CSIS Discussion Paper No. 122

A method for analyzing the segregation between point distributions

Yukio Sadahiro

September 2013

Center for Spatial Information Science, University of Tokyo
5-1-5, Kashiwanoha, Kashiwa-shi, Chiba 277-8568, Japan
Phone: +81-471-36-4310
Fax: +81-3-5841-8521
sada@csis.u-tokyo.ac.jp

Abstract
This paper proposes a new method for analyzing the segregation between point distributions.
Though there have been proposed numerous methods and measures in segregation analysis, they
have at least three deficiencies: 1) statistical significance of segregation is not evaluated, 2) aspatial
properties of points are not considered, and 3) the relationship between different dimensions ofsegregation is not fully discussed. To resolve these problems, this paper proposes a new method foranalyzing the segregation between point distributions. We introduce a general procedure ofevaluating the individual components of segregation. This procedure helps us find independent
components of segregation, and provides a means of assessing their statistical significance. To test
the validity of the proposed method, we apply it to three datasets of different sizes. The result
supports the technical soundness of the method, and provides empirical findings.

Keywords: segregation; point distributions; statistical analysis; aspatial properties

1. Introduction
Segregation is a fundamental concept in geographical information science. In geographicalcontext it refers to the spatial isolation of different types of objects. Ethnic segregation has drawn
much attention in geography, demography, and sociology (Duncan and Duncan 1955; Morgan 1983;
White 1983, 1986; James and Taeuber 1985; Massey 1985; Morrill 1995). Segregation of plant andanimal species has been widely discussed in biology and ecology (O'Neill et al. 1988; Li and
Reynolds 1993; Riitters et al. 1996; Mucientes et al. 2009).
There have been proposed numerous measures to evaluate the degree of segregation.
Though earlier measures are easy to understand and calculate, they have been often criticized for themodifiable areal unit problem (Wong 1993, 2001; Reardon and O’Sullivan 2004). Recent methods
are more sophisticated (Johnston et al. 2007, 2011; Poulsen et al. 2007; Rey and Folch 2011; Páez etal. 2012), and some measures are defined based on the location of individual objects to avoid themodifiable areal unit problem (Reardon and O’Sullivan 2004; Reardon et al. 2009; Reardon andBischoff 2011; Sadahiro and Hong 2013). However, there still remain at least three problems to besolved: 1) statistical evaluation of segregation, 2) consideration of aspatial properties, and 3)
independent dimensions of segregation. We discuss these problems successively in the following.
Statistical tests permit us to judge whether or not segregation can occur by chance. Somepapers evaluate the statistical significance of segregation by using the measures of spatial
autocorrelation. Johnston et al. (2011) and Poulsen et al. (2011) use global Moran's I to evaluate thesegregation between ethnic groups. Logan et al. (2002) and Brown and Chung (2006) adopt local
Moran's I to identify local clusters of ethnic and racial groups. However, these measures are notappropriate for the segregation between multiple groups since they were originally developed toevaluate the spatial autocorrelation of a single group of objects. Though these measures can evaluate
the segregation of one group from the others, they cannot assess the statistical significance of theentire segregation between multiple groups. Other measures of spatial correlation such as those
proposed by Wartenberg (1985) and Stephane et al. (2008) evaluate the segregation between every
pair of groups, not the entire segregation between multiple groups.
Another aspect overlooked in existing studies is the aspatial properties of points. It often
happens in the real world that one group is more similar to another than the others. Austronesian
peoples are more similar to Papuan peoples in some aspects than Indo-European peoples.
Segregation between Austronesian and Indo-European peoples may be more crucial than thatbetween Austronesian and Papuan peoples. However, such aspatial properties of points have notbeen considered explicitly in segregation analysis.
The third problem lies in the discussion of segregation dimensions. Massey and Denton(1988) defines dimensions as the primary axes for measuring the degree of segregation andadvocated five dimensions called evenness, exposure, concentration, centralization, and clustering.

Reardon and O’Sullivan (2004) claims that these dimensions are not fully distinct and proposes two
independent dimensions called spatial exposure and spatial evenness. Johnston et al. (2007) uses
principal component analysis in case studies to find two independent dimensions called separatenessand location. However, the independence between these dimensions has not yet been fully examined.
Reardon and O’Sullivan (2004) claims the independence of their dimensions without showing atheoretical proof, and empirical approach taken by Johnson et al. (2007) does not assure that the twodimensions exist in any circumstances.
To resolve the above problems, this paper proposes a new method for analyzing thesegregation between point distributions. Though our method is developed based on the information
about the location of individual points, it is also effective for aggregated spatial data that have beenfrequently used in the literature. Section 2 introduces a general procedure of evaluating theindividual components of segregation. This procedure helps us find independent components ofsegregation, and provides a means of assessing their statistical significance. To test the validity of the
proposed method, Section 3 applies it to the analysis of two synthetic datasets and one real dataset.
Section 4 summarizes the conclusions with discussion.

2. Methodology
This paper introduces two terms factors and components to refer to the causes and resultsof segregation, respectively. The latter almost correspond to dimensions discussed in existing studies
such as evenness, exposure, and clustering. Independent components refer to the components that are
mutually exclusive and collectively exhaustive, i.e., components that form the entire segregation
without overlapping. This definition is equivalent to those used in Reardon and O’Sullivan (2004)
and Johnson et al. (2007).
Sadahiro and Hong (2013) proposes a method for evaluating segregation based on thelocation of individual points. We extend their method to resolve the three problems mentioned in
Section 1. This section starts with the measurement of segregation without considering the aspatial
properties of points. We then extend the method by incorporating the aspatial properties of points.
This permits us to evaluate individual components separately, and helps us find independent
components of segregation. We finally propose a method for testing the statistical significance ofsegregation.

2.1 Measurement of segregation without considering the aspatial properties of points
Suppose a set of K types of points in region R of area T. Let Pij and zij be jth point of type i
and its location, respectively. The number of type i points and its summation are denoted by Ni and
N=N1+ N2+... +NK, respectively. The set of type i points is denoted by Pi={Pi1, Pi2, ..., PiNi}.
Reardon and O’Sullivan (2004) and Sadahiro and Hong (2013) evaluate the segregation

between points by using their density distributions. Following this line, we define Di(x) as thedensity function of Pi at location x. The set of the density functions is denoted by D={D1(x),D2(x), ..., DK(x)}. 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)

The density of type i points at x is defined as the summation of proximity functions of all the type ipoints:Di ( x ) =N i ∑ φ ( x, z ij )j∫ ∑ φ ( y, z )dy
y∈R

ij

j

.
(2)

This definition standardizes the summation of proximity functions in such a way that the integrationof a density function is equal to the number of points, i.e.,∫x∈R

Di ( x ) dx = N i .
(3)

The above definition is based on the location of individual points. If the locational data ofpoints are aggregated by spatial units, we substitute the density of points in individual units for Di(x).
The method described in the following is applicable to both aggregated and disaggregated spatial
data.
Using the density functions of points, we define the local segregation at x as
 Di ( x ) s ( x; D ) = ∑ i  ∑ Dj (x)  j∑ {D ( x )}

2

=

2

.

i

i



∑ D j ( x ) 
 j


2

(4)
We adopt this definition for its simplicity and flexibility (Sadahiro and Hong, 2013). Integrating
Equation (4), we obtain the global measure of segregation:

S ( D) =∫x∈R

D ( x ) s ( x; D ) dx∫x∈RD ( x ) dx

,

(5)
where

D ( x ) = ∑ Di ( x ) .
i(6)
The measure ranges from zero to one. A large value appears when different groups are separated
while a small value indicates the integration of different groups.

2.2 Measurement of segregation with considering the aspatial properties of points
This subsection extends the proposed measures to incorporate the aspatial properties ofpoints. Let us suppose the segregation of black, dark-grey, light-grey, and white points. Dark-grey
and light-grey points are the composites of black and white points at 70/30 and 30/70 ratios,
respectively. The properties of points are represented by the proportion of black and white, i.e., {1.0,0.0}, {0.7, 0.3}, {0.3, 0.7}, and {0.0, 1.0}.
To formalize this example, we introduce a positive K×M matrix A that represents theaspatial properties of points:

 a11
a
A =  21  aK 1

a12
a22

aK 2

 a1M  a2 M .
 
 aKM (7)

Element aim indicates the mth attribute of type i points. We assume that every set of variables {a1m,a2m, ..., aKm} is given in a standardized form, i.e.,=a=a∑ a .
∑
∑
i1

i

i2

i

iM

i

(8)
The attributes of four types of points mentioned above is represented by

1.0
0.7
A=
 0.3

 0.0

0.0 
0.3
.
0.7 

1.0 
(9)

Here, we consider the spatial distribution of attributes instead of that of points themselves.
The spatial distribution of mth attribute is given by

Am ( x ) = ∑ aim Di ( x ) .
i(10)
We replace D={D1(x), D2(x), ..., DK(x)} by {A1(x), A2(x), ..., AM(x)} to evaluate the segregation ofpoint distributions. The local and global measures of segregation become Ai ( x ) s ( x; D, A ) = ∑ i  ∑ Aj ( x ) 
 j


2

 ∑ a ji D j ( x ) = ∑ ji  ∑∑ akj Dk ( x ) 
 j k


2

(11)
and

S ( D, A ) =∫x∈RD ( x ) s ( x; D, A ) dx∫x∈RD ( x ) dx,

(12)respectively. We call the latter the overall segregation.
The measure S(D) is a special case of S(D, A). The former is obtained by defining A as the
K×K identity matrix in Equation (12):1 0  0 
0 1  0 
.
AI =  0 0  1 (13)
So far we have assumed attributes to be represented by the proportion of elements.
However, the proposed measures are effective for a wider range of properties measured on a ratio
scale. The only requirement for attributes is that they can be standardized as shown in Equation (8).

2.3 Evaluation of individual components of segregation
This subsection proposes a method for evaluating individual components of segregation.
To consider a certain component, we compare two situations where its underlying factors are present
and absent. The difference in the overall segregation S(D, A) between the two situations reflects theeffect of the factors, and consequently, provides us a means of evaluating the component.
Suppose point distributions in Figure 1a, where black and white points are uniformly
distributed. In Figure 1b, white points are uniformly distributed while black points are distributed
only on the left side. Segregation is caused by the non-uniform distribution of black points.
Consequently, the difference in S(D, A) between the two patterns indicates the effect of
non-uniformity in black points on segregation.

(b)

(a)

Figure 1 The distributions of black and white points. (a) Both black and white points are uniformly
distributed. (b) Black points are distributed only on the left side, which causes segregation between
points.

We evaluate the effect of non-uniformity in type i points by considering the situation
where type i points are uniformly distributed. The set of density function D' is given byN
D' =  D1 ( x ) , D2 ( x ) ,..., i ,..., DM ( x )  .
T


(14)
The effect of the non-uniformity in type i points is measured by∆S ( ∆∆∆', A) = S ( , A) − S ( , A) .
(15)
We then suppose Figure 2 where the different number of black and white points are
uniformly distributed. Though Figure 2a exhibits no obvious segregation, segregation occurs at a
local level in Figure 2b. Points are integrated only in the neighborhood of white points, and we canfind many places where white points are absent. Segregation occurs and increases with the differencein the number of points.

(a)

(b)

Figure 2 The uniform distributions of black and white points. (a) The same number of black andwhite points are distributed. (b) White points are fewer than black points. Segregation occurs at a
local level where white points are absent.

We evaluate the effect of the difference in the number of type i points by replacing Ni withthe average number of other types of points:

∑N
j ≠i

j

K −1

=

N − Ni.
K −1(16)

The density functions become
N − Ni
D'' =  D1 ( x ) , D2 ( x ) ,...,Di ( x ) ,..., DK ( x )  .
( K − 1) Ni




(17)
The effect is measures by∆S ( ∆∆∆'', A) = S ( , A) − S (, A) .
(18)
As seen above, evaluation of a certain component requires us to compare two situations

where its factors are present and absent. However, it is sometimes difficult to remove only relevant
factors without affecting the others. To resolve the problem, we propose an indirect method forevaluating a component of segregation.
Suppose two components C1 and C2 each of which is caused by different factors. If we canremove only the factors of C1, we can evaluate C2 even if we cannot remove the factors of C2
separately. Let S and S' be the overall segregation of two situations where the factors of both C1 andC2 are present and absent, respectively. The difference S-S' represents C1+C2. Let S'' be the overall
segregation where the factors of C1 are absent. The difference S-S'' represents C1. Consequently, we
can evaluate C2 by substituting S-S'' from C1+C2:

C2 = ( S − S ') − ( S − S '')= S ''− S '.
(19)

2.4 Primary factors and components of segregation
The proposed procedure permits us to evaluate the individual components of segregation.
This subsection discusses four primary components of segregation using this procedure: spatial
segregation, compositional segregation, qualitative segregation, and attribute segregation. They arecaused by four factors what we call spatial unevenness, compositional unevenness, qualitative
uniformity, and attribute similarity, respectively. We will discuss the above factors and components
successively in the following.
Spatial unevenness refers to the non-uniformity in point distributions, which causes spatial
segregation. To evaluate spatial segregation, we can take either absolute or relative approach.
Absolute approach evaluates absolute spatial unevenness, which refers to the fluctuation of each
distribution. Relative approach considers relative spatial unevenness, which refers to the fluctuationof the proportion of each distribution to all the distributions. Though both approaches yield the same
segregation measures, they provide different bases for statistical tests as discussed later.
Absolute approach evaluates spatial segregation by considering the situation where
absolute spatial unevenness does not exist. It is represented by the set of uniform density functions:

N 
N N
DAS =  1 , 2 ,..., K  .
T 
T T
(20)
Suppose the density distributions of points defined on a one-dimensional space shown in Figure 3a.
Figure 3b shows the situation where absolute spatial unevenness is absent. Overall segregation
reduces from S(D, A) to2

N 
S ( DAS , A ) = ∑  i  .
i  N (21)The reduction represents absolute spatial segregation:

S AS=( D, A ) S ( D, A ) − S ( DAS , A )N = S ( D, A ) − ∑  i i  N 

2

.

(22)

(a)

(b)

(d)

(c)

(e)

Figure 3 Evaluation of spatial and compositional segregations on a one-dimensional space. Figure 3ashows the density distributions of points to be evaluated while the others indicate density
distributions where some factors are absent. Absent factors are (b) absolute spatial unevenness, (c)absolute spatial unevenness and compositional unevenness, (d) relative spatial unevenness, (e)relative spatial unevenness and compositional unevenness.

Relative approach introduces another representation of overall segregation. Let r be a set

of variables indicating the ratio of Di(x) to D(x):r = {r1 ( x ) , r2 ( x ) ,..., rK ( x )}D (x) 
 D ( x ) D2 ( x ).
,,..., K= 1
D (x) 
 D (x) D (x)(23)
The set of density distributions and their overall segregation are represented by

D = {r1 ( x ) D ( x ) , r2 ( x ) D ( x ) ,..., rK ( x ) D ( x )}
(24)
and

S ( D, A ) = S ( r , D ( x ) , A ) ,(25)
respectively.
Relative approach evaluates spatial segregation by considering the situation where theproportion of every type of points is uniform in R. It is represented by

N 
N N
rRS =  1 , 2 ,..., K  .
N 
N N
(26)
Figure 3d shows the absence of relative spatial unevenness. As seen in this figure, relative approach
keeps the distribution of the total density of points. Relative spatial segregation is the reduction of
overall segregation from S(r, D(x), A) to S(rRS, D(x), A):=
S RS ( D, A ) S ( r , D ( x ) , A ) − S ( rRS , D ( x ) , A )N = S ( D, A ) − ∑  i i  N 2.

(27)
Since SAS(D, A)=SRS(D, A), we define spatial segregation as
2

N
S S=
( D, A ) S AS=( D, A ) S RS=( D, A ) S ( D, A ) − ∑  i  .
i  N (28)
Compositional unevenness refers to the variation in the number of points between different
groups, which leads to compositional segregation. Compositional unevenness is another criticalfactor of segregation as seen in Figure 2. We take the indirect method to evaluate compositionalsegregation because we cannot remove compositional unevenness without affecting the other factors.

Suppose the situation where both spatial and compositional unevenness are absent. In
absolute approach, it is represented by Figure 3c where all the density distributions are defined by
the same uniform function:

N 
 N N
DASC = 
,
,...,
.
KT 
 KT KT(29)
The reduction of overall segregation is

S ( D, A ) − S ( DASC , A ) =S ( D, A ) −

1
.
K
(30)

Relative approach changes Figure 3a into Figure 3e. The latter are represented by11 1
rRSC =  , ,...,  .
K
K K
(31)
The reduction of overall segregation is

S ( D, A ) − S ( rRSC , D ( x ) , A ) =S ( D, A ) −

1
.
K
(32)

Equations (30) and (32) represent the effect of spatial and compositional unevenness, i.e., thesummation of spatial and compositional segregations. We thus subtract spatial segregation fromthese equations to measure compositional segregation:, A ) S ( D, A ) −SC ( D=2

1− S AS ( D, A )
K

1
N 
= ∑ i  −K
i  N .

(33)
This equation indicates that compositional segregation is defined as a function of only aspatial
variables. Compositional segregation evaluates the aspatial aspect of segregation.
We then discuss the third factor of segregation what we call qualitative uniformity. It refers
to the lack of the diversity of points advocated by Sadahiro and Hong (2013). When only a few typesof points exist, points of the same group tend to cluster, and consequently, segregation of points
increases. To evaluate the effect of qualitative uniformity, we take the indirect method that assumes
the absence of spatial unevenness, compositional unevenness and qualitative uniformity. Absence ofqualitative uniformity is represented by an infinite increase in the variety of points, which ismathematically represented as infinite operation K→∞. Absence of spatial and compositional
unevenness reduces overall segregation from S(D, A) to 1/K, and absence of qualitative uniformity

further reduces it infinitely to zero. Consequently, the effect of qualitative uniformity is evaluated by

SQ ( D, A ) =

1
.
K
(34)

We call this measure qualitative segregation.
The relationship between spatial, compositional, and qualitative segregations isrepresented as

S ( D, A ) = S S ( D, A ) + SC ( D, A ) + SQ ( D, A ) .
(35)
This equation indicates that overall segregation is decomposed into three components. Spatial,
compositional, and qualitative segregations are independent components of segregation since they
comprise the entire segregation without overlapping.
The final factor of segregation is attribute similarity, which refers to the similarity in theaspatial properties of points. To measure the effect of attribute similarity, we consider the situation
where different groups have completely different properties. Since it is mathematically represented
as Equation (13), the effect of attribute similarity is measured by

S A=( D, A ) S ( D, A ) − S ( D, A I ) .
(36)
We call this measure attribute segregation. Attribute segregation is usually negative or zero sincesegregation decreases with the similarity between different groups.
We have introduced four primary components of segregation: spatial segregation,
compositional segregation, qualitative segregation, and attribute segregation. Spatial segregation
represents the spatial aspect of segregation while the others evaluate the aspatial aspect ofsegregation. Spatial, compositional, and qualitative segregations are independent components asmentioned earlier, while attribute segregation is not independent of the others.

2.5 Statistical analysis of segregation
This subsection proposes a method for evaluating the statistical significance of individualcomponents of segregation. The null hypothesis assumes a stochastic process where the factor of acomponent to be examined is absent. We derive the probability distribution of the segregation
measure by using Monte Carlo simulation and test the statistical significance of the component.
The significance of spatial segregation can be evaluated by either absolute or relative
approach. Absolute approach assumes a situation where every type of points are distributed
independently according to the uniform distribution. We calculate the probability distribution of SS(D,A) in this circumstance and evaluate the statistical significance of spatial segregation. Relative
approach considers the random permutation of points. We randomize the type of points with keeping
their location and calculate SS(D, A) to obtain its probability distribution under the null hypothesis.
Test of compositional segregation employs SC(D, A) as the statistic. Since SC(D, A) is apurely aspatial variable, we perform its statistical test within an aspatial framework. The null
hypothesis considers a situation where every point is randomly assigned to one of M types. We
calculate the probability distribution of SC(D, A) in this circumstance and evaluate the statisticalsignificance of compositional segregation.

3. Applications
To test the validity of the proposed method, this section applies it to the analysis of threedifferent datasets. We omit the indicator (D, A) used in segregation measures for simplicity hereafter.
3.1 Properties of segregation measures
This subsection investigates the properties of segregation measures using a small synthetic
dataset. We consider two types of points P1 and P2 distributed on a one-dimensional space of lengthone.
We first focus on the spatial and compositional segregations by assuming matrix A as AI.
Figure 4a shows the density distributions of P1 and P2 that gradually become uniform with keeping
their total volume from D1 to D5. Spatial segregation SS decreases with spatial unevenness as shownin Figure 4b. Measures SC and SQ remain unchanged because the proportion of P1 and P2 does notchange.

P2
P2
P1

P1
D1

D2

D3

D4

(a)

0.60

0.40

SS
SC

0.20

SQ
0.00

D1

D2

D3

D4

(b)

D5

D5

Figure 4 The relationship between spatial unevenness and segregation measures. Spatial segregation
decreases with spatial unevenness from D1 to D5.

Figure 5a shows a decrease in compositional unevenness where the total volume of P1 andP2 gradually approaches with each other. Figure 5b indicates that compositional segregation
decreases monotonically with compositional unevenness from D6 to D10. Though spatial segregation
increases, overall segregation decreases due to the decrease in compositional segregation.

P2

P2
P1

P1

D6

D7

D8

D9

D10

(a)

0.60

0.40

SS
SC

0.20

SQ
0.00

D6

D7

D8

D9

D10

(b)

Figure 5 The relationship between compositional unevenness and segregation measures.
Compositional segregation decreases with compositional unevenness from D6 to D10.

We then discuss attribute segregation by using the point distributions shown in Figure 6a.
Their attributes are represented by 2×2 matrix A shown in Figure 6b, where attribute similarity
decreases from A1 to A6. We can confirm in Figure 6b that attribute segregation SA increases with adecrease in attribute similarity. We also notice that SS+SC+SQ-SA remains unchanged from A1 to A6.
This can be confirmed by substituting Equation (35) into (36):S S ( D, A ) + SC ( D, A ) + SQ ( D, A ) − S A ( D, A ) =S ( D, A ) − S A ( D, A )= S ( D, A I )

.

(37)

P2

P1

(a)

0.60

SS

0.40

SC
SQ

0.20

SA
0.00

-0.20
0.5 0.5
0.5 0.5

0.6 0.4
0.4 0.6

0.7 0.3
0.3 0.7

0.8 0.2
0.2 0.8

0.9 0.1
0.1 0.9

1.0 0.0
0.0 1.0

A1

A2

A3

A4

A5

A6

(b)

Figure 6 The relationship between attribute similarity and segregation measures. (a) Density
distributions of points. (b) Attributes of points and segregation measures. Attribute similarity
decreases from A1 to A6.
3.2 Statistical significance of segregation measures

This subsection discusses the statistical significance of spatial and compositionalsegregations. We again suppose two types of points P1 and P2 distributed on a one-dimensional
space of length one.
We evaluate the significance of spatial segregation in point distributions shown in Figure 7.
Spatial unevenness increases from left to right while compositional unevenness decreases from up to
down. The points P1 and P2 follow the density distributions defined by1
f1 ( x ) =1+ b  x − 
2

(38)
and

1

f2 ( x ) =−b  x −  ,
2

(39)
respectively (0≤b≤2). We keep N1 as 500 while N2 varies from 100 to 500.
P2
N2=100P1

P1

P2

N2=300

P2

P2
N2=500
P1

P1

b=0.0

b=1.0

b=2.0

Figure 7 Density distributions of P1 and P2. Spatial unevenness increases from left to right while
compositional unevenness decreases from up to down. The number of P1 is 500 while that of P2
varies from 100 to 500.

We employ both absolute and relative approaches in statistical tests to evaluate the effectof different null hypotheses. The significance level of spatial segregation under a null hypothesis isdefined by∞g = ∫ g ( s ) ds ,S

(40)
where g(s) is the probability density distribution of spatial segregation under the null hypothesis. A
small value of g indicates a high significance of spatial segregation.

0.70

0.60

0.50
N2=1000.40
N2=5000.30

0.20
N2=1000.10
N2=500

0

0.1

0.2

0.3

0.4

0.5

b

Figure 8 The significance level of spatial segregation under null hypothesis. The number of P2 varies
from 100 to 500 as indicated by arrows. Solid and dotted lines indicate the significance levelevaluated by absolute and relative approaches, respectively.

Figure 8 shows the significance level of spatial segregation in absolute and relative
approaches. As seen in this figure, spatial segregation becomes more significant with an increase inspatial unevenness represented by b. The figure also shows that spatial segregation becomes moresignificant with a decrease in N2. This implies that segregation is more probable to occur when fewer
points are distributed. Comparing absolute and relative approaches, we notice that the former is lesslikely to evaluate spatial segregation as statistically significant. It is because absolute approachconsiders a wider variety of point distributions in the null hypothesis. Absolute approach considers
the random distribution of points while relative approach only randomizes the type of points withkeeping their location. A narrower range of variation in the null hypothesis lowers the requirement
for statistical significance.
We then turn to the statistical significance of compositional segregation. Figure 9 shows
the distributions of P1 and P2, where N1 is 500 while N2 varies from 100 to 500. Compositional
unevenness decreases from left to right.

P2

P2

P1

P1

N2=100N2=300N2=500Figure 9 Density distributions of P1 and P2. The number of P1 points is 500 while that of P2 points
varies from 100 to 500.

Figure 10 shows the significance level of compositional segregation under the null
hypotheses. We can confirm that compositional segregation becomes less significant with a decreasein compositional unevenness. Statistical significance of compositional segregation at 5% level
requires that the ratio of P2 to P1 is smaller than 0.75 (=375/500).

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00
250

300

350

400

450

N2

Figure 10 The significance level of compositional segregation under the null hypothesis.

3.3 Real dataset
We finally apply the proposed method to the analysis of a real dataset to test its practicalfeasibility. We examine the segregation of commercial facilities in Chiba, Japan. Chiba is located 30
kilometers away in a suburb of Tokyo. We converted the list of commercial facilities in the NTT
telephone directory into spatial data by geocoding. Figure 11 shows the density distribution ofcommercial facilities in Chiba.

TokyoChiba Station

150 -

3
5
7
10
15
20
30
50

StationRailroad

Figure 11 The density distribution of commercial facilities in Chiba [1/km2].

We classified the commercial facilities into different categories by three classification
schemes and calculated their segregation measures. Scheme C-1 classified the commercial facilities
into three categories:

Group 1: retails (clothing stores, accessory stores, grocery store, ...)
Group 2: services (beauty shops, laundries, hotels, ...)
Group 3: restaurants

These groups consist of 6236, 4872, and 5197 facilities, respectively. Figure 12 shows the proportionof three types of commercial facilities.

TokyoChiba Station- 0.20- 0.30- 0.40- 0.50- 0.60- 0.70- 0.900.90 -

StationRailroad

(a)

Tokyo

Chiba Station- 0.20- 0.25- 0.30- 0.35- 0.40- 0.50- 0.800.80 -

StationRailroad

(b)

Tokyo

Chiba Station- 0.15- 0.20- 0.25- 0.30- 0.35- 0.40- 0.500.50 -

StationRailroad(c)Figure 12 The proportion of three types of commercial facilities in Chiba. (a) Retails, (b) services,(c) restaurants.

Figure 13 shows the distribution of local segregation measure s(x; D) defined by Equation(4). Comparing Figures 11 and 13, we notice a negative correlation between the density andsegregation of commercial facilities. Segregation tends to occur where only a few facilities exist.

Tokyo

Chiba Station

StationRailroad

Figure 13 The distribution of local segregation measure s(x; D).
Table 1 shows the measures of individual components of segregation. The case study
evaluates not only the primary components but also the spatial segregation of individual categories.
We first examine the result for classification scheme C-1. Table 1a shows that only spatial
segregation is statistically significant at 5% level. Segregation of commercial facilities in Chiba is
primarily caused by their spatial unevenness. Table 1b shows that SS1 and SS2 (retails and services)
are significant while SS3 (restaurants) is not significant. This implies that restaurants are distributed
more uniformly than retails and services. Restaurants are common places for people to visit insuburban areas in Japan such as Chiba.
Scheme C-2 classified the commercial facilities into eight categories each of whichcontains 1434, 1999, 420, 1370, 1013, 3461, 1411, and 5197 facilities. The result is almost
consistent with that for scheme C-1 as shown in Table 1. Only spatial segregation is statisticallysignificant in Table 1a and SS8 (restaurants) is not significant among eight categories in Table 1b.
Scheme C-3 classified restaurants further into three subcategories: 1) Japanese restaurants,
2) Western and Chinese restaurants, 3) pubs, bars, and taverns. In this scheme, all the components
are not significant as shown in Table 1. This suggests that any type of restaurant is so common
everywhere in Chiba.

Table 1 Segregation measures of commercial facilities in Chiba. (a) Segregation measures of primary
components, (b) spatial segregation of individual categories. Asterisks indicate the statisticalsignificance at 5% level in both absolute and relative approaches.

Classification scheme

S

SS

SC

SQ

C-1
C-2
C-3

0.56440.4331
0.63130.2272*0.2446*
0.26450.00380.06350.00380.33330.12500.3333SS6(a)Classification scheme

SS1

SS2

SS3

SS4

SS5

C-1
C-2
C-3

0.0574*0.1153*0.03070.0263*0.0091*0.11080.0231
0.0964*0.0847

N/A
0.0061*
N/A

N/A
N/A
0.0881* 0.1110*
N/A
N/A

SS7

SS8

N/A
0.0423*
N/A

N/A
0.0013N/A(b)

4. Concluding discussion
This paper has proposed a new method for analyzing the segregation between point
distributions. Section 2 introduces a general procedure of evaluating individual components ofsegregation. This procedure helps us find independent components of segregation, and provides ameans of assessing their statistical significance. To test the validity of the proposed method, weapplied it to the analysis of two synthetic and one real datasets. The result supports the technicalsoundness of the method, and provides empirical findings.
This paper has several advantages over existing ones. First, this paper proposes a method
for evaluating the statistical significance of segregation. Statistical test permits us to judge whether
or not a segregation measure can be obtained by chance. Second, our method takes into accountexplicitly the aspatial properties of points. We can consider spatial and aspatial aspects of point
distributions simultaneously in segregation analysis. Third, this paper proposes a general procedureof evaluating individual components of segregation. This procedure helps us consider a wider varietyof components and find independent components as shown in Section 2.
We finally discuss some limitations of the paper and potential extensions for futureresearch.
First, we should extend our method into the spatiotemporal domain. The location andproperties of points often change over time, and so does their segregation. One method to treat thetemporal dimension is to calculate segregation measures at different times and discuss their changeover time. This approach, however, is not sensitive to the change in the spatial arrangement of points.
Segregation measures remain the same even if a drastic change occurs in point distributions. A new
approach is necessary that considers both the spatial and temporal domains simultaneously insegregation analysis.
Second, we should develop a method for analyzing the segregation of continuous variables.
Though our method partially fulfills this purpose, statistical evaluation of segregation is notstraightforward due to the difficulty in the choice of null hypothesis for continuous variables. A newstatistical method needs to be developed.
Third, segregation analysis should explicitly consider the uncertainty in spatial data.
Accuracy of spatial data has been long discussed in geographical information science (Gopal andGoodchild 1989; Hunsaker et al. 2002; Shi et al. 2002). Though there exist spatial data models that
explicitly represent locational uncertainty, they have not yet been fully incorporated into analyticalmethods. An extension of our method to this direction is an important topic for future research.

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