John R. Fieberg. 2007. Kernel density estimators of home range: smoothing and the autocorrelation red herring. Ecology 88:1059–1066.

Appendix A. Simulated movement path using Euler's method.

Starting at point (x_o, y_o) = (20,10), locations at sequential times t_i+ 1 = t_i+ t were determined by discretizing Eq. 1:

The random component of movement between two successive locations, determined by the Brownian processes (dW_x and dW_y) = , was simulated by drawing two separate normal random variables with mean 0 and variance = t. To determine if the simulated movement path was of sufficient duration to be well described by UD_true in Eq. 3, I:

1. compared the mean and variance covariance matrix of the 500,000 locations to the values in Eq. 3; and
2. computed a KDE using the 500,000 locations, using separate smoothing parameters in the x and y directions chosen to minimize the asymptotic mean integrated squared error for the MVN in Eq. 3 (Simonoff 1996): (h_xo,, h_yo) = ( _x, _y)n^-1/6 = (5, 3)500,000^-1/6. Functions in the KernSmooth library (Wand and Jones 1995, Wand 2005) of program R (R Development Core Team 2005) were used to calculate the KDE on a 150 × 150 grid. I then used Bhattacharyya’s affinity (BA), a measure of similarity between two probability distributions (Bhattacharyya 1943, Fieberg and Kochanny 2005), to compare this KDE to UD true [calculated on the same 150 × 150 grid using the function dmvnorm in the mtvnorm library (Hothorn et al. 2001)]. The BA can range from 0 to 1, with a value of 1 obtained for identical distributions.

The BA between UD_true and the KDE using all 500,000 locations was >0.99. Further, the mean and covariance matrix of the locations from the long path agreed well with and of UD_true: .

LITERATURE CITED

Bhattacharyya, A. 1943. On a measure of divergence between two statistical populations defined by their probability distributions. Bulletin of the Calcutta Mathematical Society 35:99–109.

Fieberg, J., and C. G. Kochanny. 2005. Quantification of home range overlap: the importance of the Utilization Distribution. Journal of Wildlife Management 69:1346–1359.

Hothorn, T., Bretz, F., and A. Genz. 2001. On multivariate t and Gauss probabilities in R. R News, 1:27–29.

Simonoff, J. S. 1996. Smoothing methods in statistics. Springer, New York, New York, USA.

Wand, M. 2005. KernSmooth: Functions for kernel smoothing for Wand and Jones (1995). R package version 2.22-15. R port by Brian Ripley.

Wand, M. P., and Jones, M. C. 1995. Kernel smoothing. Chapman and Hall, London, UK.