CN111680450A - A Reliability Analysis Method for Structural Systems Based on Uncertain Bayesian Networks - Google Patents

Abstract

The invention provides a structural system reliability analysis method based on a fuzzy random uncertainty Bayesian network. The steps are as follows: 1) establishing a Bayesian network of system objects; 2) assuming a positive system reliability based on subjective and objective information 3) According to the boundary envelope of the prior distribution, perform sampling and use maximum likelihood estimation to fit the equivalent Beta distribution expression of the boundary envelope; 4) Establish a system based on the existing data Likelihood function expression; 5) Multiply the likelihood function and the result of 4) to obtain the boundary expression of the posterior distribution of the system reliability; 6) Perform Metropolis‑Hastings after discarding the normalized constant part of the result in 5). Sampling, and statistics of the system edge sample points are performed on the results to obtain the boundary probability density function PDF of the posterior distribution of the system reliability; 7) Obtain the boundary cumulative probability density function CDF of the posterior distribution of the system reliability; the reliability of the present invention The analytical method is scientific, with good manufacturability, and has broad application value.

Description

A Reliability Analysis Method for Structural Systems Based on Uncertain Bayesian Networks

technical field

The invention provides a structural system reliability analysis method based on uncertainty Bayesian network, which relates to a structural system reliability analysis method based on fuzzy random uncertainty Bayesian network The reliability analysis of structural systems under the fuzzy randomness information of uncertainty and cognitive uncertainty belongs to the field of structural reliability analysis.

Background technique

In the first aspect, Bayesian networks (BN), as a member of the graphical model and analysis framework, is an important tool for dealing with uncertainty in systems science, reliability science, artificial intelligence and other fields. It has been vigorously developed for more than 20 years, and its graphical expression function makes the relationship between systems and components and the state expression more intuitive and clear. It can better solve the problem of poor information and small samples in system analysis. The system analysis based on Bayesian network in reliability engineering is mainly completed according to the composition of product structure and the establishment of fault tree.

Various types of Bayesian networks (Dynamic Bayesian Networks DBN, Object-Oriented Bayesian Networks OOBN, Qualitative Bayesian Networks, etc.) currently widely used essentially start from the prior knowledge they have mastered, and use actual measurements The data and information are used as "evidence" to update and revise the existing cognition, and then obtain the posterior cognition of the problem after considering the actual factors. From a systematic point of view, it is based on the Bayesian formula, using the conditional probability table from the bottom up, constructing the likelihood function, performing iterative calculation, and comprehensively using various effective information to obtain the posterior probability inference results. It has been widely used in the field of system reliability modeling and analysis. The accuracy of system reliability analysis results mainly depends on prior information, conditional probability and experimental data as "evidence". Traditional research on Bayesian methods often focuses on the selection of prior distributions, the acquisition of conditional probabilities, and the influence of experimental data information as "evidence" on the overall reliability analysis, while the uncertainty of the input information. There are relatively few studies, and most of the existing Bayesian methods only deal with probabilistic information, fuzzy information or based on evidence theory.

On the other hand, at present, the types of input information uncertainty are mainly divided into two categories, inherent uncertainty information (represented by probability information) and cognitive uncertainty information (represented by fuzzy information, interval information, etc.), Considering the mixture of uncertain information in complex structural systems, that is, the existence of inherent uncertainty (probabilistic information) and cognitive uncertainty (poor information or fuzzy information) at the same time, it leads to the traditional shellfish using probability or fuzzy information as input. The error and complexity of the Bayesian network analysis method increase, and the Bayesian network inference method that relies on a single inherent uncertainty and cognitive uncertainty has limitations, and cannot guarantee the accuracy and confidence. Fuzzy random variable, as an uncertainty variable with both inherent and cognitive uncertainty characteristics, is a method for describing uncertain information in the form of probability distribution and the distribution parameters are fuzzy numbers. It combines the characteristics of random distribution and fuzzy mathematics. It can represent information that contains two kinds of uncertainties at the same time, which improves the confidence of cognition to a certain extent. Fuzzy random variables have been widely used in structural reliability analysis, structural reliability assessment, life model and other fields. Including fuzzy first-order reliability method FFORM (Fuzzy FirstOrder Reliability Method) based on fuzzy random variables, fuzzy Monte Carlo simulation based on fuzzy random variable theory, life prediction model based on fuzzy random variables and interval finite element method based on fuzzy random variables Wait. However, the related algorithms for applying fuzzy random variables to structural system reliability analysis still need to be expanded.

Based on the above two aspects, the present invention proposes a structural system reliability analysis method based on a fuzzy random uncertainty Bayesian network.

SUMMARY OF THE INVENTION

(1) Purpose of the present invention

The purpose of the present invention is to use fuzzy random distribution as a priori distribution of system reliability and combine Bayesian network to analyze the reliability of the structural system for a small sample structure system with insufficient sample size, aiming to explore a new inherent and A Framework for System Reliability Analysis under Cognitive Mixed Uncertainty.

The invention firstly samples the upper bound (lower bound) of the envelope of the fuzzy random prior distribution of the Bayesian network subsystem, and uses the maximum likelihood estimation to fit the equivalent Beta distribution expression, and then uses the maximum likelihood estimation to fit the equivalent Beta distribution expression. Likelihood function, using Hastings sampling, that is, Metropolis-Hastings sampling method, is used for sampling, and the high-dimensional sampling results continue to be subjected to system variable edge sampling to obtain the upper bound (lower bound) of the equivalent posterior distribution of the system.

(2) Technical solutions

The present invention adopts following technical scheme for realizing the above-mentioned purpose:

The present invention is a structural system reliability analysis method based on uncertainty Bayesian network, namely a structural system reliability analysis method based on fuzzy random uncertainty Bayesian network, which involves a fuzzy random parameter-based reliability analysis method. The Bayesian network structure reliability output upper (lower) bound for solving the model, the implementation steps are as follows:

Step (1), establish the Bayesian network of the system object;

Step (2), assume normal fuzzy random prior distribution of system reliability according to subjective and objective information;

Step (3): According to the upper (lower) bound envelope of the prior distribution, sampling is performed and the maximum likelihood estimation is used to fit the equivalent Beta distribution parameters of the envelope, and the equivalent Beta of the upper (lower) bound envelope is obtained. distribution expression;

Step (4), establish system likelihood function expression according to existing data;

Step (5), based on the Bayesian network model, multiply the likelihood function and the equivalent Beta distribution expression of the envelope to obtain the upper bound (lower bound) expression of the posterior distribution of the system reliability;

Step (6): According to the upper bound (lower bound) expression of the posterior distribution of the system, Metropolis-Hastings sampling is performed on the part of the normalized constant that has been discarded, and the system edge sample point statistics are performed on the result to obtain an approximate system reliability. The upper bound (lower bound) probability density function PDF (Probability Density Function) of the test distribution;

Step (7), obtain the upper bound (lower bound) cumulative probability density function CDF (Cumulative Distribution Function) of the posterior distribution of the system reliability;

Wherein, in step (1) described in "establishing the Bayesian network of system objects", its specific method is:

The Bayesian network is composed of a directed acyclic graph formed by the directed edge connection of the event relationship; the directed acyclic graph is defined as G=<V, E>; the set of nodes is defined as V={X ₁ , X ₂ ,...,X _n }, the directed edge E between the nodes represents the connection relationship between events, and also corresponds to the causal relationship; in the X _i →X _j connected by the directed edge, X _i is the parent node, And X _j is a child node;

The set of parent nodes of X _i can be represented by parent(X _i ) or pa(X _i );

The directed graph G expresses the assumption of conditional independence between events. If the prior probability value of the parent node and the conditional probability distribution between it and the child nodes can be determined, the joint probability distribution of all nodes can be obtained by the following formula ;

Given hypothesis S and evidence set E = {E ₁ ,E ₂ ,...,E _n }, Bayes' theorem can be expressed as:

in,

P(S): represents the probability that H is true, also known as the prior probability;

P(S|E _i ): Indicates that the given evidence is E is the conditional probability that H is true, also known as the posterior probability;

P(E _i |S): represents the conditional probability of the occurrence of evidence E _i when the given hypothesis H is true, also known as the likelihood probability;

(1) Prior probability: the probability of occurrence of various events determined according to expert data and objective facts;

(2) Posterior probability: the probability of occurrence of events A and B, and the probability of occurrence of A when event B occurs, combined with Bayesian formula, obtain the probability of occurrence of B under the assumption that A is determined to occur;

(3) Total probability formula: If (A ₁ , A ₂ ,...,A _n ) is a group of events of E, and satisfy: P(A _i )≥0, then the total probability formula is:

A typical simple Bayesian network is shown in Figure 1(a), where X ₂ and X ₃ are parent nodes, X ₁ is a child node, and directed edges X ₂ →X ₁ , X ₃ →X ₁ form a set E, For this Bayesian network connection relation;

The Bayesian network can be given directly or obtained from the system fault tree. Figure 1(a) can describe the fault tree with the top event as X ₁ , the bottom event as X ₂ and X ₃ as shown in Figure 1(b) ( The relationship between X ₂ and X ₃ is either OR gate/AND gate/NOT gate).

Among them, in the "normal fuzzy random prior distribution of system reliability assumption based on subjective and objective information" described in step (2), "based on subjective and objective information" means based on limited known data, and then according to Subjective judgment to determine the fuzzy random prior distribution of the system;

的分布，其中f(·)为随机概率密度函数，而分布的参数

为模糊数；以模糊随机正态分布为例：设

和

分别是模糊随机变量的模糊均值和模糊标准差，则模糊随机正态分布可表示为

"Fuzzy random prior distribution" refers to the distribution based on fuzzy random variables, whose distribution is a random variable, but the distribution parameters are uncertain variables of fuzzy numbers;

The distribution of , where f( ) is a random probability density function, and the parameters of the distribution

is a fuzzy number; take the fuzzy random normal distribution as an example:

and

are the fuzzy mean and fuzzy standard deviation of the fuzzy random variables, respectively, then the fuzzy random normal distribution can be expressed as

In step (2), the specific method of "assuming the normal fuzzy random prior distribution of system reliability according to subjective and objective information" is as follows:

上界可取

下界可取

均值三角模糊数的最大可能值可取

上界可取

下界可取

之后交由专家或经验丰富的工程技术人员对以上参数进行适当的修改，即得到假设的系统可靠度先验分布。According to the existing data (the number of trials T and the number of successes N), determine the triangular fuzzy number of the mean and standard deviation of the fuzzy random distribution of the system reliability; the maximum possible value of the standard deviation triangular fuzzy number is acceptable

upper bound desirable

Nether Desirable

The maximum possible value of the mean triangular fuzzy number can be taken

upper bound desirable

Nether Desirable

After that, the above parameters are appropriately modified by experts or experienced engineers and technicians, that is, the assumed prior distribution of system reliability is obtained.

Among them, in step (3), "according to the upper (lower) bound envelope of the prior distribution, sampling and using maximum likelihood estimation to fit the equivalent Beta distribution parameters of the envelope to obtain the upper (lower) envelope. ) bounded envelope equivalent Beta distribution expression”, the specific method is as follows:

"Envelope" refers to the envelope consisting of the upper and lower boundaries of the fuzzy random distribution CDF. The construction and sampling of the envelope are as follows:

和

分别是模糊随机变量的模糊均值和模糊标准差；所有的隶属函数都假定为三角模糊数；因此，模糊平均值和标准差可以分别表示为

和

其中上标L、M和U分别为下界、中值和上界；模糊随机分布CDF上界包络及样本点由两部分组成：在

的左侧以

为分布进行抽样，在

的右侧以

为分布进行抽样，上界采样点集被定义为

相对应的，模糊随机分布CDF下界包络及样本点是在

的左侧以

为分布进行抽样，在

的右侧以

为分布进行抽样，下界采样点集被定义X _α＝{x ₁,x ₂，…,x _n}；特别地，对服从x～N(μ^M,σ^M)的分布称为名义分布；图2给出了形如

和

的模糊随机变量的包络和名义分布示意图；Assume

and

are the fuzzy mean and fuzzy standard deviation of the fuzzy random variables, respectively; all membership functions are assumed to be triangular fuzzy numbers; therefore, the fuzzy mean and standard deviation can be expressed as

and

The superscripts L, M and U are the lower bound, the median and the upper bound, respectively; the upper bound envelope and sample points of the fuzzy random distribution CDF are composed of two parts:

to the left of

Sampling for the distribution, in

to the right of

To sample for the distribution, the upper bound sampling point set is defined as

Correspondingly, the lower bound envelope of the fuzzy random distribution CDF and the sample points are in

to the left of

Sampling for the distribution, in

to the right of

^Sampling for a distribution, the lower ^bound set of sampling points is defined as X _α = _{ x ₁ , x ₂ , . 2 gives the form

and

Schematic diagram of the envelope and nominal distribution of the fuzzy random variable;

"Beta distribution" means

The probability density function is the function shown in the following equation:

对形如具有f(x；α,β)概率密度函数的变量x，称为服从参数为α,β的Beta分布(中文译称：贝塔分布)；where Γ( ) is the gamma distribution,

For a variable x with a probability density function of f(x; α, β), it is called a Beta distribution with parameters α, β (Chinese translation: Beta distribution);

和Beta分布概率密度函数构造似然函数：Maximum likelihood estimation α, β refers to the use of sampling sample point sets

Construct the likelihood function with the Beta distribution probability density function:

And solve the following equations to get α, β, then get the equivalent Beta distribution expression

Wherein, described in step (4), "establish system likelihood function expression according to existing data", its concrete practice is as follows:

According to the existing data P(E _i |H) of each node X _i , i=1,2,...n, construct the likelihood function P(EH) in the Bayesian network:

where E=(E ₁ , E ₂ , . . . , E _n ).

Among them, in step (5), "based on the Bayesian network model, the likelihood function and the equivalent Beta distribution expression of the envelope are multiplied to obtain the upper bound (lower bound) expression of the posterior distribution of the system reliability. The specific method is as follows:

"Based on the Bayesian network model" means: solving the model description according to the Bayesian network in step (1),

和步骤(4)中得到的基于当前数据构建的贝叶斯网络似然函数表达式

相乘，则系统可靠度后验分布的上界(下界)表达式"Multiplying the likelihood function and the equivalent Beta distribution expression of the envelope" means: multiplying the Beta distribution equivalent expression of the prior distribution envelope obtained in step (3)

and the Bayesian network likelihood function expression constructed based on the current data obtained in step (4)

Multiply, then the upper bound (lower bound) expression of the posterior distribution of system reliability

where E=(E ₁ , E ₂ , . . . , E _n ).

Among them, in step (6), "According to the upper bound (lower bound) expression of the posterior distribution of the system, Metropolis-Hastings sampling is performed on the part of the normalized constant that is discarded, and the system edge sample point statistics are performed on the result, Obtain the approximate upper bound (lower bound) probability density function PDF of the posterior distribution of system reliability;", the specific method is as follows:

"Truncate the normalization constant" means to discard the denominator part of P(HE) obtained in step (5), and retain the numerator expression; and then perform Metropolis-Hastings sampling on the numerator expression;

"Metropolis-Hastings sampling" means sampling as follows:

Let the numerator expression after discarding the normalization constant be π(x), and select the sampling times N;

1) Extract the sample point x _i+1 from the proposed distribution density function x～N(x _i ,0.1), where the initial value of x _i is x ₁ =0.5, σ ₁ =0.5;

2) Calculate the transition probability P(x _i ; x _i+1 )=P(x _i |x～N(x _i+1 ,0.1)), P(x _i+1 ;x _i )=P(x _{i+ 1} |x～N(x _i ,0.1))

3) Calculate the acceptance probability α _i

4) Extract u _i to U(0,1) from the uniform distribution, if u _i <α _i , then σ _i+1 = _xi+1 ; otherwise σ _i+1 = _xi ; repeat steps 1) to steps 3) N times, the n-dimensional sample point set can be obtained; for this sample point set, the sample point of the "system" dimension is extracted, which is the upper bound (lower bound) sample point {σ _i },i of the posterior distribution of the system =1,2,...,N;

"Get the upper (lower) probability density function PDF of the approximate system reliability posterior distribution" means:

i＝0,1,...,m，计算{σ_i},i＝1,2,...,N在每一区间中的样本个数{σ_Number,i},i＝1,2,...,m，则直方图为

According to the sample points {σ _i }, i=1,2,...,N, draw a frequency histogram on the interval [0,1] to approximate the upper (lower) probability density function PDF of the posterior distribution: [0,1] is evenly divided into m parts {[dx ₀ ,dx ₁ ],(dx ₁ ,dx ₂ ],...,(dx _m-1 ,dx _m ]}, where dx ₀ =0,dx _m =1,

i=0,1,...,m, calculate {σ _i }, i=1, 2,...,N the number of samples in each interval {σ _{Number, i} }, i=1, 2 ,...,m, then the histogram is

Wherein, "obtaining the upper bound (lower bound) cumulative probability density function CDF of the posterior distribution of the system reliability" described in step (7) refers to the frequency histogram obtained according to step (6), in the interval [0,1 ] to calculate the frequency accumulation, and draw the approximate CDF histogram of the upper bound (lower bound) of the posterior distribution of the system reliability according to the following formula

The flow chart of the reliability analysis method of the present invention is shown in FIG. 3 .

(3) Advantages and effects of the present invention

The structural system reliability analysis method of the present invention, combined with the respective characteristics of the Bayesian network analysis method and fuzzy random variables, proposes a Bayesian network system reliability calculation framework based on fuzzy random parameters, and its advantages and effects are:

(1) It constructs a Bayesian network solution process under inherent and cognitive mixed uncertainties, which can guide the reliability analysis of structural systems under incomplete data;

(2) Based on the characteristics of fuzzy random variables, it obtains the upper and lower bounds of the system reliability distribution according to the Bayesian network, which can provide engineering/research personnel with a posterior reference for system reliability;

(3) It is convenient and quick to calculate, can make full use of the existing data to calculate the reliability of the system, can also be implemented under unknown conditional probability, and can solve the numerical calculation of the structural system caused by the mixed uncertainty information widely existing in the project. large, low-confidence issues;

(4) The reliability analysis method of the present invention is scientific, has good manufacturability, and has wide popularization and application value.

Description of the drawings (the serial numbers, symbols and codes in the drawings are explained as follows)

Figure 1 is a typical simple Bayesian network and fault tree.

Figure 2 is a fuzzy random cumulative distribution function.

FIG. 3 is a flow chart of the reliability analysis of the system according to the present invention.

Figure 4 is the Bayesian network in Case 1.

Figure 5 shows the prior distribution of system reliability in Case 1.

Figure 6 shows the equivalent prior Beta distribution of the reliability of the case 1 system.

Figure 7 is the acceptance rate (upper bound) of the first 1000 samples of Metropolis-Hastings in case 1.

Figure 8 is the acceptance rate (lower bound) for the first 1000 samples of Metropolis-Hastings in case 1.

Figures 9a and 9b are histograms of the posterior distribution frequencies for the Case 1 system.

Figure 10 is the cumulative density of the posterior distribution frequency of the case 1 system.

Figure 11 is a fault tree for "turbine system failure".

Figure 12 is a Bayesian network for "turbine system failure".

Figure 13 is the prior distribution of the reliability of the case 2 system.

Figure 14 shows the equivalent prior Beta distribution of the reliability of the case 2 system.

Figure 15 shows the acceptance rate (upper bound) for the first 1000 samples of Metropolis-Hastings in case 2.

Figure 16 shows the acceptance rate (lower bound) of the first 1000 samples of Metropolis-Hastings in case 2.

Figures 17a and 17b are histograms of the posterior distribution frequencies for the Case 2 system.

Figure 18 is the cumulative density of the posterior distribution frequency of the case 2 system.

The serial numbers, symbols and codes in the figure are explained as follows:

In Figure 1, X ₁ , X ₂ and X ₃ are Bayesian network nodes; · are fault tree AND gates;

In Figure 2, CDF is the cumulative distribution function;

In Figure 5, CDF is the cumulative distribution function;

In Figure 6, PDF is the probability density function, and Beta distribution is beta distribution;

In Figures 9a and 9b, PDF is the probability density function, and S represents the system of case 1;

In Figure 10, CDF is the cumulative distribution function, and S represents the system of case 1;

In Figure 13, CDF is the cumulative distribution function;

In Figure 14, PDF is the probability density function, and Beta distribution is beta distribution;

Figure 17a and Figure 17b, PDF is the probability density function, S represents the system of case 2;

In Figure 18, CDF is the cumulative distribution function, and S represents the system of Case 2.

The symbols and codes involved in this manual are supplemented as follows:

Beta Distribution - Beta Distribution

FFORM(Fuzzy First Order Reliability Method)--Fuzzy First Order Reliability Method Based on Fuzzy Random Variables

PDF(Probability Distribution Function)--Probability Density Function

CDF (Cumulative Distribution Function)--cumulative distribution function

Metropolis-Hastings Sampling - Hastings Sampling

Detailed ways

The technical solution of the invention will be described in detail below with reference to calculation examples and accompanying drawings.

Case 1 (numerical case):

Assuming a Bayesian network as shown in Figure 4, there are 3 nodes in total, node S represents the system, and nodes C ₁ and C ₂ represent subsystems. When both the marginal distribution and the conditional distribution are unknown, according to the known data, the method of the present invention is used to calculate the posterior distribution P(S|C ₁ , C ₂ , S) of the system. It is assumed that the known data is shown in Table 1.

Table 1 Known data

Bayesian network node name Known data (successes/total trials) Subsystem C₁ 11/14 Subsystem C₂ 37/41 System S 8/11

A method for reliability analysis of structural system based on fuzzy random uncertainty Bayesian network of the present invention, as shown in FIG. 3 , the implementation steps are as follows:

Step (1), establish a Bayesian network of the system object; according to the description, establish a Bayesian network, as shown in Figure 4 for the schematic diagram 1 of the Bayesian network.

Step (2), assume normal fuzzy random prior distribution of system reliability according to subjective and objective information;

According to the known data system data is 8/11, we assume that the prior distribution of the system obeys the normal fuzzy random distribution, and the mean and standard deviation of the distribution parameters are assumed to be triangular fuzzy numbers, as shown in Table 2. A schematic diagram of cumulative probability 2 is shown in Figure 5.

Table 2 Systematic fuzzy random prior distribution

Step (3): According to the upper (lower) bound envelope of the prior distribution, sampling is performed and the maximum likelihood estimation is used to fit the equivalent Beta distribution parameters of the envelope, and the equivalent Beta of the upper (lower) bound envelope is obtained. distribution expression;

其中f为Beta函数表达式。而后求解如下方程组即可得到α,β，则得到等效Beta分布表达式The envelope of the fuzzy random variables generated in step (2) is constructed as shown in Figure 5, and 1000 samplings are performed according to the method explained in step (3), and the sample points are recorded as x _i (i=1, 2, .. .,1000), then the likelihood function in the likelihood estimation is

where f is the Beta function expression. Then solve the following equations to get α, β, then get the equivalent Beta distribution expression

即使与之对应的表达式。上(下)界包络等效Beta分布见图6所示。The parameters α=2.2510 and β=1.4075 of the equivalent Beta distribution of the upper bound envelope are solved; the parameters α=1.4969 and β=1.5324 of the equivalent Beta distribution of the lower bound envelope are obtained. bring in

Even the corresponding expression. The equivalent Beta distribution of the upper (lower) bound envelope is shown in Figure 6.

Step (4), establish system likelihood function expression according to existing data;

According to the data in Table 1, assuming that all nodes are binomial distribution, the system likelihood function expression is:

L(p ₁ , p ₂ , p _S )=p ₁ ¹¹ (1-p ₁ ) ³ p ₂ ³⁷ (1-p ₂ ) ⁴ p _S ⁸ (1-p 1 ₎ ³

Step (5), based on the Bayesian network model, multiply the likelihood function and the equivalent Beta distribution expression of the envelope to obtain the upper bound (lower bound) expression of the posterior distribution of the system reliability;

According to the results of steps (3) and (4), the upper bound expression of the posterior distribution of the system reliability can be obtained as:

The lower bound expression of the posterior distribution of system reliability is:

Step (6): According to the upper bound (lower bound) expression of the posterior distribution of the system, Metropolis-Hastings sampling is performed on the part of the normalized constant that has been discarded, and the system edge sample point statistics are performed on the result to obtain an approximate system reliability. upper bound (lower bound) probability density function PDF of the test distribution;

The expression after omitting the normalization constant in step (5) is:

Metropolis-Hastings sampling is performed on it, and the number of sampling times is N=10000 times. For clarity, the acceptance rates for the first 1000 times of extraction are shown in Fig. 7 (upper bound) and Fig. 8 (lower bound).

The [0,1] interval is divided into 50 parts, and the system edge frequency statistics are performed on the Metropolis-Hastings sampling results, and finally the PDF frequency table of the fitted posterior distribution is obtained in Table 3, and the corresponding posterior distribution PDF histogram is shown in Figure 9a and as shown in Figure 9b,

Table 3 System Posterior Distribution PDF Frequency Table

Step (7), obtaining the upper bound (lower bound) cumulative probability density function CDF of the posterior distribution of the system reliability;

According to the result of step (6), the frequency is accumulated in each interval, and the CDF frequency table of the system posterior distribution is obtained as shown in Table 4. The corresponding posterior distribution CDF histogram is shown in Fig. 10.

Table 4 System Posterior Distribution CDF Frequency Table

Case 2 (engineering case):

The fault tree of the aero-engine turbine system is shown in Figure 11. The top event is defined as "turbine system failure", which consists of two fault events connected in series, namely turbine disk failure and turbine blade failure, among which the turbine disk may have faults There are 2 types of turbine blades, and there are 2 types of turbine blades. The occurrence of various failure modes has corresponding failure reasons. Therefore, in order to more accurately find the basic events that reflect the failure of the rotor system, all the failure causes leading to the failure mode are defined as the bottom events of the fault tree. Among them, all child nodes are connected in series. The known data are shown in Table 5.

Table 5 Known data

A method for reliability analysis of structural system based on fuzzy random uncertainty Bayesian network of the present invention, as shown in FIG. 3 , the implementation steps are as follows:

Step (1), establish a Bayesian network of system objects; according to the case description, establish a Bayesian network, as shown in Figure 12 for Bayesian network schematic diagram 1, each node is marked with known data.

Step (2), assume normal fuzzy random prior distribution of system reliability according to subjective and objective information;

According to the known data system data is 194/195, we assume that the prior distribution of the system obeys the normal fuzzy random distribution, and the mean and standard deviation of the distribution parameters are assumed to be triangular fuzzy numbers, which are shown in Table 6. The schematic diagram 2 of the cumulative probability is shown in Figure 13.

Table 6 Systematic fuzzy random prior distribution

Step (3): According to the upper (lower) bound envelope of the prior distribution, sampling is performed and the maximum likelihood estimation is used to fit the equivalent Beta distribution parameters of the envelope, and the equivalent Beta of the upper (lower) bound envelope is obtained. distribution expression;

其中f为Beta分布函数表达式。而后求解如下方程组即可得到α,β，则得到等效Beta分布表达式The envelope of the fuzzy random variables generated in step (2) is constructed as shown in Figure 13, and 1000 samplings are performed according to the method explained in step (3), and the sample points are recorded as x _i (i=1, 2, .. .,1000), then the likelihood function in the likelihood estimation is

where f is the expression of the beta distribution function. Then solve the following equations to get α, β, then get the equivalent Beta distribution expression

即使与之对应的表达式。上(下)界包络等效Beta分布见图14所示。The upper bound envelope equivalent Beta distribution expression parameters α=56.8116, β=1.8496; the lower bound envelope equivalent Beta distribution expression parameters α=25.2457, β=1.2317. bring in

Even the corresponding expression. The equivalent Beta distribution of the upper (lower) bound envelope is shown in Figure 14.

Step (4), establish system likelihood function expression according to existing data;

According to the data in Table 1, assuming that all nodes are binomial distribution, the system likelihood function expression is:

Step (5), based on the Bayesian network model, multiply the likelihood function and the equivalent Beta distribution expression of the envelope to obtain the upper bound (lower bound) expression of the posterior distribution of the system reliability;

According to the results of steps (3) and (4), the upper bound expression of the posterior distribution of the system reliability can be obtained as:

The lower bound expression of the posterior distribution of system reliability is:

Step (6): According to the upper bound (lower bound) expression of the posterior distribution of the system, Metropolis-Hastings sampling is performed on the part of the normalized constant that has been discarded, and the system edge sample point statistics are performed on the result to obtain an approximate system reliability. The upper bound (lower bound) probability density function PDF of the test distribution;

The expression after omitting the normalization constant in step (5) is:

Metropolis-Hastings sampling is performed on it, and the number of sampling times is N=10000 times. For clarity, the acceptance rates for the first 1000 times of extraction are shown in Fig. 15 (upper bound) and Fig. 16 (lower bound).

The [0,1] interval is divided into 25 parts, and the system edge frequency statistics are performed on the Metropolis-Hastings sampling results, and finally the PDF frequency table of the fitted posterior distribution is obtained in Table 7, and the corresponding posterior distribution PDF histogram is shown in Figure 17a and as shown in Figure 17b,

Table 7 System Posterior Distribution PDF Frequency Table

Step (7), obtaining the upper bound (lower bound) cumulative probability density function CDF of the posterior distribution of the system reliability;

According to the result of step (6), the frequency is accumulated in each interval to obtain the system posterior distribution CDF frequency table as shown in Table 8. The corresponding posterior distribution CDF histogram is shown in Fig. 18.

Table 8 System Posterior Distribution CDF Frequency Table

Claims (8)

1. a structural system reliability analysis method based on uncertainty Bayesian network, namely a kind of structural system reliability analysis method based on fuzzy random uncertainty Bayesian network, it is characterized in that: its implementation steps are as follows: Step (1), establish the Bayesian network of the system object; Step (2), assume normal fuzzy random prior distribution of system reliability according to subjective and objective information; Step (3), sampling according to the upper and lower bound envelopes of the prior distribution, and using maximum likelihood estimation to fit the equivalent Beta distribution parameters of the envelopes to obtain the equivalent Beta distribution expressions of the upper and lower bound envelopes; Step (4), establish system likelihood function expression according to existing data; Step (5), based on the Bayesian network model, multiply the likelihood function and the equivalent Beta distribution expression of the envelope to obtain the upper and lower bound expressions of the posterior distribution of the system reliability; Step (6): According to the upper and lower bound expressions of the posterior distribution of the system, perform Metropolis-Hastings sampling on the part of the normalized constant that has been discarded, and perform statistics on the system edge sample points on the results to obtain an approximate system reliability posterior The upper and lower bound probability density function PDF of the distribution; Step (7): Obtain the upper bound and lower bound cumulative probability density function CDF of the posterior distribution of the system reliability. 2. a kind of structural system reliability analysis method based on uncertainty Bayesian network according to claim 1, namely a kind of structural system reliability analysis method based on fuzzy random uncertainty Bayesian network, it is characterized in that in: In step (1), the specific method of "establishing a Bayesian network of system objects" is: The Bayesian network is composed of a directed acyclic graph formed by the directed edge connection of the event relationship; the directed acyclic graph is defined as G=<V, E>; the set of nodes is defined as V={X ₁ , X ₂ ,...,X _n }, the directed edge E between the nodes represents the connection relationship between events, and also corresponds to the causal relationship; in the X _i →X _j connected by the directed edge, X _i is the parent node, And X _j is a child node; The set of parent nodes of X _i can be represented by parent(X _i ) or pa(X _i ); The directed graph G expresses the assumption of conditional independence between events. If the prior probability value of the parent node and the conditional probability distribution between it and the child nodes can be determined, the joint probability distribution of all nodes can be obtained by the following formula ;

Given hypothesis S and evidence set E = {E ₁ ,E ₂ ,...,E _n }, Bayes' theorem can be expressed as:

in, P(S): represents the probability that H is true, also known as the prior probability; P(S|E _i ): Indicates that the given evidence is E is the conditional probability that H is true, also known as the posterior probability; P(E _i |S): represents the conditional probability of the occurrence of evidence E _i when the given hypothesis H is true, also known as the likelihood probability; (1) Prior probability: the probability of occurrence of various events determined according to expert data and objective facts; (2) Posterior probability: the probability of occurrence of events A and B, and the probability of occurrence of A when event B occurs, combined with Bayesian formula, obtain the probability of occurrence of B under the assumption that A is determined to occur;

(3) Total probability formula: If (A ₁ , A ₂ ,...,A _n ) is a group of events of E, and satisfy: P(A _i )≥0, then the total probability formula is:

The Bayesian network can be given directly and obtained from the system fault tree. 3. a kind of structural system reliability analysis method based on uncertainty Bayesian network according to claim 1, namely a kind of structural system reliability analysis method based on fuzzy random uncertainty Bayesian network, it is characterized in that in: In the "normal fuzzy random prior distribution of system reliability assumption based on subjective and objective information" described in step (2), "based on subjective and objective information" means based on limited known data, and then based on subjective judgments , which determines the fuzzy random prior distribution of the system; "Fuzzy random prior distribution" refers to the distribution based on fuzzy random variables, whose distribution is a random variable, but the distribution parameters are uncertain variables of fuzzy numbers;

The distribution of , where f( ) is a random probability density function, and the parameters of the distribution

is a fuzzy number; take the fuzzy random normal distribution as an example:

and

are the fuzzy mean and fuzzy standard deviation of the fuzzy random variables, respectively, then the fuzzy random normal distribution can be expressed as

In step (2), the specific method of "assuming the normal fuzzy random prior distribution of system reliability according to subjective and objective information" is as follows: According to the existing data: the number of trials T and the number of successes N, determine the triangular fuzzy number of the mean and standard deviation of the fuzzy random distribution of the system reliability; the maximum possible value of the standard deviation triangular fuzzy number is taken as

upper bound

lower bound

The maximum possible value of the mean triangular fuzzy number can take

upper bound

Nether can take

Afterwards, experts and experienced engineers and technicians make appropriate modifications to the above parameters, that is, to obtain the hypothesized prior distribution of system reliability. 4. a kind of structural system reliability analysis method based on uncertainty Bayesian network according to claim 1, namely a kind of structural system reliability analysis method based on fuzzy random uncertainty Bayesian network, it is characterized in in: In step (3), "According to the upper and lower bound envelopes of the prior distribution, sampling is performed and the equivalent Beta distribution parameters of the envelope are fitted by maximum likelihood estimation, and the upper and lower bound envelopes are equivalently obtained. Beta distribution expression", the specific method is as follows: "Envelope" refers to the envelope consisting of the upper and lower boundaries of the fuzzy random distribution CDF. The construction and sampling of the envelope are as follows: Assume

and

are the fuzzy mean and fuzzy standard deviation of the fuzzy random variables, respectively; all membership functions are assumed to be triangular fuzzy numbers; therefore, the fuzzy mean and standard deviation can be expressed as

and

The superscripts L, M and U are the lower bound, the median and the upper bound, respectively; the upper bound envelope and sample points of the fuzzy random distribution CDF are composed of two parts:

to the left of

Sampling for the distribution, in

to the right of

To sample for the distribution, the upper bound sampling point set is defined as

Correspondingly, the lower bound envelope of the fuzzy random distribution CDF and the sample points are in

to the left of

Sampling for the distribution, in

to the right of

To sample from a distribution, the lower bound set of sampling points is defined as X _α = { x ₁ , x ₂ , ..., x _n }; in particular, a distribution obeying x ~ N(μ ^M ,σ ^M ) is called a nominal distribution ; "Beta distribution" means The probability density function is the function shown in the following equation:

where Γ( ) is the gamma distribution,

For a variable x with a probability density function of f(x; α, β), it is called a Beta distribution with parameters α, β; Maximum likelihood estimation α, β refers to the use of sampling sample point sets

Construct the likelihood function with the Beta distribution probability density function:

And solve the following equations to get α, β, then get the equivalent Beta distribution expression

5. a kind of structural system reliability analysis method based on uncertainty Bayesian network according to claim 1, namely a kind of structural system reliability analysis method based on fuzzy random uncertainty Bayesian network, it is characterized in that in: Described in step (4), "establish system likelihood function expression according to existing data", its concrete method is as follows: According to the existing data P(E _i |H) of each node X _i , i=1,2,...,n, construct the likelihood function P(E|H) in the Bayesian network:

Wherein, E=E ₁ , E ₂ , . . . , E _n . 6. a kind of structural system reliability analysis method based on uncertainty Bayesian network according to claim 1, namely a kind of structural system reliability analysis method based on fuzzy random uncertainty Bayesian network, it is characterized in in: In step (5), "based on the Bayesian network model, multiply the likelihood function and the equivalent Beta distribution expression of the envelope to obtain the upper and lower bound expressions of the posterior distribution of the system reliability", which The specific methods are as follows: "Based on the Bayesian network model" means: solving the model description according to the Bayesian network in step (1), "Multiplying the likelihood function and the equivalent Beta distribution expression of the envelope" means: multiplying the Beta distribution equivalent expression of the prior distribution envelope obtained in step (3)

and the Bayesian network likelihood function expression constructed based on the current data obtained in step (4)

Multiplying, then the upper and lower bound expressions of the posterior distribution of the system reliability

Wherein, E=E ₁ , E ₂ , . . . , E _n . 7. A kind of structural system reliability analysis method based on uncertainty Bayesian network according to claim 1, namely a kind of structural system reliability analysis method based on fuzzy random uncertainty Bayesian network, it is characterized in that in: In step (6), "According to the upper and lower bound expressions of the posterior distribution of the system, Metropolis-Hastings sampling is performed on the part of the normalized constant that is discarded, and the result is subjected to system edge sample point statistics to obtain an approximate The upper and lower bounds of the probability density function PDF of the posterior distribution of system reliability;", the specific methods are as follows: "Truncate the normalization constant" means to discard the denominator part of P(H|E) obtained in step (5), and retain the numerator expression; and then perform Metropolis-Hastings sampling on the numerator expression; "Metropolis-Hastings sampling" means sampling as follows: Let the numerator expression after discarding the normalization constant be π(x), and select the sampling times N; 1) Extract the sample point x _i+1 from the proposed distribution density function x～N(x _i ,0.1), where the initial value of x _i is x ₁ =0.5, σ ₁ =0.5; 2) Calculate the transition probability P(x _i ; x _i+1 )=P(x _i |x～N(x _i+1 ,0.1)), P(x _i+1 ; x _i )=P(x _i+1 |x～N(x _i ,0.1)) 3) Calculate the acceptance probability α _i

4) Extract u _i to U(0,1) from the uniform distribution, if u _i <α _i , then σ _i+1 = _xi+1 ; otherwise σ _i+1 = _xi ; repeat steps 1) to steps 3) N times, the n-dimensional sample point set is obtained; for this sample point set, the sample points of the "system" dimension are extracted, which are the upper and lower bound sample points of the posterior distribution of the system {σ _i }, i = 1 ,2,...,N; "Get the upper and lower bound probability density function PDF of the approximate system reliability posterior distribution" means: According to the sample points {σ _i }, i=1,2,...,N, draw a frequency histogram on the interval [0,1] to approximate the upper and lower bound probability density function PDF of the posterior distribution: the interval [ 0,1] is evenly divided into m parts {[dx ₀ ,dx ₁ ],(dx ₁ ,dx ₂ ],...,(dx _m-1 ,dx _m ]}, Among them, dx ₀ =0, dx _m =1,

Calculate the number of samples {σ _{Number, i} }, i=1, 2,...,m in each interval of {σ _i }, i=1, 2,...,N, then the histogram is

8. A kind of structural system reliability analysis method based on uncertainty Bayesian network according to claim 1, namely a kind of structural system reliability analysis method based on fuzzy random uncertainty Bayesian network, wherein in: In step (7), "obtaining the upper bound and lower bound cumulative probability density function CDF of the posterior distribution of the system reliability" refers to the frequency histogram obtained in step (6), which is performed on the interval [0,1]. Frequency accumulation calculation, draw the approximate CDF histogram of the upper and lower bounds of the posterior distribution of the system reliability according to the following formula

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