Monte Carlo and Quasi-Monte Carlo Methods for Statistics

Stanford University, Stanford CA

Investigators

Abstract

Computer simulations are used in virtually every branch of science and engineering, because some problems are simply beyond closed form mathematical solution, and because computers have become extremely fast. Against that trend, there is the constant feeling, and mathematical support, for the idea that strategically sampled values can give even better results than random ones do. Computer generated imaging is one of the largest users of simulations. Simulations of the behavior of light underly images from the economically significant motion picture and computer game industries, to problems of architectural rendering and scientific visualization. Many of the same ideas that go into choosing input points for simulations are also used to computerize difficult statistical decisions and to design industrial products. Quasi-Monte Carlo sampling is a method of producing inputs that are more evenly distributed than random points are. They are said to have low discrepancy. Most of the existing low discrepancy solutions are about placing points inside the unit cube. Some applications in computer rendering require samples from the triangle, simplex or other such spaces. Plain Monte Carlo sampling of those spaces is straightforward but low discrepancy sampling is quite different. The transformations that work for Monte Carlo may destroy the low discrepancy properties of the resulting points in the triangle. This project will develop low discrepancy sampling methods for the triangle and for tensor products of triangles. Factor analysis is about hundred years old, yet it remains problematic to even choose the number of factors to use in it. This project will develop cross-validatory methods to select the number of factors.