Moran's I

Synonyms

Moran's index; Moran coefficient

Definition

Moran's I, based on cross-products, measures value association and is calculated for n observations on a variable x at locations i, j as

$$ I=\frac{\sum\limits_i {\sum\limits_{j\ne i} {w_{ij} (x_i-\bar{x})(x_j-\bar{x})}} } {S^2\sum\limits_i {\sum\limits_{j\ne i} {w_{ij}}}}\:. $$

Where x _i denotes the observed value at location i, $\bar{x}$ is the mean of the x variable over the n locations,

$$ S^2 = \frac{1}{n}\sum\limits_i ({x_i} -\bar x)^2\:, $$

and w _ij is the element of the spatial weights matrix for locations i and j, defined as 1 if location i is contiguous to location j and 0 otherwise. Other more complicated definitions of spatial weights matrices allow for the computation of the Moran's I at various levels of proximity or distance.

Main Text

Moran's Iis one of the oldest indicators of spatial autocorrelation and is still a widely accepted measure for determining spatial autocorrelation. It is used to estimate the...