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.
Current address: Department of Anthropological Sciences, Stanford University, Stanford, California, United States of America
The authors have declared that no competing interests exist.
Parametric kernel methods currently dominate the literature regarding the construction of animal home ranges (HRs) and utilization distributions (UDs). These methods frequently fail to capture the kinds of hard boundaries common to many natural systems. Recently a local convex hull (LoCoH) nonparametric kernel method, which generalizes the minimum convex polygon (MCP) method, was shown to be more appropriate than parametric kernel methods for constructing HRs and UDs, because of its ability to identify hard boundaries (e.g., rivers, cliff edges) and convergence to the true distribution as sample size increases. Here we extend the LoCoH in two ways: “fixed sphereofinfluence,” or
Ecology is currently undergoing a revolution in terms of our ability to collect large sets of data with unprecedented precision on the position of individuals in the landscape (e.g. plusminus several meters using current GPS technology
Currently, the boundary of the HR is commonly delimited using the 95% isopleth of an unbounded UD, where the UD is typically constructed using the symmetric bivariate Gaussian (i.e. a parametric) kernel method
The reasons for omitting outlying points in estimating the size of HRs are threefold: (1) locations based on relatively inaccurate triangulation of radio collars result in imprecise location estimates (this is philosophically consistent with the parametric kernel methods, such as the radially symmetric—i.e. one parameter—bivariate Gaussian or harmonic kernels, that associate a smooth distribution with each data point); (2) HR area estimates using MCP and parametric kernel construction methods are very sensitive to outlying points
Here we describe extensions to a recently developed local convex hull (LoCoH) approach
The advantage of LoCoH's direct use of data becomes evident when constructing UDs from data influenced by idiosyncratic geometries such as geomorphological boundaries and holes (e.g. lakes or rocky outcrops) associated with the space over which animals move
In this paper, we present two modifications of the “fixed
After presenting a description of the methods and reviewing the MSHC approach (minimum spurious hole covering—see
Finally, we note that links to software for the implementation of LoCoH using ArcView/ArcGIS, or in the R Statistical package Adehabitat, or as a web application can be found at
As elaborated in more detail in Getz and Wilmers
Instead of choosing, as in the fixed
If
The above method for constructing a fixed radius LoCoH is reminiscent of fixed kernel methods that use kernels with finite support, such as the uniform or Epanechnikov kernels
The adaptive or
For relatively low values of
The actual points used in the analysis, selected at random within boundaries defined in the methods to conform with the specified isopleth rules, are plotted here in the upper row for data sets A, B, and C. For each set, the 20% isopleth surrounds the densest aggregation of points that appear as relatively black areas in each of the plots. UDs constructed using the fixed kernel leastsquares crossvalidation method for these data are illustrations in the lower row (sizes have been adjusted to provide visual correspondence—where precise estimates of the fits are given in
Kruger National Park, showing the location of the four collared buffalo used in the empirical data test of the study. The Satara and Lower Sabie regions are shown as insets 1 and 2, respectively.
Data (true area)  A (85.3 units)  B (68.0 units)  C (257.0 units) 

13.4% (8.8%, 4.6%) 

9.0% (6.7%, 2.3%) 

15  27  17 

15.0% (8.4%, 6.6%)  10.3% (5.9%, 4.4%)  8.8% (5.6%, 3.2%) 

2.0  1.0  1.75 





21.0  19  19 
GK 95%  27.3% (22.2%, 5.2%)  30.3% (2.9%, 27.4%)  20.2% (14.9%, 5.2%) 
GK 99%  20.9% (10.4%, 10.4%)  56.6% (0.4%, 56.2%)  15.0% (3.9%, 11.1%) 
GK minimum 
20.9% (10.4%, 10.4%)  22.2% (10.9%, 11.5%)  14.6% (6.1%, 8.6%) 
(isopleth) 
(99%)  (87.5%)  (98.25%) 
The best estimate is in bold type.
optimal values reflect integer resolution for
For our manufactured data sets where the boundaries of the areas are known, or in cases of field data where the boundaries of particular holes are known, values of the parameters for
For purposes of comparison we constructed UDs using symmetric bivariate Gaussian kernels. Although we sought to use the optimized value for the width parameter,
We manufactured three datasets (
We collected field data on African buffalo movements using VHF and GPS collars place on individuals from November 2000 to August 2006 in the Satara and Lower Sabie regions of the Kruger National Park. For the purposes of demonstrating the LoCoH methodology we restrict our analyses to GPS recordings of locations taken once an hour from four adult females over the following periods of times: female T13, July 15, 2005 to Oct 29, 2005; female T15, Sept 16, 2005 to Feb 16, 2006; female T7, Sept 15, 2005 to Jan 29, 2006; female T16, July 27, 2005 to October 8, 2005. These data were collected in decimal degrees and reprojected to Universal Transverse Mercator (UTM) [WGS84, Zone 36S] in ArcGIS 9. These data represent two buffalo at each of two sites in Kruger National Park: the first is the Satara region (T07 and T15) and the second is the Lower Sabie region (T13 and T16) (
For each of the datasets we constructed
We constructed images of the resulting LoCoH UDs for our optimal parameter values, as well as half and twice the optimal values.
Lastly, we examined how the total error of the UDs constructed using the different methods changed as we used different sample sizes. We generated random samples containing 1000, 800, 600, 400, and 200 points using the specifications and isopleth rules outlined earlier for each manufactured dataset. We repeated this process 15 times (this number is relatively low but suffices if we are generating estimates purely for comparative purposes among methods) as a way of generating error estimates (i.e. for a total of 75 samples per dataset). We located the optimal value of
For purposes of comparison, we generated UDs for each of the four individuals using each of the 4 different methods. Since we were uncertain over what range of values we should explore the performance of our MSHC algorithm, we initially constructed UDs using our heuristic rules for selecting
For each of the three data sets we plot in
Type I (dotted line), Type II (dashed line) and Total Error (solid line) (percentages) associated with the construction of 100% and 20% isopleths are plotted for the
Also note in
For each of the three data sets, the errors of the LoCoH models are plotted as a function of sample size for the optimal (i.e. error minimizing) values of the parameters (
The effect of sample size on the optimal (i.e. error minimizing) value of parameters,
The optimal value
Data (true area) 












14.1  13.5 (0.70)  1.41 (0.18)  2.37 (0.05)  25.5 (0.04)  24.9 (1.00) 
1000 points  31.6  14.5 (0.42)  0.56 (0.00)  1.67 (0.06)  25.7 (0.00)  23.3 (0.84) 

14.1  11.9 (0.48)  0.74 (0.08)  1.51 (0.04)  18.4 (0.12)  15.5 (0.74) 
1000 points  31.6  20.6 (0.60)  0.57 (0.00)  0.79 (0.00)  19.6 (0.00)  14.0 (0.23) 

14.1  12.7 (0.42)  1.00 (0.03)  3.10 (0.05)  19.7 (0.03)  24.4 (0.82) 
1000 points  31.6  17.3 (0.27)  0.46 (0.00)  0.79 (0.03)  19.9 (0.00)  20.7 (0.59) 
Mean values are given with standard error in parentheses calculated over 15 different samplings of the data.
In
Illustrations of UDs constructed for data set A using
Illustrations of UDs constructed for data set B using
Illustrations of UDs constructed for data set B using
For the sake of completeness and to permit visual comparisons, the fixed kernel leastsquare cross validation UDs (95^{th} percentile) are plotted for data sets A, B, and C in
Silverman's parametric kernel method
Comparisons of UD constructions using an
Collar 





Kernel  
parameter  ( 
( 
( 
( 

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
Implemented in Animal Movement Extension for ArcView 3.x
In the Satara area (
Note that the symmetric bivariate Gaussian kernel UDs have slightly jagged boundaries because they are generated from an underlying grid, while LoCoH UDs are generated directly from the polygonal elements.
In statistics, nonparametric methods always require fewer assumptions than the corresponding parametric methods. In the case of UD constructions, both parametric and LoCoH kernel methods require common assumptions about data to avoid misinterpretations that come from bias with respect to the way the data are collected. By definition, however, parametric kernel methods always involve additional assumptions about the form of the distributions governing the data that nonparametric methods do not make. Thus, although traditional kernel methods can produce UDs and HRs that follow highly irregular data, they are still based upon parametric kernels that require the investigator to specify their functional form. LoCoH kernels, on the other hand, take their form directly from the data, thereby relieving the investigator of the burden and bias associated with choosing a functional form for the kernels. Further, parametric kernel UD constructions are almost always based on noncompact (i.e. unbounded) symmetric bivariate Gaussian kernels. This implies an adhoc decision must be made on which isopleth to use in HR constructions. Although, typically, the 95^{th} percentile is used a 90^{th} percentile boundary may decrease sample size bias
Even bounded parametric kernel methods (e.g. Epanechnichov kernels) will always overshoot the data by an amount equal to the value of the kernel radius parameter
In a previous publication, we demonstrated the superiority of
In this modified case, both
There has been some confusion about the need for points to have a certain temporal properties. This issue has recently been clarified by Börger et al.
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 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
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
We thank Tabitha Graves and Shirli BarDavid 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.