While global measures permit us to test for spatial patterning over the whole study area, it may be the case that there is significant autocorrelation in only a smaller section, which is swamped in the context of the whole. Both distance statistics (Getis and Ord, 1992, 1996, Ord and Getis, 1995), and the local indicators of spatial association derived by Anselin (1995, see also Getis and Ord, 1996), resemble passing a moving window across the data, and examining dependence within the chosen region for the site on which the window is centred. The specifications for the window can vary, using perhaps contiguity or distance at some spatial lag from the considered zone or point.
There are clear connections here both to the study of point patterns -- although methods for boundary correction have not been specifically added to weighting matrix definitions yet -- and to geostatistics, since these statistics have application to the exploration of non-homogeneities in relationships between locations across the study area. They are however subject to a correlation problem, that estimated values of the local indicator for neighbouring zones or sites will be correlated with each other because they are necessarily calculated from many of the same values, recalling that neighbouring placements of the moving window will most likely overlap. Ord and Getis (1995) provide suitable adjustments to critical values of the and statistics.
By extension from the global measure presented above, Getis (1991, see also Getis and Ord, 1996, Anselin, 1995) defines:
where is the measure for location i defined in terms of the weighting matrix with elements , and captures the interaction between the attribute values at locations i and j. Getis and Ord (1996) define six different measures, the local Moran : , three local Geary-type statistics ( , , and ) with , and the and statistics with and respectively ( and differ in that includes the attribute value at location i as well as those at ). and have been shown to be asymptotically normally distributed as the number of neighbours of location i, , increases, for instance by increasing the radius d around i used to define the weighting matrix.
The uses to which local statistics have been put are to identify ``hot-spots'', to assess stationarity prior to the use of methods assuming that the data do conform to this assumption, and other checks for heterogeneity in the data series. A typical application is to plot the estimates values of a local statistic with increasing distance from a selected location i, perhaps also controlling for direction (Getis and Ord, 1996, Bivand, 1997). In addition, Anselin (1996) has suggested that a plot of against its spatial lag , termed a Moran scatterplot, particularly used with dynamic linked visualization, may assist in revealing local patterning.
Examples of the application of local statistics in relation to topics in economic geography are O'Loughlin and Anselin (1996), examining trade bloc formation -- challenging assertions made by Krugman, and by Barkley et al. (1995) and Bao and Henry (1996) in exploring the use of local indicators in assessing the appropriateness of definitions of functional economic areas. Talen and Anselin (1998) have also used these methods to evaluate the measures used to define accessibility to public playgrounds, a study in the equity of urban service delivery.