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Truncating the Singular Value Decomposition for Ill-Posed Problems

Published

Author(s)

Bert W. Rust

Abstract

Discretizing the first-kind integral equations which model many physical measurement processes yields an ill-conditioned linear regression model. Least squares estimation usually gives a sum of squared residuals much smaller than the expected value and a wildly oscillating, physically implausible estimate of the function being measured. Taken together, these two symptoms suggest that the solution has captured part of the variance that properly belongs to the residuals. One strategy for increasing the sum of squared residuals and simultaneously stabilizing the estimate is to truncate the singular value decomposition of the instrument response matrix. Conventionally this has been regarded as a rank determination problem, but in many cases, the matrix is clearly not rank-deficient. This paper suggests an alternate strategy which uses the variances of the measuring errors to specify a truncation for the elements of the rotated measurement vector rather than the singular value distribution of the matrix. It also develops some new diagnostics for the residuals which are useful not only for choosing the truncation level, but also for assessing the quality of the estimate obtained by any procedure.
Citation
Siam Journal on Numerical Analysis

Keywords

ill posed problems, linear regression, regression diagnostics, singular value decomposition

Citation

Rust, B. (1998), Truncating the Singular Value Decomposition for Ill-Posed Problems, Siam Journal on Numerical Analysis (Accessed March 1, 2024)
Created July 1, 1998, Updated June 2, 2021