Anselin (1995) Local Indicators of Spatial Association — LISA

Summary

Anselin introduces LISA — Local Indicators of Spatial Association — a general class of local statistics for exploratory spatial data analysis. A LISA does two things: for each location it measures the degree of significant spatial clustering of similar values in its neighborhood, and, summed over all locations, it is proportional to a global measure of spatial association (e.g. Moran's I). This lets a global autocorrelation statistic be decomposed into each observation's contribution, revealing local clusters / "hot spots" and spatial outliers that a single global number hides. The local Moran is developed as the leading example, with significance assessed by conditional permutation, and applied to the spatial pattern of conflict among African countries plus Monte Carlo study.

Key Claims

• Definition of a LISA. A statistic is a LISA if (i) it gives, for each observation $i$, an indicator of significant spatial association of similar values around $i$, and (ii) the sum of the local statistics over all $i$ is proportional to a corresponding global statistic of spatial association.
• Local Moran. The prototypical LISA is $I_i = z_i \sum_{j} w_{ij} z_j$, where $z_i$ are the (mean-centered/standardized) deviations and $w_{ij}$ are (row-standardized) spatial-weights; then $\sum_i I_i \propto$ the global Moran's I. Positive $I_i$ flags a location surrounded by similar values (a High-High or Low-Low cluster), negative $I_i$ a spatial outlier (High-Low or Low-High).
• Two uses. (a) Local pockets of nonstationarity / hot spots — akin to the Getis-Ord $G_i$/$G_i^*$ statistics; (b) influence diagnostics — assessing how much each location contributes to the global statistic and identifying outliers, as in the Moran scatterplot (plot of $z_i$ against its spatial lag $\sum_j w_{ij} z_j$, whose four quadrants are the HH/LL/HL/LH association types and whose slope is global Moran's I).
• Inference by conditional permutation. Because the exact distribution of a local statistic is intractable and locations overlap, significance is assessed by a conditional randomization/permutation test (permute the other values, holding $z_i$ fixed); the strong multiple-comparison problem (one test per location, with dependence) is flagged and only approximately correctable.
• Application. The local Moran maps significant clusters and outliers in African inter-country conflict, demonstrating spatial structure invisible to the single global Moran's I.

Concepts Introduced or Extended

• Spatial Autocorrelation

Entities Mentioned

• Luc Anselin

Quotes

"A local indicator of spatial association (LISA) ... the sum of LISAs for all observations is proportional to a global indicator of spatial association."

"They may be interpreted as indicators of local pockets of nonstationarity, or hot spots ... [and] used to assess the influence of individual locations on the magnitude of the global statistic and to identify outliers."

My Take

This is one of the most-used papers in all of spatial analysis, and its idea is deceptively simple: a global autocorrelation statistic is an average, and averages hide local structure, so decompose the global into per-location pieces and map them. The local Moran and the accompanying Moran scatterplot became the default exploratory tools of spatial econometrics precisely because they turn "is there spatial autocorrelation?" into "where are the clusters and where are the anomalies?" — the question applied researchers actually have. The honest limitation Anselin foregrounds is inference: overlapping neighborhoods and one test per location make the multiple-comparison problem severe and the permutation p-values only indicative, a caveat too often ignored in applied LISA maps. For the wiki it anchors a new spatial autocorrelation concept — the measurement/testing side of spatial dependence that complements the modeling side captured by the CAR and point-process pages.