16.2 LISA Principle

As pointed out in the discussion of global spatial autocorrelation, statistics designed to detect the presence of such correlation, such as Moran’s I and Geary’s c, are formulated to reject the null hypothesis of spatial randomness, typically in favor of an alternative of clustering (occasionally in favor of negative spatial autocorrelation). Such clustering is a characteristic of the complete spatial pattern, but it does not provide an indication of the location of the clusters.

An early suggestion of a test for local spatial autocorrelation was formulated by Getis and Ord (1992) (see Chapter 17). This was generalized in Anselin (1995), where the concept of a local indicator of spatial association, or LISA was introduced. A LISA is viewed as having two important characteristics. First, it provides a statistic for each location with an assessment of significance. Second, and less importantly, it establishes a proportional relationship between the sum of the local statistics and a corresponding global statistic.

As shown in section 13.4, most global spatial autocorrelation statistics can be expressed as a double sum over the i and j indices, for all pairs of observations, as \(\sum_i \sum_j g_{ij}\). The local form of such a statistic would then be, for each observation (location) \(i\), the sum of the relevant expression over the \(j\) index, \(\sum_j g_{ij}\).

More precisely, and as covered previously, a spatial autocorrelation statistic consists of a combination of a measure of attribute similarity between a pair of observations, \(f(x_i,x_j)\), with an indicator for geographical or locational similarity, in the form of spatial weights, \(w_{ij}\). For a global statistic, this takes on the form: \[\sum_i \sum_j w_{ij}f(x_i,x_j)\]. A generic expression for a local indicator of spatial association is then: \[\sum_j w_{ij}f(x_i,x_j).\] For each different measure of attribute similarity \(f\), a different statistic for global spatial autocorrelation results. Consequently, there will be a corresponding LISA for each such global statistic. First, the local counterpart of Moran’s I is considered.