A Characteristic-Hull Based Method for Home Range Estimationtgis_

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1 : Research Article A Characteristic-Hull Based Method for Home Range Estimationtgis_ Joni A Downs Department of Geography University of South Florida Mark W Horner Department of Geography Florida State University Abstract Recent literature has reported inaccuracies associated with some popular home range estimators such as kernel density estimation, especially when applied to point patterns of complex shapes. This study explores the use of characteristic hull polygons (CHPs) as a new method of home range estimation. CHPs are special bounding polygons created in GIS that can have concave edges, be composed of disjoint regions, and contain areas of unoccupied space within their interiors. CHPs are created by constructing the Delaunay triangulation of a set of points and then removing a subset of the resulting triangles. Here, CHPs consisting of 95% of the smallest triangles, measured in terms of perimeter, are applied for home range estimation. First, CHPs are applied to simulated animal locational data conforming to five point pattern shapes at three sample sizes. Then, the method is applied to black-footed albatross (Phoebastria nigripes) locational data for illustration and comparison to other methods. For the simulated data, 95% CHPs produced unbiased home range estimates in terms of size for linear and disjoint point patterns and slight underestimates (8 20%) for perforated, concave, and convex ones. The estimated and known home ranges intersected one another by 72 96%, depending on shape and sample size, suggesting that the method has potential as a home range estimator. Additionally, the CHPs applied to estimate albatross home ranges illustrate how the method produces reasonable estimates for bird species that intensively forage in disjoint habitat patches. Address for correspondence: Joni A Downs, Department of Geography, University of South Florida, 4202 East Fowler Avenue, NES 107, Tampa, FL 33620, USA. jdowns@cas.usf.edu doi: /j x

2 528 J A Downs and M W Horner 1 Introduction Home range estimation is one of the most widely applied forms of spatial analysis in animal ecology, as it provides the most basic measurement of animal space use patterns. Typically, the home range is considered the smallest spatial area in which an animal spends 95% of its activities (Worton 1987, White and Garrott 1990). Home ranges are estimated from animal tracking data, which consist of a set of points that document an individual s location over time, using a variety of statistical, and more recently, GIS-based techniques (Casaer et al. 1999, Kernohan et al. 2001, Selkirk and Bishop 2002, Sampson and Delgiudice 2006). However, Laver and Kelly (2008) noted that there is no common consensus as to the most accurate or acceptable way to compute home range estimates. Resolving this issue is paramount if results from different studies of animal space-use patterns and movements are to be both valid and comparable to one another. Kernel density estimation (KDE) and the minimum convex polygon (MCP) have been traditionally favoured for home range analysis, although both techniques have garnered much criticism in the literature during recent years. In particular, studies have noted that KDE-based results are affected by the method of bandwidth selection (e.g. Gitzen et al. 2006, Horne and Garton 2006, Fieberg 2007), are sensitive to sample size (Seaman et al. 1996, Blundell et al. 2001), and are not robust to the shape of the point pattern of locations (Downs and Horner 2008, Mitchell and Powell 2008). Numerous studies have reported that KDE systematically produces home range estimates that are positively biased (Girard et al. 2002, Hemson et al. 2005, Downs and Horner 2008). Downs and Horner (2008) also noted that KDE methods produce more biased estimates for home ranges that conform to complex shapes, such as those that are linear or perforated, as compared to those that mimic parametric statistical distributions. Similarly, MCP-based home range estimates are widely regarded as inaccurate and inconsistent, being highly sensitive to both sample size and point pattern shape (e.g. Worton 1987, Barg et al. 2005, Franzreb 2006, Righton and Mills 2006, Downs and Horner 2008). Consequently, a number of new, GIS-based home range estimation methods have emerged as possible alternatives to KDE and the MCP. These include a trajectory-based form of KDE (Downs and Horner 2007) and various hull- or polygon-based methods (e.g. Worton 1995, Casaer et al. 1999, Getz and Wilmers 2004, Bath et al. 2006). This article presents a new hull-based home range estimator that is relatively simple to construct using GIS: the characteristic hull (Galton and Duckham 2006). Characteristic hulls, special bounding polygons constructed from a set of point locations, may be well suited for home range analysis. Unlike the MCP and some other polygon-based alternatives, characteristic hulls can have concave edges, be composed of disjoint regions, and contain empty portions of unused space within hull interiors. This article first describes how characteristic hulls are generated using GIS (Duckham et al. 2008) and how they can be applied for home range analysis. Second, the characteristic hull-based method is applied to simulated animal locational data that conform to several point pattern shapes in order to test its accuracy as a home range estimator. Then, the technique is applied to satellite tracking data for black-footed albatrosses for illustration and comparison to other home range estimation methods. Finally, the applicability of the characteristic hull for home range analysis is discussed and recommendations for future research are provided.

3 A Characteristic-Hull Based Method for Home Range Estimation Methods 2.1 Characteristic Hulls In geographic information science, a bounding hull sometimes called a container or envelope is defined as some polygon (two-dimensional shape) that encloses some number of point, line, or other polygon features within its interior. Hulls are widely applied in GIS for the purpose for delineating the area occupied by a set of points (Galton and Duckham 2006). The most widely used hull is the minimum convex polygon (MCP), the smallest area polygon that contains all points of a distribution such that all internal angles are 180 degrees. In fact, the MCP was one of the first home range estimators developed in ecology (Mohr 1947). Duckham et al. (2008) developed characteristic hulls as a method to delineate polygonal regions occupied by a set of points in space that is an alternative to the MCP. Characteristic hull polygons (CHPs) are special in that they can have non-convex edges, as well as empty holes within their interiors. Additionally, they can take the form of multiple disjoint polygons. CHPs are generated from the Delaunay triangulation (DT) for a set of input points. The DT is a graph constructed by connecting neighbouring points to form triangles such that the minimum angle is the triangulation is maximised. The CHP is then extracted from the DT by removing some number of triangles from the graph. The number of potential characteristic hulls that can be generated from a set of points extends from the minimum triangle (all but the smallest triangle removed) to the minimum convex polygon (no triangles removed). Various rules can be used to determine which triangles are removed from the DT, such as some percentage of the largest triangles. Such a procedure can be easily implemented in most GIS software packages. CHPs can be used to estimate home ranges from animal locational data. Here, CHPs generated from 95% of the smallest triangles are used to represent the home range, with this chosen as a general rule, because the home range is defined as the smallest area in which an animal spends 95% of its time. In this case, the largest 5% of triangles, measured in terms of perimeter, are removed from the DT, as these areas would be expected to be the least likely occupied by the animal. Although other rules could be used, such as removing the largest 5% in terms of area, perimeter is suggested as the criteria such that large, slender triangles, which often occur at the outer boundary of the point pattern, would be removed from the DT first. Since the triangulation is constructed in such a way that minimises the number of slender triangles (i.e. those containing two small angles), it is intuitive to remove the largest of the slender triangles first. It should be noted, however, that in most cases, the largest triangles measured by perimeter would also be the largest in area. Construction of a 95% CHP from sample point locations is illustrated in Figure Application to Simulated Data Simulated animal home ranges of five general shapes were generated to test the potential use of CHPs for home range estimation. Similar to Millspaugh et al. (2004), point patterns were generated by a Poisson cluster process with radii of 0.25 or 0.45 using the S-PLUS (Insightful Corp.) Spatial Stats module (Kaluzny 1998) to serve as known animal home ranges. Patterns of 10,000 points were initially generated using this process, with 60 each designated to represent concave (single distinct point distribution with

4 530 J A Downs and M W Horner Figure 1 Characteristic hull polygon generated from the Delaunay triangulation of a set of points non-convex edges) and disjoint (multiple distinct clusters) home ranges. Additionally, convex (derived by extracting points using a MCP), perforated (points extracted from the interior of the home range), and linear home ranges (generated by extracting points associated with buffered line features) were generated from the simulated point distributions by modifying them using GIS, as per the procedures outlined in Downs and Horner (2008). These patterns were used to represent the known areas occupied by the animal, with the known home range defined as the smallest area containing 95% of the points. To test the accuracy of the characteristic hull as a home range estimator, 95% CHPs were generated using 50, 100, and 200 random points sampled from the simulated locational data. Characteristic hulls were produced using ArcGIS 9.2 (Environmental Systems Research Institute, Inc.). Mean percent bias in calculated area [(Estimated home range known home range)/known home range] was computed for each combination of home range shape and sample size. Additionally, the percent of the estimated home range that intersected the known home range was also computed to measure how well the CHPs corresponded to the known home range shape. Standard deviations were also derived for both measures of accuracy and summarised for each combination of home range area and sample size. 2.3 Application to Albatross Data CHPs were constructed using satellite tracking data for three black-footed albatrosses (Phoebastria nigripes) for illustration and comparison to other methods. The data were

5 A Characteristic-Hull Based Method for Home Range Estimation 531 previously described and analysed by Fernández et al. (2001) in terms of foraging patterns. The data used here contained 175, 256, and 278 locations for the respective individual animals. In addition to constructing 95% CHPs, home range estimates were also computed using MCP and kernel density estimation (KDE; Worton 1987) methods. MCPs were constructed using the densest 95% of the locations. KDE was applied using fixed kernels and bandwidths selected by a least-squares cross validation (LSCV) procedure (Gitzen and Millspaugh 2003), an algorithm that attempts to minimize the error associated with the density estimate when points are excluded from the dataset one by one (Silverman 1986). When the LSCV algorithm did not converge, then the reference bandwidth, which selects the optimal bandwidth under the assumption that the sample data come from a bivariate normal distribution, was used as a substitute. In all cases, the smallest area containing 95% of the computed intensity was delineated as the home range. KDE and MCP home range estimates were computed using the Home Range Tools Extension (Rodgers et al. 2005) in ArcGIS. The home range estimates were calculated and mapped using an equal-area cylindrical map projection. 3 Results 3.1 Application to Simulated Data Sample home range estimates are illustrated for each point pattern shape in Figure 2. Note how CHPs were able to retain the concave (Figure 2a), disjoint (Figure 2c), perforated (Figure 2d), and linear (Figure 2e) nature of the respective home ranges. On average, the CHP method produced home range estimates for the simulated locational data that were slightly small ( x = 8.7%, s.d. = 16.3%). Mean percent bias differed between the five home range shapes, although accuracy tended to improve with increasing sample size in all scenarios (Table 1). Unbiased estimates were produced for linear and disjoint point patterns, though the estimates were somewhat variable as measured by standard deviation, particularly at the smallest sample size of 50 points. CHPs produced home range estimates that were negatively biased by about 10 15% for concave and perforated shapes. This is illustrated in Figure 2b, where removal of triangles on the outer boundary of the DT resulted in non-convex edges. However, these estimates appeared less variable than for linear and disjoint patterns. CHPs performed relatively worse for convex shapes, underestimating known home range sizes by about 20%. Percent intersection between the known and estimated home ranges also varied among point pattern shapes. While CHPs produced more biased estimates for convex and concave shapes than the other samples, they tended to intersect more of the known home range (~90 95%). Estimates for the remaining three shapes intersected known home ranges by about 80%, indicating that some areas outside of the true home range were included in these estimates. 3.2 Application to Albatross Data The three methods used to estimate home ranges for the three black-footed albatrosses produced markedly different results (Figure 3). CHPs produced home range estimates that were on average 45 and 63% smaller than both MCP and KDE estimates, respectively. KDE produced the largest estimates, with the home range boundaries extending

6 532 J A Downs and M W Horner Figure 2 Characteristic hull polygons used to estimate home ranges for: (a) concave, (b) convex, (c) disjoint, (d), perforated, and (e) linear point patterns of simulated animal locations well beyond the observed distribution of points. Both KDE and MCP estimates included large expanses of apparently unoccupied space. The CHP method, on the contrary, produced home range estimates with concave edges and regions of empty space within the interior of the range. While the CHPs overlapped the 95% MCPs in most areas, it should be noted that in all three examples the CHPs did contain some areas not included in the MCPs. An additional feature of the CHPs is that narrow triangles kept in the home range estimates appear to link distant foraging areas that were used intensively by the albatrosses. While the actual home ranges of the albatrosses are unknown, the observation that CHPs produced smaller estimates than MCP and KDE, which chronically produce overestimates (Girard et al. 2002, Hemson et al. 2005, Downs and Horner 2008), suggests that the new method may be more accurate.

7 A Characteristic-Hull Based Method for Home Range Estimation 533 Table 1 Accuracy (mean % deviation from true value, standard deviation) of characteristic hull polygons used for home range estimation for simulated animal location data conforming to five point pattern shapes (n = 60/scenario), with the mean and standard deviation of the percent intersection between known and estimated home ranges Bias Intersection Shape n % sd % sd concave concave concave convex convex convex disjoint disjoint disjoint linear linear linear perforated perforated perforated Discussion and Conclusions The results of this study suggest that CHPs have potential as a method of home range estimation, because they can assume the form of numerous complex home range shapes, including those that are linear, have concave edges, are composed of disjoint regions, and contain areas of unoccupied space within their interiors. The application of CHPs to albatross tracking data illustrates how the method can produce reasonable home range estimates for bird species that intensively use patches of habitat that are separated by a large distance. An additional advantage of the method is that CHPs are easily constructed using GIS software, which already is a common environment for analysing animal locational data. While the results of the simulations show that CHPs, as applied in this case, did not produce perfectly accurate results, the levels of bias are a marked improvement over those reported for MCP and KDE methods applied using similar data, which over estimated home ranges by about % depending on method, sample size, and shape (Downs and Horner 2008). Additionally, the estimated home ranges overlapped the known home ranges by a large amount, which also suggests CHPs perform reasonably well. CHPs produced unbiased or slight underestimates for linear, disjoint, and perforated home ranges, shapes for which other methods traditionally produce poor estimates (Blundell et al. 2001, Getz and Wilmers 2004, Hemson et al. 2005, Row and Blouin- Demers 2006). CHPs performed relatively worse for concave and convex point patterns.

8 534 J A Downs and M W Horner Figure 3 Home range estimates for three albatross (a-c) using characteristic hull polygons (shaded), minimum convex polygons (solid line), and kernel density estimation (dotted line) These results initially suggest that, like other home range estimators, the CHP method is not robust to point pattern shape. However, this observation might be better explained by the choice of rule for deciding which triangles to remove from the DT than any inherent problems with the technique. Here, keeping 95% of the smallest triangles is suggested as an initial rule of thumb that might be comparable to other home range estimation methods. However, particularly in the case of concave and convex shapes, the results suggest that perhaps too many triangles were removed from the DT. Since MCPs produce unbiased estimates for convex home ranges (Downs and Horner 2008), then one

9 A Characteristic-Hull Based Method for Home Range Estimation 535 might assume that no triangles should be removed from the DT in these cases. Similarly, for concave patterns, removing 5% may be too many. While the 95% rule produces somewhat acceptable results at least much better than commonly applied methods future work should explore other rules for triangle removal for CHP-based home range estimation, as a fixed percentage does not appear optimal for all home range types. For instance, it may be possible to apply some type of shape index such that only triangles that exceed some specified criteria of slenderness are removed from the DT. Another partial explanation for the tendency of CHPs to slightly underestimate home ranges is that areas outside the MCP cannot be included in the home range. If animals visit any locations outside of those occupied by the sampled points, these regions are omitted from the estimated home range. This can be problematic if the animal spends a significant amount of time near the edge of the point pattern. However, proper sampling techniques can minimise this effect by ensuring the sampled locations are representative of the animal s actual home range. At this initial stage of development of the CHP-based home range estimator, it is only recommended that CHPs are used to estimate home ranges that are suspected to conform to linear, disjoint, or perforated shapes. For other types of home ranges, the method should only be applied with caution and perhaps by removing fewer than 5% of the triangles. However, especially at small sample sizes, it may be difficult to determine the actual shape of the animal s home range from the set of sample points. In this case, one might assume the expected shape from previous studies or use a more conservative method of home range estimation. Newly explored methods of home range estimation include a trajectory-based form of KDE (Downs and Horner 2007) and local nearestneighbour convex hulls (Getz and Wilmers 2004), among others, and these may serve as adequate alternatives. Finally, while this study explores the use of characteristic hulls in the context of animal home range estimation, the technique may be suitable for analysing other types of spatial point patterns, as Duckham et al. (2008) suggest. Similar GIS-based methods are commonly used for analysing crime locations (Bailey and Gatrell 1995), plant distributions (Brunsdon 1995), and traffic accidents (Okabe et al. 2009), among other applications. The results of this study suggest characteristic hulls may also be suitable for characterising some of these point patterns, if they conform to irregular shapes similar to animal home ranges. Acknowledgements This work was supported, in part, by the University of South Florida Internal Awards Program under Grant No. R The authors would like to offer special thanks to Professor Dave Anderson of Wake Forest University for use of the albatross telemetry data which made this research possible. References Bailey T C and Gatrell A C 1995 Interactive Spatial Data Analysis. Harlow, Longman. Barg J J, Jones J, and Robertson R J 2005 Describing breeding territories of migratory passerines: Suggestions for sampling, choice of estimator, and delineation of core areas. Journal of Animal Ecology 74:

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