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Multi-scale approach to measuring residential segregation and 
the case of Yaffo, Tel-Aviv

Itzhak Benenson and Itzhak Omer
Department of Geography and Human Environment, Tel Aviv University, Israel,


An approach to the individual-based description of segregation phenomena is proposed. The proposed framework is based on hierarchical representation of urban residential space and the estimation and comparison of individual spatial segregation at different levels of the hierarchy.

Keywords: residential distribution, local segregation measures, GIS, population census


The measurement of urban residential patterns provides a framework for verifying theories and models of migration and residential choice (Benenson, 1999; Portugali, 2000). With only some exceptions (Waldorf, 1992; Schnell and Benjamini 1999), the measures or segregation indices (Peach, 1975; Massey and Denton, 1988) utilise aggregate data, namely, numbers or fractions of groups over given partitions of the studied area. The resulting global measure is a single number that represents the state of an average member of a group over the entire area. Contrary to the "averaged" view, the segregation state of particular individual depends on his/her location. A person can exist simultaneously in strong segregation regarding the other residents in his/her house and in weak segregation regarding the residents of neighbouring houses. Consequently, the series of estimates representing the segregation state of individual with respect to the residents within different neighbourhoods should be constructed and applied at each specific location.

The inability of the group approach to account for spatial relations between individuals was understood more than two decades ago (Lee, 1978, Peach, 1979), although it could not be replaced until necessary geo-referenced personal data were made available (Boal, 1987). The situation changed in the mid 1990s, when population censuses in several European countries, including Israel, began including for the first time, exact geo-referencing of householders and households (ICBS, 2000). This detailed spatial data allows the development of individual-based local measures of segregation at the resolution of houses, streets, neighborhoods, and so forth.

The aim of this paper is to suggest a framework for the individual-based description of segregation phenomena. The proposed framework is based on hierarchical representation of urban residential space and the estimation and comparison of individual spatial segregation at different levels of the hierarchy. Conceptually, our approach follows the humanistic-phenomenological stream in human geography (Relph, 1976; Tuan, 1977; Sack, 1980), with its stress on personal experience and relationships. We apply the proposed approach in a study of residential patterns in the Yaffo region of Tel-Aviv.

The concept of spatial hierarchy

A human being can belong to and exist in several spatial aggregates concurrently and needs "borders" or "edges" of the place he/she perceives at a moment (Tuan, 1977; Johnston, 1991). The spatial aggregates the person perceives are, usually, hierarchically organized. The hierarchy of an individual's experience regarding the built environment can, in turn, be defined by them (Relph, 1976; Ley, 1983). For example, the level of houses can serve as the basis of a person's hierarchy; the "home area" (Lee, 1968), that is, the population of the closest neighboring houses, can be considered as the next aggregate; and so forth. We will compare two approaches to the hierarchical representation of space, which we illustrate with an example from the Yaffo residential area of Tel-Aviv.

Bottom-up approach

The units of the lowest level L0 of the hierarchy D represent individuals' locations (apartments). The units of the upper levels are defined recursively, according to the neighborhood relations between the inhabitants of the units of L0. The choice of these units depends on the researcher's objectives, although houses seem the natural choice to serve as units of level L1. The units of the next level L2 can be defined as sets of houses neighboring to the L1 - house. The definition of neighboring houses is of a principal value for constructing the bottom-up hierarchy, later we propose an approach for obtaining that definition. Neighborhoods of increasing dimensions can be considered as units of levels L3, L4, L5, until the highest possible level of the city as a whole, LC, is reached. For each individual A, therefore, we consider the series of embedded spatial units L0A, L1A, LCA as representing different levels of a nested hierarchy with A as the center. It is worth noting that according to the bottom-up hierarchy, each individual house belongs to several units at each level beginning at the level L2, while defines one of these units only.

Top-down approach

The administrative division of a city can be used to determine hierarchy D as well. Usually, the city has up to five or six levels of administrative divisions, defined by regions, electoral districts, boroughs, ZIP zones, and so forth. Each unit of a higher level (e.g. region) is subdivided into the units of a lower level (electoral district). At each level of administrative hierarchy units do not intersect and an individual belongs to one of them only.

Top-down and bottom-up hierarchies for the Yaffo residential area

The top-down hierarchy D for Yaffo (Fig. 1) contains five levels: LC, the "city" of Yaffo; L4, the statistical areas; L3, electoral districts; L2, the street cells (defined below, see the "Yaffo case" section); and, finally, L1, the level of houses. According to administrative hierarchy the neighboring houses to a houses H at level n are those located within the unit of level n that contains house H. That is, all the houses within the street cell comprise the neighborhood of each separate house within this cell at the level of cells, and so on. 
Regarding the bottom-up approach, the most important step is the transition from the level L1 to L2, where we are required have to define the immediate spatial proximate of each house H

To do so in an intuitively acceptable manner that avoids computational complexities, we use the coverage of Voronoi polygons constructed around the centroids of populated buildings (Fig. 2).

Two houses are considered as neighbors if their Voronoi polygons have a common edge. Consequently, we define first order neighborhood of a house H as a set of houses U1(H), where each G Î U1(H) satisfies the following conditions:
1. G is adjacent to H;
2. The distance between the centroids of G and H is less than a given threshold value (we set it equal to 100 m for the case of Yaffo);
3. G and H are on the same side of a street if it has two or more traffic lanes;
Higher-order neighborhoods are defined recursively. Namely, the neighborhood Uk(H) of the order k (k = 2, 3, ) is defined as Uk-1(H) plus houses next to houses of Uk-1(H):

Uk(H) = {F | F Î U1(G) AND G Î Uk-1(H)}   (1)
Figure 3 presents the resulting hierarchy.

Indices of spatial segregation

Global indices

As noted above, one can find a dozen analytically different global segregation indices used in geographic research as of the early 1950s (Geary, 1954; Duncan and Duncan, 1955). The indices can be roughly classified according to the phenomena they are meant to reveal - dissimilarity, exposure, clustering, and so on (Massey and Denton, 1988). In parallel, the indices differ according to the number of population groups they relate to. The indices of spatial autocorrelation (used primarily for recognition of homogeneous spatial clusters) relate to members of a single population group (Getis and Ord, 1996). Most other indices estimate relationships between two groups (Wong, 1997), while some indices - Moran joint count statistics for multi-colored maps (Goodchild, 1986) or Kendall t (Kendall, 1970) - operate with an arbitrary number of population groups. Each global index represents the state of an average representative of a given group with respect to a partition defined a priori. The most popular is the dissimilarity index (Peach, 1975), while we consider the Geary index of spatial heterogeneity (Goodchild, 1986) of the population distribution according to characteristic f

K1 = Sijwij||fj fi||/(s2SiSjwij)   (2),

where fi, fj are the values of the characteristic at locations i and j; weights wij define an a-priory "influence" of locations j on location i, where Sijwij = 1, and s2 denotes the variance of f.

Local indices

Geographical approaches to estimating local spatial segregation and local measures of segregation phenomena were suggested during the 1980s (Getis and Ord, 1992) and developed in parallel with geostatistics (see Cressie, 1993, for review). As a result, measures of spatial relationship applied in geographical research share the title "spatial correlation/autocorrelation" with those applied in geostatistics but differ from them formally. In this paper we continue with a discussion of geographical measures of spatial relationships and delay the comparison of geographical and geostatistical measures to the future. Local indices of spatial segregation are applied primarily to the examination of the spatial clustering of population groups. Two of them, meant to reveal the homogeneous or heterogeneous domains over the studied area, are considered below:

The index G of local autocorrelation proposed by P. Getis and K. Ord (Getis and Ord, 1992, Getis and Ord, 1996), is based on a comparison of the local average of a characteristic f to its global average. The G-version ignores the value of characteristic f at the location itself and accounts for its neighbors only, while the G*-version, which we use below, accounts for the value of f at the location i itself: 

G*i,n = R(SjÎ Un(i)wijfj <f>)/s   (3),

where <f> is a mean and s2 is a variance of f, Sijwij = 1 and R is a normalizing multiplier.
The global counterpart of Getis' indices is the average value of a characteristic over the entire city. One can easily recognize the similarity between G*i,n and a moving average over neighborhood.

It follows from (3) that positive and high values of G*i,n are obtained when the mean value of f over a neighborhood is essentially higher than the global mean and negative and relatively low when the local mean falls below the global mean. Consequently, either high positive or low negative G*i,n values identify relatively homogeneous neighborhoods.

Local Geary indices K1 and K2 (Getis and Ord, 1996; Anselin, 1995) estimate the variance of the characteristic within the neighborhood.

K1i,n = R1SjÎ Un(i)wij|fj fi|/s   (4),
K2i,n = R2SjÎ Un(i)wij(fj fi)2/s2   (5),

where the terms have the same meaning as in (3).

The K1i,n and K2i,n are always positive; the higher their value, the higher the heterogeneity of the neighborhood. One can easily recognize the similarity between K2i,n and the semivariance.

Local Getis indices correspond to the global average of the characteristic f, while local Geary indices can be considered as components of the global Geary segregation index (Anselin, 1995). Each global index can be localized either formally, by expanding global segregation indices into local components (Anselin 1995) or by simply reformulating the index's analytical expression (Benenson and Omer, 2000). Here we do not consider other local segregation indices, including the local Moran I (Anselin, 1995) and joint count statistics (Goodchild, 1986).

Statistical inference for local segregation indices

The problem of statistical inference for local segregation indices is rather complex. Our goal here is to recognize the changes in population distribution over space, while the standard geostatistic approach is based on the stationarity of the distribution variance (Cressie, 1993, Pannatier, 1996). Two approaches to statistical inference regarding local measures of segregation are considered in the literature. The first one is based on maximum likelihood estimation of the index variance and further normalizing of the index (Getis and Ord, 1996; Anselin, 1995). Such an approach is problematic unless the assumption of stationarity, which we do not want to accept, is fulfilled. In addition, maximum likelihood approach results in indices varying within non-standardized intervals; thus, the comparison of the indices calculated for different situations becomes problematic. The alternative is a resampling computational approach, based on Monte Carlo simulations (Anselin, 1995). Resampling seems better suited to the poorly defined statistical situations we are studying, especially because its relatively heavy calculations are of minor importance when using state-of-the-art hardware. The limitations of the scope of this paper prevent further discussion of the problem of statistical inference, which will be delayed to a subsequent paper.

Measuring segregation according to spatial hierarchy


Let us consider some local segregation index S. For an individual B located at unit i Î L0 and characterized by the value fB of characteristic f, we estimate the value of Si,n at location i regarding level Ln of D in the following way. First, select neighborhood Un(i) of i, which belongs to Ln, and estimate the values of fA for each individual A located within Un(i); second, calculate Si,n based on fB and the set of fA. Below, we consider the level L0 to consist either of families or of houses and map the spatial distribution of Si,n over units i of L0.

The case of Yaffo

What follows is an attempt to understand what might result from the individual-based approach when geo-referenced personal data are available. The main source of data for this study is the Israeli Census of Population and Housing for 1995 (ICSB, 2000), which data are available for supervised study at ICBS. Census data are geo-referenced at the personal level, that is, one of the fields of the personal record contains a unique identifier of the polygon representing the building a person lives in. Based on these data, we can study the ethnic residential distribution in the Yaffo region of Tel-Aviv, which is occupied by Arab and Jewish residents. The Yaffo area is about 7 km2, and its population in 1995 was about 40,000. The Jewish majority comprised about 70% of the population and the Arab minority the other 30%. Regarding ethnic distribution we based our study on the householder identity, because mixed families are extremely rare.

In addition to level of buildings, the census GIS of Tel-Aviv contains layers of streets, statistical areas, electoral districts and open spaces (ICSB, 2000). The ethnic residential distribution we investigate is based on the "religion" of the person only, defined by the six categories of the census questionnaire: "Jewish", "Moslem", "Druze", "Christian", "Foreign worker", "unknown". For the purposes of the current study, we have combined the categories of "Moslem", "Druze" and "Christian" into one entitled "Arab". The "unknown" category is considered as "Jewish", because the majority of these householders are new immigrants whose Jewish affiliation has yet to be officially recognized. The top-down and bottom-up hierarchies D of the Yaffo urban space are already presented above. Ethnicity is a binary characteristic and to avoid inconveniences we skip the level of houses and consider the level of houses as a basic level of D for the Yaffo case.

We use MapInfo GIS with a Vertical Mapper add-in as tools for the geographical analysis and ArcCad topological tools for constructing street cells. The procedures for constructing neighborhoods based on Voronoi polygons and for calculating local segregation indices are developed as MapBasic extensions of MapInfo. In the following, we consider G*i,n and K1i,n indices, set weights wij = 1/(Nn - 1), where Nn is a number of units in Un(i), and set coefficients R equal to one. To map the results of analysis, we always divide the interval of the index's variation into three classes, corresponding to high, low and intermediate values of the index.

Ethnic residential segregation according to a top-down hierarchy

To represent G*i,n we map the fraction of Arab population in a unit for each level of Yaffo's administrative hierarchy (Fig. 4). According to (3), the difference between G*i,n, and this fraction is a constant number. Going down the levels of hierarchy, one can see that the maps at the levels of statistical areas, electoral districts and street cells (Fig. 4b-d) do not add much to understanding residential segregation in Yaffo created by house-level map (Fig. 4a). In contrast, low-resolution maps provide an essentially biased representation of the situation due to the arbitrarily location of the boundaries of the administrative units and the high number of unpopulated buildings found in Yaffo. It is obvious that the bias increases when the spatial units grow (Figs. 4b-d). We do not present here maps of K1i,n, which are uninformative for units above the house levels and proceed to the bottom-up hierarchy, which begins with houses in any case. Let us note here that with the bottom-up approach, the mean size of a neighborhood of order 4 corresponds to the size of a street cell and the size of a neighborhood of order 8 corresponds to an electoral district. 

Ethnic residential segregation according to a bottom-up hierarchy

Here we compare spatial distributions of the indices G*i,n and K1i,n constructed for the level of houses versus neighborhoods of orders n = 1, 2, 4 and 8 (Figs. 5 - 6). Prior to analyzing each index separately, let us point out two important differences that stem from a comparison of Fig. 4 and Figs. 5 - 6. First, it is difficult to locate the boundaries between homogeneous areas in Fig. 4, whereas the locations of the boundaries remain constant in Figs. 5 - 6. On the maps in Figs. 5 - 6, constructed for higher levels of the hierarchy the boundaries may "disappear", but they do not actually move in space. Second, the boundaries between homogeneous/heterogeneous domains in Figs. 5 - 6 do not correspond to the boundaries between street cells, electoral districts or statistical areas, which define the administrative partition and the top-down hierarchy.

Local Getis G*

The impression we receive from the mapping of the areas of concentration of the Arab and Jewish population in Yaffo, as seen in Fig. 5, is unambiguous. We can clearly distinguish domains of homogeneous populations of Arabs or of Jews; these domains are characterized by values of G*i,n that are close to the maximum or the minimum for all n. The rest of the Yaffo area, where for some or for all n the values of G*i,n are far from the extremes found, cannot be uniquely classified because the moderate values of Getis local index can be obtained in two ways. First, they may result from relatively homogeneous Un(i), where the fractions of the Arab population in all the houses are similar and close to global fraction of Arabs in Yaffo. Second, they can be obtained for heterogeneous Un(i), where the average fraction of Arabs is close there global fraction in Yaffo. The houses for which G*i,n is far from the extremes for all or some n comprise about 20% of the populated houses in Yaffo. In order to understand the segregation situation of the individuals inhabiting these houses, we continue the analysis by means of the local Geary K1 index.

Local Geary K1

According to (4), K1i,n is low when the values of the characteristic are similar for all the houses within Un(i) (Fig. 6). Thus, in addition to homogeneous areas of extreme fractions of Arabs and Jews disclosed by extreme values of G*i,n, low values of K1i,n mark the locations where the ethnic content of the houses is similar, although it is far from the extremes of 1:0 or 0:1. To test for whether such locations exist, let us consider the relationship between the two indices (Fig. 7). It is clear from Fig. 7 that in Yaffo's mixed neighbourhoods (those having a high percentage of both Arabs and Jews) are always heterogeneous, that is, they contain houses having all possible ratios of Arab to Jewish populations including purely Jewish and purely Arab ones. 

Description of personal segregation situations by means of local segregation indices

Up to this point, we have reviewed the ability of indices to indicate domains of homogeneous and heterogeneous ethnic structure. Our basic goal, however, is to estimate the segregation situation of a single person who, during the daily activity cycle, experiences all the neighborhoods simultaneously. Let us consider a hypothetical member A of an Arab ethnic group located in house i and then describe "the segregation situation of A " in terms of the values of G*i,n and K1i,n for different n.

The simplest situation is that of individual A residing in house i, which is located within a domain having the highest or lowest values of G*i,n and close to zero K1i,n for all n. Such an individual is located within an almost homogeneous Arab environment within the house, at the level of adjacent houses and above, up to the level of everyday local activity containing local shops and religious and cultural centers - a neighborhood of order 8. High values of G*i,n indicate that A is not exposed to the Jewish population until visiting distant locations. When A is located in a house that is within an area of minimal G*i,n (and, again, close to zero K1i,n) for all n, the situation is reversed, that is, A is found within a homogeneous Jewish environment within the house and at all levels up to that of everyday activity.

The areas of either homogeneous Arab or Jewish population cover about 80% of Yaffo. Let us try to conceptualize the segregation situation if A were located in a house in the remaining 20% of Yaffo's territory. The basic properties of a person's segregation situation here are indicated by K1i,n and its variation with n (Fig. 8). A low K1i,n for all n characterizes a uniform situation, one where A perceives each house in the neighborhoods Un(i) as having equal and intermediate (due to low G*i,n) fractions of an Arab population. The non-uniform case is characterized by K1i,n fluctuating with n. Concerning Yaffo, non-uniform situations, that is K1i,n varies with n are observed at locations within the areas displaying close to zero values of G*i,n (Fig. 8). A's segregation state at such a locations depends on the range of the fluctuations in K1i,n. If the range is high, then A perceives all the possible variations of the ethnic structure that can be observed in Yaffo within the radius of everyday local activity. For the intermediate range of fluctuations, A remains within heterogeneous, but predominantly Arab (characterized by high values of G*i,n) or Jewish (low values of G*i,n) neighborhoods. 

Global estimates of Jewish-Arab segregation in Yaffo

The mean fraction of Arabs (corresponding to Getis G*i,n) over the area we consider, which is bigger than the Yaffo administrative region, equals 0.188. The global Geary index K1, calculated for houses regarding neighborhoods of order 1 equals K1 = 0.099. Taken alone, these intermediate values demonstrate that the ethnic residential distribution in Yaffo is not homogeneous, although its global heterogeneity is not high. The local measures G*i,n and K1i,n express this heterogeneity.


The purpose of this paper is to present the individual-based approach to measuring residential segregation and to apply it to a real-world situation. This approach enables the identification of areas of homogeneity according to mean, variation and other characteristic of a residential distribution and provides a multi-dimensional view of the segregation situation of a person. A socio-geographic perspective on segregation demands further investigation of the reasons for the observed phenomena. These reasons may be constraints of the built environment, the tendency of similar householders to locate in close proximity, and so forth. For example, in Yaffo the correlation between the fraction of the Arab population and the architectural style of a building is relatively high (Omer, 1996). Hence the latter can be considered a factor contributing to the persistence of homogeneous Arab or Jewish areas there.

A number of important issues await further study, including the problem of statistical inference and the relation between standard methods of geostatistics and local indices of segregation. From an applied point of view, however, the most promising avenue seems the utilization of these indices for the presentation of census data. Till now, the proposed partitioning of the city into relatively homogeneous sub-areas had been based on uniting houses according the mean values of the population characteristic there (Martin, 1998). To properly reflect the character of residential distribution we have to account for its other properties, of which local variation seems the most important.


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