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form and Eq. (4.43), the decision function in terms of the fuzzy model by fuzzy mean defuzzification becomes

(4.58)

where is the antecedent firing strength, and μ_i(x_j) is the membership of x_j in the fuzzy set A_i1. By the generalization of Eqs. (4.46) and (4.56), SVR is formulated as minimization of the following functional:

(4.59)

where C is the regularization parameter. The solution of (4.59) is used to determine the decision function f(x), and is given by (see Eq. (4.56)):

(4.60)

where are subjected to constraints 0 ≤ α_i , c is the number of support vectors, and the kernel k(x, x_i) = 〈Φ(x), Φ(x_i)〉 is an inner product of the images in the feature space. The model given by Eq. (4.60) is referred to as the SVR model.

      Motivated by the underlying concept of granularity, both the kernel in Eq. (4.60) and the fuzzy membership function in Eq. (4.59) are information granules. The kernel is a similarity measure between the support vector and the non‐support vector in SVR, and fuzzy membership functions associated with fuzzy sets are essentially linguistic granules, which can be viewed as linked collections of fuzzy variables drawn together by the criterion of similarity. Hence, [97–99] regarded kernels as the Gaussian membership function of the t‐norm‐based algebra product

(4.61)

and incorporated SVR in FM. In Eq. (4.61), x_i = [x_i1, x_i2, . . . x_id]^T denotes the support vector in the framework of SVR, but x_ij is referred as to the center of the Gaussian membership function. Parameter σ_j is a hyperparameter of the kernel, whereas it represents the dispersion of the Gaussian membership function in fuzzy set theory.

Fuzzy model based on SVR: Combining the fuzzy model with SVR, we can build a fuzzy system that can use the advantages that each technique offers, so the trade‐off could be well balanced under this combination. Such a model is developed to extract support vectors for generating fuzzy rules, so c is equal in both Eqs. (4.58) and (4.60). Sometimes there are too many support vectors, which will lead to a redundant and complicated rule base even though the model performance is good. Alternatively, we could reduce the number of support vectors and utilize them to generate a transparent fuzzy model. Simultaneously, we make the fuzzy model retain the original performance of the SVR model, and learn the experience already acquired from SVR. In such a way, an experience‐oriented learning algorithm is created. So, a simplification algorithm is employed to obtain reduced‐set vectors instead of support vectors for constructing the fuzzy model, and the parameters are adjusted by a hybrid learning mechanism considering the experience of the SVR model on the same training data set. The obtained fuzzy model retains the acceptable performances of the original SVR solutions, and at the same time possesses high transparency. This enables a good compromise between the interpretability and accuracy of the fuzzy model.

      Constructing interpretable kernels: Besides Gaussian kernel functions such as Eq. (4.61), there are some other common forms of membership functions:

      The triangle membership function:

      The generalized bell‐shaped membership function:

The trapezoidal‐shaped membership function:

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