Local Moran's I (LISA)
Local Moran's I Statistic (LISA) · Also known as: Local Indicator of Spatial Association, LISA statistic, Anselin Local Moran, local spatial autocorrelation index
Local Moran's I, introduced by Luc Anselin in 1995, is a Local Indicator of Spatial Association (LISA) that decomposes global spatial autocorrelation into location-specific contributions. For every observation it produces a signed statistic and a significance value, enabling researchers to identify spatial clusters (high-high, low-low) and spatial outliers (high-low, low-high) on a map.
Read the full method
Sign in with a free account to read this section.
Method map
The neighbourhood of related methods — select a node to explore.
+14 more
When to use it
Use Local Moran's I when you suspect that spatial clustering or outliers vary across your study area and you need to identify exactly where those patterns occur. It is appropriate for areal data (census tracts, countries, grid cells) and point data aggregated to zones, when a meaningful spatial weights matrix can be defined. Use it after a global Moran's I test has detected overall autocorrelation or when local heterogeneity is the primary research question. Do not use it when observations are truly independent (no spatial structure), when the study area has fewer than about 30 units (permutation distributions become unreliable), or when you need a regression model — in those cases combine LISA with spatial lag or spatial error regression.
Strengths & limitations
- Decomposes global autocorrelation into location-specific contributions, revealing where clustering or anomalies actually occur.
- Simultaneously identifies spatial clusters (HH, LL) and spatial outliers (HL, LH), providing richer map-based insight than global statistics.
- Permutation-based significance testing requires no distributional assumptions about the data.
- Directly interpretable output: cluster maps communicate findings to non-technical audiences.
- Computationally straightforward and widely implemented in GeoDa, R (spdep), Python (PySAL), and ArcGIS.
- Multiple testing: running a significance test at every location inflates the Type I error rate; corrections such as FDR or Bonferroni should be applied.
- Results are sensitive to the choice of spatial weights matrix; different neighbour definitions can produce different cluster maps.
- Assumes spatial stationarity of the underlying process; if the process itself shifts across the study area the statistic may be misleading.
- Requires sufficient neighbours per unit; edge locations with few neighbours have less stable estimates.
- Does not model causal relationships — it describes patterns, not mechanisms.
Frequently asked
How is Local Moran's I different from global Moran's I?
Global Moran's I produces a single statistic summarising overall spatial autocorrelation across the entire study area. Local Moran's I produces a separate statistic and p-value for each location, allowing you to map exactly where clusters or outliers occur. Local Moran's I values sum (approximately) to the global statistic.
What do the four LISA cluster types mean?
High-High (HH): a high-value location surrounded by high-value neighbours — a hotspot. Low-Low (LL): a low-value location surrounded by low-value neighbours — a coldspot. High-Low (HL): a high-value location surrounded by low-value neighbours — a spatial outlier. Low-High (LH): a low-value location surrounded by high-value neighbours — another type of outlier.
How do I choose the spatial weights matrix?
The choice should reflect the assumed spatial interaction process. Queen contiguity (shared edges or corners) is common for polygon data; distance-based or k-nearest-neighbour weights suit irregular point patterns. Sensitivity analysis — re-running the analysis with alternative weights — is recommended to check robustness.
Do I need to correct for multiple testing?
Yes. Running a hypothesis test at every location means that some will appear significant by chance. The False Discovery Rate (FDR) correction is commonly recommended in spatial analysis; Bonferroni is more conservative. GeoDa and PySAL both offer FDR-corrected cluster maps.
Can I use Local Moran's I with point data?
Yes, but point data must first be aggregated to zones (e.g., by counting points per grid cell or administrative unit) to construct a spatial weights matrix. Alternatively, kernel density surfaces can be used, though the LISA framework applies most naturally to areal data.
Sources
- Anselin, L. (1995). Local indicators of spatial association—LISA. Geographical Analysis, 27(2), 93–115. DOI: 10.1111/j.1538-4632.1995.tb00338.x ↗
- Anselin, L. (2010). Local spatial autocorrelation. In A. S. Fotheringham & P. A. Rogerson (Eds.), The SAGE Handbook of Spatial Analysis (pp. 255–278). SAGE Publications. link ↗
How to cite this page
ScholarGate. (2026, June 3). Local Moran's I Statistic (LISA). ScholarGate. https://scholargate.app/en/spatial-analysis/local-morans-i
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Geary's CSpatial analysis↔ compare
- Hot Spot AnalysisSpatial analysis↔ compare
- Local Getis-Ord Gi*Spatial analysis↔ compare
- Local Indicators of Spatial AssociationSpatial analysis↔ compare
- Moran's ISpatial analysis↔ compare
- Spatial AutocorrelationSpatial analysis↔ compare