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

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

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

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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, xi) = k(x − xi) is an admissible Mercer kernel if and only if the Fourier transform

upper F left-bracket k right-bracket left-parenthesis normal w right-parenthesis equals left-parenthesis 2 pi right-parenthesis Superscript negative italic p x slash 2 Baseline integral Underscript upper R Superscript italic p x Baseline Endscripts exp left-parenthesis minus j left-parenthesis normal w dot normal x right-parenthesis right-parenthesis k left-parenthesis normal x right-parenthesis d x

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

upper F left-bracket k right-bracket left-parenthesis normal w right-parenthesis equals left-parenthesis 2 pi right-parenthesis Superscript negative d slash 2 Baseline product Underscript j equals 1 Overscript d Endscripts StartFraction 2 left-parenthesis 1 minus cosine w Subscript j Baseline gamma Subscript j Baseline right-parenthesis Over w Subscript j Superscript 2 Baseline gamma Subscript j Baseline EndFraction

      and

upper F left-bracket k right-bracket left-parenthesis normal w right-parenthesis equals left-parenthesis StartFraction pi Over 2 EndFraction right-parenthesis Superscript d slash 2 Baseline product Underscript j equals 1 Overscript d Endscripts exp left-parenthesis minus a Subscript j Baseline bar w Subscript j Baseline bar right-parenthesis a Subscript j Baseline period

      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

upper F left-bracket k right-bracket left-parenthesis normal w right-parenthesis equals left-parenthesis 2 pi right-parenthesis Superscript negative d slash 2 Baseline product Underscript j equals 1 Overscript d Endscripts StartFraction 2 left-parenthesis cosine w Subscript j Baseline a Subscript j Baseline minus cosine w Subscript j Baseline left-parenthesis a Subscript j Baseline plus c Subscript j Baseline right-parenthesis right-parenthesis Over w Subscript j Superscript 2 Baseline c Subscript j Baseline EndFraction

      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 product Underscript a Overscript b Endscripts x 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 StartSet left-parenthesis normal x 1 comma y 1 right-parenthesis comma ellipsis comma left-parenthesis normal x Subscript n Baseline comma y Subscript n Baseline right-parenthesis EndSet subset-of German upper R Superscript d German times German upper R, 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.

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