Using the bootstrap and fast Fourier transform to estimate confidence intervals of 2D kernel densities

Abstract

Kernel density estimators are often used to estimate the utilization distributions (UDs) of animals. Kernel UD estimates have a strong theoretical basis and perform well, but are usually reported without estimates of error or uncertainty. It is intuitively and theoretically appealing to estimate the sampling error in kernel UD estimates using bootstrapping. However, standard equations for kernel density estimates are complicated and computationally expensive. Bootstrapping requires computing hundreds or thousands of probability densities and is impractical when the number of observations, or the area of interest is large. We used the fast Fourier transform (FFT) and discrete convolution theorem to create a bootstrapping algorithm fast enough to run on commonly available desktop or laptop computers. Application of the FFT method to a large (n>20,000) set of radio telemetry data would provide a 99.6% reduction in computation time (i.e., 1.6 as opposed to 444 hours) for 1000 bootstrap UD estimates. Bootstrap error contours were computed using data from a radio-collared polar bear (Ursus maritimus) in the Beaufort Sea north of Alaska.