*l here is an example of this in experiment 2 case 2, Section 9.3, p. 145

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MODEL FITTINGA N D EVALUATION

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be of order (Murata et al., 1994). Equation (5.6) is also applicable when D is replaced by a penalized deviance, discussed in the Section 5.4, p. 62. If the parametric family contains then = the number of parameters, = trace(BA- ), where A and are defined and (5.6) reduces to the AIC, else in (5.4). Moody s 1992 effective degrees of freedom pcff gives an equivalent form for see also Moody and Utans (1995) for a comparison of cross-validation and NIC model selection. Ripley (1996, 55.6) cautions that both p,ff and NIC depend on the penalty function surface having a single pronounced local minimum. If this condition is not met then the effective degrees of freedom may depend more on the local minimum found than the number of hidden layer units. See also Ripley (1996) $2.2 and 54.3 for a derivation and treatment of the NIC.

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Other methods

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Sietsma and Dow (1991) suggest interactive pruning guided by heuristics. Utans and Moody (1991) use generalized cross-validation and Squared Prediction Error (Barron, 1984) t o do model selection, but use the number of parameters the degrees of freedom. Mozer and Smolensky (1989) suggest pruning units on a measure of relevance that is an estimate of jqr)- p where p ( r ) is the penalty function with unit T removed. See also 6 (p. 69), where task-based pruning is introduced and discussed. This method relies on testing for similarities in the outputs of hidden layer units.

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PENALIZED TRAINING

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Penalization appears in linear models under the name ridge regression . If the matrix is ill conditioned5 then, writing Xmin as the smallest eigenvalue of and the true value,

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- is

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Amin

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a lower bound for the average squared

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(see Marquardt, 1970, for a discussion), and thus the distance between G and estimates may be quite poor. In these circumstances Marquardt comments that the absolute values of the u s may be too large and may change sign with negligible changes in the data. A better procedure in these circumstances is to solve LL = ( X 7 X + k l ) - X 7 y for some constant This inflates the eigenvalues of the matrix, giving a biased estimator with perhaps a lower expected mean square error. In order to rectify the tendency of MLPs to overfit the data, a number of similar penalization methods have been tried. All are methods for obtaining a smoother separating boundary between classes, and all involve some trade-off between error minimization and smoothness via the selection of a smoothing parameter. These methods have all been introduced in the context of decreasing the error of the MLP

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See Stewart (1987) for some history of the concept of matrix condition. It appears t o have developed separately within the statistics and numerical analysis communities. Within numerical analysis it can be traced through Turing (1948) and Golub and Wilkinson (1966). Piegorsch and Casella (1989) give a history of matrix conditioning and ridge regression in statistics. T h e condition number n ( A ) = 1IAJII/A- JJ a measure of the stability or sensitivity of a matrix (or is the linear system it represents) to numerical operations. If it is large then numerical operations involving the matrix may have a large error.

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PENALIZED TRAINING

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on previously unseen data by restricting the MLP in the way that it fits the training data. The methods include the following: Adding noise to the training data. Sarle (1995) coins the term smoothed ridging for this method and Bishop (1995b) shows that it is approximately equivalent to the Tikhonov regularization term

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when the amplitude of the noise is small. See Sietsma and Dow (1991) for an example; Scaling the weights down after training; Replacing the target value with the convolution of the target and a probability distribution; Adding a regularization term to the error function. This is penalized ridging (Sarle, 1995) and can be justified within a Bayesian framework. It appears in the literature in two forms. One form is the case where the penalty term is of the form 1 - tJ2 ~IIDZ*IIZ, (5.8)

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where is a differential operator. Equation (5.7) is a finite sample version of such a differential operator regularization term. Note that (5.7) and (5.8) involve derivatives of z* with respect to s. In order to fit these penalized models the derivatives with respect to the are needed, see Bishop (1993) for the computations. Ripley (1996, $4.3) notes that regularization has been most explored in approximation theory, the results are generally for least squares approximation. Poggio and Girosi (1990a,b) and Girosi et al. (1993) discuss regularization in the context of surface reconstruction from sparse data points. The approach is motivated by the recognition that the problem of function approximation from sparse data is ill-posed in the sense that the solution is not unique, and that additional constraints are needed in the form of an appropriate prior on the class of approximation functions to guarantee the uniqueness of the solution. Often the prior is one that assumes a high degree of smoothness. They show that (5.8) leads to a class of hiddenlayer networks that they call regularization networks of which radial basis function networks are a subset. If G is the Green s function (Courant and Hilbert, 1953) of where is the adjoint of then (5.8) leads to

= CwnG(IIs-snt~)

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and gives a fitted surface equivalent to a generalized spline. In particular if is then (5.8) leads to the cubic spline. An alternative form of regularization where the prior represents a knowledge other than smoothness of the fitted surface. In the case of the

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