ACTIVATION AND PENALTY FUNCTIONS in .NET

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ACTIVATION AND PENALTY FUNCTIONS
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Table 4 . 4
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The class-conditional classification rates for the least squares error ftunct,ion and the logistic activation function. The true class is shown down the table aiid the ascribed class across the table.
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Table 4.5
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The class-conditional classification rates for the cross-entropy penalty function and the softmax activation function. The true class is shown down the table and the ascribed class across the table.
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I 1 2
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3 4 0.1333 0.1302 0.0297 0.6981 0.0265 0.0246 0.7055 0.1440 0.1378 0.7025 0.1343 0.1335
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2 3 4 1 I I I 0.7046 0.1376 0.1318 0.0260
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Table 4.6
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The class-conditional classification rates for the least squares penalty function and the softmax activation function. The true class is shown down the table and the ascribed class the table.
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device, modelling data coming from (1 - c)Fl eF2, where the F2 distribution is a contaminating distribution, is common in the modeling of data with outliers. A good choice for is the Cauchy distribution. The MASS library provides a birth weight data set of 189 observations and 10 variables. The data were collected a t Baystate Medical Center, Springfield, MA, during 1986. A number of the variables are categorical or factor variables. These have t o be encoded numerically in order to fit the model. Examine the encoded model matrix to see how this is done. Using the multinom function fit a log-linear model to the birth weight data to test for association between birth weight and the the 9 explanatory variables.
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the nnet allows a single layer MLP to be fitted by specifying Fit a single layer MLP to the design matrix from Exercise 4.2. Using the
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family fit a GLM model to the birth weight data set.
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COMPLEMENTS AND EXERCISES
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Dobson (1990) gives an ulcer data set (given in the script files). Fit a Poisson GLM and a model to this data set.
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MODEL FITTING AND EVALUATION
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INTRODUCTION
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We consider the general question of the evaluation of a classifier and the questions of model selection and penalized fitting procedures for the MLP model. MLP models are rarely used in circumstances where a low value of p a t the observed data, is the desired goal. A central question in fitting and evaluating an MLP model is that of generalization. The term comes from the original biological motivation of neural networks and to generalize well means to have the capacity to apply learned knowledge to new situations. Quantitatively it can be defined the expected performance of the MLP, measured by p or by a misclassification rate or matrix. In practice the generalization ability of the MLP is often measured by the performance on a set of previously unseen data. The MLP is a powerful (Section 4.3, p. 37) distribution-free regression method. However, this very power means that it is possible to reduce p to 0 on the training data. Now, if we consider the data to have been generated by a process with a systematic and a random component then, in order to reduce p to 0 the MLP must be modeling noise, and in this case it is likely that the MLP will not generalize well. The term overfitting is used to describe learning the random component in the training set. Weigend (1994) suggests that, as an operational definition, overfitting can be said to occur when the error on an independent test set, having passed through a minimum, starts to increase (see item 8, p. 65).
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A Statistical Approach to Neural Networks Copyright @ 2007 John Wiley & Sons, Inc.
Pattern Recognition by Robert A. Dunne
MODEL FITTING AND EVALUATION
Fitting a model to data involves negotiating a trade-off between bias and variance, as a model with either an inflated bias or variance will lead to a n inflated mean squared error (see Section 4.4, p. 38; Geman et al. 1992; and Bishop 1995a $9). Such a model may not perform well on new data. For models with few parameters and strong distributional assumptions, such as linear models, model selection can be viewed bias reduction. Such models are referred to as confirmatory and rely on the value of p on the training data to validate the model. In some circumstances, such as with (near) collinearities in the variables, penalized fitting procedures are necessary. For flexible models, like the MLP, that can model a large class of densities given sufficient hidden-layer units, model selection and penalization are not clearly separable. It is rare that the correct MLP model can be selected in any meaningful sense. A more likely situation is that, whatever the correct model which defines the process generating the data is, the MLP is flexible enough to model this density. Because of this it is often the case that the MLP model will fit the training data too accurately, resulting in an overfitted model with a large variance. Given this, it is not surprising that linear models only rarely require penalized training, whereas MLP models almost invariably do. 5.2