| Informatics and Mathematical Modeling (2007) | |||||||||||||||
Abstract | |||||||||||||||
| Rivals and Personnaz (Rivals & Personnaz, 2000) mainly concerns estimation of con dence intervals (or error bars) for neural network prediction models trained by least squares, but also the use of approximate leave-one-out (LOO) cross validation error for model selection is considered. In (Hansen & Larsen, 1996) \Linear Unlearning for Cross-Validation, " Advances in Computational Mathematics, 5, 269{280, 1996, we proposed an approximation of the LOO error which in(Rivals & Personnaz, 2000), p. 473, footnote 10, is claimed to be invalid- even in the case of models which are linear in parameters. This is, however, a misrepresentation of our work and incorrect. In (Hansen & Larsen, 1996) we suggested LOO approximations for general cost functions possibly augmented by a regularization term (e.g., weight decay) for non-linear as well as linear models. In general we consider models which from the input vector x predict an output y by by = f(x � w), where f ( ) generally is a nonlinear function of the input x and the parameter vector w. In the case of linear models trained by least squares, the LOO error cf. equation (18) (Hansen &Larsen,1996) given exactly as: | |||||||||||||||
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