In statistics, Moran's I is a measure of spatial autocorrelation developed by Patrick Alfred Pierce Moran. 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
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
 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.
- The analysis of geographic differences in health variables.
- It has been used to characterize the impact of lithium concentrations in public water on mental health.
- It has also recently been used in dialectology to measure the significance of regional language variation.
- Moran, P. A. P. (1950). "Notes on Continuous Stochastic Phenomena". Biometrika 37 (1): 17–23. doi:10.2307/2332142. JSTOR 2332142.
- 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.
- Cliff and Ord (1981), Spatial Processes, London
- "The Analysis of Spatial Association by Use of Distance Statistics". Geographical Analysis 24 (3): 189–206. 3 Sep 2010. doi:10.1111/j.1538-4632.1992.tb00261.x.
- "Geospatial examination of lithium in drinking water and suicide mortality". Int J Health Geogr. 11 (1): 19. 2012. doi:10.1186/1476-072X-11-19. PMID 22695110.
- Grieve, Jack (2011). "A regional analysis of contraction rate in written Standard American English". International Journal of Corpus Linguistics 16 (4): 514–546. doi:10.1075/ijcl.16.4.04gri.