Is the Pattern Random or Clustered?
When you see a map of disease rates, crime incidents, or mineral grades, the first question is always: is the spatial pattern significant, or could it have occurred by chance? Moran's I statistic, introduced by Patrick Moran in 1950, provides a rigorous answer. It measures the overall degree of spatial autocorrelation — the tendency for nearby locations to have similar (or dissimilar) values. A significantly positive Moran's I confirms what your eye suspects: the pattern is clustered.
From Global to Local
While Moran's I gives a single global summary, Luc Anselin's LISA (1995) reveals where clustering occurs. LISA decomposes the global statistic into local contributions, identifying four distinct spatial regimes. High-high clusters (hot spots) show concentrations of high values — disease outbreak centers, mineral-rich zones, or high-crime neighborhoods. Low-low clusters (cold spots) show concentrations of low values. High-low and low-high spatial outliers mark locations that differ sharply from their surroundings — a wealthy neighborhood in a poor region, or a low-grade drill hole in an otherwise rich ore zone.
Interactive Exploration
In this simulation, each circle represents a spatial unit colored by its value (red=high, blue=low). Lines connect neighbors within the distance threshold. The global Moran's I and z-score update in real time as you adjust the spatial autocorrelation strength. Increasing ρ makes similar values cluster together; decreasing it creates a random salt-and-pepper pattern. LISA significance is shown by circle borders — thick borders indicate locally significant clusters or outliers at the 95% confidence level.
Applications in Science and Policy
Spatial clustering analysis is ubiquitous in modern research and decision-making. Epidemiologists use LISA to identify disease hot spots for targeted intervention. Criminologists detect crime clusters to optimize police patrol allocation. Ecologists find biodiversity hot spots and cold spots. Real estate analysts identify neighborhoods with unusual price patterns. Environmental scientists locate contamination clusters. In each case, the statistical significance test ensures that identified patterns are genuine rather than artifacts of random variation.