| 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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Suppose the two fuzzy sets we use to encode have the fit vectors ( 0.3, 0.7, 0.4, 0.2) and (0.4, 0.3, 0.9). Then the matrix W is obtained by using maxmin composition as follows.
0.3 [0.4 0.3 0.9] min(0.3,0.4) min(0.3,0.3) min(0.3,0.9) 0.3 0.3 0.3 W=0.7 =min(0.7,0.4) min(0.7,0.3) min(0.7,0.9)=0.4 0.3 0.7 0.4 min(0.4,0.4) min(0.4,0.3) min(0.4,0.9) 0.4 0.3 0.4 0.2 min(0.2,0.4) min(0.2,0.3) min(0.2,0.9) 0.2 0.2 0.2
Recall for the Example
If we input the fit vector (0.3, 0.7, 0.4, 0.2), the output (b1, b2, b3) is determined as follows, using bj = max( min(a1, w1j), , min(am, wmj), where m is the dimension of the a fit vector, and wij is the ith row, jth column element of the matrix W.
b1 = max(min(0.3, 0.3), min(0.7, 0.4), min(0.4, 0.4), min(0.2, 0.2)) = max(0.3, 0.4, 0.4, 0.2) = 0.4 b2 = max(min(0.3, 0.3), min(0.7, 0.3), min(0.4, 0.3), min(0.2, 0.2)) = max( 0.3, 0.3, 0.3, 0.2 ) = 0.3 b3 = max(min(0.3, 0.3), min(0.7, 0.7), min(0.4, 0.4), min(0.2, 0.2)) = max (0.3, 0.7, 0.4, 0.2) = 0.7
The output vector (0.4, 0.3, 0.7) is not the same as the second fit vector used, namely (0.4, 0.3, 0.9), but it is a subset of it, so the recall is not perfect. If you input the vector (0.4, 0.3, 0.7) in the opposite direction, using the transpose of the matrix W, the output is (0.3, 0.7, 0.4, 0.2), showing resonance. If on the other hand you input (0.4, 0.3, 0.9) at that end, the output vector is (0.3, 0.7, 0.4, 0.2), which in turn causes in the other direction an output of (0.4, 0.3, 0.7) at which time there is resonance. Can we foresee these results? The following section explains this further.
Let us use the operator o to denote maxmin composition. Perfect recall occurs when the weight matrix is obtained using the maxmin composition of fit vectors U and V as follows:
Also note that if X and Y are arbitrary fit vectors with the same dimensions as U and V, then:
A ∴ B is the notation to say A is a subset of B.
In the previous example, height of (0.3, 0.7, 0.4, 0.2) is 0.7, and height of (0.4, 0.3, 0.9) is 0.9. Therefore (0.4, 0.3, 0.9) as input, produced (0.3, 0.7, 0.4, 0.2) as output, but (0.3, 0.7, 0.4, 0.2) as input, produced only a subset of (0.4, 0.3, 0.9). That both (0.4, 0.3, 0.7) and (0.4, 0.3, 0.9) gave the same output, (0.3, 0.7, 0.4, 0.2) is in accordance with the corollary to the above, which states that if (X, Y) is a fuzzy associated memory, and if X is a subset of X, then (X, Y) is also a fuzzy associated memory.
We use the classes we created for BAM implementation in C++, except that we call the neuron class fzneuron, and we do not need some of the methods or functions in the network class. The header file, the source file, and the output from an illustrative run of the program are given in the following. The header file is called fuzzyam.hpp, and the source file is called fuzzyam.cpp.
The program details are analogous to the program details given in Chapter 8. The computations are done with fuzzy logic. Unlike in the nonfuzzy version, a single exemplar fuzzy vector pair is used here. There are no transformations to bipolar versions, since the vectors are fuzzy and not binary and crisp.
A neuron in the first layer is referred to as anrn, and the number of neurons in this layer is referred to as anmbr. bnrn is the name we give to the array of neurons in the second layer and bnmbr denotes the size of that array. The sequence of operations in the program are as follows:
A lot of the code from the bidirectional associative memory (BAM) is used for the FAM. Here are the listings, with comments added where there are differences between this code and the code for the BAM of Chapter 8.
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