High-Dimensional Adaptive Basis Density Estimation
In the realm of high-dimensional statistics, regression and classification have received much attention, while density estimation has lagged behind. Yet there are compelling scientific questions which can only be addressed via density estimation using high-dimensional data, such as the paths of North Atlantic tropical cyclones. If we cast each track as a single high-dimensional data point, density estimation allows us to answer such questions via integration or Monte Carlo methods. In this dissertation, I present three new methods for estimating densities and intensities for high-dimensional data, all of which rely on a technique called diffusion maps. This technique constructs a mapping for high-dimensional, complex data into a low-dimensional space, providing a new basis that can be used in conjunction with traditional density estimation methods. Furthermore, I propose a reordering of importance sampling in the high-dimensional setting. Traditional importance sampling estimates high-dimensional integrals with the aid of an instrumental distribution chosen specifically to minimize the variance of the estimator. In many applications, the integral of interest is with respect to an estimated density. I argue that in the high-dimensional realm, performance can be improved by reversing the procedure: instead of estimating a density and then selecting an appropriate instrumental distribution, begin with the instrumental distribution and estimate the density with respect to it directly. The variance reduction follows from the improved density estimate. Lastly, I present some initial results in using climatic predictors such as sea surface temperature as spatial covariates in point process estimation.