Nonparametric density contour estimation
Gebert, Mark Allen
Cox, Dennis D.
Doctor of Philosophy
Estimation of the level sets for an unknown probability density is done with no specific assumed form for that density, that is, non-parametrically. Methods for tackling this problem are presented. Earlier research showed existence and properties of an estimate based on a kernel density estimate in one dimension. Monte Carlo methods further demonstrated the reasonability of extending this approach to two dimensions. An alternative procedure is now considered that focuses on properties of the contour itself; procedures wherein we define and make use of an objective function based on the characterization of contours as enclosing regions of minimum area given a constraint on probability. Restricting our attention to (possibly non-convex) polygons as candidate contours, numeric optimization of this difficult non-smooth objective function is accomplished using pdsopt, for Parallel Direct Search OPTimization, a set of routines developed for minimization of a scalar-valued function over a high-dimensional domain. Motivation for this method is given, as well as results of simulations done to test it; these include exploration of a Lagrange-multiplier penalty on area and the need which arises for addition of a penalty on the "roughness" of a polygonal contour.