CSIS Discussion Paper No. 119R

Decomposition approach to the measurement of spatial segregation

Yukio Sadahiro* and Seong-Yun Hong**

July 2014

Center for Spatial Information Science, University of Tokyo
5-1-5, Kashiwanoha, Kashiwa-shi, Chiba 277-8568, Japan* sada@csis.u-tokyo.ac.jp** yun.hong@csis.u-tokyo.ac.jp

Keywords: spatial segregation, decomposition approach, measurement-1-Abstract
This paper develops a new method of evaluating segregation among point distributions. We argue
that there are three main sources of segregation: difference in the spatial arrangements of points;
imbalance in the number of points; and lack of diversity of points, and that segregation has three
distinctive components, each of which is due to one of these sources. The proposed measures
evaluate each component separately, and this approach has five desirable properties, which are notfully possessed by existing measures. To demonstrate the validity of the method, we apply it to an
analysis of three data sets. The results show that the proposed measures can evaluate the degree ofsegregation effectively from three different perspectives, and their interpretation can bestraightforward because each of the measures is linked with only one component of segregation.

-2-

1. Introduction
Segregation is a complex and multidimensional socio-spatial phenomenon. A complete, or
near-complete, understanding of this phenomenon requires us to evaluate segregation from various
perspectives, and for this reason, numerous indices have been proposed in geography, sociology, andother related fields. The index of dissimilarity (Duncan and Duncan, 1955), the Gini index, and theentropy-based indices (Theil and Finizza, 1971), for example, have been frequently used in theliterature, in part because of their ease of calculation and interpretation. These earlier indices, however,are often criticized as aspatial since they are insensitive to the arrangement of spatial units (Morrill,1991). To overcome this limitation, a number of alternative, spatial segregation indices have beendeveloped in more recent years (e.g., White, 1983; Morrill, 1991; Wong, 1993; Reardon andO’Sullivan, 2004).
While all these indices aim to evaluate segregation from different perspectives, they areoverlapping to some extent with each other. We thus need to choose an appropriate set of indices that
explain the whole picture of segregation efficiently and address different dimensions of segregation
separately. As Massey and Denton (1988) pointed out, however, there has been no consensus on whichindices are necessary and sufficient to fully describe patterns of segregation. Though several papers
have proposed criteria for segregation indices (James and Taeuber, 1985; Reardon and Firebaugh,
2002; Reardon and O’Sullivan, 2004), they do not directly discuss the relationship between the indices,and hence the criteria are not enough for choosing an appropriate set of indices.
To resolve the problem, Massey and Denton (1988) introduced five conceptually distinct
dimensions (evenness, exposure, clustering, centralization, and concentration). They performed factor
analysis on twenty segregation indices for the 1980 U.S. census data, from which they nominated fiveindices as the best indicators of the five dimensions. Reardon and O’Sullivan (2004), on the other hand,argued that the distinction between evenness and clustering is somewhat arbitrary, and proposed two
primary dimensions and two indices. Johnston et al. (2007) conducted principal component analysisof the U.S. census data for three different years and concluded that two basic dimensions are sufficientto grasp the residential segregation in the U.S. metropolitan areas. They suggested the use of principal
components’ (PC) scores derived from principal component analysis as measures of segregation.
The concept of such distinct dimensions provides us a logical basis for choosing a set ofindices. Since the dimensions of segregation are conceptually distinctive from each other, the chosen
indices should also evaluate individual dimensions separately. Each index should fully represent one,and only one dimension, and overlaps between the indices are undesirable as it not only reduces theefficiency of analysis but also makes dimensions incomparable.
However, the existing indices do not completely satisfy these properties. The five indiceschosen by Massey and Denton (1988) are only partially correlated with the corresponding conceptual
dimensions, implying that selected indices cannot completely explain the associated aspects of-3-

segregation. Moreover, the indices are correlated not only with their corresponding dimensions but
also with others. The indices overlap with each other not only empirically but also theoretically, which
prohibits us from evaluating each dimension separately. Reardon and O’Sullivan (2004), on the other
hand, did not prove that proposed dimensions are fully covered by their representative indices and thatthe two indices do not overlap. Furthermore, the lower bound of one index called the spatial
information theory segregation index is not specified; thus, it cannot be standardized into a relative
form (a revision of this index is given in Sadahiro and Hong, 2013). The representativeness of PC
scores proposed by Johnston et al. (2007) depends on the data set used in principal component analysis.
One PC that is chosen to represent a particular dimension of segregation in one data set may not beable to describe satisfactorily the same dimension in another data set. It is not assured that eachdimension is fully represented by a single PC. In addition, the interpretation of PC scores is notstraightforward, as there is no one-to-one correspondence between the PCs and the conceptual
dimensions of segregation.
As seen above, the existing sets of indices are not entirely appropriate for the measurementof distinct dimensions. To resolve the problems, this paper proposes a new set of indices for measuring
segregation. We focus on point distributions that represent zero-dimensional objects and approximate
higher-dimensional objects in the real world. Examples include individual persons, animal species,
houses, and land parcels. The proposed indices permit us to capture different aspects of segregation
with satisfying the desirable properties discussed above. In the next section, we argue that segregation
can be divided into three distinctive components, and then propose three indices, each of which
evaluates one component of segregation. Section 3 applies the proposed approach to an analysis ofthree spatial data sets. Section 4 discusses in detail the properties of proposed measures, and Section
5 summarizes the conclusions.

2. Method
Many of aspatial indices evaluate segregation in terms of spatial pattern such asconcentration, clustering, and unevenness. Spatial indices, on the other hand, often focus on thepossibility of social interaction between individuals since it permits us to consider the geographic scale
explicitly (Echenique and Fryer, 2007; Kaplan and Holloway, 2001; O'Sullivan and Wong, 2007;
Reardon and Firebaugh, 2002; White, 1983; Wong, 1993). Both spatial pattern and social interaction
give us a measure of segregation, though they take different perspectives. The former focuses on asource of segregation while the latter puts more weight on the result of segregation. This paper follows
the latter approach, i.e., a focus on the social interaction between individuals to consider the spatialaspect of segregation more explicitly.
Segregation indices should address distinctive dimensions of segregation separately without
overlap. To this end, this paper distinguishes the different sources of segregation. Existing papers-4-almost agree on that there are at least spatial and aspatial dimensions in segregation. For instance,
among the five dimensions proposed in Massey and Denton (1988), evenness and exposure are aspatial
while clustering, centralization, and concentration are spatial (Reardon and O’Sullivan, 2004).
Reardon and O’Sullivan (2004) proposed spatial evenness and spatial exposure that put more focus
on spatial and aspatial dimensions, respectively. Spatial and aspatial dimensions imply two sources ofsegregation, i.e., spatial and aspatial sources. Extending these sources, this paper considers threesources of segregation: 1) different arrangement of points, 2) imbalance in the number of points, and3) lack of diversity in points. The different arrangement of points is a spatial source of segregation
while the latter two are aspatial sources.
Each source causes segregation in a different manner. We introduce a term component torefer to an element of segregation caused by a specific source. The three sources cause three different
components of segregation, which we name the components locational, compositional, and qualitative
segregations. In the following subsection, we will explain the concepts of sources and components inmore detail, and propose the indices associated with the three components.

2.1 The three sources and components of segregation
Figure 1 shows the distribution of individuals of different ethnicities. The black and white
points exhibit almost the same spatial arrangement in Figure 1a. On the other hand, points are clearly
separated from each other in Figure 1b. The probability of contact between the black and white points
in the latter is obviously smaller than that in the former due to the different spatial configuration of thepoints. The term "locational segregation" refers to segregation due to such different arrangements ofpoints. Figure 1c displays black and white points that share the same proportional distributions, butdiffer in the number of points. One may think that segregation does not exist because the two types ofpoints share the same proportion at individual locations. However, if we consider the probability ofcontact between points, each white point has a higher chance of meeting points of the same color; thus,
they could be considered to be more segregated than those in Figure 1a. In this paper, we use the term"compositional segregation" to refer to segregation caused by such imbalance between the groups (i.e.,
point types) in terms of the number of points. In Figure 1d, there are four different types of points.
Although they are arranged in the same manner as the points in Figure 1a, the probability that a certain
type of point meets a different type of point is clearly larger than in Figure 1a. This implies that theincreasing variety of points reduces the level of segregation, and the term "qualitative segregation"refers to segregation caused by a lack of diversity of points.
The difference in the spatial arrangements of points is a spatial source of segregation,although the imbalance in the number of points and the lack of diversity of points are aspatial elements.
Thus, the locational segregation can be considered to address the spatial aspect of segregation, and thecompositional and qualitative segregation to evaluate the aspatial aspects. Previous studies have-5-

discussed the aspatial aspects of segregation primarily with respect to the imbalance in the proportionof different types of points, which corresponds to the compositional segregation. As seen above,
however, a close relationship exists between the variety of points and the level of segregation. Thus,
we advocate qualitative segregation as the third component of segregation.

2.2 Measurement of segregation
This subsection proposes a set of measures to evaluate the three components of segregation.
Suppose a bounded region R of area T in which , a set of K types of points are distributed. The
location of jth point of type i is denoted by zij. Let Ni and N be the number of type i points ant the total
number of points, respectively.
The measurement of segregation depends on the geographic scale of analysis (Wong, 1993;
Wong et al., 1999; Kaplan and Holloway, 2001; Reardon et al., 2008). Figure 1e shows the distributionof black and white points. Segregation is not observed from a usual point of view because both black
and white points are uniformly distributed. However, if we take a local view, we find segregation
between the points; no white point exists in the close neighborhood of black points, and vice versa.
White points are less probable to meet black points in Figure 1e than in Figure 1a. Segregation iscaused by the difference in the spatial arrangement of points at a local scale.
To deal with the geographic scale of segregation, we adopt what is called the surface-based
approach often used in existing papers (Reardon and O’Sullivan 2004; O’Sullivan and Wong, 2007;
Reardon et al., 2008; Spielman and Logan, 2013). The surface-based approach transforms a point
distribution into a continuous surface that indicates the degree of spatial clustering of points. The
surface represents the probability of contact between points since points are more probable to meetwhere many points are clustered.
We adopt the Gaussian kernel in converting point distributions into surfaces (Silverman,
1986):k  x, z ij , h  1e2

1  x  z ij  
2 h 



2

.
(1)

The parameter h is called the bandwidth that determines the geographic scale parameter of analysis. A
large h gives us a global view in the measurement of segregation while a small h permits us to evaluate
segregation at a local scale. The degree of spatial clustering of type i points at x is represented by thesummation of Equation (1) in a standardized form:-6-Di  x, h  

N i  k  x, z ij , h j  k  y , z , h dy
ij

y R

j

.
(2)

We call Di(x, h) the density function of type i points. It shows a large value where points are clustered.
It becomes smooth with an increase in the bandwidth h.
So far we have assumed that the location of individual points is known. However, spatial
data are often provided only in an aggregated form to keep the confidentiality of individuals. This is a
primary reason why most of segregation measures proposed in the literature assume spatial data
aggregated by spatial units such as census tracts and zip code units. Aggregated data are undesirable
since they inherently bring the Modifiable Areal Unit Problem, i.e., calculated indices depend on thespatial units used for data aggregation (Wong, 1993; Reardon and O’Sullivan, 2004). Though thisproblem is not fully avoidable, the method proposed in this paper can be applied to aggregated spatial
data. Suppose region R is divided into subregions R1, R2, ..., RV. Let us denote Tv and piv as the area ofvth region and the number of type i points in the vth region, respectively. A simple approach is tosubstitute the point density piv/Tv for Di(x) everywhere in the vth region. In this method, however, wecannot evaluate the geographic scale of segregation. A more sophisticated alternative is to assume thatpoints are uniformly distributed in every region and calculate the density functions of points based on
Equation (2):pivk  x, z , h dzzRv Tv
v
Di  x, h  pyR v zRv Tviv k  y , z , h dzdy .
Ni  

(3)
Unlike aspatial measures, we can consider the geographic scale of analysis explicitly by surface
conversion.
Using the density functions defined by Equation (2), we define a local segregation measureas D  x, h  s  x,  , h     i
 ,i  D  x, h  
2

(4)
where:D  x, h    Di  x, h  .
i

-7-

(5)
Though the entropy index has been often used as a measure of segregation in the literature, this paper
adopts the above definition because the entropy index cannot evaluate qualitative segregation(Sadahiro and Hong, 2013). The local segregation measure estimates the level of the imbalance
between different types of points at location x. It can be interpreted as the sum of the proportion ofeach type of point that needs to change for balanced proportions. Figure 2a shows the density functionsof five types of points in a 1D space. Figure 2b shows the distribution of s(x, , h) calculated fromthe density functions in Figure 2a. It has a large value if the proportion of different types of points at
location x varies widely. It becomes small when each type of point occupies a similar proportion of
the total.
We define a global measure of segregation by integrating s(x, , h) weighted by the total
density of points D(x, h) over the entire region:S  , h  1
D  x , h  s  x ,  , h  dx .
N xR(6)

This indicates the average degree of segregation over the entire region. We call it the overall
segregation measure, whose range is 1/K<S(, h)1 (a proof is given in Sadahiro and Hong, 2013).
As seen in Equation (6), it is a function of point set  and bandwidth h, which implies that the measuredepends on the geographic scale of analysis represented by h.
We then define the measures for the three components of segregation. To evaluate aparticular component, we need to consider two situations: where the source of that component ispresent and where it is absent. We first calculate the overall segregation measure S(, h) by assuming
the latter and comparing the result with the observed value; the difference represents the effect of thesource of that component.
Next we begin with the evaluation of locational segregation. The source of locationalsegregation is absent when all types of points have exactly the same spatial arrangement. In this case,the fragmented pieces of K types of points {q1, q2, …, qK} are distributed at every location of theoriginal points, where qi is a piece of Ni/N fraction of type i point (Figure 3). In Figure 3a, there are
four black points and eight white points. If we replace every point in this figure with a point comprising
1/3(=4/12) black and 2/3(=8/12) white points (Figure 3b), the black and white points have exactly thesame spatial arrangement. In this situation, the density functions of points are given by,

Di  x, h  NiD  x , h  , i .
N
(7)This equation indicates that the relative share of each type of point is constant everywhere in R.
We can confirm this in Figure 2c, which shows the density functions above applied to the-8-data in Figure 2a. The difference in the spatial arrangement of points disappears in Figure 2c. This
makes the distribution of the local segregation measure s(x, , h) uniform, as seen in Figure 2d;consequently, it reduces the overall level of segregation. Substituting Equation (7) into Equations (4)and (6), we obtain2

N 
S '     i  .
i  N (8)
The reduction of overall segregation measure indicates the degree of locational segregation:

SL , h   S , h   S ' N  S , h     i i  N 2.

(9)
SL(, h) is the locational segregation measure, and its range is 0SL(, h)<S(, h). This index
indicates the variation in the proportional share of points across locations, and has a large value when
the proportional share of points varies greatly across locations (i.e., when the different types of points
are separated from each other). The value of SL(, h) becomes zero when the proportional share isconstant in R.
We then consider compositional segregation. To evaluate this component, we again compare
two hypothetical cases. The first case is the same as the one that was considered during the evaluationof locational segregation, i.e., the absence of a difference in the spatial arrangements of points. For thesecond, we assume that both a difference in the spatial arrangement of points and an imbalance in theactual number of points do not exist. To represent this situation mathematically, suppose that the
fragmented pieces of K types of points {q1, q2, …, qK} are distributed at every location of the original
points, where each qi has the 1/K proportion of the total of that point. In this situation, the density
functions of the points are given by,

Di  x, h  D  x, h , i .
K(10)

This equation indicates that all types of points' are present in the same proportion across every locationin R (Figure 2e), and is applicable where both a difference in the spatial arrangements of points andan imbalance in the actual number of points are absent.
Figure 2f shows the distribution of the local segregation measure for this situation.
Compared with Figure 2d, it decreases everywhere in R, so does the overall segregation. This reduction
occurs because the imbalance in the number of points has vanished. Substituting Equation (10) into-9-

Equations (4) and (6), we obtain

S ''    

1
.
K
(11)

The degree of compositional segregation can be estimated by the reduction from S'(, h) to S''(, h):

S C     S '     S ''   
2

1
N 
  i  K
i  N .

(12)
SC() is the compositional segregation measure, and its range is 0SC()<S(, h). Unlike SL(, h),we omit h in the notation since Equation (12) is independent of the bandwidth h. This measure has alarge value when the proportional share of the point types is unbalanced across locations in R, andbecomes zero when all the types of points are present in the same proportion at every location.
Now we move on to the evaluation of qualitative segregation. To explore this component ofsegregation, we compare the case wherein a difference in the spatial arrangement of points and an
imbalance in the number of points do not exist (i.e., Equation (10)) with another hypothetical casewherein all three sources of segregation do not exist. To represent the latter case, we first assume thatthe fragmented pieces of K types of points {q1, q2, …, qK} are distributed at every location of theoriginal points, where each qi has a 1/K fraction of the type i points. Subsequently, we increase K
infinitely to resolve the lack of diversity of points. Under this circumstance, the density function of
the points approaches zero:Di  x, h   0, i(13)
Figure 2g shows the density functions of the points when all three sources of segregation do not exist.
The local segregation measure infinitely approaches zero everywhere in R as illustrated in Figure 2h,
as does the overall segregation. Thus, the qualitative segregation can be evaluated using the reduction
from S''() to zero:S Q     S ''     0


1
K

.

(14)
SQ() is the qualitative segregation measure. This measure represents the homogeneity of the points,
ranging from zero to one. A large value of SQ() implies that there are only a few different types ofpoints in R, whereas a small value indicates that a wide variety of points exist.

-10-The locational segregation evaluates the spatial aspect of segregation, and the compositionaland qualitative segregation deal with the aspatial aspects of segregation as mentioned previously. Thisis, in fact, reflected in the definitions of these measures. The definition of SL(, h) in Equation (9)contains the bandwidth h that determines the geographic scale of analysis, while SC() and SQ() aredefined only by aspatial variables as seen in Equations (12) and (14).
The relationship between these three measures can be expressed as:S   , h   S L   , h   SC     SQ    .
(15)This indicates that segregation can be decomposed into locational, compositional, and qualitative
segregations. The measures cover the whole range of segregation without overlap. This relationshipmakes the three measures comparable with each other so that we can tell what sources are more
influential on segregation. If, for instance, SL(, h) is larger than SC(), the difference in the spatial
arrangement of points is a more influential source than the imbalance in the number of points. If SC()is predominant among the three measures, the imbalance in the number of points is a primary sourceof segregation among the three sources.

2.3 Geographic scale of segregation
The measurement of segregation depends on the geographic scale of analysis as mentioned
earlier. This paper argues that the scale dependence occurs because segregation measured at a certain
scale consists of a collection of smaller elements observed at larger scales. This subsection discusses
the geographic scale of segregation in detail, and proposes another segregation measure.
Of the three components of segregation, we focus on locational segregation here because
the others represent aspatial aspects of segregation. Figure 4a illustrates SL(, h) as a function of the
bandwidth h. This figure is equivalent to the dissimilarity index D represented as a function of gridsize (Wong et al., 1999), the GD index proposed by Wong (2005), and the segregation profile proposed
by Reardon et al. (2008). The measure SL(, h) is usually a monotonically decreasing function of scaleparameter h, since the level of observed segregation tends to decrease with an increase in the scale ofanalysis.
The surface conversion is a kind of low-pass filter (Silverman, 1986; Sonka et al., 2014),
i.e., it conceals the spatial patterns in a point distribution with a scale smaller than the scale parameterh. The surface obtained by the surface conversion of scale h represents a collection of spatial patternsof scales larger than h in the point distribution, and consequently, the measure SL(, h) is thesummation of segregation of scales larger than h. This implies that the locational segregation can be
decomposed further into segregations of different scales, which we call differential locationalsegregations. The locational segregation observed at scale h is a collection of differential locational-11-

segregations measured at scales larger than h. This relationship is mathematically represented as

S L   , h    S DL   , u  du ,h(16)where SDL(, h) is the differential locational segregation measure of scale h. Differentiating both sidesof this equation with respect to h, we obtain

S DL   , h   dSL  , h  ,dh(17)Figure 4b presents SDL(, h) calculated from SL(, h) in Figure 4a. The measure SL(h1) in Figure 4a
is equal to the gray-shaded area in Figure 4b.
Calculation of SDL(, h) is, in a sense, a frequency resolution of SL(, h). Substituting
Equation (16) into (15), we obtainS   , h   SC     SQ      S DL   , u  du .
h(18)This equation indicates that segregation measured at scale h can be decomposed into compositionalsegregation, qualitative segregation, and differential locational segregations of scales larger than h.
Differential locational segregations of scales smaller than h are concealed in S(, h), which yields thescale dependence of S(, h).
2.4 Evaluation of spatial segregation by the entropy index
The entropy index has been often used as a measure of segregation in the literature (Theiland Finizza, 1971; White, 1986; Allen and Turner, 1989; Hårsman and Quigley, 1995; Reardon andO’Sullivan, 2004). This paper adopts different measures because the entropy index cannot evaluate
the three components of spatial segregation separately. However, if such a separation is notnecessary, we can use the entropy index within our framework. This subsection briefly describes theevaluation of spatial segregation by the entropy index.
We replace the local segregation measure s(x, , h) defined by Equation (4) with thatbased on the entropy index:Di  x, h   Di  x, h 
se  x ,  , h   1   
log
.
D
x
,
h


i   Dk  x , h 
k k

 k

(19)
The overall segregation becomes-12-

S e  D,  , h  xRD  x, h  se  x,  , h  dx

xRD  x, h  d x.

(20)
The locational, compositional, and qualitative segregation measures become

S Le  D ,  , h   S e  D ,  , h   S e  D L ,  , h Di  x, h   Di  x, h ,xR D  x, h  i   Dk  x, h  log  Dk  x, h  dx


Ni
N
k
 k
 

log i

N
N
i
 D  x, h  dx
xR

(21)

S Ce  D ,  , h   S e  D L ,  , h   S e  DC ,  , h  1 
i

,

Ni
N
log i
N
N

(22)
and

S Qe  D ,  , h   S e  DC ,  , h 
0

,
(23)

respectively. Equation (23) indicates that the entropy index conceals the qualitative segregation. The
ranges of the segregation measures are 0SLe(D, , h)<Se(D, , h) and 0SCe(D, , h)Se(D, , h) (aproof of the former is given in Appendix A4). The equalization measures can be calculated in a similar
way as that in the previous subsection.
Segregation measures defined from Equation (4) and those based on Equation (19) haveboth advantages and disadvantages. One strength of the former is that it permits us to decompose theaspatial components of spatial segregation into the compositional and qualitative segregations. The
latter is advantageous in that the entropy index has already been widely used in the measurement of
spatial segregation. Its implementation can be done with a slight modification of existing programs.

2.5 Comparison with Reardon and O’Sullivan’s paper
This paper shares an intention and a principle with Reardon and O’Sullivan (2004). Bothaim to present distinct dimensions of spatial segregation and define segregation measures based on thelocation of individual points. This subsection briefly compares the two papers in terms of segregation
dimensions and measures.
Reardon and O’Sullivan (2004) advocates the spatial exposure and the spatial evenness as-13-

the two dimensions of spatial segregation. The latter corresponds to the locational segregation
proposed in this paper. Spatial exposure contains at least the compositional segregation, but it is notclear whether it also contains the qualitative segregation since they do not explicitly discuss the spatial
exposure in terms of the multigroup segregation.
As shown in Section 2.2, the locational, compositional and qualitative segregations are
independent with each other. This implies that the spatial exposure and evenness are also independent.
We can change the former with keeping the latter, and vice versa. We should note, however, that thisdoes not assure the independency between their segregation measures. It is unknown whether their
spatial exposure index and spatial evenness indices are independent with each other.
To evaluate the spatial evenness, they advocate the spatial information theory segregationindex (for details, see Reardon and O’Sullivan (2004)). Let p and  pm be the density of points at
location p and the proportion of type m points in the local environment of p, respectively. The total
number of points is given by

T 

p R

 p dp .
(24)

The entropy index at location p isE p     pm log  pm .
m

(25)
The spatial information theory segregation index is defined by  1 1H
 p E p dp ,TE  pR(26)
where E is the overall entropy. The index corresponds to SL(D, , h) in this paper in that both compare
the spatial segregation with the overall segregation. While SL(D, , h) considers the difference evaluates their ratio.
between the segregations, H
 is notThe range of SL(D, , h) is known as 0SL(D, , h)<S(D, , h), while that of H

known. The latter can take a negative value, and we cannot compare the spatial segregation between
different patterns as mentioned in Section 1. This is presumably because the local entropy is calculated
from the local density of points while the overall entropy is based on the total number of points. The
summation of the local density is not always equal to the total number of points.
To avoid this problem, we suggest using the variables defined in the local environment
consistently to calculate segregation measures. Let  p be the density of points in the local
environment of p. We define the total volume of points T' as-14-

T' 

pR

 p dp .
(27)

Using these variables, we modify the spatial information theory segregation index as1
 p E p dp ,T ' E '  pR '  1
H

(28)
where

E '  m

pR

 pm dp

T'


log
M

pR

 pm dp

T'

.
(29)

The modified index would range from 0 to 1.

3. Applications
This section tests the validity of the proposed approach, by applying it to an analysis of three
data sets. For simplicity, we omit the indicators (, h) and () used in the notations hereafter.
3.1 Properties of the measures of segregation
This subsection explores the properties of the three segregation measures using a synthetic
data set. To focus on the relationship between the measures, we assume that point distributions are
already converted into density functions.
We first discuss the relationship between the locational and compositional segregation
measures. Figures 5a and 5b show the density distributions of two types of points in a 1D space. In
both figures, the proportion of each type of point varies in the vertical direction, while it is relatively
constant in the horizontal direction. The spatial arrangement of the points becomes more similar in
every row from columns A to E. Figures 5c and 5d present the segregation measures calculated for thedensity distributions in Figures 5a and 5b, respectively. The locational segregation measure SL
decreases from columns A to E as the difference in the spatial arrangement of the points decreases.
The imbalance in the proportion of the two types of point increases from columns 5 to 1 in Figures 5aand 5b; consequently, it raises the compositional segregation measure SC from columns 5 to 1 inFigures 5c and 5d. The qualitative segregation measure SQ is constant at 0.5 in all the patterns. The
compositional and qualitative segregation measures occupy the majority share of the overall
segregation, implying that aspatial sources are the primary source of segregation in Figures 5a and 6b.
From columns A to E in Figure 5c, SL changes when holding SC and SQ constant. In Figure 5d, on theother hand, SC changes when holding SL and SQ constant. This occurs because SL and SC do not overlap-15-with each other, i.e., they evaluate different aspects of segregation.
'Figures 5e and 5f display the modified spatial information theory segregation index H

defined by Sadahiro and Hong (2013) calculated for the density distributions shown in Figures 5a and ' is more similar to that of SL than
5b. Comparing Figures 5c-5f, we notice that the distribution of H
 ' represents the spatial aspect of segregation.
to that of S. This is reasonable, considering that H

Figures 5g and 5h display the index of dissimilarity developed by Duncan and Duncan (1955). The
two figures look similar to Figures 5e and 5f and thus the distribution of SL in Figures 5c and 5d.
Though the index of dissimilarity does not consider the spatial aspect of segregation explicitly, thisindex looks like being sensitive to the spatial arrangement of points.
Let us move on to the relationship between the compositional and qualitative segregation
measures. Figures 6a and 6c show the density distributions of points and their segregation measures,
respectively. SL is constantly zero because all the density distributions are uniform. The number oftypes of point increases from rows 1 to 5, while the imbalance in the proportion of points decreases
from columns A to E. Figure 6c demonstrates that the degree of qualitative segregation decreases withan increase in the number of types of point, as does the overall segregation. This supports our earlier
discussion in Figure 1d (i.e., that an increasing variety of points decreases segregation). We can also
confirm our observation in Figure 1c (i.e., an imbalance in the number of points increases segregation),by looking at the increase in SC and S from the columns E to A in Figure 6c.
We then discuss the relationship between the locational and qualitative segregation measures.
Figures 6b and 6d show the density distributions of points and their segregation measures, respectively.
The measure SC is constantly zero because all the types of points are present in the same proportion.
Similar to Figure 6c, Figure 6d shows that an increase in the number of types of point reduces SQ andS from rows 1 to 5. The locational and compositional segregation measures change while keeping SQ
unchanged in Figures 6c and 6d, implying that SL and SC does not overlap with SQ.
 ' . The
Figures 6e and 6f show the modified spatial information theory segregation index H

index is always equal to zero in Figure 6e, as it does not consider the lack of diversity of points as a
 ' in Figure 6f is more similar to the distribution
separate source of segregation. The distribution of H

of SL in Figure 6d, which is compatible with our observation in Figure 5.

3.2 Geographic scale of segregation
This subsection investigates the geographic scale of segregation. Figure 7a displays sevendifferent patterns of two types of points distributed on a 1D space of length 1. We convert the point
distributions into density functions by Gaussian kernel smoothing of bandwidth w, ranging from 0.001
to 1.
Figure 7b shows the relationship between the bandwidth h and the proposed segregation
measures. The scale of analysis changes from local to global with an increase of h. The vertical axis

-16-

shows SL+SC, i.e., the sum of the locational and compositional segregation measures. The
compositional and qualitative segregation measures are constant in all the patterns, as they are
independent of the scale of analysis. The latter is always 0.5(=1/2). SC is zero in patterns A to D, andit increases from pattern E to G with an increasing imbalance in the number of points, as indicated by
the arrows in Figure 7b. The locational segregation measure SL decreases monotonically with anincrease in the geographic scale, which reflects a shift in our viewpoint from local to global. SL
infinitely approaches zero, even when the points are clearly separated as shown in pattern D. It has alarge value, however, at a very local scale, even when the points are globally well mixed, as shown inpattern A. The locational segregation increases from patterns A to D, regardless of scale h, with theclustering of the white points (Figure 7b).
Figure 7c demonstrates the relationship between the bandwidth h and the differential
locational segregation measure SDL. The peak of SDL shifts from a local to a global scale from patterns
A to D. This indicates that the locational segregation in pattern A is primarily due to a small-scaledifference in the spatial arrangement of the points, while large-scale differences are more predominant
in pattern D. A similar shift of the peak is observed from patterns E to G, reflecting an increase in the
distance between the white points in Figure 7a.

3.3 Application of the analysis to a real data set
In this subsection, we use the proposed approach to analyze a real data set as a test of its
practical feasibility. As mentioned in Subsection 2.2, many segregation measures are defined based on
spatially-aggregated data. We thus use land use data in a lattice form to evaluate the performance ofsegregation measures calculated based on aggregated data.
We examine the changes in land use mixture observed in the urbanization process from 1974
to 1994 in Chiba, Japan. Chiba is adjacent to the east side of Tokyo Metropolis, the population of
Chiba had drastically increased with the expansion of the Tokyo metropolitan area during this period.
Urban sprawl spread from west to east all over Chiba, often with an undesirable land use mixture thatcaused traffic congestion and deteriorated the residential environment. The causes and processes ofthe land use mixture vary between different regions, and they have not yet been analyzed in detail.
This subsection reveals the detailed process of land use mixture in Chiba using the proposed measures
to evaluate the degree of mixture between different land uses.
We use the land use data of a 10-m resolution lattice provided by the Geospatial Information
Authority of Japan. The data were generated by recording land use categories at sample locations thatare distributed regularly at 10-m intervals on aerial photographs. To evaluate the land use mixture at a
local scale, we aggregated the data into the lattice format with 500 m resolution and calculated theoverall segregation measure S in each cell. We did not perform the kernel smoothing to focus on thechange of land use mixture rather than its spatial pattern.

-17-Figure 8 shows the distribution of the local segregation measure s(x). This measure indicatesthe imbalance in the proportion of different land use categories at location x, which usually decreases
with a change in the land use mixture. As seen in Figure 8, s(x) had decreased all over Chiba during
this period (dark shades indicate small s(x)). Land use mixture had spread in this area, especially along
railway lines. This is confirmed in Figure 9a, wherein the compositional segregation measure SC had
decreased from 0.41 to 0.34, as SC reflects the mean of s(x) of the entire region. Figure 9a shows thatthe locational segregation measure SC had slightly increased from 0.17 to 0.18. This indicates that thevariation in the pattern of land use mixture had gradually increased during this period. The spatial
information theory index shown in Figure 9a is similar to SL, which is again consistent with the result
obtained in Subsection 3.1.
We then examine the changes in land use mixture in more detail in six subregions, as shownin Figure 8. Subregions R1 and R2 experienced a rapid expansion of urban areas during the 1980s,
primarily as a result of new town construction. Subregions R3 and R4 both contain urban areas with along history, where the land use patterns were stable over the same period. Subregions R5 and R6 area mixture of urban, suburban, and rural areas. The former contains an old historical town, while thelatter is a new town located at the border between urban and rural areas.
Figure 9b-e shows the segregation measures calculated in the six subregions from 1974 to
1994. The measures showed similar changes, except those of subregion R3. These subregions
represent the overall tendency of the progress of land use mixture in Chiba. In subregion R3, SL
decreased and SC increased during this period. Having examined the land use pattern in R3 in detail,we found that the urbanization was already complete in 1974. Most of this subregion was alreadycovered by residential areas, and no significant changes occurred from 1974 to 1994. However, ingeneral, we can conclude that the land use mixture had changed all over Chiba, with an increase in thevariation in the land use mixture pattern from 1974 to 1994. Comparing Figures 9c and 9e, we do notfind a clear similarity between SL and the spatial information theory index. However, the spatial
information theory index is still closer to SL than SL and SL, which is not incompatible with the results
obtained in Subsection 3.1.

4. Properties of the proposed measures
The discussion in Section 1 suggests five desirable properties of segregation measures: 1)
each component should be fully represented by a single measure, 2) there should be a set of measuresthat addresses all aspects of segregation without overlap, 3) the measures should be consistently
defined independently of the data used in empirical studies, 4) the range of the measures should beanalytically known, and 5) interpretation of the measures should be intuitive and straightforward.
The proposed measures are defined based on different sources of segregation independently,
and they evaluate the different components of segregation. The measures have a one-to-one-18-correspondence to the components of segregation. Equation (15) indicates that they cover the whole
range of segregation without overlap. This confirms that the measures meet the first two criteria. We
should note, however, that the separability of components is not equivalent to empirical independence.
A typical example is Figures 5a and 5c, where SL increases and SC decreases simultaneously from rows
1 to 5. A correlation between measures can be observed even if the measures do not overlap with eachother. Empirical dependence is not incompatible with the separability of components.
The measures also satisfy the third and fourth criteria: the components of segregation we
propose in this paper and their associated measures are defined deductively prior to empiricalapplications, and their range is analytically known. As seen in the applications in Section 3, theinterpretation of the measures is straightforward. It is primarily because they clearly correspondent toeach component of segregation.
Reardon and O’Sullivan (2004) suggested eight criteria for segregation measures: 1) scaleinterpretability, 2) arbitrary boundary independence, 3) location equivalence, 4) population density
invariance, 5) composition invariance, 6) transfers and exchange, 7) additive spatial decomposability,
and 8) additive grouping decomposability. These criteria are satisfied by the definition of Di(x, h) ands(x, , h), and Equation (15), except scale interpretability and composition invariance. Scaleinterpretability requires a measure to become zero when all the groups share the same proportion in a
local environment. The measure S(, h) does not satisfy this criterion, as it becomes 1/K under thiscondition. However, if we interpret this criterion as being a requirement for a clear and interpretable
relationship between a measure and the degree of segregation, S(, h) satisfies this criterion since it
increases monotonically with the degree of segregation. Composition invariance, on the other hand,
was originally advocated by James and Taeuber, 1985, to which some papers, including Reardon andO’Sullivan (2004), raise objections. However, we can modify the proposed measures to satisfy it byreplacing Di(x, h) with the standardized density distribution of points i(x, h): i  x, h  Di  x , h 



y R

Di  y , h  d y.

(30)
Similar to existing measures, our measures aim to evaluate the different aspects ofsegregation. The locational segregation measure SL(, h) captures the spatial aspect of segregation,
which is also evaluated by evenness and concentration indices such as the index of dissimilarity, theGini index, the entropy-based indices, and the spatial information theory segregation index (Massey
and Denton, 1988; Reardon and O’Sullivan, 2004). Among those, the spatial information theory
segregation index often yields similar result with the locational segregation measure as seen in Section
3. The compositional and qualitative segregation measures SC() and SQ() focus on the aspatialaspect of segregation, which are similar to exposure indices including the isolation index, interaction-19-index, and the spatial exposure index. Compared with those existing indices, one strength of ourmeasures is that they satisfy the five desirable properties of segregation measures as discussed above.
The three measures evaluate the three different aspects of segregation without overlap, and they cover
the whole range of segregation. This permits us to evaluate the three components separately, andcompare the effect of three sources with each other.
The three measures are, in a sense, the minimum set of measures for evaluating the different
aspects of segregation. This paper, however, neither prohibits us from using other existing measures
nor forces us to use all the three proposed measures. Existing measures are still useful to evaluate
segregation from various perspectives, especially when the distinctiveness of measures is not essential.
One may argue that no segregation is observed in Figure 1c where black and white points share thesame spatial pattern. If social interaction is not considered to evaluate segregation, it is enough to use
only S(, h) or SL(, h).
The differential locational segregation is not identical to the geographic scale discussed inReardon et al. (2008), although both address the geographical scale of segregation. Differential
locational segregation is a subcomponent of locational segregation, while the geographic scale is anindependent dimension distinct from the spatial evenness.

5. Conclusion
This paper develops a new approach to the measurement of segregation among point
distributions. The major contribution of the paper is the development of new measures that evaluate
segregation from different perspectives. We argued that segregation can be decomposed into threecomponents that are supported by three measures respectively. The measures satisfy the five desirable
properties of segregation measures discussed in Subsection 2.2 that are not fully possessed by existing
indices. Since the three components of segregation are separately measurable, the proposed measures
can serve as a minimum set of measures that address distinct dimensions of segregation. The second
contribution of the paper is to advocate qualitative segregation. Existing studies consider the aspatialaspects of segregation primarily with respect to the imbalance in the proportion of different types ofpoints. As seen in Subsection 2.1, however, there exists a close relationship between the variety ofpoints and the level of segregation. Consideration of qualitative segregation provides us a moredetailed view of segregation. The third contribution of the paper is to introduce differential locationalsegregation. It is a representation of multi-scale aspect of segregation. Differential locationalsegregation permits us to consider the geographic scale of segregation explicitly, i.e., to decompose
locational segregation into smaller elements of different scales.
Finally, we discuss some limitations of the paper and potential directions for future research.
First, this paper does not evaluate the statistical significance of segregation measures. One
method to test the significance is to calculate the measures of spatial autocorrelation for individual-20-

types of points. Johnston et al. (2010) and Poulsen et al. (2011) used Moran's I and Getis's G statistics
to test the significance of the spatial clustering of individual ethnic groups. This method, however,does not directly evaluate the segregation of more than two types of points. A statistical test thatconsiders the segregation of multiple types of points simultaneously needs to be developed.
Second, this paper evaluates segregation of points without considering their attributes indetail. This treatment, however, may not be appropriate in the real world. For instance, segregation
between two population groups with very different socio-economic levels may have a larger impact
on society in comparison with that of two groups with similar profiles. In such a case, we should put
more weight on the former in the evaluation of segregation. To accomplish this, we should explicitly
take into account the attributes of points in segregation measurement.
Third, as Reardon (2009) stated, most of the existing studies have discussed segregation
among groups classified by nominal variables. However, groups can be categorized by other scales
such as ordinal, interval, and ratio variables. Reardon (2009) developed ordinal segregation measures,and Reardon and Bischoff (2011) used them in the analysis of income segregation in the U.S. from1970 to 2000. Extension of the proposed measures in this direction might be an important topic for
future research.

References
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portray ethnic residential segregation: a London example" Environment and Planning B 38 636658
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Reardon SF, Bischoff K, 2011, "Income inequality and income segregation" American Journal of
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33-67
Reardon SF, O’Sullivan D, 2004, "Measures of spatial segregation" Sociological Methodology 34 121162
Reardon SF, Matthews SA, O’Sullivan D, Lee BA, Firebaugh G, Farrell CR, Bischoff K, 2008, "Thegeographic scale of metropolitan racial segregation" Demography 45 489-514
Sadahiro Y, Hong S-Y, 2013, "Decomposition approach to the measurement of spatial segregation"
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White MJ, 1983, "The measurement of spatial segregation" American Journal of Sociology 88 10081018
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Wong DWS, 2005, "Formulating a general spatial segregation measure" Professional Geographer 57
285-294
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-22-

(a)

(b)

(c)

(d)

(e)

Figure 1 Point distributions representing individuals of different ethnicities.

-23-

0.5

0.0
(a)

(b)
0.5

0.0
(c)

(d)
0.5

0.0
(e)

(f)
0.5

0.0
(g)

(h)

Figure 2 Density functions of points assumed in the evaluation of segregation on a one-dimensional
space and the distribution of local segregation measure. Density functions where (a) all the threesources of segregation are present, (c) the difference in the spatial arrangement of points is absent,(e) the difference in the spatial arrangement of points and the imbalance in the number of points areabsent, (g) all the three sources of segregation are absent. Figures 2b, 2d, 2f, and 2h indicates the
distribution of local segregation measure of density functions shown in Figure 2a, 2c, 2e, and 2g,
respectively. Different textures indicate different types of points.

-24-

(a)

(b)

Figure 3 Evaluation of locational segregation. (a) Distribution of four black and eight white points.
(b) A hypothetical situation where both black and white points have the same spatial arrangement.

-25-

0.5
SL(, h1)SL(, h)

0.0
h

h1

Local

Global(a)

0.2

SDL(, h)SL(, h1)

0.0
h
Local

h1
Global(b)

Figure 4 Relationship between the locational segregation measure and the differential locationalsegregation measure. The horizontal axis indicates the window width h used in kernel smoothing.

-26-

1

1

2

2

3

3

4

4

5

5
A

B

C

D

E

A

B

C

(a)

E

(b)

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2
0.0
A

D

0.2
B

C

D

E

5

4

3

2

1

0.0
A

B

SL

C

D

SC

(c)

E

5

4

3

SL
SQ

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2
B

C

D

E

5

4

3

2

1

0.0
A

B

C

D

(e)

E

5

4

3

2

1

(f)

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2
0.0
A

1

SC

(d)

SQ

0.0
A

2

0.2
B

C

D

E

5

4

3

2

1

0.0
A

(g)

B

C

D

E

5

4

3

2

1

(h)

Figure 5 Density distributions of points (a, b), their segregation measures (c, d), the spatial
information theory segregation index (e, f), and the index of dissimilarity (g, h). Different colors
indicate different types of points in Figures 5a and 5b.

-27-

1

1

2

2

3

3

4

4

5

5
A

B

C

D

E

A

B

C

(a)

E

(b)

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4
0.2

0.2
0.0
A

D

B

C

D

E

5

4

3

2

1

0.0
A

B

SL

C

D

SC

(c)

E

5

4

3

1

SL
SC

(d)

SQ

SQ

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2
0.0
A

2

0.2
B

C

D

E

5

4

3

2

1

0.0
A

B

(e)

C

D

E

5

4

3

2

1

(f)

Figure 6 Density distributions of points (a, b), their segregation measures (c, d), and the spatial
information theory segregation index (e, f). Different colors indicate different types of points in
Figures 6a and 6b.

-28-

0.5

D
C

SL+SC

G

SC(G)

B

A

F

A
0.0
0.001

B
C

SC(F)

E
0.01

SC(E)

0.1

h

1

Local

Global
(b)

D
0.2

D

E

C
F

G

SDL

G

F
E

(a)

B

A
0.0
0.001

0.01

h

0.1

Local

1
Global(c)

Figure 7 Segregation measures of the distribution of two types of points on a one-dimensional
space. (a) Point distributions, (b) the summation of the locational and compositional segregation
measures, (c) the differential locational segregation measure. Bold solid lines indicate patterns from
A to D while bold broken lines indicate patterns from E to G.

-29-

N

0

N

5

10km

0

R1

10km

R1

R5

R2

R5

R2

R3

R3

-0.10

-0.10

-0.20

-0.20

-0.30

-0.30

-0.40
-0.50

5

-0.40

R6
R4

-0.50

-0.60

-0.60

-0.70

-0.70

-0.80

-0.80

-0.90

-0.90

(a)

R6
R4

(b)

Figure 8 The local segregation measure s(x) of land use pattern in (a) 1974 and (b) 1994.

-30-

1.00

0.50

0.80

R5

0.40

0.60

R2
R6
R1
R3
R4

0.30

0.40
1.00
0.20
0.80



H’
S

0.60

0.00
1974

R5

R2
R6
R1

0.20

R4
R3

0.10

1979

1984

1989

1994

0.00
1974

1979

(b)

0.40

1984

1989

1994

(c)

SC
SL
SQ

0.20
0.00
1974

1979

1984

1989

0.50

1.00

0.40

0.80

R5
R2
R3

1994
0.30

R5

(a)

R3
R6
R2
R4
R1

0.20
0.10
0.00
1974

1979

1984

(d)

1989

1994

0.60

R4
R6
R1

0.40
0.20
0.00
1974

1979

1984

1989

1994

(e)

Figure 9 The segregation measures of land use pattern from 1974 to 1994. (a) Segregation measuresof the entire region, (b)-(e) Segregation measures of the subregions shown in Figure 8. (b) The
overall segregation measure S, (c) the locational segregation measure SL, (d) the compositionalsegregation measure SC, (e) the spatial information theory segregation index.

-31-

Appendix A1This appendix derives the minimum value of s(x, , h) defined by Equation(4) using theLangrangian multiplier method. Let us define di(x, h) as

d i  x, h  Di  x, h  D  x, h 

.

k

k

(31)
We minimizef   di2  x, h    1   di  x, h  
i
i


(32)
with constrain d  x, h   1 ,
i

i

(33)
where  is the Langrangian multiplier. Partial differentiation of f by di(x, h) and that by  aref
 2 d i  x, h   di  x, h 
(34)
and

f
 1   d i  x, h  ,

i
(35)
respectively. Sincef 0,d i  x, h (36)
we obtaindi  x, h  


2

.
(37)

Substitution of Equation (37) into Equation (35) yields

1   d i  x, h   1 
i

K
2

 0.
(38)

-32-

Solving this equation, we have

2
.
K



(39)
Substituting the above equation into Equation (38), we obtaind i  x, h  

1
.
K
(40)

This gives the minimum of s(x, , h), that is,s  x ,  , h    d i  x , h 

2

i

.

1

K

(41)
Appendix A2This appendix derives the minimum value of SL(, h). To this end, we divide region R into
M subregions denoted by {R1, R2, ..., RM}. Let Dij be the density of type i points in Rj, whichcorresponds to Di(x, h) in Equation (2). The proportion of type i points in Rj is given by

pij 

Dij

D

.

ij

i

(42)
The total density of points in Rj is denoted by i. The total density of all the points is

T   A Rj   j ,j(43)
where A(Rj) is an operator that gives the area of Rj.
The locational segregation SL(, h) is defined as

S L  , h  1N 
A  R j   j  pij2    i T ji
i  N 221 
 N i  2
   A  R j   j pij    T 
T i  j N  .

(44)
We use the Langrangian multiplier method, where we minimize

-33-

N

N 
fi   A  R j   j p   i  T    i T   pij A  R j   j  .
N
j
j
N

2

2
ij

(45)
The constraints are

i,  pij A  R j   j 
j

Ni
T
N
(46)

Partial differentiation of fi by pij and that by  aref i
 2 A  R j   j pij   A  R j   j .
pij
(47)
and

f i N i
T   pij A  R j   j ,
 N
j
(48)
respectively. Solving

f i
 0,
pij
(49)
we obtain

pij 


2

.
(50)

This equation indicates that pij does not depend on i. We thus obtain

pij 

Ni
.
N
(51)

and the minimum of SL(, h):221  N i   N i  
S L   , h     A  R j   j      T 
T i  j N   N   .

0
(52)

Appendix A3-34-This appendix proves that the qualitative segregation is independent of the locational andcompositional segregations. To this end, we add m types of points to the point set considered in Section
2. We keep the total number of points but can modify the number of each type of points. Let Ni’ be thenumber of type i points after the addition of new points.
We first consider the independency of the qualitative segregation from the locationalsegregation. The locational segregation remains unchanged when the following equation holds for any
x:
2

2





 Di  x, h   Di '  x, h   .
i  D  x, h    i  D '  x, h  
 k

 k

 k

 k

(53)
This condition is equivalent to p   p' ,
2
i

2
i

i

i

(54)
under the constraint: p   p '  1.
i

i

i

i

(55)
We assume m=1 and pi=pi’ for i=1, ..., K-2. Equations (54) and (55) become

p K2 1  pK2  p '2K 1  p '2K  p '2K 1 ,
(56)
and

p K 1  p K  p ' K  p ' K 1  p ' K 1 .
(57)
We can rewrite pK-1 as

p K 1  p K  a(58)
without losing generality. Substituting Equations (57) and (58) into Equation (56), we solve it in termsof pK:pK   p ' K  p ' K 1   2 p ' K p ' K 1  ap ' K  ap ' K 1
2
(59)

The condition of the existence of a positive pK is-35-

4 a  p ' K  p ' K 1    p ' K  p ' K 1  .
2(60)
It is always possible to satisfy the above inequality by choosing close values of p’K and p’K+1.
Consequently, we can change the locational segregation with keeping the qualitative segregation.
We then consider the independency of the qualitative segregation from the compositionalsegregation. The compositional segregation is unchanged when
2

2

1
1
N 
 N '
i  Ni   K 2  i  Ni   K  m 2 .
(61)
Solving Equation (61) in terms of m, we obtain

K2

m

2

2

 N '
N 
K 2   i   K 2   i  1
i  N i  N K .

(62)
Since m is positive, the following inequality needs to hold:
2

2

2

1
N 
 N '
N 
i  Ni   K 2  i  Ni   i  Ni  .
(63)
It is always possible to satisfy the above inequality by slightly reducing every Ni and add a new typeof points. Consequently, we can change the qualitative segregation with keeping the compositionalsegregation.

Appendix A4This appendix derives the minimum value of the locational segregation SLe(D, , h). We use
the same setting as that of Appendix A2. The locational segregation SLe(D, , h) is written as

S Le  D,  , h  1 A  R j   j i pij log pij  i pij log pij
T j

N
N 
1 
   A  R j   j pij log pij  i T log i 
T i  j
N
N

.

(64)
We use the Langrangian multiplier method to calculate the minimum value of SLe(D, , h). We
minimize-36-fi   A  R j   j pij log pij 
j

N

Ni
N
T log i    i T   pij A  R j   j  .
N
N
j
N

(65)

with constraints

i,  pij A  R j   j 
j

Ni
T.
N
(66)

Partial differentiation of fi by pij and that by  aref i
  j A  R j  log pij  1    .
pij
(67)
and

f i N iA  R    pij A  R j  , Njrespectively. Solving

f i
 0,
pij
(68)
we obtain

pij  K 1  .
(69)This implies that pij is independent of i. From this we derive

pij 

Ni
.
N
(70)

Substitution of Equation (70) into (64) yields

S Le  D, , h  

Ni
N N
N 
1 
log i  i T log i  A  R j   j
T i  j
N
N N
N .

0
(71)

-37-