Contingent kernel density estimation.

Kernel density estimation is a widely used method for estimating a distribution based on a sample of points drawn from that distribution. Generally, in practice some form of error contaminates the sample of observed points. Such error can be the result of imprecise measurements or observation bias. Often this error is negligible and may be disregarded in analysis. In cases where the error is non-negligible, estimation methods should be adjusted to reduce resulting bias. Several modifications of kernel density estimation have been developed to address specific forms of errors. One form of error that has not yet been addressed is the case where observations are nominally placed at the centers of areas from which the points are assumed to have been drawn, where these areas are of varying sizes. In this scenario, the bias arises because the size of the error can vary among points and some subset of points can be known to have smaller error than another subset or the form of the error may change among points. This paper proposes a "contingent kernel density estimation" technique to address this form of error. This new technique adjusts the standard kernel on a point-by-point basis in an adaptive response to changing structure and magnitude of error. In this paper, equations for our contingent kernel technique are derived, the technique is validated using numerical simulations, and an example using the geographic locations of social networking users is worked to demonstrate the utility of the method.

Using tri-axial acceleration data to identify behavioral modes of free-ranging animals: general concepts and tools illustrated for griffon vultures.

Integrating biomechanics, behavior and ecology requires a mechanistic understanding of the processes producing the movement of animals. This calls for contemporaneous biomechanical, behavioral and environmental data along movement pathways. A recently formulated unifying movement ecology paradigm facilitates the integration of existing biomechanics, optimality, cognitive and random paradigms for studying movement. We focus on the use of tri-axial acceleration (ACC) data to identify behavioral modes of GPS-tracked free-ranging wild animals and demonstrate its application to study the movements of griffon vultures (Gyps fulvus, Hablizl 1783). In particular, we explore a selection of nonlinear and decision tree methods that include support vector machines, classification and regression trees, random forest methods and artificial neural networks and compare them with linear discriminant analysis (LDA) as a baseline for classifying behavioral modes. Using a dataset of 1035 ground-truthed ACC segments, we found that all methods can accurately classify behavior (80-90%) and, as expected, all nonlinear methods outperformed LDA. We also illustrate how ACC-identified behavioral modes provide the means to examine how vulture flight is affected by environmental factors, hence facilitating the integration of behavioral, biomechanical and ecological data. Our analysis of just over three-quarters of a million GPS and ACC measurements obtained from 43 free-ranging vultures across 9783 vulture-days suggests that their annual breeding schedule might be selected primarily in response to seasonal conditions favoring rising-air columns (thermals) and that rare long-range forays of up to 1750 km from the home range are performed despite potentially heavy energetic costs and a low rate of food intake, presumably to explore new breeding, social and long-term resource location opportunities.

LoCoH: nonparameteric kernel methods for constructing home ranges and utilization distributions.

Parametric kernel methods currently dominate the literature regarding the construction of animal home ranges (HRs) and utilization distributions (UDs). These methods frequently fail to capture the kinds of hard boundaries common to many natural systems. Recently a local convex hull (LoCoH) nonparametric kernel method, which generalizes the minimum convex polygon (MCP) method, was shown to be more appropriate than parametric kernel methods for constructing HRs and UDs, because of its ability to identify hard boundaries (e.g., rivers, cliff edges) and convergence to the true distribution as sample size increases. Here we extend the LoCoH in two ways: "fixed sphere-of-influence," or r-LoCoH (kernels constructed from all points within a fixed radius r of each reference point), and an "adaptive sphere-of-influence," or a-LoCoH (kernels constructed from all points within a radius a such that the distances of all points within the radius to the reference point sum to a value less than or equal to a), and compare them to the original "fixed-number-of-points," or k-LoCoH (all kernels constructed from k-1 nearest neighbors of root points). We also compare these nonparametric LoCoH to parametric kernel methods using manufactured data and data collected from GPS collars on African buffalo in the Kruger National Park, South Africa. Our results demonstrate that LoCoH methods are superior to parametric kernel methods in estimating areas used by animals, excluding unused areas (holes) and, generally, in constructing UDs and HRs arising from the movement of animals influenced by hard boundaries and irregular structures (e.g., rocky outcrops). We also demonstrate that a-LoCoH is generally superior to k- and r-LoCoH (with software for all three methods available at http://locoh.cnr.berkeley.edu).