Savo G. Glisic - Artificial Intelligence and Quantum Computing for Advanced Wireless Networks

The trapezoidal‐shaped membership function:

where b is the center of the membership functions, and parameters γ , a , and c are mean values of the dispersion for the three examples of membership functions.. Could the above functions also be used for constructing an admissible Mercer kernel? Note that they are translation invariant functions, so the multidimensional function created by these kinds of functions based on product t ‐norm operator is also translation invariant. Furthermore, if we regard the multidimensional functions as translation invariant kernels, then the following theorem can be used to check whether these kernels are admissible Mercer kernels:

A translation ‐ invariant kernel k (x, x i) = k (x − x i) is an admissible Mercer kernel if and only if the Fourier transform

is non ‐ negative [100]. For the case of the triangle and generalized bell‐shaped membership functions, the Fourier transform is respectively as follows:

and

Since both of them are non‐negative, we can construct Mercer kernels with triangle and generalized bell‐shaped membership functions. But the Fourier transform in the case of the trapezoidal‐shaped membership function is

which is not always non‐negative. In conclusion, the kernel can also be regarded as a product‐type multidimensional triangle or a generalized bell‐shaped membership function, but not the trapezoidal‐shaped one. The notation is also considered as a fuzzy logical operator, namely, the t‐norm‐based algebra product [101, 102]. The obtained Mercer kernels could be understood by means of the conjunction (and) used in the previous sections. Thus, one can assign some meanings to the constructed Mercer kernels to obtain linguistic interpretability.

Experience‐oriented FM via reduced‐set vectors: Given n training data , the goal of experience‐oriented FM is to construct a fuzzy model such as Eq. (4.58)that has a good trade‐off between interpretability and accuracy. We examine the trade‐off using the proposed algorithm with two objectives: to minimize the number of fuzzy rules and maximize the accuracy, that is, the approximation and generalization performance.

Given the good performance of SVR, it is reasonable to share the successful experience of SVR in FM. So, SVR with Mercer kernels is employed to generate the initial fuzzy model and the available experience on the training data. It is also expected that a reduction in the number of rules could make the resulting rule base more interpretable and transparent. Thus, a simplification algorithm is introduced to generate reduced‐set vectors for simplifying the structure of the initial fuzzy model, and at the same time the parameters of the derived simplified model are adjusted by a hybrid learning algorithm including the linear ridge regression algorithm and the gradient descent method based on a new performance measure. As a start, let us reformulate Eq. (4.60)through a simple equivalent algebra transform to obtain

(4.62)

where c is the number of support vectors, the parameters of the function and Θ ′ = denote the kernel parameters. Obviously, if k (x, x i) is created by a Gaussian, triangle, or generalized bell‐shaped membership function, then Eq. (4.62)is consistent with the TS fuzzy inference structure, and exhibits good performance under the optimal model selection procedures. However, c , that is, the number of support vectors, usually becomes quite large so that the fuzzy model suffers from being uninterpretable. To compensate for this drawback, c should be replaced by a smaller c f. Thus, a simplified fuzzy model is used:

(4.63)

where c ′ is the number of rules, β i= ( θ i, θ 0i, z j, Θ ) are the consequent parameters of the rule ( x j, z ij, Θ j) θ i+ θ 0i, and Θ = ( Θ 1, … , Θ d) denote the dispersion parameters of the membership functions. The consequent parameter b does not remain unchanged anymore, and it is replaced by θ 0iin order to increase the adjustment ability of each consequent.

Rather than directly extracting support vectors to generate fuzzy rules, the FM problem is solved by learning the parameters in Eq. (4.63)while considering the experience of Eq. (4.62). It is expected that Eq. (4.63)would be able to describe the input–output behavior in the same way as Eq. (4.62). However, the experience acquired heavily depends on the selection of the hyperparameters [103, 104]. Improper selection of these hyperparameters may result in bad performance and bring on useless experience and information. Here, some selection methods are suggested: