2 URBAN SEGREGATION: DEFINITIONS, TRENDS, AND
2.5 Measuring urban segregation
1991; Reardon and O'Sullivan 2004; White 1983; Wong 1993,1998). This study adopts two spatial indices proposed by Feitosa et al. (2007) to measure different dimensions and scales of segregation. The first, named generalized spatial dissimilarity index, captures the dimension evenness/clustering (section 2.11). The second, called spatial isolation index, captures the dimension exposure/isolation (section 2.11). Global and local versions of these measures are used in a complementary manner to depict segregation patterns. While global indices summarize the segregation degree of the entire city, local indices show segregation as a spatially variant phenomenon that can be displayed in maps.
The indices adopted in this study rely on the idea that an urban area comprises different localities, which are places where people live and exchange experiences with their neighbors. The intensity of these exchanges varies according to the distance among population groups, given a suitable definition of distance. The population characteristics of a locality are expressed by its local population intensity, which is calculated by using a kernel estimator. A kernel estimator is a function that estimates the intensity of an attribute in different points of the study area (Silverman 1986).
To calculate the local population intensity of a locality j, a kernel estimator is placed on the centroid of areal unit j and estimates a weighted average of population data. The weights are given by the choice of a distance decay function (e.g., Gaussian) and a bandwidth parameter (Figure 2.6). This procedure allows researchers to specify functions that formalize a hypothesis about how population groups interact across spatial features. The specification of different bandwidths, for instance, enables analyses in multiple scales: the indices are able to start from the most detailed data and generalize them for analyzing segregation in broader scales.
Figure 2.6 Gaussian kernel estimator (Feitosa et al. 2007).
The local population intensity is a geographically weighted population average that considers the distance between groups. Formally, the local population intensity of a locality j (Lj) is calculated as (Feitosa et al. 2007):
J
j j
j k N
L
1
, (2.1)
Where: Nj is the total population in areal unit j; J is the total number of areal units in the study area; and k is the kernel estimator which estimates the influence of each areal unit on the locality j.
The local population intensity of group m in the locality j (Ljm) is calculated by replacing the total population in areal unit j (Nj) with the population of group m in areal unit j (Njm) in equation (2.1):
J
j jm
jm k N
L
1
. (2.2)
2.5.1 Measuring the spatial dimension evenness/clustering
The global version of the generalized spatial dissimilarity index (D(m)) measures the average difference of the population composition of the localities from the population composition of the city as a whole. The formula of D(m) is:
J
j M
m m m
m jm j
N m N
D
1 12 1
)
(
, (2.3) Where:
j jm
jm L
L
. (2.4)
In equation (2.3), N is the total population of the city; Nj is the total population in areal unit j; m is the proportion of group m in the city; jmis the local proportion of
group m in locality j; J is the total number of areal units in the study area; and M is the total number of population groups. In equation (2.4), Ljm is the local population intensity of group m in locality j; and Lj is the local population intensity of locality j.
The index D(m) measures the proportion of people who would have to move from their localities to achieve an even population distribution. It varies from 0 to 1, where 0 stands for the minimum degree of evenness and 1 for the maximum degree.
Despite these established meanings, it is still hard to interpret the values obtained within this [0,1] interval: does a D(m) value equal to 0.6 reveal a situation of severe segregation or not? This is not a trivial question, since the values of segregation measures are sensitive to the scale of the data: indices computed for smaller areal units tend to present higher values than indices computed for larger areal units (Feitosa et al.
2007). This is called the grid problem (White 1983) and it is inherent to all segregation measures.
In the case of spatial segregation measures, as the ones presented in this section, the scale variability is also related to the bandwidth used in the computation of the measures. An index computed with a small bandwidth will have higher values than another that is computed with a large bandwidth. Because of that, it is unfeasible to establish fixed thresholds that assert whether the index results indicate a severe segregation level or not. Instead, the interpretation of global indices of segregation is more useful when relational, for example, focused on the comparison of values obtained for an urban area in different points in time. Based on that, it is possible to draw conclusions about segregation trends along the years.
The local version of the generalized spatial dissimilarity index (dj(m)) is obtained by decomposing the index D(m). It shows how much each locality contributes to the global D(m) measure of the city (Feitosa et al. 2007). The local index dj(m) can be displayed as a map and used to identify critical areas. The formula of dj(m)is:
M
m m m
m jm j
j N
m N d
12 (1 )
)
(
, (2.5)
Where: the equation parameters are the same as in equation (2.3).
2.5.2 Measuring the spatial dimension exposure/isolation
The global version of the spatial isolation index (Qm) measures the average proportion of group m in the localities of each member of the same group (Feitosa et al. 2007):
j
J jm
j m
jm
m L
L N
Q N
1
, (2.6)
Where: Njm is the population of group m in areal unit j; Nm is the population of group m in the study region, Ljm is the local population intensity of group m in locality j, and Lj is the local population intensity of locality j.
The isolation index varies from 0 (minimum isolation) to 1 (maximum isolation). The results of the index Qm depend on the overall composition of the city.
For example, if the proportion of the group m increases in the city, the index Qm tends to become higher.
The local version of the spatial isolation index (qm) can also be obtained by decomposing Qm (Feitosa et al. 2007):
j m m m im
L L N
q N
. (2.7)
Where: the equation parameters are the same as in equation (2.6).
In general, measures of segregation are useful tools for describing the phenomenon in its multiple scales and dimensions. By computing these measures to different dates, it is possible to analyze several aspects of segregation: Is the global segregation of a city increasing or decreasing? Is this trend applied to both dimensions of segregation? What is happening at smaller/larger scales? Where are the most critical areas of poverty isolation?
Measures of segregation, in particular the local ones, can be also used to explore the relationship between the segregation of social groups and other urban
indicators. For example, local indices of segregation estimated at different scales and compared with violence rates can reveal whether poor families isolated at broader scales are more vulnerable to violent events than those who are segregated at smaller scales or not segregated. Such experiments can contribute to the debate about different patterns of segregation and their impacts.
Nevertheless, despite the value of these measures, they represent only static snapshots of segregation at a certain moment. They are unable to help researchers to understand the underlying dynamics of the phenomenon or how different contextual mechanisms (such as those described in section 2.4) can lead to the emergence of specific patterns of segregation. The next chapter introduces a set of concepts and methods related to the theory of complexity that contribute to overcome this limitation.
3 URBAN SEGREGATION AS A COMPLEX SYSTEM: CONCEPTS