Accurate Least-Squares Techniques Using the Orthogonal Function Approach

摘要

With the increased availability of computers, the least-squares method of analyzing data is now widely used. This report presents the orthogonal function approach to the least-squares method. This approach is important in practice because it provides accurate numerical results, even when used on large data sets. It is also useful in theoretical considerations, particularly in examining the statistical structure of a data set. The Gram-Schmidt orthogonal function approach is first compared to the standard least-squares method. Next, properties of the approach are shown and a technique to enhance its numerical accuracy is given. Statistical aspects are then discussed. Some applications of the approach are illustrated. The first use can be described as a dominant pattern recognition technique for large data sets and the second is for time series analysis. Finally, a time-saving method used when the functions fitted to the data are polynomials is described, and a related application to Fourier analysis is included.