CSIS Discussion Paper No. 119

Decomposition approach to the measurement of spatial segregation

Yukio Sadahiro* and Seong-Yun Hong**

February 2013

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

Abstract
This paper proposes a new method of measuring spatial segregation. Many of the existingmeasures of spatial segregation assume spatial data aggregated by spatial units. This leads to a
problem called the modifiable areal unit problem, where measures are inevitably unstable andunreliable because of their sensitivity to the definition of spatial units. Another problem of theexisting measures is that the relationship between the different dimensions of spatial segregation is
not fully revealed and understood. This prevents us from extracting the distinct and independent
dimensions of spatial segregation. To resolve the problem, this paper takes a decomposition
approach to the measurement of spatial segregation. Our primary goal is to decompose the spatial
segregation into mutually exclusive and independent components. We introduce three measures ofspatial segregation each of which evaluates a different dimension of spatial segregation. To test thevalidity of the method, we apply it to three spatial datasets of different sizes. The result shows thatthe method is effective for evaluating the spatial segregation from three different perspectives.

1. Introduction
Spatial segregation is a fundamental concept in geographical information science. It refers
to the spatial isolation of different types of spatial objects. Spatial segregation of racial andoccupational groups has drawn much attention of geographers and sociologists. Spatial segregationof plant and animal species has been discussed in biology and ecology. Landuse mixture, the lack ofspatial segregation, is a critical issue in urban and regional planning.
Numerous measures have been proposed in the literature to evaluate the degree of spatial
segregation quantitatively. Most of them utilize spatial data aggregated by spatial units such as
census tracts, zip-code areas, and school districts. Among those, one of the most commonly usedmeasure is the dissimilarity index proposed by Duncan and Duncan (1955). Though its suitability
has been debated for a long time (White, 1983), the index has been widely used in racial segregation
analysis because of its ease of calculation and interpretation.
The dissimilarity index and its variants are aspatial measures in the sense that they areinsensitive to the arrangement of spatial units. Spatial measures have been developed to address thislimitation by explicitly considering the distribution of spatial units. Morgan’s distance-based
measure of segregation (1983) and White’s spatial segregation indices (1983) calculate the distance
between spatial units and incorporate it into the definition of their measures. Morrill (1991) andWong (1993) introduced an additional component to the dissimilarity index, so the degree ofsegregation can be moderated by the arrangement of spatial units.
The above measures are all defined based on aggregated spatial data. Considering thatspatial data are often provided in an aggregated form to keep the confidentiality of individuals these
unit-based approaches can be useful to some extent. This, however, causes a problem called themodifiable areal unit problem. Since spatial measures are sensitive to the definition of spatial units,
they are inevitably unstable and unreliable.
Reardon and O’Sullivan (2004) appropriately points out this problem and proposes newmeasures of spatial segregation. They define the measures directly from the location of individual
points instead of using aggregated spatial data. When spatial data are available only in an aggregatedform, the measures approximate the true values by assuming the uniform distribution of point ineach unit. This assures that the measures converge to the true values with an increase in the
resolution of spatial data and permits us to avoid the instability in the evaluation of spatial
segregation.
Another significant contribution of Reardon and O’Sullivan (2004) is that it examines indetail the five dimensions advocated by Massey and Denton (1988) and re-conceptualizes into two
fundamental dimensions. They claim that the distinction between the dimension of evenness and thatof clustering is arbitrary and propose two dimensions called the spatial exposure and the spatial
evenness as principal and distinct dimensions of spatial segregation.

Their approach, however, has several problems that remain to be resolved. First, they did
not define the spatial exposure and evenness in a formal way. Though their discussion is adequate
and persuasive, they did not indicate a precise definition of the two dimensions. Second, theindependency between the two dimensions is not fully discussed. They argue that the dimensions are
distinct, but do not give a tangible proof of the independency between exposure and evenness
measures. Third, the range of the segregation measures is not completely shown. This causes adifficulty in comparing the spatial segregation between different patterns because we cannot
standardize the measures to obtain relative and comparative ones.
To resolve the above problems, this paper proposes a new method of measuring spatial
segregation. Following the line of Reardon and O’Sullivan (2004), we discuss the spatial segregationof point objects whose exact location is assumed to be known. Our aim is to decompose the spatial
segregation into different components that are mutually exclusive and independent. Section 2 defines
several measures of evaluating the spatial segregation. Section 3 calculates the measures on threespatial datasets of different sizes. Section 4 summarizes the conclusions with discussion.

2. Method
Spatial segregation has several different aspects. Suppose the four distributions of whiteand black points shown in Figure 1. The same number of white and black points are distributed in
Figure 1a and Figure 1b, while black points are fewer than white points in Figure 1c and Figure 1d.
White and black points are uniformly distributed in Figure 1a and Figure 1c, while they are spatiallyseparated in Figure 1b and Figure 1d.
Figure 1b and Figure 1d present complete spatial segregation of white and black points.
Points are highly mixed in Figure 1a. In Figure 1c, though white and black points are globally mixed,
a local segregation is observed due to the inequality in the number of points. Many white points do
not have black neighborhoods except those located close to black ones.

(a)

(b)

(c)

(d)

Figure 1 Four distributions of white and black points.
This implies that spatial segregation depends on at least the difference in location and thatin number of points. This distinction almost corresponds to the spatial evenness and exposureadvocated by Reardon and O’Sullivan (2004). This paper, however, further discusses spatial
exposure from a different perspective. Figure 2 indicates the distributions of individuals of different
species. If a white circle moves around its neighborhood, it is more probable to meet an individual ofa different species in Figure 2b than in Figure 2a. Spatial exposure depends on the difference in not
only the number but also the diversity of points.

(a)

(b)

Figure 2 Distributions of individuals. Different symbols indicate different species.
The above observation suggests three factors of spatial segregation: locational unevenness,compositional unevenness, and qualitative homogeneity. The locational unevenness causes a
difference in location of points, while the compositional unevenness increases the difference in thenumber of points, and consequently, both increase spatial segregation. The qualitative homogeneity
decreases the variation of points, which also increases spatial segregation. We discuss these threefactors in a more formal and quantitative way in the following.

2.1 Evaluation of spatial segregation
Suppose K types of points distributed in region R. We denote jth point of type i and its
location as Pij and zij, respectively. Let Λi={Pi1, Pi2, ..., PiNi} be the set of type i points. The total
number of points is N (=N1+ N2+... + NK).
Let φ(x, zij) be a proximity function that indicates the spatial proximity between location x
and point Pij. One definition of φ(x, zij) is a distance-decay function such as(

φ ( x, z ij ) = exp −α x − z ij

)
.
(1)

We may also employ the Voronoi diagram, where points of the same type serve as generators. The
proximity function φ(x, zij) is1 if x ∈ Vij
0 otherwisef ( x, z ij ) = ,(2)where Vij is the Voronoi region of Pij.

Using the proximity function, we define the density functions of points. Though they arespatially-weighted, we simply call them density functions in the following. The density of Pij at
location x is defined as

φ ( x, ζ ij )

ζ ij ( x ) =∫y∈Rφ ( y , ζ ij ) dy.

(3)
The density function of type i points is given by

Di ( x ) = ∑ ζ ij ( x )j=∑jφ ( x, ζ ij )

∫

y∈R

.

φ ( y , ζ ij ) dy
(4)

We can confirm∫x∈R

Di ( x ) dx = N i ,(5)where Ni is the number of type i points.
We then define segregation measures. The local segregation at x is measured by2 Di ( x )  .
t (x) = ∑ i  ∑ Dk ( x ) 
 k

(6)
Integration of t(x) over R yields the overall segregation:

S=

∫

x∈R

∫

D ( x ) t ( x ) dxx∈RD ( x ) dx

,

(7)
where

D ( x ) = ∑ Di ( x ) .
i(8)
The minimum of t(x) is 1/K as shown in Appendix A1. Its maximum is 1, and it occurs only when asingle type of points exist. The range of S is thus 1/K <S≤1.
The overall segregation S is a composite of segregations caused by the three factors

mentioned earlier. We will decompose it into three components in the following.
The first component is the locational segregation. We can estimate this aspect ofsegregation by eliminating the locational unevenness in a given point distribution, calculating thedegree of segregation for it, and comparing it with that for the original point distribution. The
elimination is completed by the locational equalization, which converts the density distribution of
every type of points into the uniform distribution with keeping the total density distribution of points.
The density function of type i points becomes

Di ' ( x ) =Ni
D (x) .
N
(9)

As a result, spatial segregation reduces from S to2

N 
S ' = ∑ i  .
i  N (10)
Figure 3a illustrates the density distributions of five types of points on a one-dimensional space. The
locational equalization transforms the distributions into those shown in Figure 3b.

1
2

3
4
5

(a)

1
2

3
4
5

(b)

1
2
3
4
5

(c)

Figure 3 Elimination of locational and compositional unevenness in point distributions. (a) Density
distributions of five types of points on a one-dimensional space. Different patterns indicate different
types of points. (b) Density distributions obtained after the locational equalization. (c) Density
distributions obtained after the compositional equalization.

The locational segregation is measured by the decrease from S to S’:

S L= S − S '
=

∫

x∈R

∫

D ( x ) t ( x ) dxx∈RD ( x ) dx

2N  .
− ∑ i 
i  N (11)

Since 0<S’ and 0≤SL (a proof is shown in Appendix A2), the range of SL is 0≤SL<S. It shows a large
value when the locational equalization is very effective, that is, the spatial segregation greatly varies
between different locations. The measure SL becomes zero if the locational equalization does not atall reduce the spatial segregation. This occurs when all the density distributions of points are alreadyuniform before the locational equalization.
The second component of spatial segregation is the compositional segregation. We define
it as the segregation further reduced from S’ by eliminating the compositional unevenness in point
distributions. This elimination is performed by the compositional equalization, which equalizes the
proportion of the total density between different types of points. The compositional equalizationtransforms the density function of type i points into

Di '' ( x ) =

1
.
K
(12)

and reduces spatial segregation to

S '' =

1
.
K2
(13)

Figure 3c shows the result of the compositional equalization applied to the distributions shown in
Figure 3b. The compositional segregation is measured by

SC= S '− S ''
2

=

1 .
 Ni 
−∑i  N  K 2(14)

The range of SC is 0≤SC<S’, which can be proved in a similar way as that of Appendix A1. It shows alarge value if the number of points greatly varies between different types of points. The measure SC
becomes zero when the compositional equalization is completely ineffective, that is, all the types ofpoints have the same number of points.
The third component of spatial segregation is the qualitative segregation. We define it as
the segregation reduced from S’’ by eliminating the qualitative homogeneity of points. The
elimination is completed by increasing infinitely the types of points. As K approaches infinity, thespatial segregation infinitely reduces. Consequently, the qualitative segregation is measured by

SQ= S ''− 0
=

1
K2

.

(15)
The range of SQ is obviously 0<SQ≤1.
The locational segregation evaluates the spatial aspect of segregation while thecompositional and qualitative segregations consider the aspatial aspects. The decomposition of
spatial segregation into the three components permits us to discuss the different aspects of spatial

segregation separately. Figure 4 displays the relationship between the segregation measures. As seenin the figure, the locational, compositional and qualitative segregations are mutually exclusive. They
are also independent with each other in that we can change one aspect of segregation with preserving
the other two dimensions. For instance, the locational and compositional segregations are
independent with each other, and they are both independent of the qualitative segregation. We willshow concrete examples in the next section. The qualitative segregation is also independent of thelocational and compositional segregations. A proof is given in Appendix A3.

SL

S

SC
S’

S’’

SQ

Figure 4 Relationship between the segregation measures.
The segregation measures proposed above do not meet the criterion of composition
invariance defined by James and Taeuber (1985). Their definition claims that a measure should
remain unchanged if the distribution of spatial objects changes but its relative spatial distribution
does not change. Though some papers raise objections to this definition (Coleman et al., 1982;
Reardon and O’Sullivan, 2004), we can modify the proposed measures to satisfy it by using thestandardized density distribution of points instead of Di(x):ηi ( x ) =Di ( x )∫y∈R

Di ( y ) dy.

(16)
Since it is independent of the absolute distribution of points, the segregation measures defined basedon Equation (16) meet the criterion of composition invariance.
The segregation measures proposed above evaluate the three components of spatial

segregation by using the result of eliminating the three factors of spatial segregation successively
from the overall segregation. This implies that the measures cannot distinguish point distributions
that share the same values of segregation measures. Suppose the checkerboard patterns (Wong, 2005;
O’Sullivan and Wong, 2007) of different resolutions as shown in Figure 5. Each cell contains thesame number of white or black points. If we define the density functions of points by simply
dividing the number of points by the cell size, the two patterns show the same values of segregation
measures, and consequently, we cannot distinguish the two patterns.

Figure 5 Checkerboard patterns of different resolutions that cannot be distinguished by the
segregation measures.

One option to resolve the problem is to use a distance-decay function such as those definedby Equation (1). This enables us to take into account explicitly the spatial dimension of density
distributions, and yields different values of segregation measures between the two patterns.
Interpretation of the result, however, is not straightforward since the result heavily depends on thedefinition of distance-decay function. Its interpretation requires us to fully understand the propertiesof distance-decay function and the relationship between the function and result, especially when the
point distributions present more complicated patterns.
The above problem arises because the locational unevenness is evaluated in the aspatial
rather than the spatial dimension. The locational segregation SL is dimensionless as seen in Equation(11). To resolve the problem, we evaluate the locational equalization in the spatial dimension using
the earth moving transformation (Peleg et al., 1989; Rubner et al., 2000; Zhao et al., 2010, Sadahiro,2012). The earth moving transformation is originally a transportation problem that converts a pile ofdirt into another form by transferring the dirt between regions with the least cost. We regard thelocational equalization defined by Equation (9) as the earth moving transformation from Di(x) intoDi’(x). The solution gives the volume of transfer between locations, from which we calculate two
equalization measures useful for evaluating the locational equalization.
To discretize the density distributions of points, we divide region R by M subregions {R1,

R2, ..., RM} of the same size. The subregions should be small enough for a close approximation. Let
Dij be the density of type i points in Rj. The density of type i points transferred from Rj to Rk isdenoted by xijk (xijk≥0). The locational equalization of Di(x) is formulated as the following
optimization problem where the total volume of transfer is minimized.

Problem LE (Locational equalization):∑∑ x

minimizeijk

xijk , j , k∈Njksubject toNi
Dij − ∑ xijk + ∑ xikj =
A( R)
j
k
xijk ≥ 0 if j and k are adjacentxijk = 0 otherwise
Operator A(R) gives the area of region R. Problem LE is a linear optimization problem whose
computational complexity is O(n). It is thus solvable in a linear time.
From the solution of Problem LE, we calculate the weighted average distance of locational
equalization. Let ljk be the distance between Rj and Rk. The equalization distance is defined byLE =2∑∑∑ l jk xijk
i

j

k

∑∑∑ x

ijk

i

=

j

k

2∑∑∑ l jk xijk
i

j

k

.

Ni

∑∑ D − A ( R )
ij

i

j

(17)
This distance measure permits us to evaluate the locational equalization in the spatial dimension.
Another measure useful for evaluating the locational equalization is the ratio of the total volume oftransfer called the equalization ratio:

∑∑∑ x
R =
∑∑ D

ijk

i

j

k

E

ij

i

j

Ni

∑∑ D − A ( R ) .
ij

=

i

j

2∑∑ Dijij

(18)This measure indicates the discrepancy of the density distributions of points from the uniform
distribution. It ranges from zero to one.

2.2 Evaluation of spatial segregation by the entropy index
Existing studies of spatial segregation often adopt the entropy index as a measure of spatial
segregation (Pielou, 1977; White, 1986; Allen and Turner, 1989; Wong, 1993; Reardon andO’Sullivan, 2004). This paper, on the other hand, defines segregation measures in a different manneras shown in Equation (6). This is because the entropy index prevents us from decomposing thecompositional and qualitative segregations separately as shown later. However, if they can be treatedas a single component of segregation, we can use the entropy index within our framework. This
subsection briefly describes the evaluation of spatial segregation by the entropy index.
We replace the local segregation measure t(x) defined by Equation (6) with that based on
the entropy index:
Di ( x )  Di ( x )te ( x ) = 1 + ∑ 
log
.
x
D
(
)
i  ∑ Dk ( x )∑k k 
 k(19)
The overall segregation becomes

Se =

∫

x∈R

D ( x ) te ( x ) dx∫x∈RD ( x ) dx
(20)

Segregation measures S’ and S’’ are defined as

Se ' = 1 + ∑

Ni
N
log i
N
N

Se '' = 1 + ∑

1
1
log
K
K,

i

and

i

=0
(21)
respectively. The locational, compositional, and qualitative segregation measures becomeS Le= Se − Se '
Di ( x )  Di ( x ),∫x∈R D ( x ) ∑i  ∑ Dk ( x ) log ∑ Dk ( x ) dx


Ni
N
k
 k
 −
log i
∑
N
N
i
∫ D ( x ) dx
x∈R

(22)

SCe
= Se '− Se ''= 1+ ∑
i

Ni
N ,
log i
N
N
(23)

and

SQe = Se ''
=0

,
(24)

respectively. Equation (24) indicates that the entropy index conceals the qualitative segregation. This
prevents us from evaluating spatial segregation caused by the qualitative homogeneity shown in
Figure 2. The ranges of the segregation measures are 0≤SLe<Se and 0≤SCe≤Se (a proof of the former isgiven in Appendix A4). The equalization measures can be calculated in a similar way as that in theprevious subsection.
Segregation measures defined from Equation (6) 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.

2.3 Comparison with Reardon and O’Sullivan’s paper
This paper shares an intention and a principle with Reardon and O’Sullivan (2004). Bothaim to evaluate spatial segregation from different perspectives and define segregation measures
based on the location of individual points. This subsection briefly compares the two papers in termsof segregation dimensions and measures.
Reardon and O’Sullivan (2004) advocates the spatial exposure and the spatial evenness as
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 thespatial exposure in terms of the multigroup segregation.
As shown in Section 2.1, 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 this does not assure the independency between their segregation measures. It is
unknown whether their spatial exposure index and spatial evenness indices are independent witheach other.
To evaluate the spatial evenness, they propose the spatial information theory segregationindex defined by = 1− 1Hτ p E p dp ,TE ∫p∈R(25)
where T and E are the total number of points and the overall entropy, respectively. This corresponds
to SL in this paper in that both compare the spatial segregation with the overall segregation. While SL evaluates their ratio.
considers the difference between the segregations, H
 is not known. The latter can take a
The range of SL is known as 0≤SL<S, while that of H
negative value, and we cannot compare the spatial segregation between different patterns asmentioned 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 population of points. Unlike
Equation (5), the summation of the local density is not always equal to the total population.
To avoid this problem, we suggest using the variables defined in the local environment
consistently to calculate segregation measures. We modify the spatial information theory segregationindex as1
τ p E p dp ,T ' E ' ∫p∈R

 '= 1 −
H

(26)
where

T'=∫

p∈R

τ p dp
(27)

and
M

E ' = −∑

m =1

∫

p∈R

τ pm dp

T'

log M

∫

p∈R

τ pm dp

T'

.
(28)

The modified index would range from 0 to 1.

3. Applications

To test the validity of the method proposed above, this section applies it to an analysis ofthree datasets of different sizes.

3.1 Small synthetic dataset
This subsection aims to investigate the properties of the segregation measures. Figure 6
and Figure 7 show a small synthetic dataset of the density distributions of points on a
one-dimensional space and their segregation measures.

1
1.0
2

0.8
0.6

3

0.4
4

0.2
0.0
A

5
A

B

C

D

B

E

C

D

(a)

E

5

4

3

2

1

SL
SC
SQ

(c)

1
1.0
2

0.8
0.6

3

0.4
4

0.2
0.0
A

5
A

B

C
(b)

D

E

B

C

D

E
(d)

5

4

3

2

1

SL
SC
SQ

Figure 6 Density distributions of points (a, b) and their segregation measures (c, d). Different colors
indicate different types of points in Figure 6a and b.

1
1.0
2

0.8
0.6

3

0.4
4

0.2
0.0
A

5
A

B

C

D

B

E

C

D

(a)

E

5

4

3

2

1

SL
SC
SQ

(c)

1
1.0
2

0.8
0.6

3

0.4
4

0.2

5

0.0
A
A

B

C

D

E

(b)

B

C

D

E
(d)

5

4

3

2

1

SL
SC
SQ

Figure 7 Density distributions of points (a, b) and their segregation measures (c, d). Different colors
indicate different types of points in Figure 7a and b.

We first discuss the relationship between the locational and compositional segregations inFigure 6. To focus only on the two segregations, we consider the distributions of two types of points.
In every row from column A to E, the proportion of two types of points remains unchanged while thegradient of the boundary between two types of points changes. The gradient also changes from row 1
to 5 in Figure 6a while it remains unchanged in Figure 6b. Since the number of types of points does
not change, SQ is constant in Figure 6c and Figure 6d. In column E, the locational segregation SL isequal to zero in all the cases due to the uniform density distributions. The compositional segregation
SC increases from row 5 to 1 with the unevenness in the composition of points. This is consistent
with our earlier observation in Figure 1c, where a difference in the number of points causes a local
segregation. The locational unevenness decreases from column A to E, which results in a decrease in
SL.
The overall segregation S(=SL+SC+SQ) first decreases and then increases from row 5 to 1 incolumns from A to D in Figure 6c. This is because the locational and compositional segregations canchange simultaneously. While SL decreases monotonically from row 5 to 1, SC increases continuously.
This confirms us that we cannot distinguish the locational and compositional segregations by only
using S. The distinction requires at least two measures.

In Figure 6c, the locational segregation SL changes with keeping SC and SQ unchanged
from column A to E. In Figure 6d, on the other hand, SC changes with keeping SL and SQ from row 5
to 1. This clearly indicates that the locational and compositional segregations are independent witheach other.
We then consider the relationship between the compositional and qualitative segregations.
To exclude the effect of locational segregation, we consider the uniform density distributions inFigure 7a. The number of point types K ranges from two to six. Figure 7c shows that the qualitative
segregation SQ increases with a decrease of K from row 5 to 1, so does the overall segregation. This
supports our earlier discussion in Figure 2, that is, the locational segregation increases with thequalitative homogeneity. We also confirm our observation in Figure 1c by looking at an increase in
SC and S from column E to A, where the compositional unevenness monotonically increases.
We finally discuss the relationship between the locational and qualitative segregations. To
exclude the effect of compositional segregation, we consider the density distributions of points
shown in Figure 7b, in each of which all the distributions have the same total density. Similar to
Figure 7c, Figure 7d shows that a decrease of K increases SQ from row 5 to 1. The locationalsegregation SL increases while SC and SQ remain unchanged from column E to A. This again supports
the independency of locational segregation.

3.2 Larger synthetic datasets
We then apply the method to larger synthetic datasets to test the validity of equalization
measures. We consider five checkerboard patterns of side 32, each of which consists of 2*2, 4*4,
8*8, 16*16, and 32*32 cells as shown in Figure 8. Every cell contains the same number of either
white or black points.

Figure 8 Checkerboard patterns of white and black points. Every cell contains the same number of
either white or black points.

The result is shown in Table 1. While the segregation measures fail to distinguish thecheckerboard patterns of different resolutions, the equalization distance LE clearly detects thedifference between them. In checkerboard patterns, the locational equalization is completed bytransferring a half of points in each cell to one of its adjacent cell. Consequently, the equalization

distance is theoretically equal to the distance between the centroids of adjacent cells, though it isviolated by the boundary effect. This permits LE to distinguish the checkerboard patterns of different
resolutions.

Table 1 Segregation and equalization measures of checkerboard patterns

Resolution

SL

SC

SQ

LE

RE

2*2

0.00

0.00

0.25

10.688

0.50

4*4

0.00

0.00

0.25

4.031

0.50

8*8

0.00

0.00

0.25

1.773

0.50

16*16

0.00

0.00

0.25

1.063

0.50

32*32

0.00

0.00

0.25

1.000

0.50

We then consider the distributions of white and black points of the same number in arectangular region of 16*32 cells. Figure 9 shows the five patterns of the density distribution of
black points. Patterns P1, P2, and P3 have a single peak, while P4 and P5 have two and four peaks,
respectively.
The result is shown in Table 2. We again confirm the effectiveness of the equalization
distance LE. It increases from P1 to P3 as the peak shifts from the center to the boundary of theregion. It decreases from P1 to P4 and P5 as the distribution becomes smooth. The equalization ratio
RE can also distinguish P4 and P5 from P1, P2, and P3. This is because the total volume of transfer
necessary for flattening peaks decreases as the density distribution becomes smooth.

P1

P2

P3

P4

P5

Figure 9 Distributions of white and black points of the same number. Grey shades indicate thedensity distribution of black points.

Table 2 Segregation and equalization measures of density distributions shown in Figure 9.

Pattern

SL

SC

SQ

LE

RE

P1

0.00

0.00

0.25

10.422

0.72

P2

0.00

0.00

0.25

11.880

0.72

P3

0.00

0.00

0.25

16.350

0.72

P4

0.00

0.00

0.25

6.933

0.65

P5

0.00

0.00

0.25

5.090

0.61

3.3 Real dataset
We finally use the method to analyze a real dataset to test its practical feasibility. We
examine the urbanization process from 1974 to 1994 in Chiba Prefecture, Japan. Figure 10 shows the
population distribution of Chiba in 1994. Since Chiba is adjacent to the east of Tokyo Metropolis, its
population density is higher along railway lines in the western area that is densely inhabited bypeople working in Tokyo.

N

0

5

10km

R1

R5

R2

R3

490.000147.00074.00045.00028.000-

R6
R4

16.0007.0001.0000.000-

Figure 10 Population distribution of Chiba Prefecture, Japan in 1994. White and black broken lines
indicate subregions in which landuse mixture is analyzed in detail.

We obtained the landuse data of 10m resolution from Geospatial Information Authority ofJapan, and aggregated them into the raster format data of 500m resolution. We calculated thesegregation measures of landuse pattern in each cell and summed them up over the entire region.
Urbanization is usually accompanied with a progress of landuse mixture. The transition
proceeds from the dominance of natural landuse to the mixture of urban landuse such as residential,commercial, and industrial areas. Consequently, a decrease in the segregation measures is often

observed with the progress of urbanization.
Figure 11 shows the distribution of the overall segregation S. Note that darker shades
indicate smaller values of S where the landuse is highly mixed. Comparing Figure 11b with Figure
10, we notice that the landuse mixture and the population density are highly correlated. Thisconfirms us of the close relationship between the landuse mixture and urbanization. The urban area
expanded especially in the western area of Chiba during this period because of the rapid
concentration of population in the metropolitan area of Tokyo.

N

0

5

N

10km

0

-0.10

-0.10

-0.20

-0.20

-0.30

-0.30

-0.40

-0.40

-0.50

-0.50

-0.60

-0.60

-0.70

-0.70

-0.80

-0.80

-0.90

-0.90

5

10km

Figure 11 The overall segregation S of landuse pattern in (a) 1974 and (b) 1994.
We then examine the progress of landuse mixture in more detail in six subregions shown in
Figure 10. Subregions R1 and R2 experienced a rapid expansion of urban areas in 1980’s primarily
caused by the new town construction. Subregions R3 and R4 both contain urban areas of a long
history, where landuse pattern had been stable during the period. Subregions R5 and R6 present a
mixture of urban, suburban, and rural areas. The former contains an old historical town while thelatter is a new town located at the boundary of urban and rural areas.
Figure 12 shows the segregation measures from 1974 to 1994. The overall segregation S
decreased in all the regions, which suggests the progress of urbanization all over Chiba. Comparing
Figure 12b and Figure 12c, we notice that the decrease of S was caused by the rapid reduction of SC.
Subregions R1, R2 and R6 show a drastic decrease of SC during this period. The locationalsegregation SL, on the other hand, slightly increased in many regions. The compositional unevennessof landuse decreased while the locational unevenness increased with the progress of urbanization.

This implies that urban areas expanded scatteredly rather than uniformly in Chiba. We confirm thisin Figure 13 where the equalization distance LE increased in R1 and R2.

Figure 12 The segregation measures of landuse pattern from 1974 to 1994. (a) The overall
segregation S, (b) the locational segregation SL, (c) the compositional segregation SC.

Figure 13 The equalization distance LE from 1974 to 1994.
4. Concluding discussion
This paper has developed a new method of evaluating the spatial segregation. Using thelocational and compositional equalizations, the method decomposes the spatial segregation into three
exclusive and independent components. The locational segregation evaluates the spatial aspect ofsegregation caused by the locational unevenness. The compositional and qualitative segregations
reflect the compositional unevenness and the qualitative homogeneity, both of which are aspatial
factors of spatial segregation. The paper proposes three segregation and two equalization measures
to evaluate the three components of spatial segregation quantitatively. To test the validity of themethod, the paper applied it to an analysis of three spatial datasets of different sizes. The result

indicated that the method is effective for evaluating the spatial segregation from three different
perspectives.
Our method has several advantages over existing method. First, the method defines thethree components of spatial segregation in a formal and quantitative way. It clearly reveals the
exclusive and independent relationship between the components. Second, the paper proposes threesegregation measures whose range is analytically obtained. This permits us to compare the spatial
segregations between different patterns. Third, following the line of Reardon and O’Sullivan (2004),
the paper defines the segregation measures based on the location of individual points. This enables
us to apply the method to both point data and their aggregated data, the latter of which permits us toapproximate the segregation measures calculated from the former.
We finally discuss some limitations and extensions of the paper for future research.
First, the paper considers three components of spatial segregation. Existing papers, on theother hand, have proposed further dimensions. Massey and Denton (1988), for instance, advocates
five dimensions called evenness, exposure, clustering, centralization, and concentration. Though
they may not be independent with each other, they are still effective and convenient forunderstanding the spatial structure of segregation from various perspectives. For a better usage ofexisting dimensions, their properties need to be clarified in more detail. To this end, we should
analyze and describe the relationship between our components and existing dimensions.
Second, we should consider the spatial segregation in the spatiotemporal dimension.
Numerous moving objects exist in the real world such as human beings, animal species, vehicles andplanes. Their spatial distributions change continuously, and so does their spatial segregation.
Fortunately, a rapid progress of data acquisition tools in GIS enables us to capture the trajectories ofmoving objects. An analytical method of spatiotemporal segregation needs to be developed.
Third, this paper evaluates the similarity of different types of points equally between any
pair of different types. In the real world, however, this treatment is sometimes inappropriate. For
instance, the difference between European and Asian peoples is larger than that between Asian andPacific peoples. When we discuss their spatial segregation, segregation of European and Asianpeoples may be more critical than that of Asian and Pacific peoples. To treat such cases, we shouldtake into account explicitly the attributes of spatial objects and their differences in the evaluation of
spatial segregation.
Fourth, the behavioral approach seems useful in discussing the spatial segregation ofmoving objects. The concept of exposure (Massey and Denton, 1988; Reardon and O’Sullivan,
2004) implicitly considers the movement of spatial objects that is represented as the possibility ofcontact between objects. Assuming a more realistic behavior of objects, we can define segregation
measures that are practically more useful to describe the spatial segregation. Extension in thisdirection sounds promising.

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Appendix A1This appendix derives the minimum value of t(x) defined by Equation (6) using theLangrangian multiplier method. Let us define di(x) asdi ( x ) =Di ( x ).
∑ Dk ( x )k(29)
We minimize

f
=





∑ d ( x ) + λ 1 − ∑ d ( x ) 
2
i

i

i

i

(30)
with constrain∑ d (x) = 1 ,
i

i

(31)
where λ is the Langrangian multiplier. Partial differentiation of f by di(x) and that by λ are∂f= 2d i ( x ) − λ∂di ( x )
(32)
and

∂f
= 1 − ∑ di ( x ) ,
∂λ
i
(33)
respectively. Since∂f=0,∂di ( x )(34)
we obtaindi ( x ) =

λ
2

.
(35)

Substitution of Equation (35) into Equation (33) yields

λK

1 − ∑ di ( x ) =
1−

=
0.

2

i

(36)
Solving this equation, we have

λ=

2
.
K
(37)

Substituting the above equation into Equation (36), we obtaindi ( x ) =

1
.
K
(38)

This gives the minimum of t(x), that is,t ( x ) = ∑ {di ( x )}

2

i

.

1
=
K

(39)
Appendix A2This appendix derives the minimum value of SL. 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 ζij(x) in Equation (3). The proportion of type i points in Rj is given by

pij =

Dij

∑D

.

ij

i

(40)
The total density of points in Rj is denoted by ρi. The total density of all the points is

T = ∑ A( Rj ) ρ j ,j(41)
where A(Rj) is an operator that gives the area of Rj.
The locational segregation SL is defined as

SL
=

1
N 
A ( R j ) ρ j ∑ pij2 − ∑  i ∑T ji
i  N 221 
 N i  2=∑ ∑ A ( R j ) ρ j pij −  N  T 
T i  j


.

(42)
We use the Langrangian multiplier method, where we minimize

N

N 
∑j A ( R j ) ρ j pij2 −  Ni  T + λ  Ni T − ∑j pij A ( R j ) ρ j  .


2

fi
=

(43)
The constraints are

Ni∀i, ∑ pij A ( R j ) ρ j =
T
N
j
(44)
Partial differentiation of fi by pij and that by λ are∂fi
2 A ( R j ) ρ j pij − λ A ( R j ) ρ j .
=
∂pij
(45)
and

∂fi N i=T − ∑ pij A ( R j ) ρ j ,
∂λ N
j
(46)
respectively. Solving

∂fi
= 0,
∂pij
(47)
we obtain

pij =

λ
2

.
(48)

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

pij =

Ni
.
N
(49)

and the minimum of SL:

2
2
1 
 N i   N i  ρAR−∑ ∑ ( j ) j  N   N  T  .
T i  j


SL
=

=0
(50)

Appendix A3This 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 the number 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 forany x:
2

2





 Di ( x )  Di ' ( x )  .
∑i  D ( x )  = ∑i  D ' ( x ) 
∑ k

∑ k

 k

 k

(51)
This condition is equivalent to∑ p = ∑ p' ,
2
i

2
i

i

i

(52)
under the constraint:=p ∑=p ' 1.
∑
i

i

i

i

(53)
We assume m=1 and pi=pi’ for i=1, ..., K-2. Equations (52) and (53) become

pK2 −1 + pK2 = p '2K −1 + p '2K + p '2K +1 ,
(54)
and

pK −1 + pK − p ' K − p ' K +1 =p ' K −1 .
(55)
We can rewrite pK-1 as

pK=
pK + a−1(56)

without losing generality. Substituting Equations (55) and (56) into Equation (54), we solve it interms of pK:pK =− ( p ' K + p ' K +1 ) + 2 p ' K p ' K +1 + ap ' K + ap ' K +1
2
(57)

The condition of the existence of a positive pK is

4a ( p ' K + p ' K +1 ) > ( p ' K − p ' K +1 ) .
2(58)
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

1
N 
∑i  Ni  −=K2

2

1
 N '
∑i  Ni  − K + m 2 .
(
)
(59)

Solving Equation (59) in terms of m, we obtain

m
=

K2
2

2

 N '
N 
K 2 ∑  i  − K 2 ∑  i  +1
i  N i  N −K .

(60)
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  .
(61)
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. We use thesame setting as that of Appendix A2. The locational segregation SLe is written as

S Le
=

1
∑ A ( R j ) ρ j ∑i pij log pij − ∑i pij log pij
T j

Ni
Ni 
1 
=
∑ A ( R j ) ρ j pij log pij − T log ∑
T i  j
N
N

.

(62)
We use the Langrangian multiplier method to calculate the minimum value of SLe. We minimize

=
fi

 Ni



∑ A ( R ) ρ p log p − N T log N + l  N T − ∑ p A ( R ) ρ  .
Ni

j

j

ij

Ni

ij



j

ij

j

j

j



(63)
with constraints

Ni∀i, ∑ pij A ( R j ) ρ j =
T.
N
j
(64)
Partial differentiation of fi by pij and that by λ are∂fiρ j A ( R j )( log pij − 1 − l ) .
=
∂pij
(65)
and

∂fi N iA ( R ) − ∑ pij A ( R j ) ,=∂λ Njrespectively. Solving

∂fi
= 0,
∂pij
(66)
we obtain

pij = K 1+ λ .
(67)This implies that pij is independent of i. From this we derive

pij =

Ni
.
N
(68)

Substitution of Equation (68) into (62) yields

=
S Le

Ni
N N
N 
1 
log i − i T log i ∑ A ( R j ) ρ j∑
T i  j
N
N N
N .
=0
(69)