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We fit an MLP model to some data in 2-dimensions, so that we can visualize the decision boundary. We first load the library to make the function available to us.
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We then set up the data and plot it
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3This discussion assumes a 0/1 loss function where the cost of a correct classification is 0 and an incorrect classification is 1 . It is possible to have a cost matrix specifying the cast of misclassifying a member of C,, to Cq2. See Titterington et al. (1981) for an example with a cost matrix ascribing differential costs to various outcomes for head-injured patients.
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The plot is shown in Figure 2.2. We have designed this test example so that we need a non-linear decision region to achieve a zero misclassification error. We fit an MLP of size = 2.3.2, that is 2 input nodes, 3 hidden layer nodes and 2 output nodes.
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plots the lines defined by the rows of 0. It is only usable when the input space is of 2-dimensional. Similarly, sets up a matrix of output values for the MLP, suitable for the function to plot.
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Note that the final decision boundary is a smooth curve. The apparent roughness is caused by the discrete grid used by which could be altered (although it will then require more time for the computation).
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The lines defined by the rows of R and the 0.5 contour for the MLP outputs.
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There are a number of other things we can do to explore MLPs with this data set. You should try different starting values (just change the value of rotate the data and see what happens
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fit the model with more (or less) hidden layer units.
2.2 The Iris data set (Fisher, 1936) is available in the library. The data set consists of 4 measurements (sepal length, sepal width, petal length and petal width) on each of 50 individuals of 3 species: Iris setosa; versicolor; and virginica. Fit an MLP model to predict membership of the three classes. The help file for (MASS library) gives an example of fitting the model on 1/2 of the data and testing the model fit on the other 1/2. Tky this.
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In this chapter we review linear discriminant analysis (LDA), a classical statistical procedure with a number of desirable features. We also look a t some flexible extensions to LDA and consider the connection between LDA and the MLP model. LDA was developed in the case of two classes by Fisher (1936) and extended to multiple classes by Rao (1948). See Rao (1973, pp. 574-580) or Mardia et al. (1979) for a standard exposition. The method is often very successful and we note that in the STATLOG project (Mitchie et al., 1994), a large-scale comparison of classifiers on a wide variety of real and artificial problems, LDA was among the top three classifiers for 1 1 of the 22 data-sets1. We begin by formulating the criterion. Assume that there are Q classes indexed by q = 1 . . . . , with Np observations in the qth class, = and and
N x Q
are the usual target and data matrices. Assume that the columns of
N x P
mean centered or else set
1 NNXN
'It is quite complicated to summarize the results of this large set of trials, and there was no outright winner. However, it is fair t o say that the two simplest techniques considered, LDA and nearest neighbor, did better than many more sophisticated techniques!
A Statzstacal Approach to Neural Networks Copyright @ 2007 John Wiley & Sons, Inc.
Pattern Recognition by Robert A . Dunne
LINEAR DISCRIMINANT ANALYSIS
Define and then
T ( T 7 T ) - T 7 ,the projection matrix onto the column space of
C, = N - X 7 X ,
C =(Q - l ) - 1 ( P 7 X ) 7 ( P T X ) and , ,
(3.1)
C =(N W
- p T ) x ) 7 ( ( 1 PT)x), -
the total covariance matrix, between-classes covariance matrix (the covariance of the projection onto the column space of and the within-classes covariance matrix (the covariance of the projection onto the null space of respectively. Defining matrix of class means, = ( T 7 T ) - T 7 X , we have a t the Q x
C, =(Q C ,
=(N -
C, isassumed t o b e t h e s a m e forall theclassesand N C , = (Q-l)C,+(N-Q)C,,. The task in a LDA is to find A such that the ratio of determinants
(3.2)
is maximized (see Problem 3.1, p. 3 0 ) . This maximizes the between-classes covariance relative to the within-classes covariance in the linear space spanned by the columns of A. The columns of A define the linear discriminants also called the canonical variates . Note that there is nothing in this formulation about classifying observations. The data is projected into a linear space selected according to a criteria. In that space it may be classified or visualized desired. The way that the linear space has been selected may make the operations easier then in the original space. Equation 3.2 can be maximized by finding A such that IA7C,AI is maximized subject to A7C,A = Using a Lagrange multiplier the problem can be written as