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Sensitivity curves
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Figure 10.4 As in Figure 9.7, MLPs of size 2.2.1 were fitted to this d a t a set. Two decision boundaries are shown, one for a standard MLP, which chases the aberrant point, and one for the robust MLP, which does not.
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The LeSy-Lindberg central limit theorem will probably not be applicable unlikely to be independent and identically distributed. the coordinates are
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A ROBUST FITTING PROCEDURE FOR MLP MODELS
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We consider the effect of the robust modification of the MLP on the shape of the sensitivity curves. In Figure 10.4, we consider the same scenario in Figure 9.7 (p. 150) except that the MLP is now fitted robustly. Clearly the single point now has little effect on the final model when the robust procedure is adopted. The SC (Section 9.2, p. 144) for the parameter wz has been plotted in Figure 9.6 (p. 149) for the standard MLP. In Figure 10.5 a contour plot is given for the SC for w2 for the robustly fitted MLP. Comparing Figures 9.6 and 10.5 it can be seen that the influence of a single point in some regions of feature space will have a much larger effect for the standard MLP than for the robustly fitted MLP. Note that, unlike the standard MLP, the SC does not follow the lines denoting the limits of linear separability. As the fitted MLP model is highly dependent on the starting values, the illustrated SC only reflects the MLP model with a particular set of starting values. It would be possible to remove the effect of the starting values and get a fuller picture of the shape of the SC by averaging over a number of fits from different starting values.
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The sensit,ivkycurve lor weight wz for thk robust f i L P of size 2.2.1
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The example revisited
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We reconsider the example of Section 10.3.1 (p. 163), but now using the robust MLP, Hunt s procedure, and our robust version of Hunt s procedure. In order t o rnake the problem a little harder we now add a single additional point a t (-20, -20) in addition to the 10 outlying points. Table 10.8 shows the results of using the MLP model. Table 10.9 shows the results of an MLP of size with Hunt s procedure while Table 10.10 shows Hunt s procedure with the robust modifications. Both the robust version of Hunt s procedure and the robust MLP model perform well in this setting. Figure 10.6 shows the decision boundary for these two models in the case of 01,2 = -0.7 and aberrant points at (-4, -4). Hunt s procedure failed in this case and the decision boundary does not intersect the plot. The reason for this is the additional point at (-20, -20) (not shown in the figure) which greatly inflates the non-robust estimate of C.
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For linear models a number of diagnostic procedures are available. It is fairly standard to have access to diagnostic output such the following:
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Table 10.8
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The error rates for the robust MLP model of size
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aberrant values a t U1,2=-0.7 -4,-4 37.75
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-4,-3.5 -3.5,-4 -3,-3 -3,-1.5 -3,1.5 -1.5,-1.5 1.5,1.5 3.3
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37.72 37.75 37.75 37.74 42.40 38.88 38.00 37.75
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1,2=-0.3 01,2=0 ~ 1 , 2 = 0 . 3 32.11 28.19 25.98 32.11 28.19 25.98 32.11 28.19 25.98 32.11 28.19 25.99 28.19 26.24 32.50 31.00 27.61 35.23 31.86 29.70 34.15 28.06 25.78 31.38 32.15 28.08 25.88
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1,2=0.7 24.00 24.00 24.00 24.05 24.57 24.98 25.57 23.92 23.96
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Table 10.9
The error rates for Hunt s procedure, MLP
p l . T h e data set has
aberrant values a t al,2=-0.7 ~1,2=-0.3 -4,-4 50.36 48.03
-4,-3.5 -3.5,-4 -3,-3 -3,- 1.5 -3,1.5
- 1.5,- 1.5 1.5,1.5
50.55 50.43 50.38 49.16 41.28 50.63 39.65 39.62
47.03 47.73 48.10 41.95 35.15 47.60 35.79 35.77
1,210.7 27.25 27.22 27.23 27.59 28.17 28.62 28.86 26.71 26.72
Table 10.10
The error rates for the robustified Hunt s procedure. The data set has an additional point a t (-20,-20).
-4,-4
-4,-3.5 -35-4 -3,-3 -3,-1.5 -3,1.5 -1.5,-1.5 1.5,1.5 3,3
~,2=-O.7 1,2=-0.3 32.52 37.33 32.02 37.36 31.93 38.49 32.01 37.09 31.88 37.13 32.30 40.48 32.57 36.97 32.83 38.24 31.86 37.16
1,2=0
29.83 29.56 29.60 29.73 30.34 30.39 32.96 31.16 29.90 -
1,2=0.3 28.04 26.94 26.84 27.02 31.06 26.69 31.87 28.86 28.68
1,2=0.7 25.46 25.49 25.75 25.38 26.83 25.47 27.68 26.70 25.89