Local Indicators of Spatial Association (LISA)
Also known as: LISA, local spatial autocorrelation statistics, local Moran's I, Anselin LISA
LISA, introduced by Luc Anselin in 1995, decomposes a global spatial autocorrelation index into a location-specific statistic for every observation. It identifies where statistically significant spatial clusters and outliers occur on a map, enabling researchers to move beyond a single global summary and pinpoint the geographic sources of spatial dependence.
Key highlights
- Pinpoints exact geographic locations of significant spatial clusters and outliers, going far beyond a single global index.
- Decomposes global Moran's I so that individual locations driving the overall pattern are identifiable.
- The permutation-based inference requires no distributional assumption about the data.
- Produces intuitive, directly mappable output (cluster maps) that communicate findings to non-specialist audiences.
- Flexible: applies to any continuous or count attribute on areal or point-in-polygon data with any neighbor definition.
- Widely implemented in open-source tools (GeoDa, PySAL/esda, R spdep) with active community support.
Intuition
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How it works
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When to use it
Use LISA when you need to identify specific geographic locations of spatial clusters or outliers — for example, mapping disease hot spots, crime clusters, or regional economic inequalities — rather than just testing whether clustering exists globally. It requires georeferenced areal or point data with an attribute measured on at least an interval scale, and a meaningful neighbor definition. LISA is less appropriate for very small datasets (fewer than ~30 locations) where permutation tests lack power, for purely temporal data without a geographic component, or when the research question is about global trends rather than local heterogeneity. Be aware that running LISA on many locations simultaneously raises multiple-testing concerns; corrections such as the false discovery rate (FDR) are advisable.
Strengths & limitations
- Pinpoints exact geographic locations of significant spatial clusters and outliers, going far beyond a single global index.
- Decomposes global Moran's I so that individual locations driving the overall pattern are identifiable.
- The permutation-based inference requires no distributional assumption about the data.
- Produces intuitive, directly mappable output (cluster maps) that communicate findings to non-specialist audiences.
- Flexible: applies to any continuous or count attribute on areal or point-in-polygon data with any neighbor definition.
- Widely implemented in open-source tools (GeoDa, PySAL/esda, R spdep) with active community support.
- Results are sensitive to the choice of spatial weights matrix; different neighbor definitions can produce substantially different cluster maps.
- Running tests simultaneously for all locations inflates the type-I error rate; without multiple-testing correction some apparent clusters may be false positives.
- The permutation-based p-values are pseudo-p-values; they approximate but do not replace exact inferential guarantees.
- LISA identifies co-location of similar values but does not explain why clusters exist; causal interpretation requires additional analysis.
- Performance degrades with very small samples (n < 30) where permutation distributions are coarse.
Common pitfalls
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Applications
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Frequently asked
What is the difference between LISA and global Moran's I?
Global Moran's I produces a single statistic summarising the overall degree of spatial autocorrelation across the entire study area. LISA decomposes this global measure into a local statistic for every location, revealing where clusters and outliers are concentrated. A global index can indicate no overall autocorrelation even when strong local clusters cancel each other out.
How do I choose the spatial weights matrix?
The choice should reflect the substantive theory of spatial interaction for your phenomenon. Rook or queen contiguity is common for regular areal grids; distance-band or k-nearest-neighbor weights suit point data or irregular polygons. Always report the choice and run a sensitivity check with an alternative specification to assess robustness.
How many permutations should I use?
At least 999 permutations are standard for p < 0.05; use 9,999 if you need stable estimates near p < 0.01. More permutations reduce Monte Carlo variability in the pseudo-p-values but increase computation time linearly.
What does a High-Low outlier mean on a cluster map?
A High-Low spatial outlier is a location with a high attribute value surrounded predominantly by neighbors with low values. It stands out from its local context. This can signal a genuine anomaly (e.g., an isolated affluent enclave in a deprived region) or a data quality issue worth investigating.
Should I apply a multiple-testing correction?
Yes, especially with large datasets where many locations are tested simultaneously. Without correction, the expected number of spurious significant results grows with the number of locations. The Benjamini-Hochberg false discovery rate (FDR) procedure is widely recommended; GeoDa and PySAL both support it.
Sources
- 1.Anselin, L. (1995). Local Indicators of Spatial Association — LISA. Geographical Analysis, 27(2), 93–115.
- 2.Anselin, L. (2010). Local Spatial Autocorrelation. In A. S. Fotheringham & P. A. Rogerson (Eds.), The SAGE Handbook of Spatial Analysis (pp. 255–275). SAGE Publications.
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Cite this page
ScholarGate. (2026, June 3). Local Indicators of Spatial Association. ScholarGate. https://scholargate.app/spatial-analysis/local-indicators-of-spatial-association