The second classifier linear classifier or multiple

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The second classifier, linear classifier or multiple-class linear classifier, is used solely to solve linearly separable problems such as the one in Fig. 16. In Fig. 16, the three classes C 1 , C 2 and C 3 are linearly separable using three linear decision surfaces d 1 , d 2 and d 3 (each one separating one class from the remaining two). The equations of the linear classifier for the generalized two problem classes C 1 and C 2 are: (6) where i = 1, 2, …, N 1 ; and: (7) where d is the dimensionality of the pattern vector; j = 1, 2, …, N 2 ; N i and N j are the number of training examples from the classes C 1 and C 2 respectively; and w i is a set of weights to be calculated during training, where i = 0, 1, 2, …. Training consists of using known examples from each class (from the training set) to estimate the values of weights w 0 , w 1 , w 2 , …, w d , where d is the dimensionality of d X w w xj w xj w xj j d d – ... – > ( ) = 0 1 1 2 2 0 d X w w xi w xi w xi i d d + + – ... – > ( ) = 0 1 1 2 2 0 162 Acoustic Emission Testing F IGURE 15. Basic principle of k nearest neighbors classifier and respective piecewise linear decision surface for the exclusive or (XOR) problem. x 1 C 1 C 2 C 3 C 4 x 2 Legend C 1 , C 2 , C 3 , C 4 = training classes x 1 , x 2 = indices F IGURE 16. Basics of linear classifier and respective linear decision surfaces. + + + C 1 C 2 C 3 x 2 d 1 ( X ) = 0 d 3 ( X ) = 0 d 2 ( X ) = 0 x 1 Legend C 1 , C 2 , C 3 = training classes d 1 , d 2 , d 3 = linear decision surfaces X = unknown pattern vector x 1 , x 2 = indices
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the feature space. In other words, weight values are estimated by solving the set of ( N 1 + N 2 ) inequalities. Usually, the solution is through an iterative process. 1-6 Several generalizations to multiple-class problems exist, 1-6 among which one C linear discriminant function exists for each class. In this case, upon training and estimation of weights for all C discriminant functions, the unknown pattern X is classified to the class scoring the highest value among the C discriminant functions. 1-6 No matter which classification scheme is used, the data composing the training set should vary enough to simulate the expected variability of the real problem. Once the training phase is completed, the key questions are what the error rate is and how accurate the estimate of it is. One way to estimate the error rate is to evaluate how well the classifier sorts the examples of a testing set containing data of previously known classification. In the worst case, where data are limited, the test set might be identical with the training set. This test set would be the smallest possible: if the classifier cannot correctly classify the examples used to train it, then it is not trained. In most cases, a common test procedure is to divide the available data into two subsets — one to train the classifier and the other to test it. The proportion of data used to train and test the classifier also needs to be considered.
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