Moran's I
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.
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Definition
Moran's I is defined as
where is the number of spatial units indexed by and ; is the variable of interest; is the mean of ; and 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
Its variance equals
where
[3] Values of I range from −1 to +1. Negative values indicate negative spatial autocorrelation and positive values indicate positive spatial autocorrelation. A zero value indicates a random spatial pattern. For statistical hypothesis testing, Moran's I values can be transformed to Z-scores.
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.
Uses
Moran's I is widely used in the fields of geography and GIScience. Some examples include:
- The analysis of geographic differences in health variables.[4]
- It has been used to characterize the impact of lithium concentrations in public water on mental health.[5]
- It has also recently been used in dialectology to measure the significance of regional language variation.[6]
Sources
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- ↑ Cliff and Ord (1981), Spatial Processes, London
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