Computing Science and Statistics pp 63-72 | Cite as

# Nonnegatively Constrained Confidence Interval Estimation for Ill-Posed Problems

Conference paper

## Abstract

Let the be obtained by discretizing a system of first-kind integral equations with known functions where

*m*x*n*linear model$$\hat{y}=K\bar{x}+\hat{\epsilon}$$

$$\hat{y}_i=\int_{a}^{b}K_i(\xi)x(\xi)d\xi+\hat{\epsilon}_i,i+1,...m,$$

*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$$\phi=w^T\bar{x}\approx \int_{a}^{b}w(\xi)x(\xi)d\xi,$$

*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

Duality Theorem Ridge Regression Optimal Interval Confidence Interval Estimation Nonnegativity Constraint
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

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## Copyright information

© Springer-Verlag New York, Inc. 1992