Evaluating the Usefulness of Least Squares Regression in Petrophysics Interpretation

Abstract

This paper is to update the information provided in the author’s previous papers on evaluating the uncertainty in least squares results. It is prompted by new information that shows that the usefulness of any least squares result cannot be guaranteed by conventional statistics (such as R-squared, F-statistic, or standard error of the regression, sigma). A new method based on the singular value decomposition (SVD) of a matrix, when accompanied by a Relative Error Bound (REB) on the estimated parameters, provides the user with a tool that can better assess the usefulness of any least squares result. Another important aspect of the REB is that it provides the user of the SVD method with a powerful tool for judging which is the best among several candidate solutions. In addition, it provides the user with a numerically stable method of computing the data ordinarily provided by principal component regression by eliminating the need to perform an eigenvector-eigenvalue analysis of an ATA matrix. This is of particular interest because forming the ATA matrix is often accompanied by a loss of data. The new method also provides the user with an improved method for selection of which principal components should be retained for a given problem.