CN111680450B - Structural system reliability analysis method based on uncertainty Bayesian network - Google Patents
Structural system reliability analysis method based on uncertainty Bayesian network Download PDFInfo
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Abstract
The invention provides a structural system reliability analysis method based on a fuzzy random uncertainty Bayesian network, which comprises the following steps: 1) Establishing a Bayesian network of the system object; 2) Assuming normal fuzzy random prior distribution of the system reliability according to the subjective and objective information; 3) Sampling is carried out according to the boundary envelope of the prior distribution, and the equivalent Beta distribution expression of the boundary envelope is fitted by adopting maximum likelihood estimation; 4) Establishing a system likelihood function expression according to the existing data; 5) Multiplying the likelihood function with the 4) result to obtain a boundary expression of the posterior distribution of the system reliability; 6) Removing the normalization constant part from the result in the step 5), sampling Metropolis-Hastings, and carrying out system edge sample point statistics on the result to obtain a boundary probability density function PDF of the posterior distribution of the system reliability; 7) Obtaining a boundary cumulative probability density function CDF of the posterior distribution of the system reliability; the reliability analysis method provided by the invention is scientific, has good manufacturability and has wide popularization and application values.
Description
Technical Field
The invention provides a structural system reliability analysis method based on an uncertainty Bayesian network, relates to a structural system reliability analysis method based on a fuzzy stochastic uncertainty Bayesian network, aims to perform reliability analysis on a structural system under fuzzy stochastic information containing inherent uncertainty and knowledge uncertainty, and belongs to the field of structural reliability analysis.
Background
In the first aspect, bayesian network methods (Bayesian networks, BN), which are an important tool for dealing with uncertainty problems in the fields of system science, reliability science, artificial intelligence, etc., as a member of the graph model and analysis framework, have been developed in over twenty years, and their graphical expression functions make the relation and state expression of the system and elements more intuitive and clear. The problem of poor information and small samples in system analysis can be well solved. The system analysis based on the Bayesian network in the reliability engineering is mainly completed according to the structure composition of the product and the establishment of the fault tree.
Various Bayesian networks (dynamic Bayesian network DBN, object-oriented Bayesian network OOBN, qualitative Bayesian network and the like) widely used at present are basically based on mastered prior knowledge, and the actual measured data and information are used as evidence to update and correct the existing cognition, so that posterior cognition of the problem is obtained after actual factors are considered. From the perspective of the system, the method is based on a Bayesian formula, utilizes a conditional probability table from bottom to top to construct a likelihood function, performs iterative computation, comprehensively utilizes various effective information to obtain a posterior probability reasoning result, and is widely applied to the field of system reliability modeling and analysis. The accuracy of the system reliability analysis result mainly depends on three parts of prior information, conditional probability and experimental data serving as 'evidence'. Traditional Bayesian method researches often focus on selection of prior-detection distribution, acquisition of conditional probability and influence of experimental data information serving as 'evidence' on overall reliability analysis, but uncertainty research on input information is relatively less, and most existing Bayesian methods only process probability information, fuzzy information or are based on evidence theory.
On the other hand, currently, the types of uncertainty of input information are mainly divided into two major types, namely inherent uncertainty information (represented by probability information) and cognitive uncertainty information (represented by fuzzy information, interval information and the like), and considering that uncertainty information of a complex structural system is mixed, namely, the inherent uncertainty (probability information) and cognitive uncertainty (lean information or fuzzy information) exist simultaneously, the errors and the complexity of the traditional Bayesian network analysis method with the probability or fuzzy information as the input are increased, the Bayesian network reasoning method relying on single inherent uncertainty and cognitive uncertainty has limitations, and the accuracy and the confidence cannot be ensured. The fuzzy random variable is used as an uncertainty variable with both inherent and cognitive uncertainty characteristics, is an uncertainty information description method with probability distribution form and fuzzy number as a distribution parameter, integrates the characteristics of random distribution and fuzzy mathematics, can characterize information containing both uncertainties, and improves the cognitive confidence degree to a certain extent. The fuzzy random variable has been widely used in the fields of structural reliability analysis, structural reliability evaluation, life model and the like. The fuzzy first-order reliability method FFORM (Fuzzy First Order Reliability Method) based on the fuzzy random variable, the fuzzy Monte Carlo simulation based on the fuzzy random variable theory, the life prediction model based on the fuzzy random variable, the interval finite element method based on the fuzzy random variable and the like are included. But the related algorithms currently applying fuzzy random variables to structural system reliability analysis remain to be extended.
Based on the two aspects, the invention provides a structural system reliability analysis method based on a fuzzy random uncertainty Bayesian network.
Disclosure of Invention
Object of the invention
Aiming at a small sub-sample structure system with insufficient sample size, the invention adopts fuzzy random distribution as the prior distribution of the system reliability and combines a Bayesian network to carry out reliability analysis on the structure system, and aims to explore a new system reliability analysis framework under inherent and cognitive mixed uncertainty.
The invention firstly samples the envelope upper bound (lower bound) of fuzzy random prior distribution of a Bayesian network subsystem, fits an equivalent Beta distribution expression by adopting maximum likelihood estimation, then samples according to a likelihood function of the Bayesian network by adopting a Ha Siting sampling method, namely a Metropolis-Hastings sampling method, and continuously samples the edge of a system variable on a high-dimensional sampling result to obtain the upper bound (lower bound) of equivalent posterior distribution of the system.
(II) technical scheme
The invention adopts the following technical scheme to realize the aim:
the invention discloses a structural system reliability analysis method based on an uncertainty Bayesian network, namely a structural system reliability analysis method based on a fuzzy random uncertainty Bayesian network, which relates to a Bayesian network structure reliability output upper (lower) boundary solving model based on fuzzy random parameters, and comprises the following implementation steps:
step (1), establishing a Bayesian network of a system object;
step (2), assuming normal fuzzy random prior distribution of the system reliability according to the subjective and objective information;
step (3), sampling according to the upper (lower) boundary envelope of the prior distribution, and fitting equivalent Beta distribution parameters of the envelope by adopting maximum likelihood estimation to obtain an upper (lower) boundary envelope equivalent Beta distribution expression;
step (4), establishing a system likelihood function expression according to the existing data;
step (5), multiplying the likelihood function by the equivalent Beta distribution expression of the envelope based on the Bayesian network model to obtain an upper bound (lower bound) expression of the posterior distribution of the system reliability;
step (6), sampling the truncated normalized constant part according to an upper (lower) expression of the posterior distribution of the system, and carrying out system edge sample point statistics on the result to obtain an approximate upper (lower) probability density function PDF (Probability Density Function) of the posterior distribution of the system reliability;
step (7), obtaining an upper bound (lower bound) cumulative probability density function CDF (Cumulative Distribution Function) of the system reliability posterior distribution;
the specific method of "establishing a bayesian network of a system object" in the step (1) is as follows:
the Bayesian network is formed by directed acyclic graphs formed by connection of directed edges of event relationships; the directed acyclic graph is defined as G= < V, E>The method comprises the steps of carrying out a first treatment on the surface of the Where the set of nodes is defined as v= { X 1 ,X 2 ,...,X n Directed edges E between nodes represent the connection relationship between events and also correspond to causal relationships; x connected by directed edges i →X j Wherein X is i Is a father node, and X j Is a child node;
X i parent node set available parent (X i ) Or pa (X) i ) To represent;
the directed graph G expresses a conditional independence assumption among events, and if the prior probability value of a father node and the conditional probability distribution between the father node and a child node can be determined, the joint probability distribution of all nodes can be obtained by the following formula;
given an assumption S and evidence set e= { E 1 ,E 2 ,...,E n The bayesian theorem can be expressed as:
wherein, the liquid crystal display device comprises a liquid crystal display device,
p (S): the probability that H is true, which may also be referred to as a priori probability;
P(S|E i ): a conditional probability, also called a posterior probability, that a given evidence is that E is H true;
P(E i s): representing evidence E when a given hypothesis H is true i The conditional probability of occurrence, also called likelihood probability;
(1) Prior probability: probability of occurrence of various events determined according to expert data and objective facts;
(2) Posterior probability: knowing the probability of occurrence of event A, B and the probability of occurrence of A when B occurs, and combining a Bayesian formula to obtain the probability of occurrence of B under the condition that the assumption A determines occurrence;
(3) Full probability formula: if (A) 1 ,A 2 ,...,A n ) A set of events that is E, and satisfies: p (A) i ) And (3) not less than 0, the full probability formula is as follows:
a typical simple Bayesian network is shown in FIG. 1 (a), where X 2 And X 3 As father node, X 1 As child node, directed edge X 2 →X 1 ，X 3 →X 1 A set E is formed, and the Bayesian network connection relation is formed;
the Bayesian network can be directly given, or can be obtained from a system fault tree, with the top event of FIG. 1 (a) being depicted as X in FIG. 1 (b) 1 The bottom event is X 2 ，X 3 Fault tree (X) 2 And X is 3 The relationship is or gate/and gate/not gate).
In the step (2), the "normal fuzzy random prior distribution according to the reliability of the subjective and objective information assumption system" refers to the fuzzy random prior distribution of the system based on limited known data and then according to subjective judgment;
"fuzzy random prior distribution" refers to a distribution based on fuzzy random variables, which is a random variable, but whose distribution parameters are uncertain variables of fuzzy numbers; i.e. shaped likeWherein f (·) is a random probability density function and the parameter of the distribution +.>Is a fuzzy number; taking the fuzzy random normal distribution as an example: is provided with->And->The fuzzy mean and fuzzy standard deviation of the fuzzy random variable respectively, the fuzzy random normal distribution can be expressedIs->
The "normal fuzzy random prior distribution of the reliability of the system is assumed according to the subjective and objective information" in the step (2) is as follows:
determining the triangular fuzzy number of the mean value and the standard deviation of the fuzzy random distribution of the system reliability according to the existing data (the test times T and the success times N); the maximum possible value of the standard deviation triangle fuzzy number can be takenThe upper bound is taken->Lower bound is desirableThe maximum possible value of the mean triangle blur number is +.>The upper bound is taken->Lower bound is taken->And then, expert or experienced engineering technicians are used for carrying out proper modification on the parameters, so as to obtain the assumed prior distribution of the system reliability.
The specific method of the method is as follows:
"envelope" refers to an envelope made up of the upper and lower boundaries of a fuzzy random distribution CDF, the envelope being constructed and sampled as follows:
is provided withAnd->Respectively a fuzzy mean value and a fuzzy standard deviation of the fuzzy random variable; all membership functions are assumed to be triangular fuzzy numbers; therefore, the fuzzy average and standard deviation can be expressed as +.>Andwherein upper labels L, M and U are lower, median and upper bounds, respectively; the fuzzy random distribution CDF upper bound envelope and sample points are composed of two parts: at->To the left of->Sampling the distribution, at->To the right side of (2)For sampling the distribution, the upper bound sample point set is defined as +.>Correspondingly, the fuzzy random distribution CDF lower bound envelope and the sample point are in +.>To the left of->Sampling the distribution, at->To the right of>For sampling the distribution, a set of lower bound sampling points is definedX α ＝{x 1 ,x 2 ，…,x n -a }; in particular, for compliance with x to N (mu) M ,σ M ) Is referred to as a nominal distribution; FIG. 2 shows the shape +.>And->An envelope and nominal distribution diagram of fuzzy random variables;
"Beta distribution" means
The probability density function is a function shown in the following formula:
wherein Γ (·) is gamma distribution,the pair shape is called Beta distribution (Chinese translation: beta distribution) obeying parameters alpha, beta, as a variable x of the probability density function with f (x; alpha, beta);
maximum likelihood estimation of alpha, beta refers to the use of a set of sampled sample pointsAnd a Beta distribution probability density function constructs a likelihood function:
and solving the following equation set to obtain alpha and Beta, and obtaining the equivalent Beta distributed expression
The specific method of "establishing a system likelihood function expression according to the existing data" in the step (4) is as follows:
according to each node X i I=1, 2, n. existing data P (E i I H), a likelihood function P (EH) in the bayesian network is constructed:
wherein E= (E) 1 ,E 2 ,...,E n )。
The specific method of "multiplying the likelihood function and the equivalent Beta distribution expression of the envelope based on the bayesian network model to obtain the upper bound (lower bound) expression of the posterior distribution of the system reliability" described in the step (5) is as follows:
"based on a Bayesian network model" means: from the bayesian network solution model description in step (1),
"multiplying the likelihood function with the equivalent Beta distribution expression of the envelope" means: equivalent Beta distribution expression of the prior distribution envelope obtained in the step (3)And (4) the Bayesian network likelihood function expression constructed based on the current data obtained in the step (4)>Multiplying the upper (lower) expression of the system reliability posterior distribution
Wherein E= (E) 1 ,E 2 ,...,E n )。
The method comprises the steps of (1) sampling a truncated normalization constant part according to an upper (lower) boundary expression of a system posterior distribution, and carrying out system edge sample point statistics on the result to obtain an approximate upper (lower) boundary probability density function PDF of the system reliability posterior distribution, wherein the upper (lower) boundary expression of the system posterior distribution is described in the step (6); ", the specific method is as follows:
the term "eliminating normalization constant" means eliminating denominator of the P (HE) obtained in the step (5) and retaining the molecular expression; then, metropolis-Hastings sampling is carried out on the molecular expression;
"Metropolis-Hastings sampling" refers to sampling according to the following steps:
setting the molecular expression after eliminating the normalization constant as pi (x), and selecting the sampling times N;
1) From the proposed distribution density functions x-N (x i 0.1) sample point x is extracted i+1 Wherein x is i Initial value x 1 ＝0.5，σ 1 ＝0.5；
2) Calculating 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) Calculating the probability of acceptance alpha i
4) Extracting u from uniform distribution i U (0, 1), if U i ＜α i Sigma is then i+1 ＝x i+1 The method comprises the steps of carrying out a first treatment on the surface of the Otherwise sigma i+1 ＝x i The method comprises the steps of carrying out a first treatment on the surface of the Repeating the steps 1) to 3) for N times, so that an N-dimensional sample point set can be obtained; for this set of sample points, the sample points of the "system" dimension are extracted, i.e., the upper (lower) sample points { σ ] of the system posterior distribution i },i＝1,2,...,N；
"upper bound (lower bound) probability density function PDF of the approximate system reliability posterior distribution" refers to:
from sample points { sigma } i I=1, 2,..n, in interval [0,1]The frequency histogram is plotted above to approximate the upper (lower) probability density function PDF of the posterior distribution: interval [0,1]]Evenly divide 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, calculating { σ } i I=1, 2,.. Number,i I=1, 2, m, the histogram is +.>
Wherein, the "obtaining the upper bound (lower bound) cumulative probability density function CDF of the system reliability posterior distribution" described in the step (7) refers to performing frequency cumulative calculation on the interval [0,1] according to the frequency histogram obtained in the step (6), and drawing the upper bound (lower bound) approximate CDF histogram of the system reliability posterior distribution according to the following formula
The flow chart of the reliability analysis method is shown in fig. 3.
(III) advantages and effects of the invention
The structural system reliability analysis method combines the respective characteristics of the Bayesian network analysis method and the fuzzy random variable, provides a Bayesian network system reliability calculation framework based on the fuzzy random parameters, and has the advantages and effects that:
(1) The method constructs a Bayesian network solving process under inherent and cognitive mixed uncertainty, and can guide the reliability analysis of a structural system under incomplete data;
(2) Based on the characteristics of fuzzy random variables, the upper and lower bounds of the system reliability distribution are obtained according to a Bayesian network, and a system reliability posterior reference can be provided for engineering/scientific researchers;
(3) The method is convenient and quick to calculate, can fully utilize the existing data to calculate the system reliability, can be implemented under the unknown conditional probability, and can solve the problems of large calculation amount and low confidence of the numerical value of the structural system caused by mixed uncertainty information widely existing in engineering;
(4) The reliability analysis method provided by the invention is scientific, has good manufacturability and has wide popularization and application values.
Description of the drawings (in the drawings, the numbers, symbols and code numbers are as follows)
Fig. 1 is a typical simple bayesian network and fault tree.
FIG. 2 is a fuzzy random cumulative distribution function.
FIG. 3 is a flow chart of a system reliability analysis according to the present invention.
Fig. 4 is a bayesian network in case 1.
Fig. 5 is a case 1 system reliability prior distribution.
Fig. 6 is a case 1 system reliability equivalent a priori Beta distribution.
FIG. 7 is a plot of the 1000 receptions (upper bound) of case 1Metropolis-Hastings prior to sampling.
FIG. 8 is a plot of the 1000 receptions (lower bound) of case 1Metropolis-Hastings prior to sampling.
Fig. 9a and 9b are case 1 system posterior distribution frequency histograms.
Fig. 10 is a plot of case 1 system posterior distribution frequency cumulative density.
FIG. 11 is a fault tree of "turbine System faults".
FIG. 12 is a Bayesian network of "turbine system failures".
Fig. 13 is a case 2 system reliability prior distribution.
Fig. 14 is a case 2 system reliability equivalent a priori Beta distribution.
FIG. 15 is a plot of the 1000 receptions (upper bound) of case 2Metropolis-Hastings prior to sampling.
FIG. 16 is a plot of the 1000 receptions (lower bound) of case 2Metropolis-Hastings prior to sampling.
Fig. 17a and 17b are case 2 system posterior distribution frequency histograms.
Fig. 18 is a plot of case 2 system posterior distribution frequency cumulative density.
The numbers, symbols and codes in the figures are described as follows:
in FIG. 1, X 1 ，X 2 And X 3 Is a Bayesian network node; AND gate for fault tree;
in fig. 2, CDF is a cumulative distribution function;
in fig. 5, CDF is a cumulative distribution function;
in fig. 6, PDF is a probability density function and Beta distribution is Beta distribution;
in fig. 9a and 9b, PDF is a probability density function, S represents the system of case 1;
in fig. 10, CDF is a cumulative distribution function, S represents the system of case 1;
in fig. 13, CDF is a cumulative distribution function;
in fig. 14, PDF is a probability density function, and Beta distribution is Beta distribution;
fig. 17a and 17b, pdf is a probability density function, S representing the system of case 2;
in fig. 18, CDF is a cumulative distribution function, and S represents the system of case 2.
The symbols and codes referred to in this specification are as follows:
beta distribution-Beta distribution
FFORM (Fuzzy First Order Reliability Method) -fuzzy first order reliability method based on fuzzy random variable
PDF (Probability Distribution Function) - -probability Density function
CDF (Cumulative Distribution Function) -cumulative distribution function
Metropolis-Hastings sample-Ha Siting sample
Detailed Description
The technical scheme of the invention is described in detail below with reference to the examples and the attached drawings.
Case 1 (numerical case):
assume a shellfishAs shown in FIG. 4, the leaf network has 3 nodes in total, node S represents the system, and node C 1 And C 2 Representing the subsystem. When the edge distribution and the condition distribution are unknown, the posterior distribution P (S|C) of the system is calculated by the method according to the known data 1 ,C 2 S). It is assumed that the known data are shown in table 1.
TABLE 1 known data
| Bayesian network node names | Data (number of successes/total number of trials) |
| Subsystem C 1 | 11/14 |
| Subsystem C 2 | 37/41 |
| System S | 8/11 |
The invention discloses a structural system reliability analysis method based on a fuzzy random uncertainty Bayesian network, which is shown in fig. 3 and comprises the following implementation steps:
step (1), establishing a Bayesian network of a system object; according to the description, a Bayesian network is established, as shown in FIG. 4 for a Bayesian network diagram 1.
Step (2), assuming normal fuzzy random prior distribution of the system reliability according to the subjective and objective information;
according to the known data system data of 8/11, we assume that the prior distribution of the system obeys normal fuzzy random distribution, the mean value and standard deviation of the distribution parameters are both assumed to be triangle fuzzy numbers, the two are shown in a table 2, and a schematic diagram of the normal fuzzy random accumulation probability is shown in fig. 5.
Table 2 System fuzzy random prior distribution
Step (3), sampling according to the upper (lower) boundary envelope of the prior distribution, and fitting equivalent Beta distribution parameters of the envelope by adopting maximum likelihood estimation to obtain an upper (lower) boundary envelope equivalent Beta distribution expression;
building an envelope for the fuzzy random variable generated in step (2) as shown in FIG. 5, sampling 1000 times according to the method in the explanation of step (3), the sample point being denoted as x i (i=1, 2,.,. 1000), the likelihood function in the likelihood estimation isWhere f is the Beta function expression. Then solving the following equation set to obtain alpha and Beta, and obtaining the equivalent Beta distributed expression
Solving to obtain an upper bound envelope equivalent Beta distribution expression parameter alpha= 2.2510, beta= 1.4075; the lower bound envelope equivalent Beta distribution expression parameter α= 1.4969, β= 1.5324. Brought inEven if the expression corresponding thereto. The upper (lower) bound envelope equivalent Beta distribution is shown in figure 6.
Step (4), establishing a system likelihood function expression according to the existing data;
from the data in table 1, assuming that all nodes are binomial distributions, 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 S ) 3
step (5), multiplying the likelihood function by the equivalent Beta distribution expression of the envelope based on the Bayesian network model to obtain an upper bound (lower bound) expression of the posterior distribution of the system reliability;
according to the results of the steps (3) and (4), the upper bound expression of the posterior distribution of the system reliability can be obtained as follows:
the lower bound expression of the system reliability posterior distribution is:
step (6), sampling the truncated normalized constant part according to an upper (lower) expression of the posterior distribution of the system, and carrying out system edge sample point statistics on the result to obtain an approximate upper (lower) probability density function PDF of the posterior distribution of the system reliability; the method comprises the steps of carrying out a first treatment on the surface of the
The expression after dropping the normalization constant in the step (5) is:
Metropolis-Hastings sampling is performed on the sample, and the sampling times are N=10000 times. For clarity of presentation, the 1000 times before extraction is shown in FIG. 7 (upper bound) and FIG. 8 (lower bound).
Dividing the interval of [0,1] into 50 parts, carrying out system edge frequency statistics on the sampling result of Metropolis-Hastings, finally obtaining a fitting posterior distribution PDF frequency table shown in Table 3, corresponding posterior distribution PDF histograms shown in figures 9a and 9b,
table 3 System posterior distribution PDF frequency table
Step (7), obtaining an upper bound (lower bound) cumulative probability density function CDF of the posterior distribution of the system reliability;
frequency accumulation is carried out on each interval according to the result of the step (6), the obtained system posterior distribution CDF frequency is shown in table 4, the corresponding posterior distribution CDF histogram is shown in figure 10,
table 4 system posterior distribution CDF frequency table
Case 2 (engineering case):
the aeroengine turbine system fault tree is shown in fig. 11, the top event is defined as a 'turbine system fault', and the top event is formed by connecting 2 fault events together in series, namely a turbine disc fault and a turbine blade fault, wherein the number of possible fault modes of the turbine disc is 2, the number of possible fault modes of the turbine blade is 2, and the occurrence of each fault mode has corresponding fault reasons. Therefore, in order to more accurately find the basic event reflecting the failure of the rotor system, all the failure causes that cause the failure mode to occur are defined as the bottom event of the failure tree. Wherein all the child nodes are in a series relationship. The known data are shown in table 5.
TABLE 5 known data
The invention discloses a structural system reliability analysis method based on a fuzzy random uncertainty Bayesian network, which is shown in fig. 3 and comprises the following implementation steps:
step (1), establishing a Bayesian network of a system object; according to the case description, a bayesian network is established, as shown in fig. 12 of a bayesian network diagram 1, and each node marks known data.
Step (2), assuming normal fuzzy random prior distribution of the system reliability according to the subjective and objective information;
according to the known data system data 194/195, we assume that the prior distribution of the system obeys the normal fuzzy random distribution, and that the mean value and standard deviation of the distribution parameters are triangle fuzzy numbers, both are shown in table 6, and the normal fuzzy random cumulative probability diagram 2 is shown in fig. 13.
Table 6 system fuzzy random prior distribution
Step (3), sampling according to the upper (lower) boundary envelope of the prior distribution, and fitting equivalent Beta distribution parameters of the envelope by adopting maximum likelihood estimation to obtain an upper (lower) boundary envelope equivalent Beta distribution expression;
building an envelope for the fuzzy random variable generated in step (2) as shown in FIG. 13, sampling 1000 times according to the method in the explanation of step (3), the sample point being denoted as x i (i=1, 2,.,. 1000), the likelihood function in the likelihood estimation isWhere f is the Beta distribution function expression. Then solve for asThe alpha and Beta can be obtained by the following equation set, and then the equivalent Beta distribution expression is obtained
Solving to obtain an upper bound envelope equivalent Beta distribution expression parameter alpha= 56.8116, beta= 1.8496; the lower bound envelope equivalent Beta distribution expression parameter α= 25.2457, β= 1.2317. Brought inEven if the expression corresponding thereto. The upper (lower) bound envelope equivalent Beta distribution is shown in figure 14.
Step (4), establishing a system likelihood function expression according to the existing data;
from the data in table 1, assuming that all nodes are binomial distributions, the system likelihood function expression is:
step (5), multiplying the likelihood function by the equivalent Beta distribution expression of the envelope based on the Bayesian network model to obtain an upper bound (lower bound) expression of the posterior distribution of the system reliability;
according to the results of the steps (3) and (4), the upper bound expression of the posterior distribution of the system reliability can be obtained as follows:
the lower bound expression of the system reliability posterior distribution is:
step (6), sampling the truncated normalized constant part according to an upper (lower) expression of the posterior distribution of the system, and carrying out system edge sample point statistics on the result to obtain an approximate upper (lower) probability density function PDF of the posterior distribution of the system reliability;
the expression after dropping the normalization constant in the step (5) is:
Metropolis-Hastings sampling is performed on the sample, and the sampling times are N=10000 times. For clarity of presentation, the 1000 times before extraction is shown in FIG. 15 (upper bound) and FIG. 16 (lower bound).
Dividing the interval of [0,1] into 25 parts, carrying out system edge frequency statistics on the sampling result of Metropolis-Hastings, finally obtaining a fitting posterior distribution PDF frequency table shown in Table 7, corresponding posterior distribution PDF histograms shown in figures 17a and 17b,
table 7 system posterior distribution PDF frequency table
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Step (7), obtaining an upper bound (lower bound) cumulative probability density function CDF of the posterior distribution of the system reliability;
frequency accumulation is carried out on each interval according to the result of the step (6), the frequency of the system posterior distribution CDF is shown in a table 8, the corresponding posterior distribution CDF histogram is shown in figure 18,
table 8 System posterior distribution CDF frequency table
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Claims (8)
1. The utility model provides a structural system reliability analysis method based on uncertainty Bayesian network, which is applied to an aeroengine turbine system, wherein a top event is defined as a turbine system fault, and consists of 2 fault events which are connected in series together, namely a turbine disc fault and a turbine blade fault, wherein the number of the turbine disc faults is 2 in fault modes, the number of the turbine blades is 2, and the occurrence of various fault modes has corresponding fault reasons; in order to more accurately find out the basic event reflecting the rotor system fault, defining all fault reasons causing the fault mode to be generated as the bottom event of the fault tree; wherein, all the child nodes are in series connection;
TABLE 1 known data
The method is characterized by comprising the following steps of:
step (1), establishing a Bayesian network of a system object; according to the case description, a Bayesian network is established, and each node marks known data;
step (2), assuming normal fuzzy random prior distribution of the system reliability according to the subjective and objective information;
according to the known data system data 194/195, setting the prior distribution of the system to follow normal fuzzy random distribution, wherein the mean value and standard deviation of the distribution parameters are assumed to be triangle fuzzy numbers, and the triangle fuzzy numbers are shown in a table 2;
table 2 System fuzzy random prior distribution
Step (3)) Sampling according to the upper and lower bound envelopes of the prior distribution, and fitting equivalent Beta distribution parameters of the envelopes by adopting maximum likelihood estimation to obtain equivalent Beta distribution expressions of the upper and lower bound envelopes; constructing an envelope for the fuzzy random variable generated in the step (2), sampling 1000 times according to the method in the step (3), and marking a sample point as x i (i=1, 2,.,. 1000), the likelihood function in the likelihood estimation isWherein f is a Beta distribution function expression; and then solving the following equation set to obtain alpha and Beta, and obtaining an equivalent Beta distribution expression:
solving to obtain an upper bound envelope equivalent Beta distribution expression parameter alpha= 56.8116, beta= 1.8496; lower bound envelope equivalent Beta distribution expression parameter α= 25.2457, β= 1.2317; brought inEven if the expression corresponding thereto;
step (4), establishing a system likelihood function expression according to the existing data;
assuming that all nodes are binomial distribution, the system likelihood function expression is:
step (5), multiplying the likelihood function by the equivalent Beta distribution expression of the envelope based on the Bayesian network model to obtain upper and lower boundary expressions of the posterior distribution of the system reliability;
according to the results of the steps (3) and (4), the upper bound expression for obtaining the posterior distribution of the system reliability is:
the lower bound expression of the system reliability posterior distribution is:
step (6), sampling the truncated normalization constant part according to the upper and lower boundary expressions of the posterior distribution of the system, and carrying out statistics on system edge sample points on the result to obtain upper and lower boundary probability density functions PDF of the posterior distribution of the system reliability;
the expression after dropping the normalization constant in the step (5) is:
Metropolis-Hastings sampling is carried out on the sample, and the sampling frequency is N=10000 times;
dividing the interval [0,1] into 25 parts, and carrying out system edge frequency statistics on the Metropolis-Hastings sampling result to finally obtain a fitting posterior distribution PDF frequency table shown in Table 3;
table 3 System posterior distribution PDF frequency table
Step (7), obtaining an upper bound and a lower bound cumulative probability density function CDF of the posterior distribution of the system reliability;
performing frequency accumulation on each interval according to the result of the step (6) to obtain a system posterior distribution CDF frequency table shown in Table 4;
table 4 system posterior distribution CDF frequency table
2. A structural system reliability analysis method based on an uncertainty bayesian network according to claim 1, wherein: in step (1), the bayesian network is formed by directed acyclic graphs formed by directed edge connections of event relationships; the directed acyclic graph is defined as g=<V,E>The method comprises the steps of carrying out a first treatment on the surface of the Where the set of nodes is defined as v= { X 1 ,X 2 ,...,X n Directed edges E between nodes represent the connection relationship between events and also correspond to causal relationships; x connected by directed edges i →X j Wherein X is i Is a father node, and X j Is a child node;
X i parent node set can use parent (X i ) Or pa (X) i ) To represent;
the directed graph G expresses a conditional independence assumption among events, and if the prior probability value of a father node and the conditional probability distribution between the father node and a child node can be determined, the joint probability distribution of all nodes can be obtained through the following formula;
given an assumption S and evidence set e= { E 1 ,E 2 ,...,E n The bayesian theorem can be expressed as:
wherein, the liquid crystal display device comprises a liquid crystal display device,
p (S): the probability that H is true, also known as the prior probability;
P(S|E i ): a conditional probability, also called a posterior probability, that a given evidence is that E is H true;
P(E i s): representing evidence E when a given hypothesis H is true i The conditional probability of occurrence, also called likelihood probability;
(1) Prior probability: probability of occurrence of various events determined according to expert data and objective facts;
(2) Posterior probability: knowing the probability of occurrence of event A, B and the probability of occurrence of A when B occurs, and combining a Bayesian formula to obtain the probability of occurrence of B under the condition that the assumption A determines occurrence;
(3) Full probability formula: if (A) 1 ,A 2 ,...,A n ) A set of events that is E, and satisfies: p (A) i ) And (3) not less than 0, the full probability formula is as follows:
bayesian networks can be given directly and derived from the system fault tree.
3. A structural system reliability analysis method based on an uncertainty bayesian network according to claim 2, wherein: in step (2), the fuzzy random prior distribution refers to a distribution based on fuzzy random variables, which are random variables, but the distribution parameters are uncertain variables of fuzzy numbers; i.e. shaped likeWherein f (·) is a random probability density function and the parameter of the distribution +.>Is a fuzzy number; is provided with->And->
The fuzzy mean and fuzzy standard deviation of the fuzzy random variable respectively, the fuzzy random normal distribution can be expressed as
According to the existing data: the test times T and the success times N are used for determining the mean value of the fuzzy random distribution of the system reliability and the triangular fuzzy number of the standard deviation; maximum possible value of standard deviation triangle fuzzy numberUpper bound is->Lower bound is->The maximum possible value of the mean triangle blur number can be +.>The upper bound can take->Lower bound can take->A hypothetical system reliability prior distribution is obtained.
4. A structural system reliability analysis method based on an uncertainty bayesian network according to claim 3, wherein: in step (3), the envelope is an envelope composed of upper and lower boundaries of the fuzzy random distribution CDF, and the construction and sampling of the envelope are as follows:
is provided withAnd->
Respectively a fuzzy mean value and a fuzzy standard deviation of the fuzzy random variable; all membership functions are assumed to be triangular fuzzy numbers; thus, the fuzzy average and standard deviation can be expressed asAnd->Wherein upper labels L, M and U are lower, median and upper bounds, respectively; the fuzzy random distribution CDF upper bound envelope and sample points are composed of two parts: at the position ofTo the left of->Sampling the distribution, at->To the right of>For sampling the distribution, the upper bound sample point set is defined as +.>Correspondingly, the fuzzy random distribution CDF lower bound envelope and the sample point are in +.>To the left of->Sampling the distribution, at->To the right of>For sampling the distribution, a set of lower bound sampling points is definedX α ＝{x 1 ,x 2 ，…,x n -a }; in particular, for compliance with x to N (mu) M ,σ M ) Is referred to as a nominal distribution;
beta distribution refers to:
the probability density function is a function shown in the following formula:
wherein Γ (·) is gamma distribution,the variable x of the pair shape with f (x; alpha, beta) probability density function is called Beta distribution with obeying parameter alpha, beta;
maximum likelihood estimation of alpha, beta refers to the use of a set of sampled sample pointsAnd a Beta distribution probability density function constructs a likelihood function:
and solving the following equation set to obtain alpha and Beta, and obtaining an equivalent Beta distribution expression:
5. the method for analyzing the reliability of a structural system based on an uncertainty bayesian network according to claim 4, wherein: in step (4), according to each node X i I=1, 2, where, n existing data P (E i I H), a likelihood function P (E i H) in the bayesian network is constructed:
wherein e=e 1 ,E 2 ,...,E n 。
6. A structural system reliability analysis method based on an uncertainty bayesian network according to claim 5 and wherein: in step (5), the bayesian network model is based on: from the bayesian network solution model description in step (1),
multiplying the likelihood function with the equivalent Beta distribution expression of the envelope means: equivalent Beta distribution expression of the prior distribution envelope obtained in the step (3)And (4) the Bayesian network likelihood function expression constructed based on the current data obtained in the step (4)>Multiplying the upper and lower bounds of the posterior distribution of system reliability
Wherein e=e 1 ,E 2 ,...,E n 。
7. A structural system reliability analysis method based on an uncertainty bayesian network according to claim 6 and wherein: in the step (6), the elimination of the normalization constant means that the denominator part of the P (HE) obtained in the step (5) is eliminated, and the molecular expression is retained; then, metropolis-Hastings sampling is carried out on the molecular expression; metropolis-Hastings sampling refers to sampling according to the following steps:
setting the molecular expression after eliminating the normalization constant as pi (x), and selecting the sampling times N;
1) From the proposed distribution density functions x-N (x i 0.1) sample point x is extracted i+1 Wherein x is i Initial value x 1 ＝0.5，σ 1 ＝0.5；
2) Calculating 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) Calculating the probability of acceptance alpha i
4) Extracting u from uniform distribution i U (0, 1), if U i ＜α i Sigma is then i+1 ＝x i+1 The method comprises the steps of carrying out a first treatment on the surface of the Otherwise sigma i+1 ＝x i The method comprises the steps of carrying out a first treatment on the surface of the Repeating the steps 1) to 3) for N times to obtain an N-dimensional sample point set; for this sample point set, sample points of the system dimension, i.e., upper and lower sample points { sigma } of the system posterior distribution are extracted i },i＝1,2,...,N；
The upper and lower bound probability density functions PDF for obtaining the approximate system reliability posterior distribution are:
from sample points { sigma } i I=1, 2,..n, in interval [0,1]And (3) drawing a frequency histogram to approximate an upper bound probability density function PDF of posterior distribution: interval [0,1]]Evenly divide into m parts
{[dx 0 ,dx 1 ],(dx 1 ,dx 2 ],...,(dx m-1 ,dx m ]}，
Wherein dx is 0 ＝0,dx m ＝1,Calculation of { sigma ] i I=1, 2,.. Number,i I=1, 2, m, the histogram is
8. The method for analyzing the reliability of the structural system based on the uncertain bayesian network according to claim 7, wherein: in step (7), frequency integration calculation is performed on the interval [0,1] according to the frequency histogram obtained in step (6), and the upper-bound and lower-bound approximate CDF histograms of the system reliability posterior distribution are drawn according to the following formula
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