Strong robust generalized cross-validation for choosing the regularization parameter

Lukas, M.A. (2008) Strong robust generalized cross-validation for choosing the regularization parameter. Inverse Problems, 24 (3).

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Abstract

Let fλ be the Tikhonov regularized solution of a linear inverse or smoothing problem with discrete noisy data yi, i = 1, ..., n. To choose λ we propose a new strong robust GCV method denoted by R1GCV that is part of a family of such methods, including the basic RGCV method. R1GCV chooses λ to be the minimizer of γV(λ) + (1 − γ)F1(λ), where V(λ) is the GCV function, F1(λ) is a certain approximate total measure of the influence of each data point on fλ with respect to the regularizing norm or seminorm, and γ in (0, 1) is a robustness parameter. We show that R1GCV is less likely to choose a very small value of λ than both GCV and RGCV. RGCV and R1GCV also have good asymptotic properties for general problems with independent errors. Strengthening previous results for RGCV, it is shown that the (shifted) RGCV and R1GCV functions are consistent estimates of 'robust risk' functions, which place extra weight on the variance of fλ. In addition RGCV is asymptotically equivalent to the modified GCV method. The results of numerical simulations for R1GCV are consistent with the asymptotic results, and, for suitable values of γ, R1GCV is more reliable and accurate than GCV.