I believe that the best estimators available for estimating home ranges and home range utility distributions are kernel density estimators (Powell et al. 1997; Seaman 1993; Seaman et al. 1999; Seaman and Powell 1996; Worton 1989). Nonparametric statistical methods for estimating densities have been available since the early 1950s (Bowman 1985; Breiman et al. 1977; Devroye and Gyorfi 1985; Fryer 1977; Nadaraya 1989; Silverman 1986; Tapia and Thompson 1978) and one of the best known is the kernel density estimator (Silverman 1986). The kernel density estimator produces an unbiased density estimate directly from data and is not influenced by grid size or placement (Silverman 1986). Worton (1989) suggested that a kernel density estimator could be used to estimate home ranges of animals but little work (Worton 1995) had been published on the method as a home range estimator before Seaman's (1993; Powell et al. 1997; Seaman et al. 1999; Seaman and Powell 1996; Seaman et al. 1998) work, which is elaborated here.

Kernel estimators produce a utility distribution in a manner that can be visualized as follows. On an x—y plane representing a study area, cover each location estimate for an animal with a three-dimensional "hill", the kernel, whose volume is 1 and whose shape and width are chosen by the researcher. The width of the kernel, called the band width (also called window width or h), and the kernel's shape might hypothetically be chosen using location error, the radius of an animal's perception, and other pertinent information. Luckily, kernel shape has little effect on the output of the kernel estimators, as long as the kernel is hill-shaped and rounded on top (Silverman 1986), not sharply peaked (deduced from criticisms by Gautestad and Mysterud, personal communication). Although no objective method exists at present to tie band width to biology or to location error, except that band width should be greater than location error (Silverman 1986), objective methods do exist for choosing a band width that is consistent with statistical properties of the data on animal locations. Band width can be held constant for a data set (fixed kernel). Or band width can be varied (adaptive kernel) such that data points are covered with kernels of different widths ranging from low, broad kernels for widely spaced points to sharply peaked, narrow kernels for tightly packed points. Although adaptive kernel density estimators have been expected, intuitively, to perform better than fixed kernel estimators (Silverman 1986), this has not been the case (Seaman 1993; Seaman et al. 1999; Seaman and Powell 1996). The utility distribution is a surface resulting from the mean at each point of the values at that point for all kernels. In practice, a grid is superimposed on the data and the density is estimated at each grid intersection as the mean at that point of all kernels. The probability density function is calculated by multiplying the mean kernel value for each cell by the area of each cell.

Choosing band width is one of the most important and yet the most difficult aspects of developing a kernel estimator for animal home ranges (Silverman 1986). Narrow kernels reveal small-scale details in the data, and, consequently, tend also to highlight measurement error (telemetry error or trap placement, for example). Wide kernels smooth out sampling error but also hide local detail. The optimal band width is known for data that are approximately normal but, unfortunately, animal location data seldom approximate bivariate, normal distributions (Horner and Powell 1990; Seaman and Powell 1996). For distributions that are not normal, a band width more appropriate than that for a normal distribution can be chosen using least squares cross validation. This process chooses various band widths and selects the one that provides the minimum estimated error. Seaman (1993; Seaman and Powell 1996) found that cross-validation chooses band widths that estimate known utility distributions better than do band widths appropriate for bivariate normal distributions.

Using computer simulations and telemetry data for bears, Seaman (Seaman 1993; Seaman et al. 1999; Seaman and Powell 1996) explored the accuracy of both fixed and adaptive kernel home range estimators and compared their accuracies to the harmonic mean estimator. He used simulated home ranges that looked much like real home ranges but he knew the utility distri butions for the simulated home ranges. Seaman chose points randomly within the simulated home ranges, simulating the collection of telemetry or trapping or sighting location data, and then he estimated the simulated home ranges from the "location" data points. He then compared the kernel estimators to the harmonic mean estimator because the harmonic mean estimator was widely used into the early 1990s, it appeared preferable to most well-known nonkernel estimators (Boulanger and White 1990), and Seaman's comparisons can be extrapolated to other home range estimators through Boulanger and White's (1990) results. Seaman found that the different home range estimators varied greatly in accuracy of estimating both home range areas and utility distributions (figure 3.4).

The fixed kernel estimator, using cross-validation to choose band width,

estimate with cross-validated band width choice. (C) Adaptive kernel density estimate with cross-validated band width choice. (D) Fixed kernel density estimate with ad hoc band width choice. (E) Adaptive kernel density estimate with ad hoc band width choice. (F) Harmonic mean estimate. Modified from Powell et al. (1997).

estimate with cross-validated band width choice. (C) Adaptive kernel density estimate with cross-validated band width choice. (D) Fixed kernel density estimate with ad hoc band width choice. (E) Adaptive kernel density estimate with ad hoc band width choice. (F) Harmonic mean estimate. Modified from Powell et al. (1997).

yielded the most accurate estimates of home range areas and had the smallest variance. These estimates averaged 0.7 percent smaller than the true areas of the simulated home ranges, whereas the adaptive kernel estimates averaged about 25 percent larger than true. The harmonic mean estimator overestimated true home range area by about 20 percent. The cross-validated, fixed kernel estimator also estimated the shapes of the utility distributions the best (figure 3.4). Figure 3.2 depicts the utility distribution isoclines for the home range of an adult female black bear. In addition, for simple, simulated home ranges, the fixed and adaptive kernel estimators generate consistent 95 percent home range areas with as few as 20 location estimates (Noel 1993; Seaman et al. 1999). However, the harmonic mean estimator requires 125 location estimates or more.

The adaptive kernel estimators performed slightly worse than the fixed kernel estimators in all of the tests, apparently through overestimation of peripheral use (Seaman 1993; Seaman et al. 1999; Seaman and Powell 1996). Adaptive kernel estimators also appear sensitive to autocorrelation within data sets. The amount of kernel variation can be adjusted for adaptive kernel estimators, but Seaman has found no consistent or predictable pattern of adjustment that minimizes error for these estimators (Seaman et al. 1999). Consequently, the best estimators at present are fixed kernel estimators with band width chosen via least-squares cross-validation (Seaman 1993; Seaman et al. 1999; Seaman and Powell 1996).

Kernel estimators share three shortcomings with most other home range estimators. First, they ignore time sequence information available with most data on animal locations (White and Garrott 1990). All estimators assume that all location data points are independent and that time sequence information is irrelevant. Future kernel estimators will incorporate brownian bridges between consecutive location estimates, with the heights, widths, and shapes of the bridges dependent on the time and distance between locations, as developed by Bullard (1999). Second, kernel estimators estimate the probability that an animal will be in any part of its home range; therefore, they sometimes produce 95 percent home range outlines that have convoluted shapes or disjunct islands of use. For example, figure 3.5 shows the 95 percent fixed kernel home range for an adult female black bear, bear 61, whom I studied in 1983-1985. Bear 61's home range in 1983 nearly surrounds a large area not designated as her home range. Surely, this bear was familiar with the surrounded area and included it on her cognitive map; however, she chose not to use that area regularly in 1983. In other years, she did use that area (figure 3.1). The fixed kernel estimate of bear 61's home range accurately quantifies the

Figure 3.5 The 95% fixed kernel home range for adult female black bear 61 in 1983. Bear 61's home range nearly surrounds a large, central area not designated as her home range. The thin dotted line marks the study area border.

area that she used in 1983 but may not accurately define the area that she actually considered to be her home range in 1983.

Third, related to the second problem, kernel estimators estimate the probability that an animal will be in any part of its home range but do not estimate how important that part of the home range is to the animal. For researchers asking questions related to time or studying animals for whom time and importance coincide, no problem exists because home range estimators provide probabilities for use of space or, alternatively, probabilities for extent of time in given areas. This aspect of home range estimators is a problem for researchers interested in the underlying importance of habitats or landscape characteristics when time and importance may not coincide. Some parts of a home range that are used little may be very important because they contain a limiting resource needed only at low levels of use. If time and importance do not coincide, kernel estimators (and all other estimators) do not estimate importance accurately.

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