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

iweight kernel (this is an unexplained discrepancy between Seaman and Powell, 1996 and
Silverman, 1986).
Given a true density function for some distribution of data (Figure 4.3.2.1.a.), various values of h
(the smoothing parameter) are used to obtain density estimates. For example, Figure 4.3.2.1.b.
shows the density estimate given a smoothing parameter smaller than the optimum (or true)
value. The difference between the estimated and true density is evaluated as the sum of the area
of deviation of the estimate from the true density. Figure 4.3.2.1.c. shows that difference given a
smoothing parameter slightly larger than the optimum value.
a b c
Figure 4.3.2.1. Least-Squares Cross-Validation with the true density (black line) (a) and an over-
(yellow line) (b) and under-estimate (blue line) (c) of the smoothing parameter.
Finally, with highly oversmoothed data, as with a smoothing parameter that is considerably larger
than optimum (Figure 4.3.2.2.a.), the difference between the density estimate and the true density
is large. The loss function can be considered to be the difference in area between the estimate
and the truth. In this case it is the area difference, but in home range estimation, with bivariate
data, the loss function would be the difference in volume between the two surfaces. In reality this
loss function is the integrated square error (integrated since we are dealing with a density, and
square error, since we want to incorporate error in both over- and under-estimation). It is intuitive
that we want the smallest deviation from the truth, and thus, when we plot the integrated square
error for various smoothing parameters (Figure 4.3.2.2.b.), we search for the absolute minimum
on the curve. At this point we find the associated smoothing parameter, which becomes our
estimate of the optimum smoothing parameter.
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