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.
KeywordsDuality Theorem Ridge Regression Optimal Interval Confidence Interval Estimation Nonnegativity Constraint
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© Springer-Verlag New York, Inc. 1992