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Experiment 1
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We consider a small data set consisting of 10 points in 5 points in each of 2 classes (Figure 9.1) and we fit an M L P model of size 1 . 1 . 1 (so that there are four weights)
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Figure 9.1 Two fitted curves with a 10-point data set and 1 additional outlying point of either class 0 or class 1. We can readily calculate the empirical IC for the 4 parameters under additional observation of either class. See the sub-plot for the 1Cs for an observation of class 1.
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We add an additional point, an outlier, and refit the model from the same starting weights, to see what effect this has on the fitted model. When we add, respectively, a point from class 1 and a point from class 0, we get a very different outcome, both shown in Figure 9.1. In this instance, the asymmetric nature of the ICs leads to
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noticeably different results in the two cases . Note that the only effect that the outlier has is to change the shape of the fitted sigmoid (as shown in Figure 9.1), and that the decision boundary remains at approximately the same point and the classification accuracy remains the same. The inset plot in Figure 9.1 shows the empirical IC functions for an observation of class 1, while Figure 9.2 shows the same but with target values of 0.9 and 0.1. Note that in Figure 9.2 the shape has changed considerably and that it is now clearly discernible that the i s for 211 and 212 redescend to 0 in one direction and to a constant value in the other direction. This is described a partially redescending IC function and discussed in Section 8.7 (p. 139).
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Figure 9.2 The empirical ICs for the four weights. Note that the ICs for weights are partially redescending.
After one iteration the empirical IC and the SC are very similar in shape. However after repeated iterations the shapes of the two curves diverge due to the iterative fitting procedure. Figure 9.3 gives the shape of the SCs after 2000 iterations. Note that despite its more complex shape we still see a central region of active influence (Section 8.5.4, p. 136) and then convergence to a constant (perhaps zero) --t f w .
Experiment 2
We consider a simple problem with two classes and two features. The classes are linearly separable and were chosen from uniform distributions on rectangular regions (Figure 9.4). The procedure of the experiment was to progressively move one point, and note the effect on the separating hyperplanes. For each movement of the point, the MLP was refitted using the same starting weights. For comparison with the MLP classifier, we also calculated Fisher s linear discriminant function ( 3, p. 19). As the LDF gives a hyperplane onto which the data are projected, we plot the hyperplane given by the normal vector to the LDF and passing through the grand mean of the classes. Clearly both the LDF
2The same starting values were used in each case.
Sensitivity Curves for a 1.1.1 MLP
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taraet values of 0.1 and 0.9.2000 iterations
The sensitivity curves for the 4 weights F i g u r e 9.3 calculated after 2000 it,erations. Note that the iterative nature of the MLP means that the sensitivity curve presents a inore complex picture than the empirical IC function.
Behaviour of MLPs versus Discriminant Functions - ExDeriment2
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F i g u r e 9.4 The MLP and LDF decision boundaries with one outlying point. In plot 3 t.he h4LP follows the aberrant point whereas in plot 5 it is quite insensitive t o it. The LDF is increasingly influenced by the point its Mahalanobis distance is increased. Plot 1 also shows the limits of linear separability.