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Figure 38 Feedforward Neural Network Classi cation Boundary Illustration For classi cation problems, the task of hidden units is to form the decision boundaries to separate di erent classes Figure 38 illustrates the boundaries for a three-class problem Solid lines represent boundaries For this arti cial problem ten boundaries exist Since each hidden unit implements one boundary, ten hidden units are required to perform the classi cation as illustrated in the gure Less hidden units can be used, but at the cost of an increase in classi cation error Also note that in the top left corner there are misclassi cations of class , being part of the space for class This problem can be solved by using three additional hidden units to form these boundaries How can the number of hidden units be determined without using any prior knowledge about the input space This very important issue is dealt with in 7, where the relationship between the number of hidden units and performance is investigated In the case of function approximation, assuming a one-dimensional function as depicted in Figure 39, ve hidden units with sigmoid activation functions are required to learn the function A sigmoid function is then tted for each in ection point of the target function The number of hidden units is therefore the number of turning points plus one In the case of linear activation functions, the hidden units perform the same task However, more linear activation functions may be required to learn the function to the same accuracy as obtained using sigmoid functions
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Training of NNs starts on randomly selected initial weights This means that each time a network is retrained on the same data set, di erent results can be expected, since learning starts at di erent points in the search space; di erent NNs may disagree, and make di erent errors This problem in NN training prompted the development of
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Figure 39 Hidden Unit Functioning for Function Approximation ensemble networks, where the aim is to optimize results through the combination of a number of individual networks, trained on the same task In its most basic form, an ensemble network as illustrated in Figure 310 consists of a number of NNs all trained on the same data set, using the same architecture and learning algorithm At convergence of the individual NN members, the results of the di erent NNs need to be combined to form one, nal result The nal result of an ensemble can be calculated in several ways, of which the following are simple and e cient approaches: Select the NN within the ensemble that provides the best generalization performance Take the average over the outputs of all the members of the ensemble Form a linear combination of the outputs of each of the NNs within the ensemble In this case a weight, wn , is assigned to each network as an indication of the credibility of that network The nal output of the ensemble is therefore a weighted sum of the outputs of the individual networks The combination of inputs as discussed above is sensible only when there is disagreement among the ensemble members, or if members make their errors on di erent parts of the search space Several adaptations of the basic ensemble model are of course possible For example, instead of having each NN train on the same data set, di erent data sets can be used One such approach is bagging, which is a bootstrap ensemble method that creates individuals for its ensemble by training each member network on a random redistribution of the original training set [84] If the original training set contained
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