Nonnegatively Constrained Confidence Interval Estimation for Ill-Posed Problems

  • Bert W. Rust
Conference paper

Abstract

Let the m x n linear model
$$\hat{y}=K\bar{x}+\hat{\epsilon}$$
be obtained by discretizing a system of first-kind integral equations
$$\hat{y}_i=\int_{a}^{b}K_i(\xi)x(\xi)d\xi+\hat{\epsilon}_i,i+1,...m,$$
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
$$\phi=w^T\bar{x}\approx \int_{a}^{b}w(\xi)x(\xi)d\xi,$$
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.

Keywords

Assure Convolution Lution Estima Lawson 

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References

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Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • Bert W. Rust
    • 1
  1. 1.Center for Computing and Applied MathematicsNational Institute of Standards and TechnologyGaithersburgUSA

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