Kernel Home Range Estimation for ArcGIS, using VBA - Fish and ...

Silverman (1986) first described kernel density estimation for the layperson. As a result, much of
the subsequent kernel home range literature rests heavily on this work. Silverman describes
kernel estimation as follows (Figure 4.1.1. adapted from Silverman, 1986). Density estimation in
its simplest form is a histogram representation of the variable in question (i.e. the x or y
coordinate of a set of locations) (Figure 4.1.1.a.). The naïve estimator is a histogram constructed
such that each point falls in the centre of a sampling interval (bin) (Figure 4.1.1.b.). In this
method, the sampling intervals overlap, and the points that are included in any interval are
weighted according to a uniform distribution. The kernel estimator is an improvement of this
naïve estimator in that it replaces the uniform weighting function with a kernel function. This
kernel function is a probability density function with a specified distribution (Figure 4.1.1.c.).
Equation 1 defines the kernel density estimator:
a b c
Figure 4.1.1. Progression of density estimation from simple histogram techniques to smooth
kernel estimation (Adapted from Silverman, 1986). In these examples, the abcissa (x axis)
could be x coordinate for a set of locations, that in the animal spends more time or is seen
more in the center of the distribution in the x direction.
( x − X )
n
⎡ ⎤ ⎧ i
f ˆ 1
( x)
= ⎢ ⎥
( ) ∑ K ⎨
⎣ nh ⎦ = ⎩ ⎭ ⎬⎫
2
i 1 h
(1)
K is the kernel that determines the shape of the distribution that is placed over each of the points.
h is the smoothing factor (also known as the smoothing parameter, bandwidth and window width),
which controls the search radius or width or the kernel. n is the number of location estimates
(points) used in the analysis. x and X refer to the vectors of the coordinates of the evaluation
point and all other points, respectively. Various kernels have been described (Silverman, 1986),
but kernel analysis has been shown to be relatively insensitive to the choice of kernel (Silverman,
1986). Seaman and Powell (1996) use, as an example, the Biweight Kernel, K B2B from Silverman
(1986). Equation 2 defines the biweight kernel:
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