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. 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. Moran's I is defined as where N {displaystyle N} is the number of spatial units indexed by i {displaystyle i} and j {displaystyle j} ; x {displaystyle x} is the variable of interest; x ¯ {displaystyle {ar {x}}} is the mean of x {displaystyle x} ; w i j {displaystyle w_{ij}} is a matrix of spatial weights with zeroes on the diagonal (i.e., w i i = 0 {displaystyle w_{ii}=0} ); and W {displaystyle W} is the sum of all w i j {displaystyle w_{ij}} . The value of I {displaystyle I} can depend quite a bit on the assumptions built into the spatial weights matrix w i j {displaystyle w_{ij}} . 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 k {displaystyle k} 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 expected value of Moran's I under the null hypothesis of no spatial autocorrelation is