86
R O G E R A . P O W E L L
animal movement must be a simplification, so Gautestad and Mysterud’s
model does simplify animal movements. It does incorporate multiscale aspects
of movement and appears to be a better model than, say, random walk models. Nonetheless, the multiscale random walk model still lacks important characteristics of true animal movements, and may thereby cause equation 3.1 to
give a false prediction.
Even if equation 3.1 is false, the fractal utility distribution based on 1/C may
still provide insight into use of space by animals. Unfortunately, by calculating
C for each cell in a grid, one loses multiscale information that is available from
an entire data set. In addition, 1/C provides no insight into estimated use of
interstitial cells because it is only a transformation of the frequencies per cell
(n 1/2 instead of n). Finally, Vandermeer’s (1981) cautions concerning grid
dimensions must be addressed. One gains equal insight by calculating kernel
home ranges and examining the probabilities for animals to be in cells of different sizes (scales), and kernel estimators are free of grid size constraints.
Fractal approaches to animal movements may provide new insights into
animals’ home ranges, but their utility is still uncertain.
KERNEL ESTIMATORS
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,
R O G E R A . P O W E L L
animal movement must be a simplification, so Gautestad and Mysterud’s
model does simplify animal movements. It does incorporate multiscale aspects
of movement and appears to be a better model than, say, random walk models. Nonetheless, the multiscale random walk model still lacks important characteristics of true animal movements, and may thereby cause equation 3.1 to
give a false prediction.
Even if equation 3.1 is false, the fractal utility distribution based on 1/C may
still provide insight into use of space by animals. Unfortunately, by calculating
C for each cell in a grid, one loses multiscale information that is available from
an entire data set. In addition, 1/C provides no insight into estimated use of
interstitial cells because it is only a transformation of the frequencies per cell
(n 1/2 instead of n). Finally, Vandermeer’s (1981) cautions concerning grid
dimensions must be addressed. One gains equal insight by calculating kernel
home ranges and examining the probabilities for animals to be in cells of different sizes (scales), and kernel estimators are free of grid size constraints.
Fractal approaches to animal movements may provide new insights into
animals’ home ranges, but their utility is still uncertain.
KERNEL ESTIMATORS
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,