| C++ Neural Networks and Fuzzy Logic |
by Valluru B. Rao
M&T Books, IDG Books Worldwide, Inc.
ISBN: 1558515526 Pub Date: 06/01/95
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In this chapter, you we will learn about fuzzy sets and their elements, both for input and output of an associative neural network. Every element of a fuzzy set has a degree of membership in the set. Unless this degree of membership is 1, an element does not belong to the set (in the sense of elements of an ordinary set belonging to the set). In a neural network of a fuzzy system the inputs, the outputs, and the connection weights all belong fuzzily to the spaces that define them. The weight matrix will be a fuzzy matrix, and the activations of the neurons in such a network have to be determined by rules of fuzzy logic and fuzzy set operations.
An expert system uses what are called crisp rules and applies them sequentially. The advantage in casting the same problem in a fuzzy system is that the rules you work with do not have to be crisp, and the processing is done in parallel. What the fuzzy systems can determine is a fuzzy association. These associations can be modified, and the underlying phenomena better understood, as experience is gained. That is one of the reasons for their growing popularity in applications. When we try to relate two things through a process of trial and error, we will be implicitly and intuitively establishing an association that is gradually modified and perhaps bettered in some sense. Several fuzzy variables may be present in such an exercise. That we did not have full knowledge at the beginning is not a hindrance; there is some difference in using probabilities and using fuzzy logic as well. The degree of membership assigned for an element of a set does not have to be as firm as the assignment of a probability.
The degree of membership is, like a probability, a real number between 0 and 1. The closer it is to 1, the less ambiguous is the membership of the element in the set concerned. Suppose you have a set that may or may not contain three elements, say, a, b, and c. Then the fuzzy set representation of it would be by the ordered triple (ma, mb, mc), which is called the fit vector, and its components are called fit values. For example, the triple (0.5, 0.5, 0.5) shows that each of a, b, and c, have a membership equal to only one-half. This triple itself will describe the fuzzy set. It can also be thought of as a point in the three-dimensional space. None of such points will be outside the unit cube. When the number of elements is higher, the corresponding points will be on or inside the unit hypercube.
It is interesting to note that this fuzzy set, given by the triple (0.5, 0.5, 0.5), is its own complement, something that does not happen with regular sets. The complement is the set that shows the degrees of nonmembership.
The height of a fuzzy set is the maximum of its fit values, and the fuzzy set is said to be normal, if its height is 1. The fuzzy set with fit vector (0.3, 0.7, 0.4) has height 0.7, and it is not a normal fuzzy set. However, by introducing an additional dummy component with fit value 1, we can extend it into a normal fuzzy set. The desirability of normalcy of a fuzzy set will become apparent when we talk about recall in fuzzy associative memories.
The subset relationship is also different for fuzzy sets from the way it is defined for regular sets. For example, if you have a fuzzy set given by the triple (0.3, 0.7, 0.4), then any fuzzy set with a triple (a, b, c) such that a ≤ 0.3, b ≤ 0.7, and c ≤ 0.4, is its fuzzy subset. For example, the fuzzy set given by the triple (0.1, 0.7, 0) is a subset of the fuzzy set (0.3, 0.7, 0.4).
Consider two fuzzy sets, one perhaps referring to the popularity of (or interest in) an exhibition and the other to the price of admission. The popularity could be very high, high, fair, low, or very low. Accordingly, the fit vector will have five components. The price of admission could be high, modest, or low. Its fit vector has three components. A fuzzy associative memory system then has the association (popularity, price), and the fuzzy set pair encodes this association.
We describe the encoding process and the recall in the following sections. Once encoding is completed, associations between subsets of the first fuzzy set and subsets of the second fuzzy set are also established by the same fuzzy associative memory system.
The neural network for a fuzzy associative memory has an input and an output layer, with full connections in both directions, just as in a BAM neural network. Figure 9.1 shows the layout. To avoid cluttering, the figure shows the network with three neurons in field A and two neurons in field B, and only some of the connection weights. The general case is analogous.
Figure 9.1 Fuzzy associative memory neural network.
This figure is the same as Figure 8.1 in Chapter 8, since the model is the same, except that fuzzy vectors are used with the current model.
Encoding for fuzzy associative memory systems is similar in concept to the encoding process for bidirectional associative memories, with some differences in the operations involved. In the BAM encoding, bipolar versions of binary vectors are used just for encoding. Matrices of the type XiT Yi are added together to get the connection weight matrix. There are two basic operations with the elements of these vectors. They are multiplication and addition of products. There is no conversion of the fit values before the encoding by the fuzzy sets. The multiplication of elements is replaced by the operation of taking the minimum, and addition is replaced by the operation of taking the maximum.
There are two methods for encoding. The method just described is what is called maxmin composition. It is used to get the connection weight matrix and also to get the outputs of neurons in the fuzzy associative memory neural network. The second method is called correlationproduct encoding. It is obtained the same way as a BAM connection weight matrix is obtained. Maxmin composition is what is frequently used in practice, and we will confine our attention to this method.
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