Moran's I

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Moran's I is a measure of spatial autocorrelation. Like autocorrelation, spatial autocorrelation means that adjacent observations of the same phenomenon are correlated. However, autocorrelation is about proximity in time. Spatial autocorrelation is about proximity in (two-dimensional) space. Spatial autocorrelation is more complex than autocorrelation because the correlation is two-dimensional and bi-directional.

Moran's I is defined as

I = \frac{N} {\sum_{i} \sum_{j} w_{ij}} \frac {\sum_{i} \sum_{j} w_{ij}(X_i-\bar X) (X_j-\bar X)} {\sum_{i} (X_i-\bar X)^2}

where N is the number of spatial units indexed by i and j; X is the variable of interest; \bar X is the mean of X; and wij is a matrix of spatial weights.

The expected value of Moran's I is

E(I) = \frac{-1} {N-1}

Its variances equals

Var(I) = \frac{NS_4-S_3S_5} {(N-1)(N-2)(N-3)(\sum_{i} \sum_{j} w_{ij})^2}

where

S_1 = \frac {1} {2} \sum_{i} \sum_{j} (w_{ij}+w_{ji})^2
S_2 = \frac {\sum_{i} ( \sum_{j} w_{ij} + \sum_{j} w_{ji})^2} {1}
S_3 = \frac {N^{-1} \sum_{i} (x_i - \bar x)^4} {(N^{-1} \sum_{i} (x_i - \bar x)^2)^2}
S_4 = \frac {(N_2-3N+3)S_1 - NS_2 + 3 (\sum_{i} \sum_{j} w_{ij})^2} {1}
S_5 = S_1 - 2NS_1 = \frac {6(\sum_{i} \sum_{j} w_{ij})^2} {1}


Negative (positive) values indicate negative (positive) spatial autocorrelation. In practice, values greater than 2 or smaller than -2 indicate spatial autocorrelation that is significant at the 5% level.

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.

This statistic was developed by Patrick A.P. Moran.[1]

[edit] Sources

  1. ^ Moran, P.A.P. (1950), Notes on Continuous Stochastic Phenomena, Biometrika, 37, 17-33.