Nonnegatively Constrained Confidence Interval Estimation for Ill-Posed Problems
Let the m x n linear model
be obtained by discretizing a system of first-kind integral equations
with known functions K i(ξ), unknown function x(ξ) and an m-vector ŷ of measurements corrupted by random errors є drawn from a distribution with zero mean vector and known variance matrix. Consider the problem of estimating linear functions of the form
where w(ξ) is an averaging function designed to elicit some desired information about x(ξ). For such problems, the least squares solution is a highly unstable function of the measurements, and the classical confidence intervals are too wide to be useful. The solution can often be stabilized by imposing physically motivated, a priori non-negativity constraints on x. This paper will show how to extend the classical confidence interval estimation technique to accommodate these nonnegativity constraints and how to compute the resulting much-improved confidence intervals.
KeywordsAssure Convolution Lution Estima Lawson
Unable to display preview. Download preview PDF.
- W. R. Burrus, Utilization of A Priori Information by Means of Mathematical Programming in the Statistical Interpretation of Measured Distributions, Ph.D. Thesis, Ohio State University, ORNL-3743, Oak Ridge National Laboratory, June 1965.Google Scholar
- W. R. Burrus, B. W. Rust, and J. E. Cope, Constrained interval estimation for linear models with ill-conditioned equations, in Information Linkage between Applied Mathematics and Industry II, A. L. Schoenstadt, et al., eds., Academic Press, New York, 1980, pp. 1–38.Google Scholar
- B. W. Rust and D. P. O’Leary, Confidence intervals for discrete approximations to ill-posed problems, submitted to Jour. Comp. Graph. Statistics, (in review).Google Scholar
© Springer-Verlag New York, Inc. 1992