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Next: The Modifiable Areal Unit Up: Exploratory spatial data analysis Previous: Global measures of spatial

Local indicators of spatial association

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 tex2html_wrap_inline863 and tex2html_wrap_inline865 statistics.

By extension from the global measure tex2html_wrap_inline855 presented above, Getis (1991, see also Getis and Ord, 1996, Anselin, 1995) defines:


where tex2html_wrap_inline869 is the measure for location i defined in terms of the weighting matrix with elements tex2html_wrap_inline873 , and tex2html_wrap_inline875 captures the interaction between the attribute values at locations i and j. Getis and Ord (1996) define six different measures, the local Moran tex2html_wrap_inline881 : tex2html_wrap_inline851 , three local Geary-type statistics ( tex2html_wrap_inline885 , tex2html_wrap_inline887 , and tex2html_wrap_inline889 ) with tex2html_wrap_inline853 , and the tex2html_wrap_inline863 and tex2html_wrap_inline865 statistics with tex2html_wrap_inline897 and tex2html_wrap_inline899 respectively ( tex2html_wrap_inline863 and tex2html_wrap_inline865 differ in that tex2html_wrap_inline865 includes the attribute value at location i as well as those at tex2html_wrap_inline909 ). tex2html_wrap_inline863 and tex2html_wrap_inline865 have been shown to be asymptotically normally distributed as the number of neighbours of location i, tex2html_wrap_inline909 , 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 tex2html_wrap_inline807 against its spatial lag tex2html_wrap_inline927 , 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.

next up previous
Next: The Modifiable Areal Unit Up: Exploratory spatial data analysis Previous: Global measures of spatial

Roger Bivand
Fri Mar 5 08:30:34 CET 1999