1 Introduction An example of class boundaries is shown in Fig. 1.6. Unlike one-against and pairwise formulations, there is no unclassifiable region Class 1 91(x)>92(×) lass 2 91(x)>g3x) g2(x)>g2(x) Class 3 Fig. 1.6 Class boundaries by all-at-once formulation 1.2 Determination of Decision Functions Determination of decision functions using input-output pairs is called train- ing In training a multilayer neural network for a two-class problem, we can determine a direct decision function if we set one output neuron instead of two. But because for an n-class problem we set n output neurons with the th neuron corresponding to the class i decision function, the obtained func- tions are indirect. Similarly, decision functions for fuzzy classifiers are indirect because membership functions are defined for each class Conventional training methods determine the indirect decision function hat each training input is correctly classified into the class designated by the associated training output. Figure 1. 7 shows an example of the decision func- tions obtained when the training data of two classes do not overlap. Assuming that the circles and squares are training data for Classes l and 2, respectively, even if the decision function g2(x)moves to the right as shown in the dotted curve, the training data are still correctly classified. Thus there are infinite possibilities of the positions of the decision functions that correctly classif8 1 Introduction An example of class boundaries is shown in Fig. 1.6. Unlike one-against-all and pairwise formulations, there is no unclassifiable region. x1 x2 0 Class 2 g1 (x) > g2 (x) Class 3 g2 (x) > g3 (x) g1 (x) > g3 (x) Class 1 Fig. 1.6 Class boundaries by all-at-once formulation 1.2 Determination of Decision Functions Determination of decision functions using input–output pairs is called train￾ing. In training a multilayer neural network for a two-class problem, we can determine a direct decision function if we set one output neuron instead of two. But because for an n-class problem we set n output neurons with the ith neuron corresponding to the class i decision function, the obtained func￾tions are indirect. Similarly, decision functions for fuzzy classifiers are indirect because membership functions are defined for each class. Conventional training methods determine the indirect decision functions so that each training input is correctly classified into the class designated by the associated training output. Figure 1.7 shows an example of the decision func￾tions obtained when the training data of two classes do not overlap. Assuming that the circles and squares are training data for Classes 1 and 2, respectively, even if the decision function g2(x) moves to the right as shown in the dotted curve, the training data are still correctly classified. Thus there are infinite possibilities of the positions of the decision functions that correctly classify