Indicators of spatial association

Indicators of spatial association are statistics that evaluate the existence of clusters in the spatial arrangement of a given variable. For instance if we are studying cancer rates among census tracts in a given city local clusters in the rates mean that there are areas that have higher or lower rates than is to be expected by chance alone; that is, the values occurring are above or below those of 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 Moran's I, 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;
and $S_0 = \sum_{i}{\sum_{j}{W_{ij}}}$ .

The matrix W 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 first law of geography, which states that Everything depends on everything else, but closer things more so - in other words, the law implies a spatial distance decay function, such that even though all observations have an influence 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 the statistic should differ over space.

But if there is no global autocorrelation or no clustering, we can still find clusters at a local level using local spatial autocorrelation. The fact that Moran's I is a summation of individual crossproducts is exploited by the "Local indicators of spatial association" (LISA) to evaluate the clustering in those individual units by calculating Local Moran's I for each spatial unit and evaluating the statistical significance for each I_i. From the previous equation we then obtain:

I_i = \frac{Z_i}{m_2} \sum_j W_{ij} Z_j

where:

m_2= \frac{\sum_i Z_i^2 }{N}

then,

I= \sum_i \frac{I_i}{N}

I is the Moran's I measure of global autocorrelation, I_i is local, and N is the number of analysis units in the map.

LISAs can for example be calculated in GeoDA, which uses the Local Moran's I, proposed by Luc Anselin in 1995.

References

Anselin, L. (1995). "Local indicators of spatial association – LISA". Geographical Analysis, 27, 93-115.
Anselin, L. (2005). "Exploring Spatial Data with GeoDATM: A Workbook". Spatial Analysis Laboratory. p. 138.