DISCUSSION

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It is clear that in ill-posed problems, such as the given example, the need for a penalization term is paramount. Fixed-slope training appears to work well a penalization method. While its implementation is more complex than that of weight decay, for example, it has an intuitive geometric interpretation in terms of the slopes of the activation functions, and of the fitted surface. Fixing the slope of the activation function makes the MLP model more resistant to the influence of isolated points.

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FIXED-SLOPE TRAINING

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Figure We consider the data set from Example 2 in Section 9.3.2, p. 146. As in the decision 9, a single point has been moved and the resulting effects noted boundaries, which are calculated for the linear discriminant function (LDF), a standard MLP niodel and a fixed-slope MLP. An inspection of the figure shows that neither t h e LDF nor t.he standard MLP are resistant t o an aberrant d a t a point, although their non-resistant a point in class "+" by virtue of how behavior is quite different. T h e LDF is influenced far that. point is (in Mahalanobis distance) from the mean of class whereas the standard MLP has a non linear influence funct,ion t h a t has a pronounced ridge along the lines of linear The fixed slope MLP is separability. See particularly the difference between plots 4 and resistant to the moving point.

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DISCUSSION

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A comparison of several classification techniques on the simulated scene. For each classifier the error rate on the image is given.

II classifier 11 Linear discriminant functions

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Quadratic discriminant functions Nearest neighbor MLP (of size 4.2.3), a cross-entropy penalty function, softmax outputs MLP, as above, with weight decay X = 0.1

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error rate 0.2536 0.2552 0.2320 0.2600 0.2216 0.2776 0.2208 0.2192 0.2224

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MLP, as above, with fixed slope, unconstrained MLP, as above, with fixed slope y = 6 MLP, as above, with fixed slope y = 5 MLP. above. with fixed slope = 4 ILILP, as above, with fixed slope y = 3

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Adby, P. R. and Dempster, M. A. H. (1974). Chapman and Hall, London. Ahrned, S. and Tesauro, G. (1988). Scaling and generalization in neural networks: a case study. In Touretzky kt al. (1988), pp. 3-9. Aires, F., Prigent, C., and Rossow, W. B. (2004). Neural network uncertainty assessment using bayesian statistics: A remote sensing application. 16(11):2415-2458. Aitchison, J. and Dunsmore, I. R. (1975). bridge University Press, Cambridge. Almeida, L. and Wellekens, C., editors (1990). 4 12, Berlin. Cam-

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Alpaydin, E. (1991). Gal: Networks that grow when they learn and shrink when they forget. Preprint, International Computer Science Institute, Berkeley, CA. Aniari, S., Murata, N., Muller, K.-R., Finke, M., and Yang, H. (1997). Asymptotic statistical theory of overtraining and cross-validation. on 8(5). Previously, University of Tokyo Techreport: METR 9506, 1995. Anderson, J. A . and Rosenfeld, E., editors (1988). M I T Press, Cambridge, MA.

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A Statistical Approach to Neural Networks for Pattern Recognition by Robert A . Dunne Copyright @ 2007 John Wiley & Sons, Inc.

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