Applications of Local Point Patterns Statistics in Plant Ecology

The potential usefulness of the local second-order point pattern statistic, the neigh-borhoodL_i(t), or local spatial statistics in general, has been highlighted recently (Fortin and Dale 2005, Perry et al. 2006). Perry et al. (2006) state that “while the

global tests suggest there is spatial segregation and at what scale(s), the local tests can explicitly show where this is occurring.” And yet there have been very few appli-cations of them since Getis and Franklin (1987) was published. Apparently it is still an idea that is ahead of its time.

Camarero et al. (2005) mapped the values of the local statistic to show where tree seedlings of a relict pine population were aggregated or repulsed at selected scales. This was one line of evidence supporting their conclusion that frequent short-distance dispersal events induced the primary spatial clustering of seedlings in safe sites, while wind turbulence caused rare medium-distance dispersal events, result-ing in clusterresult-ing at multiple scales. As part of a detailed study of subalpine forest succession based on forests reconstructed over time using dendrochronological (tree ring) data, Donnegan and Rebertus (1999) calculated a weighted bivariate neighbor-hoodL_i(t)from the number of spruce and fir neighbors surrounding a target adult limber pine to account for shading and other interactions between the species. They then used this index of clumping in a logistic regression model of pine survival and found that mortality was highest when pines were surrounded by many spruces and firs at lag distance of 2 m. Potvin et al. (2003), studying habitat selection by deer, used high values of the local K-function (at 0.5–2 km distance) based on observed locations of deer to map areas of animal concentrations, and found those patterns to be consistent with those derived from habitat selection indices and kernel estimators.

Dale and Powell (2001) presented a new method of second order point pattern analysis based on circumcircles, and included a spatially explicit mapping of pos-itive and negative residual scores to locate events that are members of clusters vs.

gaps. In one example this method was used to show that regions of high spruce seedling density and high tree density did not coincide, suggesting that local con-ditions for germination were as important to establishment as high density seed source.

Although they did not use localL_i(t), Shi and Zhang (2003) applied local spatial statistics or “local indicators of spatial association” (LISA) (Anselin, 1995; Ord and Getis, 1995) to forests. These are area pattern, rather than point pattern statistics, applied to sample data, usually measurements of a continuous variable for a point or area. In this study several LISAs including local Moran’sI, local Geary’sCand the localG^∗_i statistic, were derived from size measurements of individual trees and compared to traditional forestry competition indices used in models of tree growth.

They performed quite well as predictors and were also useful for identifying clusters of trees of similar size.

Wells and Getis (1999) also used a local statistic G^∗_i (Ord and Getis, 1995) applied to measurements of tree age made for individuals, to identify the locations of clusters of old or young trees. In their study Torrey pine trees (Pinus torreyana), a rare, endemic California pine species, were mapped in three 1-ha plots. They located clusters of younger trees and found greater clustering in young stands, consistent with establishment of cohorts following episodic disturbance (fire). This local area statistic was used in lieu of the local point pattern statistic, neighborhoodL_i(t), and without discussion of the earlier work, although Wulder and Boots (1998) identify second order neighborhood analysis (Getis and Franklin 1987) as an early

Fig. 9.1 Locations of 440 Torrey pine trees (Pinus torreyana) in the East Grove area of Torrey Pines State Reserve, La Jolla, CA, USA (E. Santos and J. Franklin, unpublished data). Map on left shows tree locations scaled by size (DBH, trunk diameter at 1.3 m height), and center map shown the tree locations scaled by the local value of L(t) at lag of 14 m (see Fig. 9.2). Negative values shown as squares, positive values as circles. Map on right shown tree locations scaled by the values of local Gi* (see text) where neighborhood contiguity is based on a lag distance of 25 m (maximum nearest neighbor distance used to avoid islands). These analyses were carried out using the spatstat package in the R statistical environment (R Development Core Team, 2004)

formulation if the Gi* statistic – the Getis model (circa 1984) – which it is. Recently, my student and I initiated a project to map the entire mainland population of Torrey pines (over 5,000 trees) and measure the size and condition of each tree for con-servation monitoring purposes. For illustration, I show the distributions of all 440 trees in the 5.69-ha East Grove area (encompassing one of Wells and Getis’ sites), of localL_i(t), and of localG^∗_i (9.1). GlobalL(t)indicates significant clumping peak-ing at about 14-m lag distance, and Moran’sIindicated positive spatial association of tree size at roughly the same scale (9.2). Figure 9.1 shows that regions of high tree density mainly comprise clumps of small trees, consistent with the previous observations of Wells and Getis (1999).

9.7 Conclusion

Judging from its citation patterns, the paper by Getis and Franklin (1987) continues to influence the practice of spatial point pattern analysis in plant ecology. However, most practitioners continue to apply global analyses, while local spatial statistics are mainly advocated by specialists in methodological papers. Therefore, the legacy of this paper is, in part, accidental. Although written to introduce local spatial statistics to ecologists, it is most often cited with reference to global point pattern statistics, perhaps because of its clear summary of classic work in this area, and perhaps also because of the visibility of the journal in which it appeared.

20

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Fig. 9.2 Global_L(t)for the trees shown in Fig. 9.1 at lags of 2–100 m showing significant clump-ing at all scales and a peak in_L(t)at 10–18 m; (b) Moran’s I as a measure of spatial autocorrelation of tree size (DBH, see Fig. 9.1 caption) where neighborhood contiguity is based on a lag distance of 10 m, indicating significant positive spatial association of tree size at lag 1 (10 m). These analy-ses were carried out using the splancs package (Rowlingson and Diggle, 1993) in the R statistical environment (R Development Core Team, 2004)

Those specialists considering methodology have focused their recent efforts on improved edge corrections and automated methods for delineating homogeneous subareas. They have advocated the application of the neighborhood density func-tion to explore pattern as a funcfunc-tion of discrete ranges of distances, and I would also advocate fitting alternative models of clustering or inhibition (certainly not a new idea). Perhaps citation patterns, like sausage- and law-making, should not be exam-ined in such detail, but the citing of Getis and Franklin for reasons other than the local spatial statistic they introduced, in a majority of cases, is perhaps indicative of a gap that still exists between theory and practice in spatial analysis.

Where local spatial statistics have been applied to point patterns in forestry and ecology they have proven very useful in identifying individuals that are members of clusters or that fall within gaps, and in locating groups of individuals that share some characteristic (such as similar size). They have been related empirically to, e.g., regeneration patterns, growth characteristics, competition and mortality. Method-ologically, measures of local context have been used delineate areas over which the assumption of stationarity is valid. There is considerable future opportunity for both exploratory spatial data analysis and hypothesis testing in spatial ecology using global and local methods.

Acknowledgements I thank E. Santos for the use of unpublished data from her graduate research, and the Torrey Pines Association and D.S. Smith, California State Parks, for supporting her study.

My NSF grant on spatial inference and prediction from species data (0452389) supported the writ-ing of this chapter, and I thank my lab readwrit-ing group on spatial ecology for their feedback in Fall 2006. This chapter was greatly improved by the comments of S.J. Rey and B. Boots, and I thank S.J. Rey and L. Anselin for providing me the privilege of contributing to this book.