Global Moran's I — Measuring Overall Spatial Autocorrelation
Also known as: Moran's I, global spatial autocorrelation index, Moran index, GMI
Global Moran's I is the most widely used single-number summary of spatial autocorrelation across an entire study area. It compares the attribute value at each location with values at neighbouring locations using a spatial weights matrix, and returns a statistic ranging from −1 (perfect dispersion) through 0 (spatial randomness) to +1 (perfect clustering). A significance test determines whether the observed pattern is stronger than random chance.
Key highlights
- Provides a single, easily communicated index of the overall degree of spatial clustering or dispersion.
- Well-grounded theoretically with exact and permutation-based inference under multiple distributional assumptions.
- Applicable to both polygon (areal) and point data through flexible spatial weights matrix specifications.
- Widely implemented in standard software (GeoDa, R spdep, ArcGIS, Python PySAL), making results reproducible and comparable.
- Directly usable as a model-specification diagnostic: significant I in OLS residuals flags the need for spatial regression.
Intuition
This section is available to Pro members. Upgrade to Pro
How it works
This section is available to Pro members. Upgrade to Pro
When to use it
Use Global Moran's I as a first diagnostic step whenever you analyse spatially referenced data and want to know whether the values are spatially clustered, dispersed, or random across the whole study area. It is appropriate for continuous or ordinal attributes measured at areal units (countries, districts, census tracts) or at point locations with a defined neighbourhood structure. It is particularly valuable before fitting any spatial regression model, because significant positive autocorrelation in regression residuals indicates that OLS assumptions are violated. Do not use it when you only care about where clusters occur (use Local Moran's I or Getis-Ord Gi* for that), when the spatial weights matrix cannot be meaningfully defined, or when the variable is categorical — in those cases different methods apply.
Strengths & limitations
- Provides a single, easily communicated index of the overall degree of spatial clustering or dispersion.
- Well-grounded theoretically with exact and permutation-based inference under multiple distributional assumptions.
- Applicable to both polygon (areal) and point data through flexible spatial weights matrix specifications.
- Widely implemented in standard software (GeoDa, R spdep, ArcGIS, Python PySAL), making results reproducible and comparable.
- Directly usable as a model-specification diagnostic: significant I in OLS residuals flags the need for spatial regression.
- Being a global summary, it cannot reveal where clusters or outliers are located — a large positive I may hide local pockets of negative autocorrelation.
- Results are sensitive to the choice of spatial weights matrix (contiguity definition, distance threshold, or k); different choices can yield different conclusions.
- For very large datasets the computational cost of the full n×n weights matrix can be prohibitive unless sparse representations are used.
- The normal-approximation inference is unreliable for small samples (n < 30) or when the attribute is heavily skewed; permutation inference is preferred.
Common pitfalls
This section is available to Pro members. Upgrade to Pro
Applications
This section is available to Pro members. Upgrade to Pro
Frequently asked
What is the difference between Global Moran's I and Local Moran's I?
Global Moran's I returns a single value summarising autocorrelation across the entire study area. Local Moran's I (a LISA statistic) decomposes this into location-specific values, identifying which individual areas contribute clusters or spatial outliers. Use Global Moran's I to answer 'Is there clustering?' and Local Moran's I to answer 'Where is it?'
How do I choose the spatial weights matrix?
The choice should reflect your theory about which locations influence each other. For polygon data, queen contiguity (shared boundary or vertex) is common. For point data, k-nearest neighbours or an inverse-distance kernel are typical. Sensitivity analysis — running the test with several plausible W matrices — is good practice to check how robust your conclusions are.
Should I use the normal-approximation or permutation inference?
Permutation inference (randomisation test) is generally preferred because it makes no distributional assumptions about the attribute. It works by repeatedly shuffling values across locations and computing I each time to build an empirical null distribution. The normal approximation is convenient for large n when the attribute is approximately normal.
What does a significant negative Moran's I mean?
A significantly negative I indicates spatial dispersion: high values tend to be surrounded by low values and vice versa, producing a checkerboard pattern. This is less common than positive autocorrelation in social and environmental data but can occur in competitive spatial systems, such as market territories or certain ecological niches.
Can I apply Global Moran's I to regression residuals?
Yes, and this is one of its most important uses. A significant I in OLS residuals indicates that spatial dependence has not been adequately captured by the predictors, violating the independence assumption and biasing inference. The remedy is typically to re-specify the model as a spatial lag or spatial error model.
Sources
- 1.Moran, P. A. P. (1950). Notes on continuous stochastic phenomena. Biometrika, 37(1/2), 17–23.
- 2.Cliff, A. D., & Ord, J. K. (1981). Spatial Processes: Models and Applications. Pion.ISBN 0850860814
You have read it. What now?
Cite this page
ScholarGate. (2026, June 3). Global Moran's I. ScholarGate. https://scholargate.app/spatial-analysis/global-morans-i