If we look more closely at the Moran I index we see that it can be disaggregated to provide a series of local indices. We start with the standard formulation we provided earlier:

Examining the computation of this “global” index provided earlier in Figure 5‑31B, we can see that each row in the computation contributes to the overall sum (Figure 5‑35A), with the covariance component being 16.19 and overall Moran I value 0.03167. Each row item could be standardized by the overall sum of squared deviations and adjusted by the number of cells, n, as shown. The total of these local contributions divided by the total number of joins, ΣΣwij, gives the overall or Global Moran I value. These individual components, or Local Indicators of Spatial Association (LISA), can be mapped and tested for significance to provide an indication of clustering patterns within the study region (Figure 5‑36). The calculations require some adjustment in order to provide Global Moran I and LISA values in their commonly used form. The first adjustment is to the weights matrix. It is usual to standardize the row totals in the weights matrix to sum to 1. One advantage of row standardization is that for each row (location) the set of weights corresponds to a form of weighted average of the neighbors that are included, which also enables different models to be readily compared. Row standardization may also provide a computationally more suitable weights matrix. This procedure alters the Global Moran I value when rows have differing totals, which is the typical situation. For the example above the effect is to increase the computed Moran I to 0.0736. The second adjustment involves using (n‑1) rather than n as the row multiplier, and then the sum is re-adjusted by this factor to produce the Global index value. These changes are illustrated in Figure 5‑35B and Figure 5‑36 and correspond to the computations performed within GeoDa, R and SpaceStat (Anselin, personal communication). ArcGIS offers the option to have no weights matrix standardization, row standardization (dividing rows by row totals) or total weights standardization (dividing all cells by the overall cell total). The Geary C contiguity index may also be disaggregated to produce a LISA measure, in the same manner as for the Moran I. Another measure, known as the "G" statistic, due to Getis and Ord (1992, 1995), may also be used in this way. The latter is supported within ArcGIS as both a global index and in disaggregated form for identifying local clustering patterns, which ArcGIS considers this as a form of hot-spot analysis.

Figure 5‑35 Local Moran I computation

A. C*W: Adjustment by multiplication of the weighting matrix, W |

B. C*W1: Adjustment by multiplication of the row-adjusted weighting matrix, W1 |

The ArcGIS implementation of the G statistic facilitates the use of simple fixed Euclidean distance bands (threshold distances) within binary weights (as described below), but also permits the use of the Manhattan metric and a variety of alternative models of contiguity such as: inverse distance; inverse distance squared; so-called Zones of Indifference (a combination of inverse distance and threshold distance); and user-defined spatial weights matrices. Unlike the Moran I and Geary C statistics, the Getis and Ord G statistic identifies the degree to which high or low values cluster together.

The standard global form of the Getis and Ord G statistic is:

whilst the local variant is

The weights matrix, W, is typically defined as a symmetric binary contiguity matrix, with contiguity determined by a (generalized) distance threshold, d. A variant of this latter statistic, which is regarded as of greater use in hot-spot analysis, permits i=j in the expression (so includes i in the summations, with wii=1 rather than 0 in the binary weights contiguity model) and is known as the Gi* statistic rather than Gi. Note that these local statistics are not meaningful if there are no events or values in the set of cells or zones under examination.

The expected value of G under Complete Spatial Randomness (CSR) is:

where n is the number of points or zones in the study area and W is the sum of the weights, which for binary weights is simply the number of pairs within the distance threshold, d. The equivalent expected value for the local variant is:

where n is the number of points or zones within the threshold distance, d, for point/region i. Closed formulas for the variances have been obtained but are very lengthy. They facilitate the construction of z-scores and hence significance testing of the global and local versions of the G statistics. For the local measure both computed values and z-scores may be useful to map.

The Rookcase add-in provides support for all three LISA measures with variable lag distance and number of lags. Further discussion of these methods is provided in several other sections of this Guide: in Section 5.2, Exploratory Spatial Data Analysis (ESDA); in Section 5.6.5, Spatial filtering models; and in Sections 4.6, Grid Operations and Map Algebra and Chapter 6, Surface and Field Analysis, relating to grid files, since there are parallels between several of the concepts applied.