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 and multi-directional.

Definition

Moran's I is defined as
where is the number of spatial units indexed by and ; is the variable of interest; is the mean of ; is a matrix of spatial weights with zeroes on the diagonal ; and is the sum of all.

Defining weights matrices

The value of can depend quite a bit on the assumptions built into the spatial weights matrix. 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.

Expected value

The expected value of Moran's I under the null hypothesis of no spatial autocorrelation is
At large sample sizes, the expected value approaches zero.
Its variance equals
where
Values of I usually range from −1 to +1. Values significantly below -1/ indicate negative spatial autocorrelation and values significantly above -1/ indicate positive spatial autocorrelation. For statistical hypothesis testing, Moran's I values can be transformed to z-scores.
Moran's I is inversely related to Geary's C, but it is not identical. Moran's I is a measure of global spatial autocorrelation, while Geary's C is more sensitive to local spatial autocorrelation.

Uses

Moran's I is widely used in the fields of geography and geographic information science. Some examples include: