The use of linear algebra in modeling the probabilities of predicted future occurrences

Singular Value Decomposition (SVD) and similar methods can be used to factor matrices into subspaces which describe their behavior. In this paper we review the SVD and generalized singular value decomposition (GSVD) and some of their applications. We give particular attention to how these tools can be used to isolate important patterns in a dataset and provide predictions of future behavior of thes patterns. A major focus of this project is the examination of a component resampling method described by Michael Dettinger which provides estimates of probability distributions for small sets of data. We tested the results of using both the SVD and the GSVD for Dettinger's method. Similarly to Dettinger, we found that the method had a tendency to give probability distributions a Gaussian shape even when this did not seem to be represented in the original data. For some data sets, however, both using the SVD and GSVD provided what appear to be reasonable probability distributions. There was not a significant difference in how well original probability distributions were estimated when using Dettinger's original method or the modifications with the reduced SVD or the GSVD. Using Dettinger's method rather than a simple histogram always provided a higher resolution of information, and was sometimes capable of matching the shape of the original probability distributions more closely.