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2.6. FUZZIFIED NEURAL NETWORKFuzzy sets that are obtained with the fuzzy cluster analysis method discussed in the above sections can be modified and/or adapted to newer circumstances with a fuzzified neural network training. For this purpose, fuzzified neural networks are reviewed here briefly (Ishibuchi, Morioka, and Türks¸en,1995). The inputs, weights, and biases of the standard feedforward neural network can be extended to fuzzy sets in a natural manner. In this chapter, the fuzzification of neural networks means just this extension. Therefore, the fuzzification does not change the neural network architecture. That is, the fuzzified neural network has the same network architecture as the standard neural network. Let us denote fuzzy sets by uppercase letters, e.g., A, B, ..., and real numbers by lowercase letters, e.g., a, b, ..., respectively. Then the input-output relation of the fuzzified neural network can be written for a fuzzy input vector Fp = (Fp1, Fp2, ..., where Wji and Wkj are fuzzy weights, and [fancy theta]j and [fancy theta]k are fuzzy biases. 2.6.1. INPUT-OUTPUT RELATION The input-output operations are defined by the extension principle of Zadeh (1975). This means that the fuzzy arithmetic (Kaufmann and Gupta, 1985) is employed for calculating the input-output relation of the fuzzified neural network. The following addition, multiplication, and nonlinear mapping of fuzzy numbers are used for defining our fuzzified neural network:
where A, B, and G are fuzzy sets, µ*(·) denotes the membership function of each fuzzy set, and [wedge] is the minimum operator. The above operations of fuzzy sets are numerically performed on level sets, i.e., [alpha]-cuts. The [alpha]-level set of a fuzzy set F is defined as:
where µF(x) is the membership function of F, and R is the set of real numbers. Because level sets of fuzzy sets are closed intervals (Kaufmann and Gupta, 1985), we can denote F[alpha] as:
where From interval arithmetic (Alefeld and Herzberger, 1983), the above operations of fuzzy numbers in (a)-(c) can be rewritten for a-level sets as follows:
If the [alpha]-level sets of B are nonnegative, i.e., if
The input-output relation of our fuzzified neural network is numerically calculated for the [alpha]-level sets of fuzzy inputs, fuzzy weights, and fuzzy biases. The input-output relation for the [alpha]-level sets can be written as We can see that the [alpha]-level sets of the fuzzy inputs Fpi are mapped to the [alpha]-level sets of the fuzzy outputs Opk. This means that the [alpha]-level sets of the fuzzy outputs Opk can be calculated by interval arithmetic on the [alpha]-level sets of the fuzzy inputs, fuzzy weights, and fuzzy biases. These calculations are performed based on the relations in (a)-(b) (b). 2.6.2. An Example As an example of the fuzzified neural network, let us consider a three-layer network with two input units, two hidden units, and a single output unit. Using this simple example, we illustrate the input-output relations of the fuzzified neural network defined above. The fuzzy outputs from the input units are the same as the fuzzy inputs Fp1 and Fp2. Therefore, the total fuzzy inputs Op1 and Op2 to the hidden units are calculated as follows: [cap gamma]p1 = Fp1 · W11 +Fp2 · W12 + [fancy theta]1 [cap gamma]p2 = Fp1 · W21 + Fp2 · W22 + [fancy theta]2 Then, the fuzzy outputs Op1 and Op2 from the hidden units are obtained as Op1 = f([cap gamma]p1) Op2 = f([cap gamma]p2) These two fuzzy outputs are then fed to the output unit. The total fuzzy input [cap gamma]p to the output unit is: Op = Op1 · W1 + Op2 · W2 + [fancy theta] Finally, the fuzzy output from the output unit (i.e., the fuzzy output from the fuzzified neural network) is calculated as: Op = f([cap gamma]p) These formulations of the input-output relations of each unit are almost the same as the standard feedforward neural network. The only difference is that fuzzy arithmetic is used with fuzzy sets in the fuzzified neural network, while real number arithmetic is used in the standard neural network. The numerical calculations in the formula above are performed for the [alpha]-level sets of fuzzy sets using interval arithmetic. It is suggested that the results of the interval arithmetic be computed with 100 level sets, i.e., [alpha] = 0.01, 0.02, ..., 1.00.
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