Local Indicators of Spatial Association

Local Indicators of Spatial Association (LISA)
Consists of a series of statistics that evaluate the existence of local clusters in the spatial arrangement of a given variable. For instance if we are studying cancer rates among census tracts in a given city the existence of local clusters in the rates indicates that there areas that have higher (or lower) rates than it is expected by chance i.e. above (or below) the values that may occur had the variable had a random distribution in space.

Global Spatial Autocorrelation
Global spatial autocorrelation is a measure of the overall clustering of the data. One of the statistics used to evaluate global spatial autocorrelation is the Moran's I, which is defined by:
 * $$ I= \frac{\frac{N}{S_{0}} \sum_{i}{\sum_{j}{W_{ij}Z_{i}Z_{j}}}}{\sum_{i}{Z_{i}^{2}}} $$

Where $$Z_{i} $$ is the deviation of the variable of interest with respect to the mean. $$W_{ij}$$ is the matrix of weights that in some cases is equivalent to a binary matrix with ones in position i,j whenever observation i is a neighbor of observation j, and zero otherwise. This matrix is required because in order to address spatial autocorrelation and also model spatial interaction, we need to impose a structure to constrain the number of neighbors to be considered. This is related to Tobler’s Law which states that “Everything depends on everything else, but closer things more so”, in other words, the law is implying a spatial decay function, such that even though all observations have incidence on all other observations, after some distance threshold that influence can be neglected.

Global versus Local
Global spatial analysis or global spatial autocorrelation analysis yields only one statistic to summarize the whole study area, in other words, global analysis assumes homogeneity. If that assumption does not hold, then having only one statistic does not make sense as that statistic may change over space. Changes may come from spatial heterogeneity processes.

What if there is no global autocorrelation or if there is no clustering, can we still find clusters at a local level? Local spatial autocorrelation as opposed to global, evaluates the presence of local clusters. Note that the Moran's I is a summation of individual crossproducts and that property is exploited by the LISA to evaluate the clustering of in those individual units by calculating Local Moran's I for each spatial unit and evaluating the statistical significance for each Ii. From previous equation it is obtained:
 * $$ I_{i}= \frac{Z_{i}}{m_{2}} \sum_{j}{W_{ij}Z_{j}} $$

where:
 * $$ m_{2}= \sum_{i}{Z_{i}^{2}} $$

then,
 * $$ I= \sum_{i}{\frac{I_{i}}{N}} $$

where $$N$$ is the number of observations. LISAs are calculated in GeoDA using the Local Moran's I, proposed by Luc Anselin in 1995.