Sliced inverse regression (SIR) and principal Hessian directions (PHD) are two newly established dimension reduction techniques. Their properties have been studied under general regression settings. This research involves the application of these methods to several problems arising from the following areas:
(1) Monte Carlo and Bayesian computation, (2) nonlinear time series, (3) semiparametric modeling, (4) multivariate outcome and functional data analysis, and (5) uncertainty analysis of mathematical/physical/computer models.
These settings are more complicated. In (1), importance sampling and rejection sampling are considered. The importance weights can be treated as the outcome values. The goal becomes the study of finding a simplified relationship between the weights and the sampled points. Dimension reduction can help find the modes of posterior distributions. Proper formulation of the problem is also needed in (2)-(4) before applying SIR/PHD.
Dimensionality is an issue that can arise in every scientific field. Generally speaking, the difficulty lies on how to visualize a data set involving several variables or the shape of a function with several arguments. SIR/PHD are domain-free dimension reduction tools that can be conveniently applied in areas where mining data for information is crucial. For this research, the primary examples include ground water modeling, recovering signals from convoluted data sequences in digital communication, and analyzing nonlinear economic time series data.
