Moran's I

Moran's I, developed by Patrick Moran in 1950, is a global spatial autocorrelationSpatial AutocorrelationSpatial autocorrelation measures the degree to which values at nearby locations are similar (positive) or dissimilar ... statistic that measures the overall degree of spatial clusteringSpatial ClusteringSpatial clustering groups geographic features based on their spatial proximity and optionally their attribute similar... in a georeferenced dataset. It evaluates whether the values of a variable observed at nearby locations are more similar (positive spatial autocorrelation), less similar (negative spatial autocorrelation), or unrelated to their spatial proximity.

Moran's I is computed by comparing the deviation of each observation from the mean with the deviations of its neighbors, as defined by a spatial weights matrix. The statistic ranges from approximately -1 to +1. A value near +1 indicates strong positive spatial autocorrelationSpatial AutocorrelationSpatial autocorrelation measures the degree to which values at nearby locations are similar (positive) or dissimilar ... (clustering of similar values), a value near -1 indicates strong negative autocorrelation (checkerboard pattern of dissimilar values), and a value near 0 suggests a spatially random pattern. Inference is typically performed using a z-score derived from the expected value and variance under the null hypothesis of spatial randomness, or through permutation testing.

Epidemiologists use Moran's I to test whether disease rates cluster geographically, providing evidence for localized environmental or behavioral risk factors. Economists apply it to income and unemployment data to detect regional economic disparities. Criminologists assess whether crime rates exhibit spatial dependence to justify spatially targeted interventions. Environmental scientists test pollution or temperature measurements for clustering. Real estate analysts examine whether property values are spatially autocorrelated, informing hedonic pricing models.

Moran's I is a global statistic that provides a single summary value for the entire study area. For identifying the locations of individual clusters and outliers, analysts use its local counterpart, Local Moran's I (a LISA statistic). Geary's CGeary's CGeary's C is a global measure of spatial autocorrelation that uses squared differences between neighboring values to ... provides an alternative global measure that is more sensitive to local variations. Together, these measures form the core toolkit for exploratory spatial data analysis.