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LoCoH: Nonparameteric Kernel Methods for Constructing ... - NCEAS

LoCoH: Nonparameteric Kernel Methods for Constructing ... - NCEAS

........................................................... Table 3. Comparison of the areas in km 2 estimated for the four buffalo GPS collar sets of data (n points) by the 95% and 100% isopleths for nonparametric (LoCoH) and parametric kernel a methods. .................................................................................................................................................. Collar n HR k-LoCoH r-LoCoH a-LoCoH Kernel parameter (k1) (r1) (a1) (â) T07 27 95% 190 268 166 173 244 89 100% 289 (53) 321 (3475) 236 (32383) 253 (44850) T13 25 95% 96 114 89 95 142 72 100% 238 (51) 144 (1470) 211 (28156) 224 (35000) T15 28 95% 126 146 128 153 121 46 100% 276 (53) 156 (1555) 205 (35684) 257 (72000) T16 16 95% 56 46 55 55 84 75 100% 118 (41) 49 (678) 90 (23401) 90 (23401) For the k, r, anda-LoCoH methods the parameters used, as described in the text, are the heuristic values k1, r1,anda1and the MSHC value â (given in parentheses). a Implemented in Animal Movement Extension for ArcView 3.x [29] using Silverman’s ad-hoc method for selecting the smoothing parameter h [14]. doi:10.1371/journal.pone.0000207.t003 important biological information is contained in spatiotemporal autocorrelations of data points [6,31,32]. It is important to note, however, that an assumption necessary to ensure the construction of adequate unbiased UDs is that the data points have been collected suitably often to obtain a representative sample of points over time to cover all modes of behavior. If this is not the case, then we have to be careful how we interpret the resulting UDs. In particular, as the sampling intensity decreases, say to twice or four times a day, it becomes increasingly likely that sparse, but regular sampling may coincide with particular activities (e.g. sleeping, drinking, eating) and result in UDs biased towards these activities. Moreover, the scale at which the utilization can be interpreted will still depend on the frequency of data points, even for extremely regularly spaced points. For example, our buffalo data, collected at hourly was still too sparse relative to rate of movement of individuals with regard to identifying small physical obstacles on the landscape, including a small hill that we know is not utilized by buffalo in the Kruger National Park. As with any numerical method that draws directly upon data, LoCoH HR estimates and UD constructions are only as good as the data they rely upon to carry out the numerical computations. If these data are particularly noisy, then holes will be filled and sharp boundaries blurred. Fortunately, the resolution of GPS data is sufficient to accurately assess the location of sharp boundaries to within a couple of meters when information is collected at appropriately high frequencies (i.e. as they relate to the rate at which individuals move along the boundaries of their range). Assuming high quality data, the great advantage of LoCoH over parametric kernel methods is that LoCoH estimates convergence to true values with increasing sample size. This allows one to study the convergence properties by comparing estimates using a tenth, quarter, half, and all the data. If half the data, for example gives an REFERENCES 1. Matosevic M, Salcic Z, Berber SA (2006) Comparison of Accuracy Using a GPS and a Low-Cost DGPS. IEEE Transactions on Instrumentation and Measurement 55: 1677–1683. 2. Holden C (2006) Inching toward movement ecology. Science 313: 779–782. 3. Burt WH (1943) Territoriality and home range concepts as applied to mammals. J Mammalogy 24: 346–352. 4. Kie JG, Baldwin JA, Evans CJ (1994) CALHOME Home Range Analysis Program User’s Manual. Fresno, California: United States Forest Service Pacific Southwest Research Station. 5. Lawson EJG, Rodgers AR (1997) Differences in home-range size computed in commonly used software programs. Wildlife Society Bulletin 25: 721–729. estimate, within a desired tolerance of the estimate obtained by all the data (e.g. 1% or 0.1%), then one can be confident about the precision of the estimate. Of course, one can also carry out bootstrapping procedures to obtain standard errors [33]. In the end, if one is interested in detecting features in the environment that may influence the way animals utilize space, there is not substitute for using several different methods—both parametric and nonparametric—to construct maps overlaid on geographical and physical feature maps and visually comparing and inspecting the results to identify features that may be attracting, repealing, or excluding individuals. In summary, LoCoH methods are superior to bounded and, especially, unbounded parametric kernel methods for constructing UDs and HRs because they directly draw upon the actual spatial structure of data that may well be influenced by hard boundaries and irregular exclusionary areas in the environment. Also, our analysis indicates that the a-LoCoH nonparametric kernel method is generally superior to the other methods that we considered, both in constructing UDs and in estimating the size of HRs. ACKNOWLEDGMENTS We thank Tabitha Graves and Shirli Bar-David for editorial comments; as well as Craig Hay, Justin Bowers, Julie Wolhuter, Robert Dugtig, Augusta Mabunda, Khutani Bulunga, and the veterinary staff at the Kruger National Park for their assistance with the field data collection. Author Contributions LoCoH Conceived and designed the experiments: WG SF. Performed the experiments: SF AL. Analyzed the data: WG SF AL SR. Contributed reagents/materials/analysis tools: PC. Wrote the paper: WG PC SF SR CW. 6. De Solla SR, Bonduriansky R, Brooks RJ (1999) Eliminating autocorrelation reduces biological relevance of home range estimates. J. Animal Ecology 68: 221–234. 7. Jennrich RI, Turner FB (1969) Measurement of non-circular home range. J Theoretical Biology 22: 227–237. 8. Ford RG, Krumme DW (1979) The analysis of space use patterns. J Theoretical Biology 76: 125–157. 9. Baker J (2001) Population density and home range estimates for the Eastern Bristlebird at Jervis Bay, south-eastern Australia. Corella 25: 62–67. 10. Fieberg J, Kochanny CO (2005) Quantifying home-range overlap: The importance of the utilization distribution. J Wildlife Management 69: 1346–1359. PLoS ONE | www.plosone.org 10 February 2007 | Issue 2 | e207

11. Horne JS, Garton EO (2006) Likelihood cross-validation versus least squares cross-validation for choosing the smoothing parameter in kernel home-range analysis. J Wildlife Management 7: 641–648. 12. Matthiopoulos J (2003) Model-supervised kernel smoothing for the estimation of spatial usage. Oikos 102: 367–377. 13. Millspaugh JJ, Nielson RM, McDonald L, Marzluff JM, Gitzen RA, et al. (2006) Analysis of resource selection using utilization distributions. J Wildlife Management 70: 384–395. 14. Silverman BW (1986) Density estimation for statistics and data analysis. London: Chapman and Hall. 176 p. 15. Worton BJ (1987) A review of models of home range for animal movement. Ecological Modelling, 38: 277–298. 16. Worton BJ (1989) Kernel methods for estimating the utilization distribution in home-range studies. Ecology 70: 164–168. 17. Worton BJ (1995) A convex hull-based estimator of home range size. Biometrics 51: 1206–1215. 18. Seaman DE, Powell RA (1996) An evaluation of the accuracy of kernel density estimators for home range analysis. Ecology 77: 2075–2085. 19. Karatzoglou A, Smola A, Hornik K, Zeileis A (2004) ‘‘Kernlab – An S4 Package for Kernel Methods in R.’’ J Statistical Software 11: 1–20. (URL http:// www.jstatsoft.org/v11/i09.). 20. Kenward RE, Clarke RT, Hodder KH, Walls SS (2001) Density And Linkage Estimators Of Home Range: Nearest-Neighbor Clustering Defines Multinuclear Cores. Ecology 82: 1905–1920. 21. Börger L, Francon N, De Michele G, Gantz A, Meschi F, et al. (2006) Effects of sampling regime on the mean and variance of home range size estimates. J Animal Ecology 75: 1393–1405. 22. Birk RJ, Stanley T, Snyderc GI, Hennig TA, Fladelande MM, Policelli F (2003) Government programs for research and operational uses of commercial remote sensing data. Remote Sensing of Environment 88: 3–16. LoCoH 23. Getz WM, Wilmers CC (2004) A local nearest-neighbor convex-hull construction of home ranges and utilization distributions. Ecography 27: 489–505. 24. Ryan SJ, Knechtel C, Getz WM (2006) Range and habitat selection of African buffalo in South Africa. J Wildlife Management 70: 764–776. 25. Redfern JV, Ryan SJ, Getz WM (2006) Defining herbivore assemblages in Kruger National Park: A correlative coherence approach. Oecologia 146: 632–640. 26. Anderson DP, Forester JD, Turner MG, Frair JL, Merrill EH, et al. (2005) Factors influencing female home range sizes in elk (Cervus elaphus) in North American landscapes. Landscape Ecology 20: 257–271. 27. Burgman MA, Fox J (2003) Bias in species range estimates from minimum convex polygons: implications for conservation and options for improved planning. Animal Conservation 6: 19–28. 28. Hemson G, Johnson P, South A, Kenward R, Ripley R, et al. (2005) Are kernels the mustard? Data from global positioning system (GPS) collars suggests problems for kernel home-range analyses with least-squares cross-validation. Journal of Animal Ecology 74: 455–463. 29. Hooge PN, Eichenlaub B (1997) Animal movement extension to ArcView. ver. 1. Anchorage, AK, USA: Alaska Science Center - Biological Science Office, U.S. Geological Survey. 30. Watts DP, Mitani JC (2000) Boundary patrols and intergroup encounters in wild chimpanzees. Behaviour 138: 299–327. 31. Blundell GM, Maier JAK, Debevec EM (2001) Linear home ranges: effects of smoothing, sample size, and autocorrelation on kernel estimates. Ecological Monographs 71: 469–489. 32. Cushman SA, Chase M, Griffin C (2005) Elephants in space and time. Oikos 109: 331–341. 33. Efron B, Tibshirani RJ (1993) An introduction to the bootstrap. London: Chapman & Hall. 436 p. PLoS ONE | www.plosone.org 11 February 2007 | Issue 2 | e207

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