9404594 Owen This project considers modern computationally intensive statistical methods, focussing on problems of numerical quadrature in high dimensions and neural networks in noisy settings. The work on quadrature will develop hybrids of equidistribution methods and Monte Carlo methods, in order to combine the best features of each. Equidistribution methods commonly provide more accurate estimates of integrals, as borne out by asymptotic calculations and some examples in the computational physics literature. Monte Carlo methods, make it easier to assess the accuracy of an estimated integral. The hybrid is formed by randomizing within a class of equidistribution methods developed by Faure and Niederreiter. It is expected that the resulting methods will produce accurate answers whose accuracy can be reliably gauged from the same data used to generate them. Artificial neural networks are widely used to predict and classify responses based on a set of predictors. They are better able to estimate complicated structures than many traditional statistical tools. They are also more prone to finding structures when given purely random data to train on. The problems considered here are guaging how much structure a neural network will learn in a noisy setting, and constructing networks that find less structure in the noise while remaining sensitive to true structure. The integrals considered here may be thought of as averages of one "output" quantity as perhaps ten or twenty "input" quantities vary over their possible values. These averages are of interest in problems from chemistry, physics, finance and statistics. One approach to calculating these averages is based on picking a list of representative input settings, evenly spread through the possible input values, and then averaging the corresponding output values. For many problems this method is quite accurate, but on any given problem it can be hard to tell exactly how accurate the answer is. A second approach uses a randomly chosen list of in put settings. This approach is usually less accurate but there are ways of using the randomness to make probabilistic accuracy statements about the answer. The proposed research combines these ideas by taking a representative list of input settings and randomly scrambling it in a way that preserves the representativeness but should still allow probabilistic statements of accuracy to be made. Artificial neural networks are often used in statistical problems such as predicting what group an object belongs to, given some measured features of it, or predicting an output number given some input numbers. They are called neural networks based on an analogy between their structure and that of a brain. They are usually trained on a set of data containing the true inputs and outputs and in many problems are effective at learning to predict future outputs from future inputs, even when the input-output relationship is very complicated. The proposed work is to study the extent to which artificial neural networks mistakenly learn random patterns from data in which the inputs are irrelevant to the outputs, and to identify which sorts of neural networks are less prone to this problem.
