4_K_Functions

# 4_K_Functions - NOTEBOOK FOR SPATIAL DATA ANALYSIS Part I...

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NOTEBOOK FOR SPATIAL DATA ANALYSIS Part I. Spatial Point Pattern Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 I.4-1 Tony E. Smith 4. K-Function Analysis of Point Patterns In the Bodmin Tors example above, notice from Figure 14a (p.20) that the clustering structure is actually quite different from that of the Redwood Seedling example in 12a (p.12). Rather than small isolate clumps, there appear to be two large groups of points in the northwest and southwest, separated by a large empty region. Moreover, the points within each group are actually quite uniform with clear subclusters. These observations suggest that the pattern of tors exhibits different structures at different scales . Hence the objective of the present section is to introduce a method of point pattern analysis that takes such scale effects into account, and in fact allows “scale” to become a fundamental variable in the analysis. 4.1 Wolf-Pack Example To motivate the main ideas, we begin with a new example involving wolf packs. A map is shown in Figure 1a below representing the relative locations of wolf packs in a portion of the Central Arctic Region in 1998. 1 The enlarged portion in Figure 1b is a schematic map depicting individual wolves in four of these packs. Fig.1a. Map of Wolf Packs Fig.1b. Enlarged Portion At the level of individual wolf locations in Figure 1b, there is a pattern of isolated clumps that bears a strong resemblance to that of the Redwood seedlings above. 2 Needless to say, this pattern would qualify as strongly clustered . But if one considers the larger map in Figure 1a, a different picture emerges. Here, the dominant feature is the remarkable uniformity of spacing between wolf packs. Each pack establishes a hunting territory large enough for its survival (roughly 15 to 20 km in diameter), and actively discourages other 1 This map is based on a more detailed map published in the Northwest Territories Wolf Notes , Winter 1998/99. See also http://www.nwtwildlife.rwed.gov.nt.ca/Publications/wolfnotes/wolf32.htm . 2 The spacing of individual wolves is of course exaggerated to allow a representation at this scale. 0 50 km Wolf packs

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NOTEBOOK FOR SPATIAL DATA ANALYSIS Part I. Spatial Point Pattern Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 I.4-2 Tony E. Smith packs from invading its territory. 3 Hence this pattern of wolf locations is very clustered at small scales , and yet very uniform at large scales . But if one were to analyze this wolf-location pattern using any of the nearest-neighbor techniques above, it is clear that only the small-scale clustering would be detected. Since each wolf is necessarily close to other wolves in the same dens, the spacing between dens would never be observed. In this simple example one could of course redefine wolf dens to be aggregate “points”, and analyze the spacing between these aggregates at a larger scale. But there is no way to analyze multiple scales using nearest neighbors without some form of re-aggregation.
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4_K_Functions - NOTEBOOK FOR SPATIAL DATA ANALYSIS Part I...

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