Savo G. Glisic

Artificial Intelligence and Quantum Computing for Advanced Wireless Networks


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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, βi = (θi , θ0i , zj , 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 (xj, zij, Θj)θi + θ0i , and Θ = (Θ1, … , Θd) denote the dispersion parameters of the membership functions. The consequent parameter b does not remain unchanged anymore, and it is replaced by θ0i in order to increase the adjustment ability of each consequent.

      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, xi) is written as k′(x, xi), 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 (xj, zij, Θj) is replaced by k(x, zi) 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

      If we let the consequent parameter θ0i be G(x0i), 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 rho greater-than 0 comma rho squared equals double-vertical-bar sigma-summation Underscript i equals 1 Overscript c Endscripts theta prime Subscript i Baseline upper Phi left-parenthesis normal x Subscript i Baseline right-parenthesis minus sigma-summation Underscript i equals 1 Overscript c prime Endscripts theta Subscript i Baseline upper Phi left-parenthesis normal z Subscript i Baseline right-parenthesis double-vertical-bar squared, 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