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The problem of learning a logical function can be recast. a geometric problem by encoding {TRUE, FALSE} {l,O}. The figure shows decision boundaries that iinplernent the function OR and AND. The XOR function would have to have points A aiid C on one side of the line and B and D on the other. It is clear that no single line can achieve t.hat, although a set of lines defining a region or a non-linear boundary can achieve it. metric XOR problem any more than the perceptron can. However, as far no statistician has ever shown a lot of concern about this fact.
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Consider two logical variables A and that can take values in the set {TRUE, FALSE}. The truth values of the logical functions AND, OR, and XOR are
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Using a layered structure of perceptron shown in Figure 2.1 (p. 10) overcame this problem and lead to a resurgence in interest in this research area. These are the multi-layer perceptrons (MLPs) that are the topic of this work. They required a different learning algorithm to the single perceptron and require that f be a differentiable function. It was the development of such algorithms that was the first step in their use. This has appeared several times in the literature, common early references being Werbos (1974) and Rumelhart et al. (1986). Already a large number of questions are apparent: such what if there are more than two classes;
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what if the classes are not linearly separable - but there is a non-linear decision boundary that could separate them;
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do we want a classifier that performs well on this data set or on a new, as yet unseen, data set Will they be the same thing
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These questions will be consider in the ensuing chapters. David Hand has recently written a interesting paper entitled Classifier Technology and the Illusion of Progress (Hand, 2006). The progress that he is questioning is the progress of sophisticated techniques like MLPs, support vector machines (Vapnik, 1995; Burges, 1998), and others. He shows that for many examples, the decrease in classification error using sophisticated techniques is only marginal. It may be so small, in fact, that it may be wiped out by the vagaries and difficulties i n attendance with real word data sets I think that David Hand s caution should be taken to heart. Sophisticated techniques should be used with caution and with an appreciation of their limitations and idiosyncrasies. If there is a good data model available, for example, an understanding that the data are Gaussian, then there may be no justification for using an MLP model. MLP models have not always been used with an appreciation of their characteristics. The fact that MLPs can be used in a black box fashion, and seem to produce reasonable results without a lot of effort being put into modeling the problem, has often led to them being used in this way. It appears that MLPs were being used on hard problems, such as speech recognition and vision, long before any real groundwork was done on understanding the behavior of the MLP as a classifier*. This has led to debate in the literature on such elementary points as the capabilities of MLPs with one hidden layer5, and a lack of understanding of the possible roles of hidden layer units in forming separating boundaries between classes. However, such understanding can be readily arrived at by considering the behavior of the MLP in simple settings that are amenable both to analytic and graphical procedures. In this book the simplest case of two classes and two variables is often used as an example and some points that have been debated in the literature may be amongst the first things that an investigator will notice when confronted with a graphical representation of the output function of an MLP in this simple setting. The aim of this book is to reach a fuller understanding of the MLP model and extend it in a number of desirable ways. There are many introductions and surveys of multi-layer perceptrons in the literature (see below for references); however, none should be necessary in order to understand this book, which should contain the necessary introduction. Other works that could usefully be consulted to gain insight into the MLP model include Cheng and Titterington (1994), Krzanowski and Marriott (1994), Bishop (1995a), Ripley (1996) and Haykin (1999). We use a number of examples from the area of remote sensing to illustrate various approaches. Richards and Jia (2006) is a good introduction to this problem area while Wilson (1992) and Kiiveri and Caccetta (1996) discuss some of the statistical issues involved. Once again, this work should be entirely self contained - with as much of the problem area introduced in each example as is needed for a full appreciation of the example. Multi-layer perceptrons have been used in the analysis of remotely sensed data in Bischof et al. (1992), Benediktsson et al. (1995)
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Early exceptions to this tendency to use MLPs without investigating their behavior are Gibson and Cowan Lee and Lippmann (1990) and Lui (1990). T h e situation has been changing markedly in recent years and many of the lacunae in the literature are now being filled. 5That is, are they capable of forming disjoint decision regions; see Lippmann (1987).
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