文章当中所用公式

Decoding

\(Y_{k} = [Y_{k}^{-}, Y_{k}^{+}] = \frac{1}{\sum_{j=1}^{h}\lambda_{jk}}(\lambda_{1k}\otimes [0,a_{1}^{opt}]\oplus \cdots\oplus\lambda_{2k}\otimes [a_{1}^{opt},a_{2}^{opt}]\oplus\cdots\oplus\lambda_{hk}\otimes [a_{h-1}^{opt},1])=\mathop{\bigoplus}\limits_{i=1}\limits^{h}\frac{\lambda_{ik}}{\sum_{j=1}^{h}\lambda_{jk}}\otimes l_{i}^{opt}\)

\(y=f(x_{1},x_{2})=0.6+2x_{1}+4x_{2}+0.5x_{1}x_{2}+25\sin(0.5x_{1}x_{2})\)

spec

\(spec_{M} = \exp^{-\frac{1}{N} \sum_{k=1}^{N}|Y_{k}^{+} - Y_{k}^{-}|}\)
\(sp = \frac{1}{N}\sum_{k=1}^{N}\exp(-|y_{k}^{+}-y_{k}^{-}|)\)

实验用

\(y = x_{1}x_{2}\exp^{-x_{1}^{2}-x_{2}^{2}}, x_{1},x_{2}\in[-2,2]\)
\(y=0.6\sin(\pi x)+0.3\sin(3\pi x) + 0.1\sin(5\pi x)， x\in [-1, 1]\)
\(y=\sqrt{x}+\cos(x)^2 -2, x\in [0 ,1]\)
\(y=0.6+2x_{1}+4x_{2}+0.5x_{1}x_{2}+25\sin(0.5x_{1}x_{2}), x_{1}\in [-4,6], x_{2}\in [-2,4]\)

RMSE performance measure: root mean squared error

\(J_{\text{RMSE}} = \sqrt{\frac{\sum_{k=1}^{N}(\hat{y}_{k}-y_{k})^{2}} {N}}\)
\(J_{\text{RMSE}} = \sqrt{\frac{1}{N}\sum_{i=1}^{N}|\frac{Y_{k}^{+}-Y_{k}^{-}}{2}-y_{k}|}\)

参照函数的对称函数

定义 \(I=(\alpha,\delta)_{L}\)表示对称模糊数I，其隶属函数可表示为
\(\mu_{I}(x)=L\left(\frac{x-\alpha}{\delta} \right ),\delta>0\)
称\(L(z)\)为参照函数，其中\(z=\frac{x-\alpha}{\delta}\),\(\alpha\)为对称中心，\(\delta\)称为模糊宽度或者模糊幅度。

三角模糊数和抛物线模糊数
当p=1,时，为三角模糊数，当p=2，时，为抛物线模糊数。
\(L(x)=\left\{\begin{matrix} 1-\frac{|x-\alpha|^{p}}{\delta^{p}} & -\delta\leq x-\alpha\leq \delta \\ 0 & -\infty <x-\alpha<-\delta, \delta<x-\alpha<\infty \end{matrix}\right.\)

下面是余弦模糊数
\(L(x)=\left\{\begin{matrix} \cos\left( \frac{\pi(x-\alpha)}{2\delta} \right)& -\delta\leq x-\alpha\leq \delta \\ 0 & -\infty <x-\alpha<-\delta, \delta<x-\alpha<\infty \end{matrix}\right.\)

三维点状热力图scatter heatmap
https://blog.csdn.net/justKidrauhl/article/details/82492986