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A rulebased ad hoc method for selecting a bandwidth in kernel homerange analyses
Animal Biotelemetryvolume 1, Article number: 13 (2013)
Abstract
Background
An important issue in conducting kernel homerange analyses is the choice of bandwidth or smoothing parameter. To examine the effects of this choice, telemetry data were collected at high sampling rates (843 to 5,069 locations) on 20 North American elk, Cervus elaphus, in northeastern Oregon, USA, during 2000, 2002, and 2003. The elk had their collars replaced annually, hence none were monitored for more than a single year. True home ranges were defined by buffering the actual paths of individuals. Fixedkernel and adaptivekernel estimates were then determined with reference bandwidths (h _{ ref }), leastsquares crossvalidation bandwidths (h _{ lscv }), and rulebased ad hoc bandwidths designed to prevent undersmoothing (h _{ ad hoc }). Both raw data and subsampled sparse datasets (1, 2, 4, 6, 12, and 24 locations/elk/day) were used.
Results
With fixedkernel and adaptivekernel analyses, reference bandwidths were positively biased (including areas not part of an animal’s home range) but performed better (lower bias, closer match between estimated and true home ranges) with increasing sample size. Leastsquares crossvalidation bandwidths were positively biased with very small sample sizes, but quickly became negatively biased with increasing sample size, as homerange estimates broke up into disjoint polygons. Ad hoc bandwidths outperformed reference and leastsquares crossvalidation bandwidths, exhibited only moderate positive bias, were relatively unaffected by sample size, and were characterized by lower Type I errors (falsely including areas not part of the true home range). Ad hoc bandwidths also exhibited lower Type II errors (failure to include portions of the true home range) than did leastsquares crossvalidation bandwidths, although reference bandwidths resulted in lowest Type II error rates. Autocorrelation indices increased to about 150 to 200 locations per elk, and then stabilized. Bias of fixedkernel analyses with ad hoc bandwidths was not affected by autocorrelation, but did increase with irregularly shaped home ranges with high fractal dimensions.
Conclusions
The rulebased ad hoc bandwidths, specifically designed to prevent fragmentation of estimated home ranges, outperformed both h _{ ref } and h _{ lscv }, and gave the smallest value for h consistent with a contiguous homerange estimate. The protocol for choosing the ad hoc bandwidth was shown to be consistent and repeatable.
Background
A basic principal in animal ecology is that species, populations, and individuals have finite limits in use of space. Species and populations are delineated by geographical ranges, and individuals are described as having a home range. Burt’s definition of home range is widely used: ‘…that area traversed by the individual in its normal activities of food gathering, mating and caring for young’ [1]. Although plotting animal locations is straightforward, and is subject primarily to measurement errors, estimating the size of the home range is often dependent on a number of assumptions, which are often either not tested or if they are tested, are often determined to be false [2].
Kernel techniques for estimating the density of a utilization distribution (UD) of a random sample of locations for an individual animal were first proposed by Worton [3]. Kernel analyses are commonly used in statistical density estimation and have the advantage of being nonparametric [4]. They are used not only with single variables, but in bivariate space as well, with the distributions of the x and y coordinates representing animal locations [3].
Although Worton [3] used the terms ‘utilization distribution’ and ‘home range’ synonymously, a distinction can be made between the two concepts. Early attempts to quantify the home range of an animal involved drawing polygons around the outermost set of locations. Such techniques result in a contiguous polygon delineating the ‘area traversed by the individual’ [1], including crucial travel corridors in which an animal spends limited amounts of time, but these fail to portray the intensity of space use within the polygon [5]. Conversely, kernel techniques provide a UD, that is, a threedimensional probability density map showing which portions of the total home range home are used most frequently [5]. Alternatively, the estimate of the UD can be sliced to reveal a twodimensional (2D) surface (for example, by taking a 95% volume contour), which is the equivalent of a traditional definition of a home range. Such 2D slices may not be contiguous but rather disjoint, being composed of multiple polygons that more accurately indicate intensity of space use [6]. To capture littleused but important areas such as travel corridors, the 2D slice may be constrained to a single, contiguous polygon [5].
The starting point in kernel analyses is to construct a bivariate kernel estimate of a probability density function around each data point (animal location). A standard normal distribution is often used, although kernels can take on other shapes such as triangular, rectangular, or parabolic [4]. The functional shape and width of the kernel is determined by the smoothing parameter or bandwidth, denoted by h. Once probability density functions are in place, a grid structure is placed over the entire field, and volumes under the functions are summed over individual locations.
The choice of a smoothing parameter is a key decision in homerange analyses involving UDs, and the initial value is often obtained from the data themselves, although there is no a priori way to choose the best value for h. Silverman [4] and Worton [3] suggested a method of constructing an optimum h for large sample sizes if the data were assumed to be normally distributed. Referred to as h _{ opt } (and occasionally, an ad hoc choice of h) by Worton [3], it is optimal only if the assumption of bivariate normality is met, and will be denoted here as the reference bandwidth h _{ ref }. If animal locations are clumped rather than normally distributed, h _{ ref } will oversmooth the data, and the estimate of homerange size will be positively biased [3].
A different approach is to choose a bandwidth that minimizes the leastsquares crossvalidation score, h _{ lscv } [3, 7]. In most instances, h _{ lscv } is less than h _{ ref }, and is often only a small proportion of the latter. Although mathematically appropriate [3], h _{ lscv } frequently results in undersmoothing, and gives an estimate of the home range that consists of multiple polygons. In extreme instances, such an estimate will generate polygons around each small cluster of points, or even individual points.
A further smoothing issue is whether to use the same h for all points (global bandwidth), resulting in a fixedkernel analysis, or to allow h to vary as a function of local point densities (local bandwidths), yielding an adaptivekernel analysis. The localbandwidth approach allows for larger kernels (greater smoothing) associated with locations, often at the edge of the animal’s distribution, where locationpoint densities are lower. This approach assigns more uncertainty to sparsely distributed locations near the edge of the home range [3].
An assumption of both kernel analyses and parametric approaches is that data points are independent. However, animal locations are collected sequentially, and the extent to which the assumption of independence is violated is a function of sampling rate [8]. Sampling rates are rapidly increasing with newer telemetry technologies, such as those based on global positioning systems [9]. Little information is currently available on how autocorrelation interacts with estimation choices in kernel analyses to bias resulting estimates. Moreover, the ability to assess bias and hence performance of different kernel techniques is ultimately dependent on defining the true home range of an animal, an issue that has not received much attention.
The objectives of this study were to define the true home ranges for 20 female North American elk, Cervus elaphus, from northeastern Oregon, USA (Figure 1), based on periodic location data collected at high sampling frequencies, yielding the actual paths of individuals by connecting the locations test the efficiency of kernel analyses using both global and local bandwidths based on h _{ ref } and h _{ lscv }, and to suggest and test a new approach to choosing a smoothing parameter or bandwidth when conducting kernel homerange analyses.
Results
Estimates of bias in kernel analyses were affected by the individual animal (F _{19,700} = 6.93, P<0.0001), sampling frequency (F _{5,714} = 28.95, P<0.0001), and technique for choosing a bandwidth (F _{2,717} = 969.64, P<0.0001), but not by choice of fixed versus adaptive kernel (F _{1,718} = 0.44, P>0.10). Kernel analyses with a bandwidth that minimized the leastsquares crossvalidation score (h _{ lscv }) exhibited positive bias with a sampling frequency of one location per day, then a severely increasing negative bias with increasing sampling frequency (Figure 2, Figure 3). This negative bias was a result of the estimated home range breaking up into multiple polygons as sample size increased (Figure 3). The effect of using a global bandwidth (fixed kernel) versus a local bandwidth (adaptive kernel) with h _{ lscv } was significant only at 6 (P = 0.0014) and 12 (P = 0.0021) locations per day. The proportion h _{ lscv }/h _{ ref } decreased with increasing sampling frequency ($\overline{X}$ ± SD = 0.77 ± 0.31, 0.42 ± 0.18, 0.21 ± 0.08, 0.14 ± 0.04, 0.11 ± 0.006, 0.10 ± 0.0003 at 1, 2, 4, 6, 12, and 24 locations per elk per day, respectively, and 0.10 ± 0.0001 (raw data)).
Kernel analyses with the reference bandwidth (h _{ ref }) exhibited a consistent positive bias as a function of sampling frequency, although the bias declined somewhat with larger sample sizes (Figure 3). Bias using h _{ ref } was generally not affected by choice of fixed versus adaptive kernel, with a significant difference (P = 0.0347) seen only when using raw data (Figure 3). In a manner similar to h _{ ref }, h _{ ad hoc } resulted in a slight positive bias in the estimation of the size of home range, although the bias was more stable with respect to sampling frequency. Bias using h _{ ad hoc } was not affected as a function of fixed versus adaptive kernel (all a priori combinations P>0.10) (Figure 3).
Type I error (including area in the estimate that was not part of the animal’s home range) varied as a function of the individual animal (F _{19,700} = 123.81, P<0.0001), sampling frequency (F _{5,714} = 260.37, P<0.0001), and method of choosing a bandwidth (F _{2,717} = 1,143.39, P<0.0001), but less so on the choice of fixed or adaptivekernel approaches (F _{1,718} = 81.86, P<0.0001) (Figure 4). All techniques exhibited large Type I errors when a sampling frequency of 1 location per day was used, whereas the use of h _{ lscv } quickly resulted in a decrease in Type I errors with larger sample sizes, and effectively eliminated them at frequencies of four or more locations per day. This pattern was a result of the breakup of the estimate of home range into multiple polygons (Figure 2). Type I errors also decreased with sampling frequency when using h _{ ref } , but remained relatively constant when using h _{ ad hoc } (Figure 4).
Specific a priori comparisons indicated that the choice of fixed versus adaptive kernel had a significant effect on Type I errors when using h _{ lscv } at a sampling frequency of 1 location per day (P<0.0001), h _{ ref } at 1, 2, 4, 6, and 12 locations per day (P < 0.0001) and at 24 locations per day (P = 0.0006), and the raw data (P = 0.0226) (Figure 4). No significant differences (P > 0.10) between fixed and adaptive kernels occurred at any sampling frequency when using h _{ ad hoc } (Figure 4).
Type II errors (failing to capture area in the estimate that was part of the animal’s home range) were affected by the individual animal (F _{19,700} = 24.48, P<0.0001), sampling frequency (F _{5,714} = 92.57, P<0.0001), and method of choosing a bandwidth (F _{2,717} = 2,050.50, P<0.0001), but less by the choice of fixed or adaptive kernels (F _{1,718} = 6.95, P = 0.0086) (Figure 4). When using h _{ lscv }, significant differences existed between fixed and adaptive kernels at two (P = 0.0335), four (P = 0.0121), and six (P = 0.0415) locations per day (Figure 4). All other a priori comparisons of fixed versus adaptive kernels within a sampling period or bandwidth selection technique were not significant (all P>0.10) (Figure 4). Type II errors increased sharply with h _{ lscv } as the estimates of home range polygons became fragmented, but use of either h _{ ref } and h _{ ad hoc } resulted in Type II errors that remained relatively stable at less than 200 hectares as a function of sample size (Figure 4).
The elk locations used in this study were not independent, but exhibited serial autocorrelation. The autocorrelation index of Swihart and Slade [8] increased with sampling frequency, reaching an asymptote of 2 to 3 at between 100 to 200 locations, corresponding to sampling frequencies of 4 to 6 locations per day (Figure 5a). Fixedkernel analyses using h _{ ad hoc } as bandwidth indicated that bias did not differ as a function of autocorrelation index (Figure 5b), and hence, sampling frequency. Bias did increase with increasing fractal dimension of the true home range (Figure 5c). As home ranges became more irregular in shape, the bias in homerange estimates increased.
Discussion and conclusions
Kernel analyses are widely used in estimating home ranges and UDs of animals, but they have some disadvantages. Choice of initial bandwidth largely determines the resulting estimates of homerange size (Figures 2, 3). A reference bandwidth (h _{ ref }) assumes bivariate normality, although samples of animal locations are frequently not normally distributed. Animals often use space in a clumped or multimodal manner, and h _{ ref }, in assuming a unimodal normal distribution, assigns high variance to the data when they are actually distributed more tightly around two or more modes. The result is oversmoothing of data, and an inflated estimate of homerange size. Conversely, a bandwidth that minimizes the leastsquares crossvalidation score (h _{ lscv }) often undersmoothes location data, and the resulting homerange estimate breaks up into disjointed polygons [5, 6], resulting in negative bias in the estimate of homerange size (Figure 3) and large Type I errors (Figure 4).
Why should an estimate of the home range for an animal be contiguous? One reason is philosophical; such a distribution matches Burt’s definition of home range as ‘that area traversed by the individual in its normal activities of food gathering, mating, and caring for young [1]. Disjoint or separate core areas, such as those defined by a 60% kernel analysis, do not violate this definition, although an estimate of the entire home range that consists of multiple polygons does. For many purposes, such as estimating the intensity of spatial use of habitats [10], disjoint polygons are appropriate. Consequently, the terms ‘utilization distribution’ and ‘home range’ are not synonymous, with only the former being a legitimate description of disjoint spatial distributions. However, the biggest disadvantage to disjoint homerange polygons resulting from the use of h _{ lscv } is that the degree of fragmentation is highly dependent on sample size (Figure 2), which is an undesirable property when analyzing animal location data sampled at high frequencies with new and emerging telemetry technologies [9, 11].
Given the disadvantages of kernel techniques, what available analytical options are essential to minimize bias and error? One issue is that as sampling frequency increases, so does serial autocorrelation. White and Garrott [2] argued that autocorrelation itself was not as much of an issue as was insuring that the sampling was evenly spread over the time period of interest. De Solla et al. [12] also recommended maximizing the number of observations using constant time intervals, arguing that such a protocol increases the biological relevance of homerange estimation. In the current study, bias was not influenced by degree of autocorrelation when using h _{ ad hoc } choice of bandwidth (Figure 5b). Type I and II errors associated with h _{ ad hoc } also appeared to be independent of sampling frequency (Figure 3). Given an appropriate choice of bandwidth such as h _{ ad hoc }, autocorrelation is not a concern. However, use of h _{ lscv }, is fraught with pitfalls associated with sampling frequency, autocorrelation, bias, and Type I and II errors. The issue is not whether an assumption of independent data has been violated, but rather how robust is a specific choice of bandwidth to such violations. This study indicates that kernel analyses using h _{ ad hoc } can be robust under these conditions, and supports previous recommendations [2, 12].
Likewise, the shape of the kernel itself may not be a crucial issue. Wand and Jones [13] noted that that efficiency of various kernel shapes varied by less than 10%. Most computer programs currently use a standard normal distribution for the kernel probability density function [14]. However, other shapes are possible, including uniform or triangular kernels [4, 13]. Some older programs use a parabolashaped Epanechnikov kernel to avoid having to evaluate the volume under the extended tails of a bivariate normal distribution [15]. It should be noted that computationally, it is not possible to conduct a strict 100% volume analysis with a standardnormal kernel; the tails of the kernel must be truncated at some point by requesting a volume of less than 100%. In some computer programs, this modification may be done automatically, for example at 99.9%, in a manner not transparent to the user. Although not tested in this study, it has been suggested that choice of kernel shape is not of major concern [13].
The advantages and disadvantages of using global versus local bandwidths in kernel homerange analyses has been the subject of debate, as has the choice of h [7]. Worton [3] favored a local bandwidth (adaptive kernel) using h _{ lscv }, but also suggested that a global bandwidth (fixed kernel) using h _{ ref } also produced valid estimates. Worton [16] later argued that although the choice of h was very important, the choice of global versus local application of that bandwidth was less so. Seaman and Powell [17] and Seaman et al. [18] reported that global use of h _{ lscv } resulted in little bias in homerange estimates, but that localbandwidth approaches overestimated areas of distribution, and thus should not be used. The results from the current study are consistent with Worton [16]; the choice between global versus local bandwidths is inconsequential in terms of bias (Figure 3), Type I, and Type II errors (Figure 4). Conversely, in this study the use of h _{ lscv } resulted in rapidly increasing negative bias and Type I errors in homerange estimates with increasing sample size (Figures 2, 4). Similar concerns have been raised by Hemson et al. [19].
Different computer programs have limits on how small h _{ lscv } can be as a function of h _{ ref }. Home Range Extension (HRE) places a minimum value of h _{ lscv } at 0.1025 h _{ ref } [14], a floor unlikely to have a pronounced effect on the calculation of h _{ lscv } in this study. However, another commonly used program (Animal Movement Extension, http://alaska.usgs.gov/science/biology/spatial/gistools/index.php/, accessed 29 January 2013) will not allow a value for h _{ lscv } of less than 0.9662 h _{ ref }, in effect implementing an incorrect definition of h _{ lscv } (= 0.9662 h _{ ref }) in many analyses (A. Rodgers, personal communication).
The current study indicates that implementation of h _{ ad hoc }, specifically designed to prevent fragmentation of estimated home ranges, in either a global or local context, outperformed both h _{ ref } and h _{ lscv }. Use of an arbitrary value for h such that h is less than or equal to h _{ ref } to improve model fit while preventing fragmentation of homerange estimates has been reported for domestic cattle, Bos taurus (h = 0.8 h _{ ref }) [20], mule deer, Odocoileus hemionus (h = 0.8 h _{ ref }) [21], and whitetailed deer, Odocoileus virginianus (h = 0.7 h _{ ref }) [22]. The protocols used in this current study were similar, but rather than select an arbitrary value for the bandwidth, the smallest value for h that was consistent with a contiguous homerange estimate was chosen. These protocols are consistent and repeatable, and have been used in other studies [23, 24].
With emerging telemetry techniques, large numbers of data on animal location can be collected at high sampling frequencies [9]. The technique of plotting the buffered path of an individual [25, 26], similar to that performed in this study to define true home ranges, may provide a useful estimate of the total area used by an animal. However, further research into perceptual ranges of different species [27, 28] will be required refine the distance by which animal paths should be buffered. Conversely, for the foreseeable future, kernel approaches will remain useful for the analysis of spatial use by animals, not only for use with sparse datasets, but most importantly for determining intensity of use within a home range.
Methods
Study area
This study was conducted at the US Forest Service’s Starkey Experimental Forest and Range (hereafter referred to as ‘Starkey’), located 35 km southwest of La Grande (45°13’N, 118°31’W) in the Blue Mountains of northeastern Oregon, USA (Figure 1). The forest is situated between 1,122 and 1,500 meters in elevation, and supports a mosaic of coniferous forests, grasslands, and riparian areas that typify the summer range for elk in the Blue Mountains [29]. A network of narrow, irregular drainage channels in the project area creates a complex and varied topography [30, 31].
Starkey consists of 10,125 hectares enclosed by a 2.4m high fence that prevents immigration or emigration of resident elk and other large herbivores [29]. The largest division within Starkey is a main study area if 7,762 hectares, from which data for this research were obtained (Figure 1). Details of the study area and facilities are available elsewhere [29, 32–34].
Determining animal locations
As part of ongoing research at Starkey on North American elk, mule, and domestic cattle, an automated radio telemetry system was developed based on rebroadcast long range navigation (LORAN)C signals in the late 1980s to collect location data on these ungulates [29]. For the current study, data were collected each November during 2000, 2002, and 2003. Periods of data collection coincided with the ability, dictated by the needs of other studies, to reduce the total number of animals being monitored, and thereby increase the sampling frequencies of study animals (Table 1). To avoid lack of independence in data resulting from individuals traveling together in herds, an associationmatrix approach was used [35]. Each year, a random subsample of four locations per day was drawn for each radiocollared elk. A temporal threshold of 1 day and a spatial threshold of 183 m were used. Deposition of fecal pellet groups by elk and mule deer in open grasslands in a forestgrassland mosaic declined at more than 183 m away from forested edges [36]. Hence, 183 m was judged a reasonable approximation for the perception threshold of the North American elk [27]. Data from one animal in each pair that was located within 183 m of each other 50% or more of the time on any given day were eliminated, thereby arriving at a final sample size of 20 female elk for the 3 years of this study (Table 1). No individual was monitored for more than 1 year.
Ethic approval
Protocols were approved by the Institutional Animal Use and Care Committee at Starkey Experimental Forest and Range [37].
The female elk in this study were (mean ± SD) 6.9 ± 2.85 years of age (range 3 to 14 years). Mean elapsed times between observations were 36.90 ± 5.22 minutes (n = 3 elk) in 2000, 8.67 ± 0.92 minutes (n = 9) in 2002, and 8.95 ± 0.87 minutes (n = 8) in 2003. Numbers of locations per individual ranged from 843 to 1,089 in 2000, and from 3,615 to 5,069 in 2002 to 2003 (Table 1).
Finally, to test the performance of different techniques for estimating homerange size using sparse datasets, data were subsampled by choosing at random 1, 2, 4, 6, 12, and 24 locations per elk per day. Techniques for homerange estimation were then applied to each dataset of reduced sampling frequency in addition to raw data. Moreover, bivariate serial autocorrelation and crosscorrelation between 2 points, but among 3 or more points in the raw and sparse datasets were estimated with a measure described by Swihart and Slade [8].
To test the accuracy of location data obtained from individual elk, each year a radio collar was placed at a known location and its position monitored regularly, along with the study animals. Based on approximately 3,000 locations determined each year for the fixed collars, the estimated error (mean ± SD) was 35.3 ± 35.9 m, comparing favorably with a previous estimate of 52.8 ± 5.87 m (mean ± SE) [38].
Analyses of home ranges
The true home range of an animal was defined by assuming first that the actual path followed by an individual was a straight line between each pair of successive locations. Elapsed times between locations were relatively short, particularly in 2002 and 2003, hence this is likely to be an accurate portrayal. Again, using a perception threshold of 183 m, the path of each elk was buffered by that amount, and then any lacuna within the resulting polygon were removed [25] to arrive at the true home range. In addition, the fractal dimension of each home range was estimated to give a measure of the irregularity of its shape.
HRE [14] for ArcView (ESRI, Redlands, CA, USA) was used to estimate elk home ranges. The 95% volumetric kernel analyses [3] were calculated using a variety of techniques, including both a global bandwidth (fixed kernel) and local bandwidth (adaptive kernel), all with a default resolution (70 × 70 cell grid) option in HRE [14]. Three different methods were used in choosing an initial bandwidth. The first was to use the reference bandwidth, h _{ ref }; the second was to use the bandwidth that minimized the crossvalidation score, h _{ lscv }; and the third was based on an ad hoc approach.
Silverman stated that ‘a natural method for choosing a smoothing parameter is to plot out several curves and choose the estimate that is most in accordance with one’s prior ideas about the density’ [4]. In the current study, the goal was to delineate a single, contiguous polygon representing a complete home range as described by Burt [1]. Therefore, the reference bandwidth (h _{ ref }) was sequentially reduced in 0.10 increments (0.9 h _{ ref }, 0.8 h _{ ref }, 0.7 h _{ ref }, …0.1 h _{ ref }). This rulebased h _{ ad hoc } was the smallest increment of h _{ ref } that: 1) resulted in a contiguous rather than disjoint 95% kernel homerange polygon, and 2) contained no lacuna within the home range. When sequentially reducing h _{ ref }, lacuna occasionally appeared that subsequently disappeared at successively smaller values of h _{ ref }. However, Once an estimate of the home range fractured into two or more polygons, the process of searching for h _{ ad hoc } was halted. In most instances, h _{ lscv } < h _{ ad hoc } < h _{ ref }, although h _{ ad hoc } < h _{ lscv } < h _{ ref } was considered. Conversely, we did not allow h _{ ad hoc } to be greater than h _{ ref } when the estimate of the home range was fragmented at h _{ ref }, but accepted the fragmented estimate instead. Note that the definition of h _{ ad hoc } used in the current study should not be confused with the discussion of h _{ ref } as an ad hoc choice by Worton [3]. This ad hoc choice of a bandwidth has previously been used to delineate home ranges in coyotes, Canis latrans [23], and in pronghorns, Antilocapra americana [24].
Finally, the various estimates of elk home ranges were compared with what were previously defined as true home ranges. Differences in size between the estimates and the true home range (% bias) and Type I (area included as part of the estimate, which was not part of the true home range) and Type II errors (area within the true home range, which was not included within the estimate), were examined. For kernel analyses, statistical tests were conducted with a general linear model in SAS software (SAS Institute, Cary, NC, USA) [39] with main factors including individual animal (n = 20), initial bandwidth (n =2: global, local), bandwidth selection technique (n = 3: h _{ ref }, h _{ lscv }, h _{ ad hoc }), and sampling frequency (n = 6: 1, 2, 4, 6, 12, and 24 locations per day plus raw data), along with interactions between the main factors. Total sample size was thus 720 records (20 × 2 × 3 × 6). Bias was transformed with a squareroot arc sin function to ensure additivity of treatment effects [40], and specific a priori comparisons were made with leastsquares means [39]. The relationship between percentage bias of the various homerange estimates as functions of degree of autocorrelation between among of individual elk and the fractal dimension of the true home range was examined.
Abbreviations
 2D:

Twodimensional
 LORAN:

Long range navigation
 HRE:

Home Range Extension
 UD:

Utilization distribution.
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Acknowledgements
I thank Alan A. Ager for assistance with preparing raw data files for ArcView GIS, and R. T. Bowyer, A.R. Rodgers, and J. K. Young for comments on previous versions of this manuscript.
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Keywords
 Adaptive kernel
 Bandwidth
 Cervus elaphus
 Fixed kernel
 Home range
 North American elk
 Smoothing parameter