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By David S. Touretzky (Editor)

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What is more important, is the choice of an appropriate training scheme, and the remainder of this book will focus on the development and comparative evaluation of different such schemes. In earlier work, [27], [30], presented in Chapter 5, the DSM model was applied, as it arises naturally along the line of derivation adopted here and presented above. Later, it was superseded by the GM model, mainly for didactic purposes. Since Gaussian mixture models are well-known in the statistics community and are increasingly being employed in the neural network community, the introduction of new, but similar transfer functions only adds unnecessary intricacy that is not helpful in communicating the advantages and disadvantages of different training schemes.

Raining Scheme An error function E for a mixture model is derived from a maximum likelihood approach. The derivation of a gradient descent scheme is performed for both the DSM and the GM networks, and leads to a modified form of the backpropagation algorithm. However, a straightforward application of this method is shown to suffer from considerable inherent convergence problems due to large curvature variations of the error surface. A simple rectification scheme based on a curvature-based shape modification of E is presented.

XN-I,XN}, where for convenience in the following notation initial observations are taken at negative times. Also, recall that Xt = (Xt, Xt-I, ... , Xt-m+t). Then P(Dlq) ~olds gIves = P(X-m+l, ... ,XN-l,xNlq) = P(xo)P(xllxo,q) ... 12) due to the Markov property. Taking the logarithm on both sides thus In (P(Dlq)) = ~ln (P(XtIXt-l,q)) +In (P(Xo)) = ~ln (P(xtlXt-1,q)) + C. 13) The independence of P(xo) from q follows again from the fact that the input distribution is not modelled by the network. 10).

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