A comprehensive decision method of reliability probability distribution model based on the fuzzy support vector machine

Abstract

A fuzzy support vector machine kernel regression decision method of reliability probability distributions is presented aiming at the complexity of reliability probability distributions and disadvantage of the other regression model. The comprehensive decision model of probability distributions is built by the network design and feature extraction of the fuzzy support vector machine algorithm. A example is give for inward stress probability distribution type of a stem structural member by the model, the recognition result is Weibull distribution, the total recognition rate achieves 98.75%. The fuzzy support vector optimized algorithm has strong ability of nonlinear mapping and functional approach, it avoids availably partial minimum and overfitting, and gains high precision by comparing the numerical value of the network output with the numerical value of experiment.

Keywords

Probability distribution fuzzy support vector machine comprehensive decision reliability analysis

1. Introduction

Reliability analysis is important research subject in the field of mechanical engineering. In reliability analysis, it is difficult to decide probability distributions of design parameters, various hypothesis and mathematic model are presented to decide probability distributions of design parameters [1, 2]. According to stress-strength interference theory, the probability distribution law of all parameters must be definited to calculate structural member reliability and execute reliability design. Traditional method is to obtain a quantity of datum by loadable testing for parts or structural member, and then probability distribution type and probability distribution parameters are decided by processing the datum using probability theory, the method will take a long time and lack of accuracy. In this paper, a comprehensive decision method of the reliability probability distribution model based on the fuzzy support vector machine of probability distribution laws is presented aiming at traditional method disadvantage. The comprehensive recognition model of the probability distribution is built by fuzzy-SVM algorithm realization, network design and feature extraction, inward stress probability distribution type of a stem structural member is recognized by the model, the recognition result is Weibull distribution, the fuzzy support vector optimized algorithm has a good generalization ability and clustering ability by comparison between the network recognition result and recursive analysis, it avoids availably partial minimum and overfitting, and gains high precision by comparing the numerical value of network output with the numerical value of experiment. It provides an important new and feasible approach for reliability analysis.

2. Fuzzy support vector mixing algorithm

Fuzzy support vector machine (F-SVM) is built by combining fuzzy algorithm and SVM based on statistical learning theory, the kernel function provides important module for support vector machines [3, 4, 5, 6].

Given samples $S=\{(x_{i},y_{i}),i=1,2,\cdots,n\}$ , if its output is fuzzy, it isn’t able to be solved by the conventional kernel function method because its corresponding optimum classification super surface isn’t able to be solved. Our strategy is that regression problem is translated into limited fuzzy classification problem, fuzzy samples sets are obtained by Eq. (2).

$\displaystyle S_{F}=\{(x_{i},y_{i}\pm\delta_{yi},s_{i});z_{j}\},$

(1) $\displaystyle s.t.\quad{i=1,2,\cdots,n,j=1,2,\cdots,m,m\leqslant n.}$

Where $(x_{i},y_{i}\pm\delta_{yi})$ is fuzzy input samples, $\delta_{yi}$ is positive real number, $\pm\delta_{yi}$ is left and right spread width of $y_{i}$ , $s_{i}$ is fuzzy membership function of the samples, $z_{i}=\pm 1$ is $j$ kinds output.

The original space $L$ is mapped into the character space $H$ by mapping $\Phi$ , let the centre of $k$ kind of samples is $\overline{x_{k}}$ , then generalized degree membership distance of sample $x_{kt}$ in the kinds of samples is defined by Eq. (2) [7].

$\displaystyle d_{k}(t)=\sqrt{K(\bar{x},\bar{x})+K(\bar{x},x_{kt})+K(x_{kt},x_{% kt})}$ (2)

The fuzzy membership degree of sample $x_{kt}$ is defined Eq. (3).

$\displaystyle s_{k}(t)=d_{k}(t)/\max d_{k}$ (3)

The original space $L$ is mapped into the character space $H$ by mapping $\Phi$ , let $y_{F}=y\pm\delta_{y}$ , it will lead to weight value $w$ and threshold value $b$ be fuzzy, namely:

$\displaystyle w_{F}=w\pm\delta_{w},b_{F}=b\pm\delta_{b}$

Where $\delta_{w}$ and $\delta_{b}$ are left and right spread width of $w$ and $b$ . The equation of generalized fuzzy optimum classification hyper plane is discribed by Eq. (4).

$\displaystyle\langle w_{F}\Phi(x)\rangle+sy_{F}+b_{F}=0$ (4)

Restriction expression is obtained by introducing to relax variable $\xi$ on the condition of non-linear and inseparable.

$\displaystyle[z_{j}(\langle w_{F}\Phi(x)\rangle+sy_{F}+b_{F})-1]+\xi_{j}\geqslant 0$ (5) $\displaystyle s.t.\quad{j=1,2,\cdots,m}$

The original quadratic programming problem is translated into fuzzy decision-making restriction programming as follows:

$\displaystyle\min\left\{\Phi(w,s,C,\xi)=\left[\frac{1}{2}\left(\|w\|^{2}+s^{2}% +C\sum\limits_{j=1}^{m}{\xi_{j}}\right)\right]\right\}$ $\displaystyle s.t.\quad P\left[\frac{1}{2}\left(\|w\|^{2}+s^{2}+C\sum\limits_{% j=1}^{m}{\xi_{j}}\right)\leqslant\eta\right]\geqslant\tau_{1}$ $\displaystyle P\{1-z[\langle w_{f}\Phi(x)\rangle+sy_{F}+b_{F}]\geqslant\gamma% \}\leqslant\tau_{2}$

Where $C=f[d(j)]$ is a penalty factor, it is the function of generalized fuzzy membership distance $d$ , and error classification samples are controlled by it in inseparable cases, so compromise disposal between proportional and algorithm complexity of error classification samples is realized, $f$ can set to linear function $C=cd(j)$ , $\xi\geqslant 0$ is relax variable; $\eta\geqslant 0$ and $\gamma\geqslant 0$ are setting threshold value; $\tau_{1}\geqslant 0$ , $\tau_{2}\geqslant 0$ is confidence level; $P\{\cdot\}$ is corresponding probability measure that represent affair $\{\cdot\}$ .

Lagrange multipliers $\alpha$ , $\beta$ are introduced to expression, Eq. (6) is obtained.

$\displaystyle\min[L(w,b,\xi)]=\min\left\{\frac{1}{2}\|{w_{F}}\|^{2}+C\sum% \limits_{i=1}^{n}{s_{i}\xi_{i}}-\sum\limits_{i=1}^{n}{\alpha_{i}\xi}-\sum% \limits_{i=1}^{n}\beta_{i}\{\varepsilon+\xi_{i}\}+1\right.\!\!\!\!\!\!\!\!\!\!% \left.\phantom{\sum\limits_{i=1}^{n}}-z_{i}[\langle w_{F}\varphi(x_{i})\rangle% +s(i)y_{F}(i)+b_{F}(i)]\right\}$ (6)

According to the saddle point theorem, the partial derivative is calculated to $w,b,\xi$ respectively, let its value equal to zero, and the result is substituted into Eq. (6), Eq. (2) is obtained:

$\displaystyle\min\left\{\frac{1}{2}\sum\limits_{i,j=1}^{n}\left(\sum\limits_{h% =1}^{k}z_{ih}\alpha_{ih}-\alpha_{ih}\right)\left(\sum\limits_{h=1}^{k}z_{jh}% \alpha_{jh}-\alpha_{jh}\right)\langle\varphi(x_{i})\varphi(x_{j})\rangle-\sum% \limits_{i,j=1}^{n}\right.$ (7) $\displaystyle\left.\left[z_{j}\left(\sum\limits_{h=1}^{k}(\alpha_{ih}-\alpha_{% jh})\langle\varphi(x_{i})\varphi(x_{j})\rangle\right)+b_{F}-2\right]-\sum% \limits_{i=1}^{n}\sum\limits_{h=1}^{k}[(\alpha_{ih}-Cs_{ih}+\beta_{ih})\xi_{ih% }]\right\}$

Restriction condition is:

$\displaystyle\sum\limits_{\textit{svm}}\alpha_{\textit{svm}}=\sum\limits_{i=1}% ^{n}\sum\limits_{h=1}^{k}z_{ih}\alpha_{ih}$ (8)

Decision function is obtained:

$\displaystyle f(x)=\arg\max\left[\sum\limits_{\textit{svm}}\left(\sum\limits_{% h=1}^{k}z_{h}\alpha_{h}-\alpha_{\textit{svm}}\right)\langle\varphi(x)\varphi(x% _{i})\rangle+b\right]=\arg\max\left[\sum\limits_{\textit{svm}}\left(\sum% \limits_{h=1}^{k}z_{h}\alpha_{h}-\alpha_{\textit{svm}}\right)K(x,x_{i})+b\right]$ (9)

Where $K$ is kernel function. Common kernel functions have linear kernel $K(x,x_{i})=\langle xx_{i}\rangle$ , multi-lays perceptron kernel $K(x,x_{i})=\tanh(v\langle xx_{i}\rangle+c)$ , polynomial kernel $K(x,x_{i})=(\langle xx_{i}\rangle+1)^{2}$ and radial basis kernel $K(x,x_{i})=\exp(-((x-x_{i})/\sigma)^{2})$ and so on.
3. FUZZY-SVM comprehensive decision model of parameter probability distribution type

3.1 Feature extraction

Probability distribution decision is a classification of probability distribution types, distribution types include mainly normal distribution D1, exponential distribution D2, Weibull distribution D3, logarithm normal distribution D4. Feature parameter must be extracted firstly before the classification, namely, feature information is obtained from output datum of simulation system [8]. Output datum are described by the experiential distribution, let $e(\sigma)$ is experiential distribution, where $\sigma$ is parameter, $\sigma_{k}(k=1,2,\cdots,N)$ is the k-th parameter, then

$\displaystyle e(\sigma)=\left\{{\begin{array}[]{l}0(\sigma<\sigma_{1})\\ k/N(\sigma\leqslant\sigma_{k})\\ 1(\sigma>\sigma_{N})\\ \end{array}}\right.$ (10)

Let $F(\sigma,\beta)$ is the corresponding theoretical distribution that is the nearest to the experiential distribution in distribution classification, $\beta$ is distribution parameter. Then $\beta$ most optimized solution $\beta^{\ast}$ that the nearest parameter is obtained from the minimum value of Eq. (11)

$\displaystyle\frac{1}{N}\sum\limits_{k=1}^{N}[e(\sigma_{k})-F(\sigma_{k},\beta% )]^{2}.$ (11)

Let

$\displaystyle U_{i}=\frac{C}{N}\sum\limits_{k=1}^{N}{[e(\sigma_{k})-F_{i}(% \sigma_{k},\beta^{\ast})]^{2}},$

$i=1,2,3,4$ . Where $F_{i}(\cdot)$ is the i-th standard classification distribution, $N$ is the sample total number, $C$ is proportional constants.

Probability distribution feature is obtained by Eq. (12) [9]

$\displaystyle H_{i}=\exp(U_{i})\quad i=1,2,3,4$ (12)

Table 1

Feature value of probability distribution simulation datum

Feature	Normal	Exponential	Weibull	Logarithm normal
	Distribution type
H1	0.9542	0.1966	0.4288	0.3920
H2	0.6528	0.9621	0.4330	0.4290
H3	0.4127	0.5029	0.9869	0.2431
H4	0.2967	0.1128	0.1324	0.9883

Table 2

Classification accuracy rate of probability distribution (%)

Distribution type
Normal	Exponential	Weibull	Logarithm normal	Total accuracy rate
98	99	99	99	98.75

3.2 Probability distribution classification design

Classification topology structure is shown in Fig. 1, four feature parameter D1–D4 is used as F-SVM network input layer, probability distribution type D1–D4 is used as output, input and input neural unit number is four [10]. Extracted feature value and goal value $D=$ [ $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ ] ${}^{T}=$ [1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1] is given from four typical simulation output datum in Table 1, and it is used as training sets to learn F-SVM network. Recognition rate that is obtained by testing is shown Table 2.

Table 3
Stress and distribution function value (stress value: 10 ${}^{7}$ Mpa)

Stress	Frequency	Distribution	Stress	Frequency	Distribution	Stress	Frequency	Distribution
		function			function			function
5.52	1	0.025	5.94	5	0.325	6.14	2	0.850
5.50	1	0.050	5.95	7	0.500	6.17	2	0.900
5.64	2	0.100	5.97	4	0.600	6.19	2	0.950
5.87	2	0.150	6.02	5	0.725	6.21	1	0.975
5.93	2	0.200	6.12	3	0.800	6.24	1	1.000

Table 4

Probability distribution classification result of the pole stress

Feature value	0.324115	0.00000	0.87580	0.34640
Recognition result	0.0009	0.0001	0.9900	0.0020

Figure 1.

F-SVM network topology structure of probability distribution classification.

4. An example of probability distribution classification

4.1 F-SVM classification

Probability distribution of inward stress of a pole arm is classified by the comprehensive decision model, the pole inward stress is random variable, 50 groups of parameters are generated according to distribution law of each factor, each group of parameter corresponding maximal stress in the dangerous section is calculated by finite element, and 40 groups of parameters are used as samples, these samples is arranged from small to large, and corresponding stress distribution function value is calculated according to experience distribution [9, 10], the result is shown in Table 3.

The pole stress distribution of in the dangerous section is recognized by the above trained F-SVM decision model, classification result of stress $S$ is shown Table 4. The result shows that the pole stress distribution of in the dangerous section is Weibull distribution.

4.2 Result verification

The comprehensive decision method of reliability probability distribution model based on the fuzzy support vector machine is compared with regression analysis of probability distribution, the regression correlation coefficient of normal distribution, exponential distribution, Weibull distribution, logarithm normal distribution are 0.8669, 0.7534, 0.9730, 0.8510 respectively.

Calculating datum and linear regression are shown in Fig. 2, it is clear that stress distribution confirms best to Weibull distribution and very little likelihood to the exponential distribution.

Figure 2.

Regression fitting of Weibull distribution.

So the regression analysis result is consistent with F-SVM decision model classification, the comprehensive decision method of probability distribution is accurate.

5. Conclusion

Probability distribution type of reliability parameter in the reliability analysis is classified to use F-SVM decision model in this paper, F-SVM has good learning ability and rapid convergence speed, the classification result is accurate by simulation experiment, total recognition rate reach 98.75%. It provides a feasible and useful method for the reliability analysis.

Footnotes

Acknowledgments

This work is supported by Natural Science Foundation of Zhejiang Province in China (LY14F020013), and supported by technology plan public project of Zhejiang Province in China (No. LGG18F040002). The authors thank the members of the reliability research team for their support of this work.

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