles County.

Probable future research lies in the following areas. One is to explore the
correlation between type I and type II variance, either through analytical or
empirical investigations. As suggested in the text, these two types of variance,
though different, are unlikely to be unrelated. It will thus be interesting to discover
and prove that intrinsic interconnection in the future research.

39

Second, while the issue of self segregation is brought up in the end of the case
study, there is admittedly little discussion in this paper about individual’s
neighborhood preference and their corresponding moving behavior, which however
directly shapes the observed pattern of racial residential segregation. Given the
same dataset used in this study, it is possible to develop certain hypotheses
regarding individual’s neighborhood preference and their relocation behavior.
Such hypotheses can then be tested against empirical data.

Third, figure 2.1 in the review section speculates on integration between the field
of spatial analysis and that of segregation studies. In-house digital mapping and
spatial statistical analysis based on geographic information system are a kind of
standard practice in the former field, while the latter traditionally involves an
abundance of onsite participatory methods, such as survey, focus group, and
ethnography. It will be interesting to combine these seemingly different methods
into an integrated research agenda, when proper.
40

Chapter 3:
White Flight in Los Angeles County, 1960-1990:
A Model of Fuzzy Tipping

3.1 Introduction
Notwithstanding geographical and historical variation, racial residential
segregation has been a persistent phenomenon in major US cities (Clark 1988b;
Massey and Denton 1993). Mainly viewed as a social problem, residential
segregation between different ethnicities has complex reasons and extensive
implications. Mathematical modeling provides a rigorous basis for understanding
racial residential segregation. Substantial efforts have been made in this area (Clark
1988a; Farley and Frey 1994; Farley et al. 1994; Fossett and Waren 2005; Krieger
1971; Logan et al. 2002a; Morgon 1983; Schelling 1971).

One of the classic segregation models is by Thomas Schelling (1971). In
Schelling’s hypothetical model, there is a critical point of minority immigration
into a White dominated neighborhood. Before that critical point, the inflow of non-
Whites does not affect the White natives. However, a catastrophic change occurs
immediately beyond that critical point and results in the full evacuation of White
natives. As the subject neighborhood tips from White to non-White, the critical
point where White flight occurs is called the tipping point. Thus Schelling’s model
41

is also called the tipping model (Krieger 1971). Through his tipping model,
Schelling suggested that even a modest extent of individual preference for racially
homogeneous neighborhood (i.e., White in Schelling’s model) may result in
complete residential segregation over time in the macro scale.

This paper delivers a nonlinear econometric test of Schelling’s tipping model,
using the Los Angeles County Union Census Tract Data Series (version 1.01)
(Ethington et al.2000). A fuzzy set function of right shoulder sigmoidal form is
selected for two purposes. One is to capture the unobserved complexities of
tipping behavior. The literature suggests that tipping may not be exclusively about
race. Rather, tipping may be partially induced by some race related factors such as
income disparity (Sethi and Somanathan 2004). Given that ethnicity is the only
variable controlled in this study, it is necessary to reinterpret Schelling’s original
tipping concept from a fuzzy set perspective. That is, the extent of White flight is
fuzzily determined by, or, a fuzzy membership function of, the proportion of non-
Whites in subject neighborhoods.

The second purpose of using the fuzzy set apparatus is to account for the
heterogeneity of functional form. The preference curve presented in Schelling
(1971)’s theoretical model has been found to exhibit several alternative shapes in
practice (Clark 1988a). This implies that Schelling’s tipping model, built upon his
assumption about preference structure, is likely to follow multiple functional forms
42

also. In addition, uncertainty about functional form is also a typical operational
issue in econometrics. As to be demonstrated below, a sigmoidal fuzzy set
function is modestly flexible to represent multiple functional forms that are
potentially explanatory regarding the tipping mechanism.

Regression results in this study suggest that tipping—in a fuzzy set version—exists
in Los Angeles County from 1960 to 1990. And the fuzzy tipping mechanism can
explain modest percentage (68.77 % -- 97.61%) of increase in White population in
each of the 1,649 census tracts during any one decade between 1960 and 1990.
The regression results also shed light on the value range of the tipping point and the
velocity of White flight both. The former may be considered a threshold measure
of acceptance to the composition of non-Whites in a local tract. The latter
measures the varying sensitivity of the White natives to the inflow of non-White
residents. Regression results suggest a steadily decreasing extent of White flight.
This finding is consistent with the mainstream segregation literature (e.g., Farley et
al.1994).

The remainder of this paper is organized as follows. Section 3.2 addresses the
segregation and tipping literature in brevity. Section 3.3 elaborates the fuzzy
tipping model as the kernel of this paper. Section 3.4 introduces an ad hoc gravity
model which aims at resolving spatial dependence and heterogeneity inherent to
43

the data used in this study. Section 3.5 presents and discusses the regression
results. Second 3.6 sums up the conclusions.

3.2 Segregation as Neighborhood Preference
Thomas Schelling’s (1971) tipping model has drawn mounting attentions recently.
For instance, Card et al. (2008) confirm the existence of tipping across
neighborhoods in major US metropolitan areas, based on data collected at the
census tract level from 1970 to 2000. According to their article, the distribution of
tipping points ranges from 5% to 20% in terms of the non-White share of local tract
population.

In an earlier study, Clark (1988a) has also examined the empirical validity of
Schelling (1971). Instead of directly focusing on the tipping mechanism, Clark
looks into Schelling’s assumption regarding the structure of individual’s preference
for neighborhood racial characteristics. Schelling (1971) used two curves to
represent different preferential structures by Blacks and Whites. One purpose of
Schelling’s tipping model was indeed to demonstrate that an extreme degree of
racial segregation can emerge, even if Whites only have a slightly stronger
preference for local racial homogeneity than Blacks. This point is generally
confirmed by Clark (1988a). However, Clark notes that the shape of preference
curve quite varies when assessed with data from different US cities. This suggests
that the preference curve assumed in Schelling (1971) may follow multiple
44

functional forms in practice. For conceivable reasons this also implies that more
than one functional form may be associated with Schelling’s tipping model.

Another relevant piece of work is by Sethi and Somanathan (2004), who
decompose the effect of income disparity and that of racial antipathy underlying
residential segregation in general and the tipping phenomenon in particular. They
argue that income disparity between different ethnicities partially causes tipping,
but tipping may persist even with converging income level across ethnicities.

Thomas Schelling’s tipping model becomes a hot spot of academic inquiries
probably because of an important transition in the real world. The civil rights
movement in the 1960s essentially overthrew the institutional basis of racial
segregation, mainly by the Caucasians against African Americans (Farley and Frey
1994). This can be no more evident, given the fact that Americans just elected a
Black president, for the first time in the country’s history. However, this does not
mean that racial segregation has fully evaded. In fact, the current era witnesses a
higher frequency of voluntary and spontaneous segregation, or the so called self
segregation, which reflects individual preference for ethnic and cultural
homogeneity (Logan et al. 2002a; Logan et al. 2002b; Marcuse and Kempen 2002;
Squires and Kubrin 2005).

45

In addition, the huge influx of Latino and Asian population is another feature of the
contemporary US urban demography. The arrival of Latino and Asian immigrants
has greatly altered the dichotomous pattern of segregation between Whites and
Blacks (Clark 1992; Myers 2007). While the new immigrants are not constrained
by formal segregation policies, racial residential segregation between Whites and
non-Whites persist (Krysan 2002; Logan et al. 2002b). “White flight”, which refers
to the phenomenon that Caucasian residents retreat from racially integrated
neighborhoods, is an important reflection of the continuing ethnic segregation due
to neighborhood preference at the individual level. The empirical phenomenon of
White flight can be emulated with Schelling’s tipping model.

3.3 A Model of Fuzzy Tipping
Figure 3.1: White Flight in Schelling’s Tipping Model

46

Figure 3.1 above characterizes White flight from the perspective of Schelling’s
tipping model. X
(t)
in figure 1 measures the proportion of non-White residents in a
given neighborhood at time t. Let W
(t)
denote White population at time t in the
same neighborhood. And let F
(t,t+1)
denote White residents who live in the subject
neighborhood at time t but have moved out in the next period, t+1. Y
(t,t+1)
follows
the below specification:

Y
(t,t+1)
= F
(t,t+1)
/ W
(t)
. (3.1)

For obvious reasons, both X
(t)
and Y
(t,t+1)
must fall into the range, [0,1], as shown in
figure 3.1. Now assume that the Schelling style tipping point, λ, equals 0.5. It is
easy to get Y
(t,t+1)
as a function of X
(t)
, as illustrated with the asterisked line in figure
3.1 above and as suggested in equation (3.2) below.

0 for X
(t)
< 0.5
Y
(t,t+1)
= (3.2)
1 for X
(t)
>= 0.5

Equation (3.2) can be viewed as a crisp set membership function of White flight.
The crossover point lies at the tipping point, X
(t)
= λ = 0.5. If both the laws of
excluded middle and of contradiction in the traditional Boolean algebra can be
47

violated (Robinson, V 2003), a fuzzy set membership function can measure the
degree (between zero and one) of White flight in a series of neighborhoods, each of
which has a certain composition of non-White residents at a given time. This thus
constitutes a fuzzy set translation of Schelling’s tipping model.

Y
(t,t+1)
= {1/ (1+e
β* ( λ - x
(t)
)
)| β > 0} (3.3)

Equation (3.3) above is the fuzzy set function adopted in this paper. Dashed line in
figure 3.2 below is a graphic illustration of equation (3.3), when β = 10 and λ = 0.5.
By comparison with figure 1, the crisp set membership values (i.e, 0 and 1) only
exist at the two extremes of the dashed curve in figure 3.2.

Figure 3.2: Two Versions of White Flight

Note: solid line: crisp set; dashed line: fuzzy set)
48

Two parameters control the curve shape of equation (3.3). λ in equation (3.3) is a
crossover point. When any value on the x-axis is over λ, the corresponding value
on the y-axis must be above 0.5 and the function curve becomes concave to the y-
axis. In reverse, when any value on the x-axis is below λ, the corresponding value
on the y-axis must be below 0.5 and the functional curve becomes convex to the y-
axis.

β is a measure of speed at which the curve approaches its either endpoint from (λ,
0.5). The larger the β, the quicker the approaching process is. It should be noted
that β is a unique element of this fuzzy tipping model, as in a crisp tipping model β
is simply assumed to be exceptionally large such that either results in discontinuity
in the functional curve or an extremely steep slope around the tipping point (Card
et al. 2008. p. 184). For example, if β is infinitely large and λ =0.5, equation (3.3)
will immediately reveal itself as Schelling’s crisp tipping model as per figure 1.
Thus Schelling’s crisp tipping model is essentially a special case of the fuzzy
tipping model presented here. The yes-or-no question in the former is asked in a
more specific fashion in the latter, regarding the particular degree of yes-or-no.
Later in this paper, the results of this fuzzy tipping model will be compared with
that of Card et al. (2008).

As a single function, equation (3.3) is actually highly flexible to account for
different functional forms of fuzzy tipping, given different values of β and λ. Figure
49

3 below illustrates this advantage. One thing needs to be noted: Even when λ is
negative, the value of Y
(t,t+1)
generated by equation (3.3) still has meaning. For
example, the dotted curve on the top left side of the below figure 3.3 starts from
Y
(t,t+1)
= 0.7311. This simply means that around 73% of the White natives will
move out of the local neighborhood, even though the neighborhood is completely
White at the present.

Figure 3.3: Different Curve Shapes based on Equation (3.3), given β and λ

50

Now suppose that there are N neighborhoods, each of which exhibits a specific
proportion of non-White residents at time t. Let X
(t)
i
denote this variable, with t
meaning time t and i standing for the i
th
neighborhood. Also, each of the N
neighborhoods sees a specific proportion of local Whites leaving at time t+1. Let
Y
(t, t+1)
i
denote this second variable, in the same notation style.

Hence the situation of White flight in each of the N neighborhoods between time t
and t+1 may be represented with one point in figure 3.2, i.e., (X
(t)
i
, Y
(t, t+1)
i
). Under
the assumption that White residents in all of the N neighborhoods conceive a same
fuzzy tipping function and have full demographic information about the local
neighborhood,
9
a nonlinear least squares regression suffices to assess the value of β
and λ.

9
As pointed out later in the conclusion, these two assumptions are admittedly draconian and can be
fully released if an agent based modeling framework is adopted instead of regression. This however
means a shift of modeling purpose from identifying an overarching behavioral structure towards
spelling out the process of interaction among individual agents. Since this study focuses on the
former aspect, it is indeed necessary to make these two assumptions.

51

3.4 Data and an Ad Hoc Gravity Model
The Los Angeles County Union Census Tract Data Series, version 1.01 (Ethington
et al. 2000) is used in this study.
10
This dataset contains information for X
(t)
i
in the
fuzzy tipping model, given a debatable assumption of a census tract as a
neighborhood (the validity of this assumption is to be further discussed later). The
dataset however does not provide direct information on Y
(t, t+t)
i
, which measures
the degree of White flight. Table 3.1 below summarizes the basic descriptive
statistics.

One attribute noteworthy is that the dataset contains spatial panel data. One can
assume the independence of observations neither in a fixed decade nor at a fixed
census tract. In this study, the temporal dependence is minimized through
regressing data between two consecutive census years only, i.e., between 1960 and
1970, 1970 and 1980, and 1980 and 1990, respectively. Therefore, the fuzzy
tipping model aforementioned is tested for three times, one per decade.

10
This demographic dataset covers 1, 656 census tracts from 1940 to 1990. Due to the subdivision
of census tracts in the latter decades, the coverage and number of census tracts vary decade by
decade. The current dataset reflects the designation of census tracts in 1990. Demographic data are
interpolated in any census tract that did not exist before 1990. Please refer to Ethington et al. (2000)
for detailed interpolation criteria. After excluding missing and inconsistent data, the dataset contains
usable demographic data in 1, 649 census tracts from 1960 to 1990.
52

49

The problem of spatial dependence and the related heterogeneity issue require
techniques of spatial econometrics (Griffith 1988, 1996, 2003; Haining 2003;
LeSage 1999). While there are many ways to infer spatial relations, the best
approach is always to spell out the spatial behavior of the subjects (Odland 1988).
To do so, a simple ad hoc gravity model is utilized to, a) use the decline rate of
White population as a proxy of White flight in each census tract, and b) spell out
the spatial interaction between White flights from different census tracts.

Let ∆W
(t,t+1)
i
denote change in White population at the i
th
neighborhood. Note that
∆W
(t,t+1)
i
contains two opposing elements. One is the inflow of Whites into the i
th

neighborhood, between time t and t+1. Let M
(t,t+1)
i
denote this first element. The
other is the outflow of Whites between time t and t+1. Let F
(t,t+1)
i
denote this
second element. Given the steady decrease of White population in Los Angeles
County since 1960 (see figure 3.4 below), it is reasonable to assume the primary
source of M
(t,t+1)
i
to be intra-county migration.
11

11
This assumption however does not necessarily preclude the possibility that there is an unobserved
variable, which accounts for White migration to and from the outside of Los Angeles County. Yet
this unobserved variable can be deemed purely exogenous and be introduced as a part of random
interference as included in the error terms.
50

Figure 3.4: White Population in Los Angeles County, 1940-1990

Let F
(t,t+1)
j
denote White flight from the j
th
census tract between time t and t+1.
Suppose that part (rather than all) of F
(t,t+1)
j
will remain in Los Angeles county. Let
R
(t,t+1)
j
denote the number of these intra-state White migrants. Further suppose that
part (rather than all) of R
(t,t+1)
j
will be “attracted” to the i
th
census tract. Let R
(t,t+1)
j
→ i
denote the number of these White migrants to the i
th
census tract. R
(t,t+1)
j → i
is
assumed to follow a simple gravity model:

R
(t,t+1)
j → i
= { F
(t,t+1)
j
* ((1- X
(t)
i
)/ (D
ij
)
p
)
| p >= 0 ∩ X
(t)
i
< X
(t)
j
∩ (1- X
(t)
i
) <= (D
ij
)
p
∩ i ≠ j } (3.4)
51

D
ij
in equation (3.4) above stands for the geographic distance between the centroid
of the i
th
census tract and that of the j
th
tract. D
ij
raised to non-negative power p
comprises a conventional term of spatial weight (McMillen and McDonald 1997;
Thrall 2002). [(1- X
(t)
i
)/ (D
ij
)
p
] can be viewed as a measure of attractiveness of the
i
th
tract to White migrants from the j
th
tract between time t and t+1. Given the
theoretical insights of tipping, White movers would only seek a census tract of a
lower proportion of non-Whites than where they came from. Thus, X
(t)
i
must be
less than X
(t)
j
. Aggregating R
(t,t+1)
j → i
gets M
(t,t+1)
i
, a proxy of White migration into
the i
th
census tract between time t and t+1. Thus,

M
(t,t+1)
i
= {∑
j
N-1
R
(t,t+1)
j → i
| i ≠ j} (3.5)

Since ∆W
(t,t+1)
i
is made up of two elements, M
(t,t+1)
i
and F
(t,t+1)
i
, it is easy to get,

∆W
(t,t+1)
i
= {∑
j
N-1
(F
(t,t+1)
j
* ((1- X
(t)
i
)/ (D
ij
)
p
)) - F
(t,t+1)
i
+ ε
| p >= 0 ∩ X
(t)
i
< X
(t)
j
∩ (1- X
(t)
i
) <= (D
ij
)
p
∩ i ≠ j} (3.6)

According to equation (3.1) mentioned at the outset of this paper, F
(t,t+1)
is the
product of W
(t)
and Y
(t,t+1)
. Recall that Y
(t,t+1)
stands for the proportion of White
natives in a given neighborhood at time t to have left in the next time, t+1. And
Y
(t,t+1)
can be measured with the fuzzy tipping function ( i.e., equation (3.3)).
Therefore, the regression equation turns out to be,
52

∆W
(t,t+1)
i
= {∑
j
N-1
(Y
(t,t+1)
j
* W
(t)
j
* (1- X
(t)
i
) / (D
ij
)
p
) - Y
(t,t+1)
i
* W
(t)
i
+ ε
| p >= 0 ∩ X
(t)
i
< X
(t)
j
∩ (1- X
(t)
i
) <= (D
ij
)
p
∩ i ≠ j} (3.7)

To test equation (3.7), a number of operations need to be carried out in Matlab to
process spatial data and operate nonlinear regressions. Specific procedures are as
follows:

A census tract map of shapefile format contained in the Los Angeles County Union
Census Tract Data Series is firstly read into Matlab, using the “shape_read” file
developed by Le Sage (2003). The file execution saves the coordinate values of
each centroid of 1,649 census tracts from the source shapfile. The descriptive
statistics of these coordinate values are shown in the first two rows of table 3.2.

There is a scale issue with the distance data. According to the third row of table
3.2, all the observations of distance generate data of more than five digits. While
this will not bias the estimate of the regression equation, it poses as a precision
problem since p serves as power over distance. In other words, p tends to exhibit a
minuscule positive value of many digits before zero. To resolve this scale issue,
pair-wise distance between the centroids of 1, 649 census tracts are logged against
10. In this way, the scale of spatial weight in equation (3.7) becomes relatively
comparable with the numerators. The last row of table 3.2 reports the descriptive
statistics of pair-wise distance after log.
53

54

Also note that with conventional statistical package, it is difficult to test regression
equation as shown in equation (3.7). First, equation (3.7) is nonlinear. Thus the
usual linear least squares toolkits are barely useful. Second and more important, the
inter-census-tract migration modeled within equation (3.7) requires regressing the
relations both within and between observations; conventional econometrics only
handles the former. Third, equation (3.7) involves several constraints over the
parameters to be calculated. Standardized regression procedures included in
popular statistical packages are not flexible enough to reflect these constraints.

In contrast, Matlab is a relatively satisfactory platform to test such complicated
regression equation as equation (3.7). Designed by the MathWorks, Matlab has a
wide range of applications that require automated computations. Good
customizability is an important feature of Matlab, with m-file as the basic
execution unit that can be revised and shared with the other people. A series of m-
files can be grouped into a folder as a toolbox to solve a specific kind of
computation problem, for instance, nonlinear regression.

In this study, two m-files designed by the MathWorks and bundled with Matlab 6.5
are used to execute nonlinear regression. The two files are ‘nlinfit.m’ in the Stats
toolbox and ‘lsqcurvefit.m’ in the Optim toolbox. The former exercises simple
nonlinear least squares curve fitting, using the Gauss-Newton (G-N) algorithm. In a
55

simplest definition, the G-N algorithm involves repeated linear least squares
estimation of a given nonlinear functional form (Ediger 1990; Seber and Wild
1989).

‘lsqcurvefit.m’ uses the Levenberg-Marquardt (L-M) nonlinear least squaress
algorithm, which is partially based on the G-N method but more powerful in the
situation that the initial parameter estimates are far from the best fits (Glanz and
Slinker 1990, Chapter 10; Marquardt, 1963). An operational advantage of
‘lsqcurvefit.m’ over ‘nlinfit.m’ is that the former includes a specification of the
lower and upper bounds of the parameter values to be calculated; ‘nlinfit.m’ does
not have that option. This operational advantage of ‘lsqcurvefit.m’ later becomes
decisive in examining the regression results.

It should be noted that the “best fit” found using the either algorithm is locally
optimal. There can be multiple sets of parameter values that achieve globally
maximal R square, but only the locally bests are identified, due to the search range
of an algorithm. The modeler is required to input the initial parameter values as a
benchmark and as a starting point of search. While the initial parameter estimates
do not need to be accurate, they do affect the curve fitting process.

The determination of the starting values is subjective but not arbitrary. The modeler
is supposed to input a set of values reflecting the nature of the research subject. In
56

this study, the initial parameter estimates are decided as λ = 0.5, β = 1, and p =2.
The intuition is as follows. First, if Schelling’s tipping model is roughly valid as
Clark (1988a) claimed, the tipping point, λ, should fall into the range between zero
and one. Therefore, selecting the mid point, 0.5, seems to be a reasonable choice.
Second, β = 1 simply means that there is presumably no multiplier effect over the
pure tipping effect. Third, the setting of p =2 follows the classic Newton’s law of
universal gravitation that underlies the gravity model (Thrall 2002).

It should be noted that, in this study, two nonlinear least squares, respectively
following the Gauss-Newton (G-N) and the Levenberg-Marquardt (L-M)
algorithm, are exercised upon data for each of the three decades between 1960 and
1990, mainly for the purpose of cross validation.

3.5. Results and Discussions
3.5.1 General Results

Table 3.3 below reports the major regression results. Three points are clear
according to table 3. First, regressions using either the Gauss-Newton (G-N) or the
Levenberg-Marquardt (L-M) algorithm generate quite high R square (i.e., 68.77 %
-- 97.61%). This finding suggests that the functional form specified in equation (7)
has a high explanatory power and quite accords with the empirical observations.

57

Table 3.3: R Square and Coefficient Values
1960--1970 1970-1980 1980-1990
Algorith
m
G-N L-M G-N L-M G-N L-M
R Square 0.9331 0.7402 0.9328 0.6877 0.9761 0.9017
λ
SE (λ)
-34.6006
(983.8647
)
0.3653
(0.0952
)
0.5072
(0.0345
)
0.5086
(0.0777
)
0.7787
(0.0340
)
0.7811
(0.0705
)
β
SE (β)
-0.0143
(0.4046)
2.6391
(0.8869
)
2.2445
(0.2600
)
2.2301
(0.5799
)
4.0375
(0.4654
)
4.0041
(0.9466
)
p
SE (p)
2.2043
(0.0123)
2.2406
(0.0282
)
2.4152
(0.0199
)
2.4140
(0.0443
)
2.4938
(0.0721
)
2.4907
(0.1459
)

Second, in most cases, the coefficient values calculated with the G-N and L-M
algorithms are very similar. The only exception is the gap between the result of the
first regression (using G-N algorithm) and that of the second (using L-M
algorithm), both regarding the 1960--1970 period. One probable cause of this
inconsistency is that the first regression generates a β value (i.e., -0.0143) beyond
the constraint (i.e., β >0) specified after equation (7). Operationally, this violation
is due to the lack of boundary control associated with the G-N algorithm. But this
should not be considered a methodological inferiority of the G-N algorithm per se.
Notice that all of the three regressions using the G-N algorithm produce higher R
square, or better fit than those using the L-M algorithm. However, a high R square
cannot guarantee the validity of the regression results. Therefore, the results of the
first regression are not acceptable despite the desirable goodness of fit of the
regression.

58

59

After table 3.3, table 3.4 above presents the indicators that are necessary to check
potential multicolinearity. Although in each of the five valid tests, λ and β have a
negative correlation around minus 0.8, the minuscule VIF values presented in table
3.4 roughly clear the suspicious redundancy between λ and β.

3.5.2 Spatial Autocorrelation

Notice that all of the six regressions in this study are exercised upon data observed
in 1,649 census tracts which exhibit a fixed geographic pattern. When observations
from neighboring census tracts tend to exhibit similar or divergent residual values,
there is a case of positive or negative spatial autocorrelation correspondingly.
Spatial autocorrelation is a ubiquitous problem in regressions involving location
specific data. As to be shown below, spatial autocorrelation exists in this study,
too. However, it must be noted that spatial autocorrelation would not bias the
estimates of the coefficient values, which would still center around the real
coefficient values. Nonetheless, spatial autocorrelation leads to inflated R square
and underestimated standard error for each of the coefficients.
In this study Geary’s C is calculated to check potential spatial autocorrelation of
residuals from the six regressions. The formula of Geary’s C may be defined as
follows (Getis et al. 2004; Odland 1988).

C = ((N – 1) / 2 ∑∑ D
ij
) * (∑∑ D
ij
(O
i
- O
j
)
2
/ ∑ (O
i
- Ō)
2
) (3.8)
60

N in equation (3.8) above accounts for the number of observations. D
ij
is defined as
distance between observation O
i
and O
j.
Ō

stands for the mean of all observations.
The value range of Geary’s C is between zero and two. If C shows a value
between zero and one, observations have positive correlation space wise. The
closer the C value is to zero, the stronger the positive spatial autocorrelation is. If
C equals to one, observations are not correlated space wise. If C is above one,
observations are negatively correlated space wise. The extreme of negative spatial
autocorrelation is when C equals to two (Getis et al. 2004; Odland, 1988).

In this study, equation (3.8) above is firstly applied to test residuals from each of
the six regressions. For instance, residual from the i
th
census tract is fed into
equation (3.8) as an observation, i.e., O
i
. Since distance between the centroids of
any pair of 1, 649 census tracts are readily available, it is easy to calculate Geary’s
C for residuals from each of the six regressions. The second row of table 3.5 below
presents this result.

61

Table 3.5: Spatial Autocorrelation Tested with Geary’s C
1960--1970 1970-1980 1980-1990
Algorithm G-N L-M G-N L-M G-N L-M
Geary’s C
for DV
6.6742*10
-4

8.1112*10
-4

0.0015

Geary’s C
for residual
6.634
*10
-4

6.6903
*10
-4

8.5449
*10
-4

8.4377
*10
-4

0.0016 0.0016
Adjustment
(%)
0. 6 0.24 5.35 4.03 6.67 6.67

According to table 3.5, residuals from each regression tend to be positively
correlated in a strong spatial sense, given the tiny values of Geary’s C. This
suggests that there are some spatial relations in the dependent variable (i.e.,
∆W
(t,t+1)
i
in equation (3.7)) not yet modeled. However, recall the gravity model
included in equation (3.7). How much spatial relation does this gravity model
account for? The last row of table 3.5 gives out the answer, through comparing the
difference between the value of Geary’s C for residuals and that for the dependent
variable. It is clear that the gravity model only emulates a small percentage (0.24%
-- 6.67%) of spatial correlation regarding the change in White population in 1,649
census tracts over four census years.

The strong spatial autocorrelation of residuals is perhaps for two reasons. First, the
moving away of some White residents from a census tract may be unrelated to the
proportion of non-Whites in the locality. Instead, the Whites leave for such other
reasons as housing price and property tax, both of which may follow certain spatial
patterns. Spatial autocorrelation in this case may be considered pure
62

autocorrelation that is beyond the coverage of the fuzzy tipping model delivered
here.

Second, recall that White immigration from the outside of Los Angeles County is
considered a stochastic disturbance in this model. However, in practice there can
be a number of census tracts receiving net White immigration from the other
counties, states, or countries. These census tracts may be near to each other and
thus exhibit positive spatial autocorrelation that cannot be adjusted with the gravity
model included in equation (3.7).

63

3.5.3 A Longitudinal Comparison of White Flight

Figure 3.5: White Flight in Los Angeles County, 1960-1990 (L-M method)

Figure 3.5 above illustrates structural shift in the phenomenon of White flight from
1960 to 1990 in Los Angles County, according to the fuzzy tipping model tested in
this paper using the L-M algorithm. Number on the x-axis represents the
proportion of non-White residents in any census tract in 1960, 1970, and 1980,
respectively for each of the three functional curves. Number on the y-axis
measures the proportion of White movers among all White residents in each of the
three decades between 1960 and 1990.

64

Figure 3.5 conveys three clear messages. First, the mechanism of tipping—in a
fuzzy set version—does exist in Los Angeles County from 1960 to 1990. For each
of the three decades tested, a tipping point, λ, can be found in the range between
zero and one. Second, for any value on the x-axis in figure 3.5, White flight
measured by the corresponding value on the y-axis is always lowest in 1990 and
highest in 1960.

Third, there is an evident increasing trend in the value of tipping point from 1960
to 1990. As stated, a tipping point may be considered a threshold, after which the
curve shape transforms from convex to concave (in reference with the y-axis) and
over half of the White natives would move out of the local census tract in the next
period. The fact that the tipping point keeps moving right bound towards one
seems to suggest a steadily increasing degree of mix overall between Whites and
non-Whites.

While the value of λ keeps increasing over decades, there is no such clear trend in
the value of β. Notice that β for the decade between 1980 and 1990 is 4.0041, of a
substantial increase by comparison with 2.2301 and 2.6391 in the previous
decades. This finding has the following implications. By comparison with the
previous decades, in the decade between 1980 and 1990, White residents tend to be
more sensitive to increase in the proportion of non-White neighbors in the census
tract, once that proportion exceeds the corresponding tipping point, 0.7811.
65

Contrarily, White residents tend to be less sensitive than they were in the previous
decades, to increase in the proportion of non-White neighbors in the local census
tract, if that proportion is still below the tipping point. The same logic can be
applied to explain β for the previous decades. Notwithstanding the complicacy and
variation in β, one finding is certain: Overall the degree of White flight had been
decreasing from 1960 to 1990.

3.5.4 Comparing the Results of Fuzzy and Crisp Tipping Model

As aforementioned, Schelling’s crisp tipping model can be understood as a special
case of the fuzzy tipping model delivered here. In other words, crisp tipping, if it
exists, should be identifiable through testing the fuzzy tipping model. For this
reason, it is useful to compare the results of fuzzy tipping model with the outcomes
of a crisp tipping model and see if they converge.

Card et al. (2008) apply a method of regression discontinuity to model crisp
tipping. Note that tipping in their model is not strictly crisp: “the function…may
not be strictly discontinuous…but only steeply sloped in the range of [tipping
point]” (Card et al. 2008: 184). However, as stated above, Card et al. (2008) do not
specifically model the speed of tipping, β, which is however included in this study
as a parameter de facto controlling the degree of fuzziness in fuzzy tipping. In this
vein Card et al. (2008) may still be considered a crisp tipping model given its lack
of any fuzzy control.
66

According to Card et al. (2008), tipping point in Los Angele-Long Beach PMSA
(Primary Metropolitan Statistical Area), by geography largely equivalent to Los
Angeles County (US census, 1999), is about 15% share of non-White per census
tract during 1970 and 1980. The corresponding number is estimated to be around
50% according to this study. The measurement gap is apparent and is most likely
due to the fact that β turns out to be quite small, i.e., 2.2301, in the fuzzy tipping
model and thus very much smoothes the tipping curve. This comparison implies
that crisp tipping may have never happened in Los Angles County from 1970 to
1980 as suggested by Card et al. (2008).

3.6. Conclusions
A fuzzy tipping model has been built and tested against a demographic census
dataset covering 1,649 census tracts in Los Angeles County from 1960 to 1990.
Results from nonlinear least squares regressions indicate the existence of fuzzy
tipping mechanism in the study context. While the operation of nonlinear
regressions per se may contribute to the methodology literature, this study has more
significant theoretical and empirical implications.

Theoretically, findings from this study, after Card et al. (2008), again confirm the
conceptual validity of Thomas Schelling’s thesis regarding the tipping mechanism.
It is however interesting to note that some results of Card et al. (2008) differ quite
substantially from the according outcomes of this study. This difference can be
67

traced back to divergent understandings and assumptions about tipping speed. In
Card (2008) and Schelling (1971) original, the tipping speed is assumed to be
extremely fast. This assumption is substantially relaxed in the fuzzy tipping model,
wherein the speed of tipping is allowed to vary and specifically controlled with a
parameter, β.

Empirically, regression results suggest that the extent of White flight had been
steadily decreasing from 1960 to 1990. This finding is consistent with a general
conclusion made in the mainstream segregation literature (e.g., Farley et al., 1994).
That is, by comparison with the previous decades, there has been a steady
progression towards racially integrated urban settlement in the post 1960 US.
Nevertheless, it must be cautioned that the results of this study do not necessarily
reflect any attitudinal change or preferential shift in the individual level. Note that
this study is based on data from Los Angeles County, which has received a huge
number of non-White immigrants in the recent decades and exhibits an exceptional
degree of racial diversity compared with most other places in the US (Clark 1992;
Myers 2007). There is thus likely a unique Los Angeles effect, whereby White
residents have to accept racial integration as a consequence of the general
demographic transition ongoing in the region. That said, one should exercise
discretion when trying to generalize the results of this study to other circumstances.

68

Future research potentially lies in two areas. First, a major conceptual limitation of
this study concerns the deliberate confusion between census tract and
neighborhood. While the former does not convincingly amount to the latter (and it
often does not), ubiquitous data unavailability forces modelers to roughly equalize
the two concepts. However, it has been widely recognized that the results of
multivariate statistical analysis are very sensitive to change in the spatial or areal
unit of observations, leading to the famous “modifiable areal unit
problem”(MAUP) (Openshaw, 1984). While there is not yet a systematic solution
to MAUP, Fotheringham and Wong (1991) suggested using multilevel analysis to
calibrate the results of multiple rounds of investigations. Given that block level
demographic data are increasingly acquirable nowadays, it may be possible to test
this fuzzy tipping model using block level data. Different regression results may
emerge and tell a very different story about White flight.

Second and perhaps more important from the modeling perspective, an agent based
modeling framework is likely to better reflect the realism of the phenomenon of
White flight. Recall that all of the White movers in this study are assumed to have
full demographic information about the local (when they flee) and alien (when they
seek new settlement) tracts in Los Angeles County. This is in fact a typical
Tieboutian (1956) scenario, whereby individual is assumed to have access to
complete information, act rationally, and can move freely. While the rapid
development of modern communication and transportation technologies makes the
69

above assumption less draconian, it is still very doubtful that every individual in
practice can gain full information, make one-shot rational decision, and move
without cost. Instead, people in real life act step by step and may get into emergent
situations that result in emergent transitions in their behaviors. Conventional
regression methods can hardly handle such emergent phenomenon. This
insufficiency of regression methodology justifies the using of agent based
simulation, which captures the path dependent nature of individual behavior in
particular and human civilization in general (Kohler 2000).

70

Chapter 4:
Beyond Preference: Modeling Segregation under Regulation

4.1 Introduction
As a longstanding American urban phenomenon, residential segregation by race
involves both preferential and institutional reasons. However, scholarship on this
subject tends to fragment, eliciting a quite clear divide between the preference-
based versus the institution-orientated perspectives. For instance, Schelling’s
(1971) classic segregation models build entirely upon individual preference
regarding the community level racial composition, yet with little attention paid to
the local regulatory context. This feature seems to largely sustain in Schelling’s
contemporary counterparts (Card et al. 2008; Chen et al. 2003; Fossett and Waren
2005). On the other hand, since the seminal work by Massey and Denton (1993),
inquiries about the institutional causes of racial segregation have mushroomed,
primarily relying on more qualitative methods coupled with in-depth case studies
(Squires and Kubrin 2005; Squires and O'Connor 2001). This paper comprises an
effort to move beyond the aforementioned preference-institution divide. Revisiting
Schelling’s (1971) classic spatial proximity model, this study explores the
segregation outcomes in an artificial municipality
12
which regulates individual

12 given that the rights to regulate local housing is mostly vested with municipal government in the
United States (see e.g., Fischel 2000).
71

agent actions for the sake of collective housing interest in an ideal Pigovian style
(Pigou 1920).

The relation between racial residential segregation and local housing regulation has
for long been debated. A typical controversy is about the impact of zoning on
local ethnic diversity. Some latest evidence identifies financially motivated zoning
as a significant factor in excluding, though covertly, ethnic minorities from White
dominated residential suburbs (Rothwell and Massey 2009). Meanwhile
researchers like Fischel (2004 p. 331) consider zoning “far more an income-based,
class issue”: Though yielding racially exclusionary effects, zoning in itself is a
kind
13
of planning regulation intended to preserve the values of local real estate
properties in the collective economic interests of local homeowners (Fischel 2004;
Mckenzie 1994; Teaford 1997).

To address such debates as the one instanced above, this paper revisits Schelling’s
(1971) seminal spatial proximity model, using a cellular automata approach. Like
in the Schelling original, a gridded municipal urban space resembling a
checkerboard is assumed to contain a finite number of locations, each represented
by a cell. Every cell either accommodates a Black or a White family. A

13 Besides zoning there are several alternative forms of local housing regulation, such as charging
impact fees. These regulatory interventions are however similar to zoning in terms of being
financially motivated for the local homeowners’ collective economic interests (see. e.g., Ihlanfeldt
and Shaughnessy, 2004).
72

household’s satisfaction with housing, measured in terms of utility, only depends
on the racial composition of its neighbors, who live upon the nearest surrounding
cells. When a household is unsatisfied it always seeks to move to another location.
In Schelling (1971) relocation is essentially free insofar as the space allows. In the
revised model, however, any relocation imposing a net social cost upon the whole
municipal community is prohibited under regulation. This essentially could be
framed as an ideal Pigovian (1920) scenario, wherein the externalities of individual
action are perfectly explicit and internalized through intervention, making the
aggregate social cost tantamount to the accounted private cost; action stops when
the cost is too high.

According to the results from a large number of cellular automations, a
municipality which governs individual agents using an ideal Pigovian mode of
regulation eventually tends to become less efficient though more ethnically
integrated, compared with the laissez faire modeled in the Schelling original.
Aside from its conceivable policy implications, this study illustrates the particular
relevance of cellular automata, as a kind of micro simulation method, to urban
planning research in general and segregation studies in particular, since the method
allows us to appreciate the interactions between agent preferences and the
constraining institutional structures.

73

The remainder of this paper is organized as follows. The next section 4.2 reviews
Schelling’s (1971) spatial proximity model and points out that the model can be
modified to account for Pigovian local regulation as a stylized institutional factor.
This is followed by the set up of cellular automation in section 4.3, with the
simulation results reported and analyzed afterwards in section 4.4. The penultimate
section 4.5 discusses the policy and intellectual implications of this study.
Conclusions are summed up lastly in section 4.6 along with speculations on future
research.

4.2 Schelling and Pigou
4.2.1 Schelling’s Spatial Proximity Model

The phenomenon of residential segregation by race not only draws urban planners’
attention, but also interests the economists. Noble laureate Thomas Schelling
(1971) is one of the first mainstream economists who have looked into the
dynamics of segregation. From a game-theoretic perspective Schelling (1971)
modeled segregation in a spatially explicit fashion. Schelling found that ethnic
integration would be eventually improbable, even if individual agents are generally
indifferent to racial difference but simply avoid becoming ethnic minority in their
local neighborhoods.

Figure 4.1 below illustrates the structure of private racial preference in Schelling’s
spatial proximity model. The two vertical axes in figure 4.1 respectively measure,
by utility (U), the degree of a White and a Black household’s satisfaction with
74

housing when living in a neighborhood. The horizontal x axis shows the percentage
of neighbors who are African Americans.

Figure 4.1: Private Racial Preference in Schelling’s Spatial Proximity Model

Schelling (1971) assumed that a household, whether Black or White, would dislike
itself becoming any degree of racial minority within a neighborhood and would
want to move away immediately if that happened. Otherwise a household should
stay happily at its current place and remain indifferent to its neighbours’ ethnic
identity. Let U = 0 denote dissatisfaction and U = 1 stand for satisfaction. Hence
the utility function is binary for both a White and a Black household:

75






≥
≤
=
% 50 , , 0
% 50 , , 1
x when
x when
U
w
(4.1)





≤
≥
=
% 50 , , 0
% 50 , , 1
x when
x when
U
b
(4.2)
Equation (4.1) and (4.2) are represented respectively by two lines in figure 4.1, one
marked with blank and the other with solid diamonds. The dotted line in figure 4.1
stands for a catastrophic transition in utility from 0 to 1 or vice versa.

Schelling (1971) defined neighborhood explicitly in spatial terms. He firstly
employed a linear model, wherein every point upon the line has eight neighboring
points, four on each side. Schelling then moved to a two dimensional checkerboard
model, in which a neighborhood is normally made up of nine cells that form a 3*3
square. This rule however does not apply to neighborhoods in the urban periphery
which may contain less than nine cells given boundary constraint.

76

Figure 4.2: An Illustration of Neighborhood in Schelling (1971)

For an illustrative example, in figure 4.2 above, there is a 3*3 square containing
nine white cells, numbered from 1 to 9. For cell 9 in the center of this white
square, it has eight neighbors, including all of the cells surrounding it. Note that
this neighborhood is unique for cell 9 given the cell’s relative location. In fact
every cell in Schelling’s checkerboard model perceives its neighborhood in
reference to the cell’s own location; no cell shares exactly the same neighborhood
with another cell.

Schelling (1971) also allowed any discontented households to move freely to
wherever they want, as long as the space is available. It should be noted that
Schelling (1971) did presume a certain number of vacancies in the checkerboard so
that an unsatisfied household could relocate to one of the preferred untaken spots
and in the mean time leave a new vacancy available for the rest unhappy agents.

2 3 4
1 9 5
8 7 6
77

4.2.2 The Pigovian Housing Regulations

Schelling’s free moving scheme exemplifies a typical oversight of institutional
arrangements in most standard segregation models. Contrary to what is emulated
in Schelling (1971), a household’s movement, in reality, is often restricted, mostly
by the local municipal authorities through various forms of regulatory control. One
primary example is residential zoning. In many affluent suburban communities the
local zoning ordinances often mandate minimum lot size and single family
occupancy (Fischel 2000; Fischel 2004). In addition to zoning a local authority
may also enact other types of housing bylaws or land use covenants to govern the
residents within its jurisdiction (Schuetz 2008). More price based interventions
include charging a one shot upfront impact fee toward a new residential
development (Ihlanfeldt and Shaughnessy 2004). Otherwise if a municipality is
managed by a local homeowner association, a new participant is often required to
deposit a considerable amount of membership dues (Mckenzie 1994; Teaford
1997). There are many other examples showing the variety of local housing
regulations in the real world.

Notwithstanding the diverse forms of housing regulations, the economists consider
most of these interventions essentially Pigovian, in the sense of aiming to
internalize the negative externalities that private housing deals can cause to the
local community as a whole (Fischel 2000; Ihlanfeldt and Boehm 1987; Mills
1979). For example anti-density zoning and impact fees mainly confront
78

individual housing transactions which may induce overcrowding and burden the
community, for instance, in terms of delivering water or sewage service to all of
the local residents. In an economic sense, the above case shows that a private party
transaction can discount the utility of those who are not directly involved in and
thus external to the deal, whereby an issue of negative externality arises and a net
aggregate social cost may take place. Anti-density zoning thus happens when the
local planning authority tries to internalize the negative externalities of
immigration, by restricting the supply of land and raising the housing price by an
according amount for potential homebuyers.

Compared with zoning impact fee is even more direct by charging the newcomers a
sum of payment to offset the negative externalities of moving-in. Interventions in
the both cases are essentially Pigovian, because under the regulations a concerned
individual actor has to take the social cost into private account. Other modes of
local housing regulation are mostly similar in this regard, although practically
speaking, the actual exercise of any intervention is always contingent upon a
variety of political, cultural or even coincidental factors and thus much more
complex than ascribed in the ideal Pigovian model (Pigou 1920).

Notwithstanding the complexity of reality, it is indeed possible and reasonable to
incorporate a stylized Pigovian intervention into Schelling’s classic spatial
proximity model of racial segregation. Note that the same externality issue as
79

aforementioned actually also lurks in Schelling’s model. Since every household in
Schelling (1971) cares about the racial make up of its neighbourhood, a
household’s utility might be affected adversely when the old neighbours move
away and new families come in, which may result in an undesirable change in the
racial composition of local neighbourhood (Pancs and Vriend 2007). If the entire
community ends up with a net loss in housing welfare due to the relocations of a
few individual agents, a Pigovian regulatory intervention becomes justifiable, for
the same reason that zoning and impact fees, for instance, are rationalized in
practice.

4.2.3 Toward an Integrated Modeling Approach

To date it remains controversial how the variegated local housing regulations can
affect the pattern of racial residential segregation. The debate particularly revolves
around the impact of zoning on local ethnic diversity. On one hand critics like
Sager (1969) and Seitles (1998) have fiercely charged exclusionary zoning as de
facto racial discrimination in the housing market. Their key allegiance is that the
US housing market is imbued with racially discriminatory private preference.
Zoning but covertly institutionalizes such private discrimination in the name of
protecting communal interest. On the other hand researchers like Fischel (2004)
insist that zoning not directly target race, since lower income renters and
homebuyers of any ethnic background can be priced out of the market due to such
ordinance as density control.
80

The above controversy is partly attributable to the gap between what is modeled
and what is happening in the reality. Conventional segregation models tend to
place an overwhelming weight upon private preference regarding the community
level racial composition, while largely neglecting the local regulatory context. A
more integrated approach would not only look at the individual’s racial preference,
but also account for the institutional factors that affect agent’s housing behavior.

Recent developments in micro social simulation, such as cellular automata and
agent-based modeling, make the above task technically feasible. For instance,
Webster and Wu (2001) used a cellular simulation to justify localized land use
control from a Coasian efficiency perspective. Heikkila and Wang (2009) also
deliver an agent-based model of polycentic urban form and illustrates the implicit
yet critical influence of social institution on urban spatial structure. In a similar
spirit, the next section presents a cellular automata approach whereby individual
agent action in Schelling’s spatial proximity model is regulated for the sake of
collective housing welfare.

4.3 A Cellular Automata Approach
A cellular automata model is coded in Visual Basic for this paper after a substantial
modification and secondary programming based on Teknomo’s (2001) original
codes. The specific modeling steps are elaborated below.

81

4.3.1 Defining Neighborhood

The cellular model in this paper builds upon an artificial municipal space which
contains N*N cells as land parcels available for housing. The definition of
neighbourhood mostly follows the Schelling (1971) original, except for one
difference. While Schelling (1971) allows a certain number of cells to be vacant,
in the revised model, however, all cells are occupied, either by a Black or a White
household. This setting is necessary because the revise model involves a moving-
by-swapping algorithm as detailed below.

4.3.2 The Moving Algorithm

Since there is no vacant cell available this time, a discontented household would
have to move by exchanging its current location with another cell. Adapted from
Zhang (2010) a moving-by-swapping algorithm is employed in this paper.
.

Conceivably it would be barely meaningful for two households from the same
racial group to swap their locations, since the both agents conceive the same utility
function and a trade would make neither better off; both would remain unhappy. In
contrast a transaction between two racially different households would at least
make neither worse off and possibly increase the total payoffs.
14
Given equation
(4.1) and (4.2), the equation (4.3) below holds for any individual location:

14 Note that no trade will happen if the both parties are satisfied already.

82






= =
= =
⇒ = +
0 , , 1
0 , , 1
1
b w
w b
b w
U if U
U if U
U U (4.3)

Equation (4.3) has the following intuition: A household would always end up
satisfied by dealing with an unhappy household, insofar as the two households are
from different ethnic groups. Note that this does not necessarily mean the former
would have to be better off, since an agent can be contented already before it
switches the home. On the other hand, a discontented household can never become
any further worse off; swapping location is nevertheless possible (yet not
guaranteed) to make the unsatisfied happy again. In summary an exchange
between two racially different households must be risk free and may entail a Pareto
improvement (Zhang 2010).

For the aforementioned reason two households from different racial backgrounds
would always agree to a deal, as long as at least one of them is unsatisfied with the
existing condition. Now assume that spatial proximity decides the priority in
making deal.
15
According to equation (4.1) and (4.2), a discontented family must

15
An implicit assumption in this model is about the vision of individual agent. An agent is assumed
to only see the immediately adjacent peers, in the same way as it delineates its own neighborhood.
However one may also consider a trade between two faraway agents made up of multiple rounds of
local transactions as the two agents move closer and closer throughout the process of spatial
simulation.
83

have more than half of its neighbors from the alternative ethnic group, so the
former has to make a multiple choice to decide which neighbor gets the priority.
This requires designing a sorting protocol. In this model a household is supposed
to start sorting from the cell immediately left to it and then move clockwise until it
finds the first possible dealmaker. For an illustrative example, a household upon
cell 9 in figure 2 would search from cell 1 to 8, following the numerical sequence
clockwise, until it identifies a suitable neighbor to trade the locations and the
sorting stops thereafter.

4.3.3 The Rule of Transition without Regulation

As there are only two kinds of households inhabiting the municipal space, every
cell can only exhibit two possible racial states, or formally, } , { b w s ∈ , where w
denotes White and b stands for Black. Also note that an unhappy agent can always
find a racially different neighbor who is willing to exchange the locations. Given
these a Markovian transition function for any single cell may be expressed as the
follows:






= =
= =
=
+
0 , , 0 , ,
0 , , 0 , ,

, ,
, ,
1 ,
t j
b
t i
w
t j
w
t i
b
t i
U or U if b
U or U if w
s (4.4)

84

1 , + t i
s in the above equation (4.4) denotes the state of cell i at time t +1, which firstly
depends on the utility of a household occupying cell i at time t. Given the
aforementioned moving-by-swapping algorithm, an unhappy household will
always transfer its current home place to a neighbor from the other racial group.
Otherwise a happy family would simply stay. Another possibility is that a
household, whether happy or not, has to swap its home with a nearby unhappy
agent who is racially different and taking cell j at time t.
16
This would also trigger
a transition in cell i’s racial state from time t to t+1. Last the sojourn between t and
t + 1 is defined as the length of time during which each and every cell (totaled
N*N) upon the model space has to update its ethnic state for once as per equation
(4.4)

4.3.4 The Pigovian Regulation

Regulation in this model targets the aggregate housing utility within the
municipality averaged by the number of households. The following equation (4.5)
assesses the average housing utility, denoted as ψ, at time t across the municipal
space.

) * /(
,
*
1
,
N N U
t i
s
N N
i
t
t i
∑
=
= ψ (4.5)

16 A caveat is that cell j’s relative location to cell i is not fixed in a spatial sense. Instead one may
assume that an agent upon cell j always gets the priority to deal with an agent upon cell i, as long as
there is a deal to make.
85

In lieu of maximizing ψ
t
, regulation in this model aims to ensure that no location
swapping would ever discount ψ
t
. This is arguably a closer representation of the
actual local housing regulations, which are often exercised on a case by case basis
but seldom involve strict strategic mathematical optimization (Simon 1982).

To formalize this type of regulation, let
t j i , ↔
ξ denote the net social profit or
aggregate efficiency gain that a location swap between two households respectively
upon cell i and cell j at time t can produce for the entire local municipality. As
indicated in the equation (4.6) below,
t j i , ↔
ξ is made up of two parts. One part is
the total private pay-offs for the two households involved in the exchange, denoted
as
t j i , ↔
Ρ . The other part is,
t j i , ↔
Ε , the externality this transaction imposes upon the
other local households which are not directly engaged in the deal.
t j i , ↔
Ε < 0
suggests a negative externality induced by the private transaction.

t j i t j i t j i , , , ↔ ↔ ↔
Ε + Ρ = ξ (4.6)

Further more if
t j i t j i , , ↔ ↔
Ε + Ρ < 0, or in other words,
t j i , ↔
ξ < 0, the transaction
entails an overall inefficiency or a net social cost and will be annulled under
Pigovian regulation for the sake of infringing collective interest. This may also be
understood as a scenario wherein the two private parties cannot make sufficient
86

profits to afford a Pigovian impact fee, for example, that is intended to make up for
the induced negative externality. Otherwise the exchange will be endorsed. In this
vein, the following relationship is readily deducible:

) * /(
,
1 1
1
N N
t j i
N
i
N
j
t t
↔
= =
+ ∑ ∑
+ = ξ ψ ψ , only if, 0
,
≥
↔ t j i
ξ (4.7)
t
t t
∀ ≥ ⇒
+
ψ ψ
1

Equation (4.7) suggests that the regulation tends to constantly improve or, at least,
would never depreciate the overall housing utility in a municipality. To ensure this
condition a backtracking algorithm is applied when coding the model in Visual
Basic (Gurari 1999).

87

4.3.5 The Rule of Transition under Regulation

Incorporating Pigovian regulation into the model requires the rule of transition
aforementioned in equation (4.4) to be accordingly modified as illustrated in
equation (4.8) below:






≥ = =
≥ = =
=
↔
↔
+
0 , , 0 , , 0 , ,
0 , , 0 , , 0 , ,

, , ,
, , ,
1 ,
t j i t j
b
t i
w
t j i t j
w
t i
b
t i
and U or U if B
and U or U if W
s
ξ
ξ
(4.8)

Mind that
t j i , ↔
ξ above only exists when there is at least one unhappy agent either
taking cell i or cell j at time t. Otherwise no transaction would happen between two
satisfied agents given the model set up.

4.3.6 The Stop Conditions

Because cellular automata is a simulative approach, some stop conditions need to
be specified in advance. In the case of this model a key interest lies with the
average housing utility, ψ
t
. If the value of ψ
t
remains constant for N iterations, or
formally,

t N t N t
ψ ψ ψ = = =
- + +
...
1
(4.9)
88

then the simulation is assumed to reach a stable endpoint at time t. Otherwise the
simulation should also stop after a sufficiently large number of iterations, which
equals to the total number of cells involved in the model:

t = N*N (4.10)

4.3.7 The Initial Segregation Pattern

An initial racial residential pattern needs to be placed upon the municipal space as
a starting point of the cellular automation. Like in the Schelling original, two types
of distribution are randomized as the initial patterns. The first type involves a
random distribution of an equal number of White and Black households. In the
second case there is even no control on the ratio between the White and Black
population. However it should be noted that, in the both cases, the number of
Blacks and Whites, once after a simulation has initiated, would remain fixed until
the end of the automation.

4.3.8 Assessing the Segregation Pattern

Recall Schelling’s (1971) key discovery: Segregation would always arise, even
though individual agents do not necessarily prefer so. Will this discordance
between “Micromotives and Macrobehaviours” (Schelling 1978) persist in a
regulated Schellingian model? To answer this question, one needs to monitor

89

the segregation pattern. For this purpose an indicator is adapted from Moran’s to
specifically measure the overall degree of segregation:
17

| )] * /( /[ )] * /( [ | | |
,
*
1
, ,
*
1
,
*
1 1
N N X X N N X X I
t i
N N
i
t i t i
N N
i
t k
N N
i
n
k
t ∑ ∑ ∑∑
= = = =
- - = (4.11)

where






=
=
=
b s if
w s if
X
t i
t i
t i
,
,
,
, , 1
, , 0
(4.12)

and cell k is one of cell i’s neighbors, including but not limited to the
aforementioned cell j. n is the total count of neighbors and for most cells, n = 8. n
might be less than eight for cells at the municipal boarders.

Given the statistical properties of Moran’s I , the value of
t
I falls in the range of [-
1, 1] (Odland 1988). In the context of this model, 1 =
t
I , if the entire municipality
is dominated by either White or Black households. In another extreme, 1 - =
t
I , if a
household, wherever it lives within the municipality, always finds all of its

17 The more conventional segregation indices such as the dissimilarity index are not suitable in this
case, wherein the boundary of neighborhood is self-referenced (given a cell’s own location) rather
than delineated exogenously as in Massey and Denton (1989), for example.
90

neighbors to be from the other ethnic group. Last, 0 =
t
I , if every household is
located in a racially integrated neighborhood, with half of the neighbors being
White and the other half being Black. Given what is stated above, | |
t
I , the
absolute value of
t
I , has a value range as [0, 1]. A larger value in | |
t
I tends to
signal a racially segregated pattern overall, whereas a smaller value suggests the
municipality to be relatively more integrated in racial terms.

4.4 The Results of Cellular Automation
The model results are presented below in two fashions. The first involves a case
study of sample simulations, by comparing the simulation outcomes with the initial
statuses and monitoring the trajectories of transition gauged with two major
indicators, respectively, ψ
t
and | |
t
I . The primary goal here is to replicate the
classic findings by Schelling (1971).

The second way of presentation concentrates on the general difference that the
Pigovian regulatory control can make to the simulation outcomes. ψ
t
and | |
t
I are
still the target measures. Their values at the end of 200 simulations, half involving
regulation and half not, are compared using paired sample T test, to see whether
there is a significant difference in their population means.

91

4.4.1 A Case Study of Simulation Results

Figure 4.3 below illustrates a sample comparison between two simulations that
both start from the same random pattern shown in figure 4.3(1), when N = 10 and
an equal number of Black and White households (i.e., 50 each) are involved in the
model. The simulation involving no regulatory control stops when t = 29 and ends
up with the pattern shown in figure 4.3(2). Although the average utility has
increased from ψ
0
= 0.600 to ψ
29
= 0.950, the final pattern becomes much more
racially segregated, given the rise from | |
0
I = 0.055 to | |
29
I = 0.452. This result
resonates with Schelling’s (1971) classic finding that segregation would eventually
dominate notwithstanding individuals’ general indifference to their neighbours’
racial identities.

Compared with figure 4.3(2), figure 4.3(3) displays a much more racially
integrated residential pattern. The simulation involving regulation stops at t = 28,
with ψ
28
= 0.900 and | I
28
| = 0.200. Although the final residential pattern is still
more segregated than the initial one, the exercise of regulation has prevented
further segregation which otherwise would become the pattern as shown in figure
4.3 (2).

92

Figure 4.3: A Sample Comparison between Two Simulations, N = 10

Figure 4.4 and 4.5 below compare the trend of ψ
t
and | I
t
| between the
aforementioned two simulations. The horizontal axes in the both figures count
iterations, t, while the vertical axes respectively show the value of ψ
t
and | I
t
|. In
figure 4.4 the trend of ψ
t
for the unregulated simulation is marked with blank
squares. For the regulated simulation, the marks are solid squares. Conceivably
the former simulation eventually attains a higher level of collective utility, though
with a bit of fluctuations in the value of ψ
t
over the course. Yet also notice that ψ
t

has never declined in the second simulation but rather kept rising until it stabilizes
at the level of 0.900. This finding seems to confirm the analytical insights carried
in equation (4.7) above.

93

Figure 4.4: Comparing the Trends of ψ
t
between Two Simulations

In figure 4.5 the trend of | I
t
| for the unregulated simulation is marked with blank
triangles. For the regulated simulation, the marks are the solid triangles. The both
trends consist of ups and downs, while the trajectory for the regulated simulation
seems a bit more stable and eventually results in a much lower level of segregation.

Figure 4.5: Comparing the Trends of | I
t
| Between Two Simulations

94

A large number of trial simulations, including those with a random number (i.e.,
not necessarily 50:50) of Blacks and Whites, are also experimented with. The
pattern summarized in the above case study however seems fairly robust. Almost
all of the automations end up with a more segregated pattern and yet also reaches a
higher level of aggregate utility compared with the initial status. This finding is
essentially consistent with that by Schelling forty years ago.

Nevertheless Schelling did not run a simulation that involves regulatory control, so
he hadn’t got an opportunity to compare the results of regulated versus unregulated
simulations. The above case however suggests that the two types of simulations
may engender very different consequences, whether in terms of utility or in terms
of the spatial pattern of segregation. While the structural differences illustrated in
figure 4.4 and 4.5 above seem to be quite commonplace and almost constantly
recurring, there is indeed a small number of observations that do not follow suit.
Hence an important question arises, that whether or not there is a generalizable
difference in ψ
t
and | I
t
| between the regulated and non-regulated simulations? The
next section addresses this question using inferential statistics.

4.4.2 Comparing the General Impacts of Regulation versus Non-regulation

Since the computer randomizes the initial pattern for every simulation and
essentially makes each simulation an independent data generation procedure, the
automation results can be regarded as random and independent observations that
95

are deployable for some standard inferential statistical tests. This readily allows for
comparing the general impacts of housing regulation versus no regulation in a more
systematic and rigorous fashion than individual case studies.

To generate data 100 unregulated simulations and 100 regulated automations are
run in a pairwise fashion upon a computer, firstly based on some random initial
states involving 50 White and 50 Black agents. The simulation outcomes are
compared using paired sample T test and the test results are summarized in the
following tables. Table 4.1 and 4.2 below basically suggest that a simulation
involving Pigovian regulation tends to generate a significantly lower level of
average utility than an automation without intervention. In contrast table 4.3 and
4.4 suggest the regulation modeled in this paper tends to result in a significantly
more integrated racial residential pattern compared with the laissez faire emulated
in the Schelling (1971) original.

96

Table 4.1: Comparing the Outcome Average Utility (50 Whites and 50 Blacks)

Table 4.2: Difference in the Mean of Average Utility (50 Whites and 50 Blacks)
Paired differences in outcome average utility (ψ
t
)

Mean Std. Deviation T df
Sig.
(2-tailed) Std. Error Mean
0.016 .028 5.796 99 .000 .003

Table 4.3: Comparing the Outcome Degree of Segregation
(50 Whites and 50 Blacks)

Table 4.4: Difference in the Mean Degree of Segregation
(50 Whites and 50 Blacks)
Paired differences in the outcome degree of segregation (| I
t
|)

Mean Std. Deviation T df
Sig. (2-
tailed) Std. Error Mean
0.051 .103 4.924 99 .000 .010

Regulated? Observations Mean Std. Deviation Std. Error Mean
no 100 .954 .015 .002 outcome average
utility (ψ
t
)
yes 100 .938 .022 .002
Regulated? observations Mean Std. Deviation Std. Error Mean
no 100 .434 .087 .009 outcome
degree of
segregation
(| I
t
|)
yes
100 .383 .100 .010
97

Almost the same results arise even if the number of Whites and Blacks are allowed
to be uneven at the beginning of the simulations.
18
Table 4.5 and 4.6 below are the
counterparts of table 4.1 and 4.2 above, while the ratio between Whites and Blacks
is purely stochastic this time. Again the exercise of regulation seems to result in a
significantly lower level of average utility compared with no regulation. Similarly
table 4.7 and 4.8 further below convey essentially the same message as table 4.3
and 4.4 above. That is, the Pigovian regulation tends to prevent racial segregation
which otherwise would be inevitable in a void of regulatory control.

Table 4.5: Comparing the Outcome Average Utility
(Random Black/White Ratio)

Table 4.6: Difference in the Mean of Average Utility
(Random Black/White)
Paired differences in outcome average utility (ψ
t
)

Mean Std. Deviation T df
Sig.
(2-tailed) Std. Error Mean
0.025 .032 7.722 99 .000 .003

18 However the magnitude of N must remain fixed and in this case N = 10 for all the simulations.
Otherwise the simulation results would become incomparable. For instance the value of | I
t
| would
be systematically smaller for larger N, according to Fotheringham and Wong’s (1991) seminal study
on the modifiable areal unit problem of spatial autocorrelation.
Regulated? observations Mean Std. Deviation Std. Error Mean
no 100 .956 .022 .002 outcome
average utility
(ψ
t
)
yes
100 .931 .021 .002
98

Table 4.7: Comparing the Outcome Degree of Segregation
(Random Black/White)

Table 4.8: Difference in the Mean Degree of Segregation
(Random Black/White)
Paired differences in the outcome degree of segregation (| I
t
|)

Mean Std. Deviation T df
Sig.
(2-tailed) Std. Error Mean
0.088 .119 7.384 99 .000 .012

4.5 Discussion
If the keyword featuring Schelling (1971) is “unintended consequence”, this seems
to be the case too for this paper. One may expect the Pigovian regulation modeled
above to result in a more efficient residential pattern which makes the largest
majority satisfied with their housing circumstances. Nevertheless the automation
results show some contrary evidence to this. One may not expect the Pigovian
regulation to address any racial issues. Yet this seems to be a matter of fact given
the simulation outcomes. This section takes a closer look and some further
thoughts at these unintended consequences.

Regulated? observations Mean Std. Deviation
Std. Error
Mean
no 100 .434 .106 .011 outcome
degree of
segregation
(| I
t
|)
yes
100 .346 .092 .010
99

4.5.1 The Inefficiency of Pigovian Regulation

The Pigovian regulation modeled in this paper proves to be inefficient, ironically,
by the fact that in the end it tends to produce less collective utility than the laissez
faire. A potential reason for the inefficiency of Pigovian regulation is associated
with equation (4.7) and related to the issue of path dependence (Arthur 1994).
According to equation (4.7), the average collective utility, ψ
t
, would never decline
in the course of any simulation insofar as there is regulatory control. However one
should note that this is neither a necessary nor sufficient condition to maximize the
overall utility at the end of the simulations. In fact figure 4.4 suggests that a higher
level of utility can be reached eventually without regulation, even though the level
of utility may fluctuate during the course of simulation. This contrasts with the
inefficient results from the ideal Pigovian regulation modeled in this paper which
forbids any efficiency loss at any time. This may prompt us to realize that, for the
sake of path dependence, sometimes temporary “pains” have to be taken along the
path for eventual “gains”; no pains no gains!
100

4.5.2 Racial Integration under Pigovian Regulation

The outcomes of cellular automations clearly suggest that ethnic integration is
possible, even though not intended, under an ideal Pigovian regulation. This seems
to directly defy such civic lawyers’ arguments as the one by Seitles (1998), for
example, that most local zoning tends to exclude ethnic minorities in a covert and
yet substantive way. Nonetheless, with regard to this point, there are a couple of
caveats noteworthy about the model set-up as well as results presented in this
paper.

First, individual agents in this model are all supposed to be generally indifferent to
race as per equation (4.1) and (4.2). In reality, however, strong individual racial
preference or private racial discrimination may still exist even though the
general public tends to condemn any sort of racial prejudice in the current era
(Farley and Frey 1994; Logan et al. 2002a).

Second, as stated above, the Pigovian regulation in this model favors the local
residents and tends to discourage moving. In other words if the initial pattern is an
integrated one, the regulation, in effect, tends to preserve the status quo compared
with the free moving mechanism set in the original Schelling (1971) model. Recall
that in this paper all of the initial states result from computerized randomizations.
The probability for a computer to generate the first group of neighboring cells
101

which all accommodate same-race households is actually very small, given
equation (4.13) below:

) * /( ) ( * )... 2 * /( ) 2 ( * ) 1 * /( ) 1 ( Pr n N N n r N N r N N r - - - - - - = (4.13)

r in equation (4.13) stands for the subtotal population within a particular racial
group, such as Black or White. N*N gives the aggregate municipal population. n
above counts the number of neighbors for a household living upon a specific cell.
Given the definition of neighborhood in this study n = 8 for most cells that are not
located at the boarders. Suppose 50 Whites and 50 Blacks upon a 10*10 municipal
space, as studied above in the results section. In that case, Pr, the probability of
utter racial segregation in the first neighborhood randomly generated by a
computer, according to equation (4.13), is less than 0.003, a very minuscule figure.
Conceivably the probability for each and every neighborhood to be utterly
segregated would actually be even smaller than that.

However the segregation patterns observed in the real-world neighborhoods are by
no means the products of mechanic randomizations. In fact, partly for historical
reasons, most urban neighborhoods in America still tend to be racially
homogenous, whether in the downtowns or the suburbs (Squires and Kubrin 2005;
Wang 2011). Exercising a regulatory intervention based on such initial states can
barely lead to racial integration, since there is a lack of racial diversity from the
102

very beginning. This implicates the necessity of cultivating ethnically mixed
neighborhoods as a desirable initial condition. In reality a number of measures
have been taken, for example, by helping lower income ethnic minorities move into
White dominated middle class communities through rent control, affordable
housing and some socially inclusionary zoning schemes. While such egalitarian
efforts have been made in many US cities, its utilitarian outcome in terms of
housing market efficiency is still quite controversial (Zheng et al. 2007).

4.5.3 Reconciling Preference and Institution

On top of the policy implications discussed above, intellectually the cellular
automation conducted in this study exemplifies an integrated perspective in the
study of racial residential segregation. Existing inquiries tend to either emphasize
individual’s racial preference or address the societal and institutional factors of
segregation. The two seemingly divergent perspectives, however, can be
reconciled meaningfully through proper technical manipulations, as demonstrated
in this paper. While micro simulation is not the only apparatus employable for the
above purpose, it provides a platform which has conceivable
potential to help us explore residential racial segregation under variegated
institutional conditions.

In a broader sense, the micro simulation approach demonstrated in this study may
also inform the entire discipline of urban planning. In his recent paper titled
103

“Should planners start playing computer games”, Devisch (2008) posits that
seemingly game-like micro simulation models may enable planners to better
understand the complex process of social and spatial evolution. Compared with the
traditional and more deterministic methods, micro simulation builds upon a unique
sensitivity to such realistic issues as myopic agent behavior, bounded rationality,
path dependency, social and spatial interactions, all of which are particularly
relevant to the practice of urban planning (Rittel and Webber 1973; Schon 1983).

4.6 Conclusions
Employing a cellular automata approach, this paper revisits Schelling’s (1971)
spatial proximity model of racial segregation. Like in the Schelling original, racial
integration appears to be systematically untenable insofar as the agents are allowed
to move freely between neighborhoods. However, the simulation results also
suggest that a Pigovian regulation to preserve collective housing welfare may
alleviate the degree of eventual segregation, which otherwise would be much more
substantial under the laissez fair emulated in Schelling (1971). Results from
inferential statistical tests confirm the above findings to hold general validity.
These discoveries prompt us to reflect about policy endeavors in practice, including
those unintended welfare and racial consequences of various forms of local housing
regulation. On top of that this paper showcases an integrated modeling approach
which reconciles the preference- based and institution-orientated perspectives in
the contemporary segregation research.
104

While this study demonstrates a micro simulation method applied to a particular
urban development problem, breakthroughs are expected, at least, in two
directions. Firstly, more realism needs to be brought about by linking the sort of
spatial process model in this paper with a large amount of empirical socioeconomic
data, especially by using such apparatuses as geographic information systems (GIS)
(Heikkila 1998). In the case of this paper inferential statistical tests are only
applied upon pseudo data generated by computerized cellular automations.
However the same thing could be done with actual demographic data that are
becoming more and more accessible to the general public. This hopefully could
resolve an aforementioned problem about the unrealistic initial segregation patterns
modeled in this and other similar studies.

Secondly the kind of model shown in this paper can be greatly refined and
calibrated if the coding process becomes more participatory. Open-source
programming is not only beneficial in technical terms, but can also address some
key conceptual issues. For instance in both this and Schelling’s (1971) original
paper the definition of neighborhood is quite draconian. Although every family
defines its own neighborhood in reference to the household’s own location, the way
of self-mapping is indeed identical for every one. However Coulton et al.’s (2001)
empirical study suggests that people tend to perceive their neighborhood geography
in very different fashions. Such cognitive heterogeneity is accountable using an
online GIS platform which allows users to map their own neighborhood.
105

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Abstract (if available)

Abstract Contemporary urban spatial analysis arguably features an emphasis on spatial structure, whereas understudying the background agent actions. This dissertation is intended to address such oversight, by presenting three independent essays on the spatial analysis of residential segregation by race in urban America. Combining the three essays, this dissertation demonstrates a series of methodological considerations and innovations to rediscover agency in modern quantitative urban spatial analysis. Aside from its methodological orientation, this dissertation implicates a number of issues related to planning theory as well as public policy.

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Asset Metadata

Creator Wang, Yiming (author)

Core Title From structure to agency: Essays on the spatial analysis of residential segregation

School School of Policy, Planning, and Development

Degree Doctor of Philosophy

Degree Program Policy, Planning, and Development

Publication Date 04/26/2011

Defense Date 03/03/2011

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Los Angeles,modelling,OAI-PMH Harvest,quantitative,residential segregation,spatial analysis,Urban

Place Name California (states), Los Angeles (counties), USA (countries)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Heikkila, Eric J. (committee chair), Ethington, Philip J. (committee member), Gordon, Peter (committee member)

Creator Email WangY53@cf.ac.uk,yiming.wang@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3774

Unique identifier UC1445733

Identifier etd-Wang-4449 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-448349 (legacy record id),usctheses-m3774 (legacy record id)

Legacy Identifier etd-Wang-4449.pdf

Dmrecord 448349

Document Type Dissertation

Rights Wang, Yiming

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu