Moran’s I

Written by Leandro Gonzalez


Moran’s I is a spatial statistic used to measure spatial autocorrelation (See formula below). Spatial autocorrelation is characterized by a correlation that occurs among samples that are geographically close (e.g. If we find a given type of bush in an area, we are more likely to find a second specimen of that same type of bush closer rather than farther away). The main idea behind spatially autocorrelation data is that values are not independent over space. This concept is based on the first law of geography proposed by Waldo Tobler: “Everything is related to everything else, but near things are more related than distant things.” By measuring spatial autocorrelation, we can determine how spatial patterns occur in our dataset or study area. Moran’s I is a global measure of spatial autocorrelation which takes the entire dataset and produces a single output value. To discover if spatial autocorrelation changes throughout the study area, one would employ a local measure of spatial autocorrelation (e.g. LISA).

Moran’s I formula

where is the number of spatial units indexed by and ; is the variable of interest; is the mean of ; and is an element of a matrix of spatial weights.

The Moran’s I measures the relationship between the values found for each feature in a dataset to the mean value of the dataset. Based on this approach, the Moran’s I tool provides an output telling us if the values in our dataset are clustered, dispersed, or random. When the difference in values among features that are geographically close to each other is smaller than the difference among all values throughout the dataset, the values are clustered. If the differences in values that are close to each other are similar to the differences in values throughout the entire dataset, the values are dispersed. Lastly, if some differences in values among features that are geographically close are smaller while others are greater than the difference in values throughout the entire dataset, the values approximate a random distribution. The Moran’s I score ranges from -1 (dispersed) to 1 (clustered). A value of 0, or very close to 0, refers to random distributions. Typically, Moran’s I is computed for point patterns or adjacent polygons.

Moran’s I score Distribution – Spatial pattern
Score > 0 Clustered
Score = 0 Random
Score < 0 Dispersed

ArcGIS allows users to calculate Moran’s I scores. This tool can be found on the Spatial Analysis toolbox (Figure 1). You can define neighboring areas based on adjacency, a user specified distance, or the distance to all features in the dataset.

The output graphic also provides Z-score to determine the level of confidence in that the pattern (whether random, clustered, or dispersed) did not occur by chance.

The Moran’s I tool does not give any information about the composition of clusters (high or low values). To obtain information about the composition of clusters, the researcher can use the General G-statistic. This tool is also available in the Spatial Analysis toolbox from ArcGIS.

It is recommended to use the Moran’s I tool before using any local regression analysis (e.g. geographically_weighted_regression). Using the Moran’s I tool will or will not confirm if spatial autocorrelation exists or not throughout the dataset.


Graphic with Moran’s I index and Z-score (see figure 1 above).

Successful Rangeland Uses

Augustine DJ, Booth DT, Cox SE, and Derner JD. 20120. Grazing Intensity and Spatial Heterogeneity in Bare Soil in a Grazing-Resistant Grassland. In RANGELAND ECOLOGY & MANAGEMENT. 65(1): 39-46. Looked at spatial patterns in bare soil in relation to stocking.

Sankey TT and Germino MJ. 2008. Assessment of juniper encroachment with the use of satellite imagery and geospatial data. In RANGELAND ECOLOGY & MANAGEMENT. 61(4):412-418. Attempted to determine if spectral mixture analysis could be used to estimate changes in Rocky Mountain juniper.

Ni SX, Lockwood JA, Wei YC, Jiang JJ, Zha Y, and Zhang HL. Spatial clustering of rangeland grasshoppers (Orthoptera : Acrididae) in the Qinghai Lake region of northwestern China. In AGRICULTURE ECOSYSTEMS & ENVIRONMENT. 95(1): 61-68. Used Geostatistical analyses to examine the pattern of spatial distribution of rangeland grasshoppers in the basin of Qinghai Lake in northwestern China.

References and Resources

McGrew, J. and C. Monroe. 2009. An Introduction to Statistical Problem Solving in Geography. 2nd edn. Long Grove, IL: Waveland Press, Inc.

Mitchell, A. 2009. The ESRI Guide to GIS Analysis. Volume 2: Spatial Measurements and Statistics. Redlands, CA: Esri Press.

Moran, P. 1950. Notes on Continuous Stochastic Phenomena. In Biometrika. 37(1-2): 17-23.

Comments are closed.