Environment and Planning B: Planning and Design 1998, volume 25, pages
327
- 343
A perimeter-based clustering index for measuring spatial
segregation: a cognitive GIS approach
C-M Lee
Wharton Real Estate Center, University of Pennsylvania, 3600 Market Street, Philadelphia,
PA 19104-2648, USA; e-mail: leecm@wharton.upenn.edu
D P Culhane
School of Social Work, University of Pennsylvania, PA 19104-2648, USA;
e-mail: dennis@cmhpsr.upenn.edu
Received 25 November 1996; in revised form 2 May 1997
Abstract. Many efforts have been made to develop segregation indices that incorporate spatial
interaction based on the contiguity concept. Contiguity refers to how similar the concentration of
the subject of interest in one areal unit is to that in adjacent areal units. However, highly segregated
situations are typically considered to be isolated sections or enclaves rather than smoothly formed
peaks of concentration in space. Therefore, highly segregated enclaves may not exhibit contiguity. In
this paper, a new index to measure the degree of clustering is developed and it is compared with the
existing indices of concentration or segregation. The proposed clustering index (7
C
) tends to give more
weight to 'enclaveness' rather than contiguity alone. This may be a good property for those cases in
which the primary concern of an investigator is the formation of enclaves of a socioeconomic subject,
including minority populations, poverty, crime, epidemics, and mortgage red-lining. Additionally, its
property of robustness to the citywide rate allows us to perform properly an intercity comparison of a
given subject by index score even when the citywide rate varies significantly, unlike the other measures.
1 Introduction
Numerous efforts have been made to develop a proper index to measure the spatial
segregation of a population group.
(1)
Though each index characterizes somewhat different
aspects of a spatial distribution, one can distinguish two types of indices: measures
ignoring spatial interaction between areal units; and measures incorporating spatial
interaction.
The problem with measures of segregation that lack spatial interaction components,
including the dissimilarity index, the Gini coefficient, and the entropy index (Theil,
1972),
is well illustrated in the case of the 'checkerboard problem', described by White
(1983).
Several efforts have been made to develop segregation indices that incorporate
spatial interaction, including the index of spatial proximity (White, 1986) and the
distance-based index of dissimilarity (Morgan, 1982). In general, these measures include
spatial interaction by distance or binary adjacency between two areal units. Recently
Wong (1993) formulated a new segregation index, which uses the length of the common
boundary of two areas as an indicator of the degree of social interaction between the
residents of the two areas.
Spatial-interaction measures in geography, and segregation measures incorporating
spatial interaction in sociology are similar in concept. However, spatial-interaction
measures in geography are based only on distribution in physical space, whereas the
segregation measures take account in population distribution overlaid on physical
space along with the distribution of physical space
itself.
The spatial-interaction segrega-
tion indices in sociology are derived from Dacey (1968) and Geary (1954), where they
have been labeled 'contiguity' measures.
See Massey and Denton (1988) for the existing measures.
328
C-M Lee, D P Culhane
Contiguity refers to how similar the concentration of the subject of interest in one
areal unit is to that in adjacent areal units. If the figures for adjoining areal units are
generally closer than those for the areal units not adjoining, this condition yields a con-
tiguous distribution of the subject of interest (Dacey, 1968; Geary, 1954). This contiguity
aspect of spatial distribution has been well developed into a field of spatial statistics
known as spatial autocorrelation. In the last twenty years, a number of instruments for
testing for and measuring spatial autocorrelation have appeared (Anselin, 1988). To
geographers, the best-known statistics are Moran's /, and, to a lesser extent, Geary's c
(Cliff and Ord, 1973).
In some cases (Massey and Denton, 1988), the contiguity measures in geography
are interpreted as clustering indices, with some modifications. However, a high degree
of clustering does not always represent a high degree of contiguity. For example, one can
imagine a spatial distribution pattern in which one subject of interest forms isolated
enclaves which have visible boundaries. That distribution is not supposed to yield a
high degree of contiguity, as the difference at the boundaries of the enclaves reduces
the overall degree of contiguity. In the real world, one is generally concerned about
isolated enclaves of a population group which are recognized by both high concentra-
tion and separateness, rather than the spatial contiguity of their distribution alone. For
point data in the natural space, there are some measures which use nearest-neighbor
methods to describe the degree of clustering (Ripley, 1981). However, for areal data in
urban space, overlaid with population, one needs to have a different measure of
clustering rather than the existing contiguity measure of segregation.
In this paper, a new index to measure the degree of clustering is developed and then
compared with the existing indices of segregation. In section 2, clustering is defined in
an operational way, and in sections 3 and 4 a method for calculating the new clustering
index and its properties are discussed. In section 5, four existing indices to be compared
with the new clustering index are discussed briefly, and the clustering index and the four
other indices are compared in two hypothetical settings including binary distribution in
a regular lattice, and semicontiguous distribution in a regular lattice. In section 6, the
five indices are compared in a real-world application, the five boroughs of New York City.
2 Operational definition of clustering
When the spatial distribution of a subject of interest on a map is examined, viewers
tend to draw arbitrary boundaries of clusters and define a set of clusters cognitively,
whether it is a point distribution or an areal data distribution. This cognition could be
said to have three attributes: the total size of the clusters, their shape, and the closeness
between them. Here, a clustering index is derived based on these three attributes.
In order to draw the boundaries of clusters, an objective way to define clusters is
needed. In an urban setting, the probability of occurrence of a subject in an areal unit
depends on population rather than the size of the areal unit. For example, all other
factors being equal, the expected number of the poor in a census tract depends on the
number of people residing in the tract rather than the physical size of the tract. Once
the rate of an object group to population in each tract is determined, the next issue is
how to define concentration of the object group. One popular way of defining concen-
tration is the location quotient (Q
l
).
The location quotient is a device frequently used to identify specialization, concen-
tration, or the potential of an area for selected employment, industry, or output
indicators (Bendavid-Val, 1983; Chen, 1994). It refers to a ratio of the fractional share
of the subject of interest at the local level to the ratio at the regional
level.
When a local
Q
l
in a region is greater than 1, the locality has a higher concentration of the subject of
interest relative to the other localities of the region combined. For example, a census
A clustering index for measuring spatial segregation
329
(a) (b)
(d) (e)
^H More clustered
(f)
Less clustered
Figure 1. Size, shape, and adjacency of
subclusters:
(a) small
size,
p = 8; (b) large size, p = 16;
(c) regular shape, p = 10; (d) irregular shape, p = 14; (e) adjacent, p = 14; (f) separated,
p = 16.
tract or a block group may be equivalent to a locality, and a city to a region. Thus, Q
1
may be used to identify census tracts that contain a higher percentage share of a
subject of interest than a city as a whole, and which have a Q
x
value greater than 1.
Here, adjacent areal units showing a high concentration of the subject form a few
clusters on the map.
Once clusters are obtained, one needs to quantify the size, shape, and closeness of
the clusters. A measure that combines these three factors is the total perimeter of the
clusters. When shape and adjacency of the clusters are the same, the total perimeter (P)
of the clusters is a proper measure of the total size [see figures 1(a) and
1(b)].
When the
size and adjacency of the clusters are constant, circular shapes have the minimum
possible values [see figures 1(c) and
1(d)].
When the size and shape of the clusters are
the same, two adjoining clusters have a smaller total perimeter than two separated
clusters [see figures 1(e) and
1(f)].
Therefore, one can measure the degree of clustering
by assessing how small the total perimeter of the clusters (the concentrated areas of a
subject of interest) is, where the concentrated areas are selected by Q
1
.
3 Calculation of a clustering index
Based on the operational definition of clustering, one can develop a clustering index.
In order to illustrate the process for calculating the clustering index, a hypothetical city
space is assumed as in figure 2(a) (see over), where P is the population of each census
tract, and x
is
the number of an object group in the tract. As a first step, every census tract
is divided into two groups: highly concentrated census tracts, and less concentrated
census tracts based on Q
l
[see figure
2(b)].
When two highly concentrated tracts are
adjacent to each other, the common boundary lines are deleted and the two polygons
of the tracts are merged to form one polygon [see figure
2(c)].
This merging process
continues and finally a few polygons result, which represent highly concentrated areas or
clusters [see figure
2(d)].
The more adjacent the highly concentrated tracts are, the more
common boundaries are erased, and the smaller the ratio of the sum of the perimeters
330
C-M Lee, D P Culhane
! (90
(30,
4J
(50,
3;
6)
(60
f(20
U40
(110
9)
. 1)
6)
13)
(90
^(20
4)
3)
j
:4/3
J3/5
6/9
9/4
PM
2
13/1
3/b
6/4
1
4/9
Total perimeter: 33 (61
—
28)
(excluding the boundary of the study area)
Total population: 490
Total object group: 49
The numbers in parentheses are
(P,
x)
(a)
(b)
f
(c)
Total perimeter of merged polygons: 23
(excluding the boundary of the study area)
Clustering index =
1
- 23/33 = 0.30
(d)
Figure 2. The merging process used to calculate the clustering index: (a) population and object
group; (b) calculating the location quotient; (c) identifying concentrated tracts; (d) merging con-
centrated tracts.
of the merged polygons to the sum of the perimeters of the original tracts will be. In our
case,
the boundaries of the study area are not included in the calculation.
In this concept, the clustering index can be denoted as follows:
r = 1
EEIWA
where // is a binary value for tract / (1 if Q
l
^ 1; 0 if Q
l
< 1); and b
tj
is the length of
the common boundary between census tracts i and j (0 if tracts i and j are not
connected or / = j). If a pair of adjacent tracts have the same / value (either 1 or 0),
|/
;
- lj\ becomes 0, and their common boundary b
tj
does not count in the numerator in
the equation. Only the boundary between a pair of adjacent tracts which have different
/ values (high concentration versus low concentration) remains.
One advantage of this measure for an irregular polygon layout is that the degree of
proximity between polygons is automatically taken into account during the merging