Moran's I

The white and black squares are perfectly dispersed so Moran's I would be −1. If the white squares were stacked to one half of the board and the black squares to the other, Moran's I would be close to +1. A random arrangement of square colors would give Moran's I a value that is close to 0.

In statistics, Moran's I is a measure of spatial autocorrelation developed by Patrick Alfred Pierce Moran.[1][2] 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

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

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 $w_{ij}$ is an element of a matrix of spatial weights.

The expected value of Moran's I under the null hypothesis of no spatial autocorrelation is

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

Its variance equals

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

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. Values range from −1 (indicating perfect dispersion) to +1 (perfect correlation). A zero value indicates a random spatial pattern. For statistical hypothesis testing, Moran's I values can be transformed to Z-scores in which values greater than 1.96 or smaller than −1.96 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.

Sources

1. ^ Moran, P. A. P. (1950). "Notes on Continuous Stochastic Phenomena". Biometrika 37 (1): 17–23. doi:10.2307/2332142. JSTOR 2332142. edit
2. ^ Li, Hongfei; Calder, Catherine A.; Cressie, Noel (2007). "Beyond Moran's I: Testing for Spatial Dependence Based on the Spatial Autoregressive Model". Geographical Analysis 39 (4): 357–375. doi:10.1111/j.1538-4632.2007.00708.x.