Moran's I

In statistics, Moran's I is a measure of spatial autocorrelation developed by Patrick Alfred Pierce Moran. Spatial autocorrelation is characterized by a correlation in a signal among nearby locations in space. Spatial autocorrelation is more complex than one-dimensional autocorrelation because spatial correlation is multi-dimensional (i.e. 2 or 3 dimensions of space) and multi-directional. Global Moran's I is a measure of the overall clustering of the spatial data. It is defined as where is the number of spatial units indexed by and ; is the variable of interest; is the mean of ; are the elements of a matrix of spatial weights with zeroes on the diagonal (i.e., ); and is the sum of all (i.e. ). The value of can depend quite a bit on the assumptions built into the spatial weights matrix . The 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 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. The idea is to construct a matrix that accurately reflects your assumptions about the particular spatial phenomenon in question. A common approach is to give a weight of 1 if two zones are neighbors, and 0 otherwise, though the definition of 'neighbors' can vary. Another common approach might be to give a weight of 1 to nearest neighbors, 0 otherwise. An alternative is to use a distance decay function for assigning weights. Sometimes the length of a shared edge is used for assigning different weights to neighbors. The selection of spatial weights matrix should be guided by theory about the phenomenon in question. The value of is quite sensitive to the weights and can influence the conclusions you make about a phenomenon, especially when using distances.