Moran’s I

Synonyms

Moran coefficient; Moran’s index

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

$$\displaystyle{ I = \frac{\sum \limits _{i}\sum \limits _{j\neq i}w_{ij}(x_{i} -\bar{ x})(x_{j} -\bar{ x})} {S^{2}\sum \limits _{i}\sum \limits _{j\neq i}w_{ij}} \,. }$$

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

$$\displaystyle{ S^{2} = \frac{1} {n}\sum \limits _{i}(x_{i} -\bar{ x})^{2}\:, }$$

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...