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2.6.3. Learning Algorithm

In this section, we describe a general learning algorithm for the fuzzified neural network. This learning algorithm can be applied to the adjustment of the fuzzy weights of various shapes.

When the fuzzy input vector is presented to the fuzzified neural network as defined above, the actual output vector is obtained as the fuzzy vector Let us assume that a fuzzy vector is given as the target vector corresponding to the fuzzy input vector Fp. That is, we assume that the input-output pair (Fp, Tp) is given for the supervised learning of the fuzzified neural network.

The aim of the learning of the fuzzified neural network is to decrease the difference between Op and Tp. That is, it is desired that the following equality holds approximately:

Tpk [congruent to] Opk for k = 1, 3, ..., no

A cost function to be minimized in the learning of the fuzzified neural network should measure the difference between the fuzzy target vector Tp and the fuzzy output vector Op. First, we define a cost function

where

In the cost functions epk[alpha], and are the squared errors for the lower and the upper limits of the [alpha]-level sets, respectively. Those squared errors are weighted by the value of [alpha]. Therefore, the squared errors for a large value of a have a large effect on the learning of the fuzzified neural network.

The cost functions epk[alpha]'s for the [alpha]-level set can be summed up over the nO output units of the fuzzified neural network as:

Therefore, the cost function ep[alpha] is the squared error between the [alpha]-level set of the fuzzy target vector Tp and the fuzzy output vector Op.

The cost functions ep that measure the difference Tp and Op can be defined by summing over all values of [alpha] as follows:

In computer simulations, one could use five or ten values of [alpha].

When N input-output pairs, (Fp, Tp), p = 1, 2, ..., N, are given as training data, we have the following cost function for these training data:

3. UNIFIED FUZZY MODEL

Zadeh's proposal (1975) of a linguistic approach as the model of human thinking introduced the "fuzziness" into systems theory. Fuzzy systems are the outcome of such a linguistic approach in our attempt to obtain more flexibility and better capability of handling and processing uncertainties of complicated and ill-defined systems. This idea was further developed by Tong (1979), Pedrycz (1984), Sugeno and Yasukawa (1993), and Yager and Filev (1994).

In the most general form, the encoded knowledge of a Multi-Input Multi-Output (MIMO) system can be interpreted by fuzzy models consisting of IF-THEN rules with multi-antecedent and multiconsequent variables. However, conceptually, a system with multiple independent outputs can be presented as a collection of Multi-Input Single-Output (MISO) fuzzy systems such that for a system with s outputs, each multiconsequent rule is broken into s single-consequent rule. Modeling and inference is more straightforward for MISO fuzzy systems. That is the reason why the literature concentrates on multi-input, single-output rules as a generic presentation of fuzzy systems.

For these reasons, we will consider in this section a fuzzy rule base that may be obtained with methods described in Section 2 as follows:

where X1, ..., Xr are input variables, and Y is the output variable, Aij, i = 1, ..., n, j = 1, ..., r, and Bj, i = 1, ..., n are fuzzy sets of the universes of discourse X1, X2, ..., Xr and Y, respectively. It is clear that a fuzzy rule base is the first essential component of fuzzy system models.

The second essential component of fuzzy system models is the reasoning mechanism, i.e., the manner in which the information is processed in a fuzzy model. In current methods of fuzzy system models, the connectives of the inference mechanism are selected a priori before the identification procedure without any theoretical basis. In order to improve the objectivity of the fuzzy modeling, one needs to introduce a unified parameterized formulation to the reasoning process in fuzzy systems. For this purpose, four reasoning parameters, p, q, [alpha], and [beta], are introduced the values of which will cause a continuous range of variation in the reasoning mechanism. Therefore, we are no longer restricted to stay at the extremes in any step of the reasoning process. Consequently, rather than selecting the connectives a priori for fuzzy system models, it is desirable to have the reasoning process adjust the above parameters with supervised training using system input-output data, such that the objective would be to minimize performance index errors. Hence, the selection of the connectives would be in agreement with system behavior patterns. In order to reduce the computational effort, a fast computational algorithm is introduced for the calculation of the parameterized family of triangular functions (Emami, Türks¸en,and Goldenberg, 1996).

3.1. APPROXIMATE REASONING

In approximate reasoning, we have essentially two methods:

  1. First Infer Then Aggregate, FITA
  2. First Aggregate Then Infer, FATI (Türks¸enand Tian, 1993)

In FITA method, we need to:

  1. Determine fuzzy aggregation of antecedents in each rule (AND connective)
  2. Determine implication relation for each individual rule (IF-THEN connective)
  3. Make inference with each rule of the set of rules, using the observed input
  4. Aggregate the fuzzy outputs obtained in (3) above to get the final fuzzy output (ALSO connective)
  5. Defuzzify the final fuzzy output

    Whereas in FATI method, steps (3) and (4) are modified as follows:

  6. Aggregate all the rules of the rule base to determine the overall rule (ALSO connective)
  7. Make inference with the overall rule using the observed input

It can be shown that the two methods of inference with a rule set, i.e., FATI and FITA, always give the same fuzzy output for a singleton input (Emami, Türks¸en,and Goldenberg, 1996). Moreover, we propose a reasoning formulation that combines Mamdani's and formal logical approximate reasonings, i.e., two mypoic reasoning approaches. This then forms the foundation of a unified parameterized fuzzy reasoning formulation. It should be recalled that in the introduction section, we have identified Mamdani's reasoning as the case (1) and formal logical reasoning as the case (2). Both of these are in Type I theory and hence are myopic.


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