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

Artificial Intelligence and Quantum Computing for Advanced Wireless Networks

left-parenthesis x Subscript j Baseline comma z Subscript italic i j Baseline comma normal upper Theta Subscript j Baseline right-parenthesis right-bracket"/>

where c′ is the number of rules, β_i = (θ_i , θ_0i , z_j , Θ) are the consequent parameters of the rule g Subscript i Baseline left-parenthesis normal x comma beta Subscript i Baseline right-parenthesis equals sigma-summation Underscript i equals 1 Overscript c prime Endscripts product Underscript j equals 1 Overscript d Endscripts mu Subscript i (x_j, z_ij, Θ_j)θ_i + θ_0i , and Θ = (Θ₁, … , 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_0i in 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:

1 The regularization parameter C can be given by following prescription according to the range of output values of the training data [103] : , where and σy are the mean and the standard deviation of the training data y, and n is the number of training data.

2 v ‐SVR is employed instead of ε ‐SVR, since it is easy to determine of the number of support vectors by the adjustment of v. The adjustment of parameter v can be determined by an asymptotically optimal procedure, and the theoretically optimal value for Gaussian noise is 0.54 [104]. For the kernel parameter Θ′, the k ‐fold cross‐validation method is utilized [104].

Reduced‐set vectors: In order to share the experience, we are interested in constructing Eq. (4.63) such that the original Eq. (4.62) is approximated. In the following, k(x, x_i) is written as k′(x, x_i), considering its kernel parameters Θ′ in Eq. (4.62). Similarly, in Eq. (4.63), product Underscript j equals 1 Overscript d Endscripts mu Subscript i (x_j, z_ij, Θ_j) is replaced by k(x, z_i) according to the kernels constructed in the previous paragraph. Then, let G(x) equals sigma-summation Underscript i equals 1 Overscript c Endscripts theta prime Subscript i Baseline k prime left-parenthesis normal x comma normal x Subscript i Baseline right-parenthesis minus sigma-summation Underscript i equals 1 Overscript c prime Endscripts theta Subscript i Baseline dot k left-parenthesis normal x comma normal z Subscript j Baseline right-parenthesis plus b . With this we have

(4.64) equation

If we let the consequent parameter θ_0i be G(x_0i), we have

StartAbsoluteValue f Subscript upper S upper R upper V Baseline left-parenthesis normal x right-parenthesis minus f Subscript italic upper F upper M Baseline left-parenthesis normal x right-parenthesis EndAbsoluteValue less-than-or-equal-to max Underscript i equals 1 comma period period comma c prime Endscripts bar upper G left-parenthesis normal x right-parenthesis minus upper G left-parenthesis normal x Subscript 0 i Baseline right-parenthesis bar

less-than-or-equal-to max Underscript i equals 1 comma period period comma c Superscript prime Baseline Endscripts StartBinomialOrMatrix StartAbsoluteValue left-parenthesis nabla Subscript upper Phi left-parenthesis normal x right-parenthesis Baseline upper G left-parenthesis normal x right-parenthesis dot left-parenthesis upper Phi left-parenthesis normal x right-parenthesis minus upper Phi left-parenthesis normal x Subscript 0 i Baseline right-parenthesis right-parenthesis right-parenthesis EndAbsoluteValue Choose plus StartAbsoluteValue sigma-summation Underscript j equals 1 Overscript c Superscript prime Baseline Endscripts theta Subscript j Baseline left-parenthesis k prime left-parenthesis normal x comma normal z Subscript j Baseline right-parenthesis minus k prime left-parenthesis normal x Subscript 0 i Baseline comma normal z Subscript j Baseline right-parenthesis minus k left-parenthesis normal x comma normal z Subscript j Baseline right-parenthesis plus k left-parenthesis normal x Subscript 0 i Baseline comma normal z Subscript j Baseline right-parenthesis right-parenthesis EndAbsoluteValue EndBinomialOrMatrix period

For a smaller upper bound, we assume that Θ′ = Θ. Then, according to the Cauchy–Schwartz inequality, the right side of the above inequality is simplified to

max Underscript i equals 1 comma period period comma c Superscript prime Baseline Endscripts double-vertical-bar nabla Subscript normal upper Phi left-parenthesis normal x right-parenthesis Baseline upper G left-parenthesis normal x right-parenthesis double-vertical-bar double-vertical-bar normal upper Phi left-parenthesis normal x right-parenthesis minus normal upper Phi left-parenthesis normal x Subscript 0 i Baseline right-parenthesis double-vertical-bar equals rho max Underscript i equals 1 comma period period comma c Superscript prime Baseline Endscripts StartRoot k left-parenthesis x comma x right-parenthesis plus k left-parenthesis x Subscript 0 i Baseline comma x Subscript 0 i Baseline right-parenthesis minus 2 k left-parenthesis x comma x Subscript 0 i Baseline right-parenthesis EndRoot

where , giving

bar f Subscript upper S upper R upper V Baseline left-parenthesis normal x right-parenthesis minus f Subscript italic upper F upper M Baseline left-parenthesis normal x right-parenthesis bar less-than-or-equal-to rho max Underscript i equals 1 comma period period comma c Superscript prime Baseline Endscripts StartRoot 2 minus 2 k left-parenthesis normal x comma normal x Subscript 0 i Baseline right-parenthesis EndRoot period

If we use the notation ‖·‖_∞ as double-vertical-bar d left-parenthesis normal x right-parenthesis double-vertical-bar equals sup Underscript normal x element-of upper R Superscript d Baseline Endscripts bar d left-parenthesis normal x right-parenthesis bar comma we can write

(4.65) double-vertical-bar f Subscript upper S upper V upper R Baseline minus f Subscript upper F upper M Baseline double-vertical-bar Subscript infinity Baseline less-than StartRoot 2 EndRoot rho period

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