Global Spatial Autocorrelation

Global spatial autocorrelation statistics are used to test for clustering of the data and identify global positive or negative spatial autocorrelation patterns.

Inference is performed through a computational permutation approach by randomly reshuffling the values of the variable to different locations. The observed statistic is compared to a reference distribution under the null hypothesis of spatial randomness.

All the examples on this page assume that the Guerry dataset is loaded and a polygon contiguity spatial weights object has been built and row-standardized.

using SpatialDatasets
using SpatialDependence
using StableRNGs
using Plots

guerry = sdataset("Guerry")
W = polyneigh(guerry)

Moran's I

Moran's I (Moran, 1948) is the most used global spatial autocorrelation statistic. It is computed as:

\[I = \frac{N}{S_0}\frac{\sum_{i}\sum_{j}w_{ij}(x_{i} - \bar{x})(x_{j} - \bar{x})}{\sum_{i}(x_{i} - \bar{x})^{2}}\]

with $S_0$ being the sum of the spatial weights, $S_0 = \sum_{i}\sum_{j} w_{ij}$.

Moran's I can be computed with the moran function. By default, $9,999$ permutations are calculated for the inference. It is possible to specify a different number of permutations with the permutations optional parameter. For reproduciibility, it is possible to specify a custom random number generator with the rng optional parameter.

moran(guerry.Litercy, W, permutations = 9999, rng = StableRNG(1234567))

Moran's I test of Global Spatial Autocorrelation
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Moran's I: 0.7176053
Expectation: -0.0119048

Randomization test with 9999 permutations.
 Mean: -0.0125941
 Std Error: 0.0707896
 zscore: 10.3150637
 p-value: 0.0001

The interpretation of Moran's I depends on its value and significance:

Geary's c

Geary's c (Geary, 1954) is a global spatial autocorrelation statistic that focuses on dissimilarity. It is computed as:

\[c = \frac{N - 1}{2S_0}\frac{\sum_{i}\sum_{j}w_{ij}(x_{i} -x_{j})^2}{\sum_{i}(x_{i} - \bar{x})^{2}}\]

Geary's I can be computed with the geary function. By default, $9,999$ permutations are calculated for the inference. It is possible to specify a different number of permutations with the permutations optional parameter. For reproduciibility, it is possible to specify a custom random number generator with the rng optional parameter.

geary(guerry.Litercy, W, permutations = 9999, rng = StableRNG(1234567))

Geary's c test of Global Spatial Autocorrelation
--------------------------------------------

Geary's c: 0.2502018
Expectation: 1.0

Randomization test with 9999 permutations.
 Mean: 1.0004114
 Std Error: 0.0722804
 zscore: -10.3791501
 p-value: 0.0001

The interpretation of Gery's c depends on its value and significance:

Reference distribution

The random permutation operation results in a reference distribution for the statistic under the null hypothesis of spatial randomness. If the Plots.jl package is loaded, it is possible to plot the reference distribution together with the observed statistic (vertical red line) using the plot function:

Ilitercy = moran(guerry.Litercy, W, permutations = 9999, rng = StableRNG(1234567))
plot(Ilitercy)

Moran Scatter Plot

The Moran Scatter Plot (Anselin, 1996) is a scatterplot with the variable in the horizontal axis and its spatial lag on the vertical axis.

If the Plots.jl package is loaded, the plot function can be used to plot a Moran scatter plot. By default, the values of the variable are z-standardized, but it is possible to build the plot without standardizing by setting the optional parameter standardize to false.

plot(guerry.Litercy, W)

In the Moran scatter plot, observations are located in four quadrants, depending on the value of the attribute and the value of their neighbors with respect to the mean: