CN111680450A - Structural system reliability analysis method based on uncertainty Bayesian network - Google Patents
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Abstract
The invention provides a structural system reliability analysis method based on fuzzy random uncertainty Bayesian network, which comprises the following steps: 1) establishing a Bayesian network of system objects; 2) according to the main and objective information, the normal fuzzy random prior distribution of the system reliability is assumed; 3) sampling according to the boundary envelope of prior distribution, and fitting an equivalent Beta distribution expression of the boundary envelope by adopting maximum likelihood estimation; 4) establishing a system likelihood function expression according to the existing data; 5) multiplying the likelihood function by the result of 4) to obtain a boundary expression of the posterior distribution of the system reliability; 6) carrying out Metropolis-Hastings sampling after a normalization constant part of the result in the step 5) is removed, and carrying out system edge sample point statistics on the result to obtain a boundary probability density function PDF of system reliability posterior distribution; 7) obtaining a boundary accumulative 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 random uncertainty Bayesian network, aims to perform reliability analysis on a structural system under fuzzy randomness information containing inherent uncertainty and recognition uncertainty at the same time, and belongs to the field of structural reliability analysis.
Background
In the first aspect, a Bayesian network method (BN) is used as a member of a graph model and an analysis framework, is an important tool for processing uncertainty problems in the fields of system science, reliability science, artificial intelligence and the like, and has been developed vigorously in more than twenty years, and the graphical expression function of the Bayesian network method enables the relationship and state expression of a system and elements to be more intuitive and clear. The problem of poor information and small samples in system analysis can be solved well. 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 a fault tree.
Various currently widely used bayesian networks (dynamic bayesian network DBN, object-oriented bayesian network OOBN, qualitative bayesian network, etc.) essentially start from grasped prior knowledge, and use the actually measured data and information as "evidence" to update and correct the existing cognition, thereby obtaining the posterior cognition of the problem after considering the actual factors. From the point of view of physique, 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 at present. The accuracy of the system reliability analysis result mainly depends on three parts of prior information, conditional probability and experimental data serving as evidence. In the traditional research on the Bayesian method, the selection of prior distribution, the acquisition of conditional probability and the influence of experimental data information as evidence on the overall reliability analysis are often focused, while the uncertainty research on input information is relatively less, and most of the existing Bayesian methods only process probability information, fuzzy information or are based on evidence theory.
On the other hand, at present, the types of uncertainty of input information are mainly divided into two main types, namely inherent uncertainty information (represented by probability information) and cognitive uncertainty information (represented by fuzzy information and interval information), and in consideration of the fact that uncertainty information of a complex structure system is mixed, namely, the inherent uncertainty (probability information) and the cognitive uncertainty (poor information or fuzzy information) exist at the same time, so that errors and complexity of a traditional Bayesian network analysis method taking probability or fuzzy information as input are increased, and a Bayesian network inference method depending on single inherent uncertainty and cognitive uncertainty has limitations and cannot guarantee accuracy and confidence. The fuzzy random variable is an uncertain variable with both inherent and cognitive uncertainty characteristics, is an uncertain information description method with a probability distribution form and simultaneously distributed parameters as fuzzy numbers, integrates the characteristics of random distribution and fuzzy mathematics, can represent information containing two uncertainties simultaneously, and improves the cognitive confidence to a certain extent. The fuzzy random variable has been widely applied in the fields of structural reliability analysis, structural reliability evaluation, life models and the like. The method comprises a fuzzy first-order Reliability method FFORM (fuzzy first order Reliability method) based on fuzzy random variables, fuzzy Monte Carlo simulation based on fuzzy random variable theory, a life prediction model based on fuzzy random variables, an interval finite element method based on fuzzy random variables and the like. However, the related algorithm for applying fuzzy random variables to the reliability analysis of the structural system still needs to be expanded.
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
Objects of the invention
The invention aims to search a new system reliability analysis framework under inherent and cognitive mixed uncertainty by adopting fuzzy random distribution as system reliability prior distribution and combining a Bayesian network to carry out reliability analysis on a structural system aiming at a small subsample structural system with insufficient sample size.
The method comprises the steps of firstly sampling the upper (lower) boundary of the envelope of fuzzy random prior distribution of a Bayesian network subsystem, fitting an equivalent Beta distribution expression by adopting maximum likelihood estimation, then sampling by adopting a Halstein sampling method, namely Metropolis-Hastings sampling method, according to the likelihood function of the Bayesian network, and continuously sampling the system variable edge of a high-dimensional sampling result to obtain the upper (lower) boundary of the equivalent posterior distribution of the system.
(II) technical scheme
In order to achieve the purpose, the invention adopts the following technical scheme:
the invention discloses a structural system reliability analysis method based on an uncertain Bayesian network, namely a structural system reliability analysis method based on a fuzzy random uncertain 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), according to the main and objective information, the normal fuzzy random prior distribution of the system reliability is assumed;
sampling according to the upper (lower) boundary envelope of the prior distribution, and fitting the 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;
multiplying the likelihood function and the equivalent Beta distribution expression of the envelope based on a Bayesian network model to obtain an upper bound (lower bound) expression of the posterior distribution of the system reliability;
step (6), carrying out Metropolis-Hastings sampling on the part with the normalization constant omitted according to an upper bound (lower bound) expression of the posterior distribution of the system, and carrying out system edge sample point statistics on the result to obtain an approximate upper bound (lower bound) probability Density function PDF (probability Density function) of the posterior distribution of the system reliability;
step (7), obtaining an upper bound (lower bound) accumulative probability density function CDF (cumulative Distribution function) of the posterior Distribution of the system reliability;
wherein, the "bayesian network for establishing system objects" in step (1) is implemented by:
the Bayesian network is formed by directed acyclic graphs formed by connecting directed edges of event relations; directed acyclic graphs are defined as G ═ V, E>(ii) a Where the set of nodes is defined as V ═ X1,X2,...,XnAn oriented edge E between the nodes represents the connection relation between events and also corresponds to a causal relation; x connected by directed edgesi→XjIn, XiIs a parent node, and XjIs a child node;
Xican use parent (X) in the parent node seti) Or pa (X)i) To represent;
the directed graph G expresses the conditional independence assumption between events, and if the prior probability value of the parent node and the conditional probability distribution between the parent node and the child node can be determined, the joint probability distribution of all nodes can be obtained by the following formula;
given the assumption S and the evidence set E ═ E1,E2,...,EnThe bayes theorem can be expressed as:
wherein,
p (S): the probability that H is true is expressed and can be called prior probability;
P(S|Ei): the conditional probability that E is true for a given evidence is denoted as H, also called a posterior probability;
P(Ei| S): representing evidence E given that assumption H is trueiConditional probability of occurrence, also known as likelihood probability;
(1) prior probability: the probability of occurrence of various events is determined according to expert data and objective facts;
(2) posterior probability: the probability of occurrence of the event A, B and the probability of occurrence of A when B occurs are known, and the probability of occurrence of B under the condition that A is determined to occur is obtained by combining a Bayesian formula;
(3) the total probability formula: if (A)1,A2,...,An) A set of events that is E, and satisfies: p (A)i) If the total probability formula is more than or equal to 0, the total probability formula is as follows:
a typical simple Bayesian network is shown in FIG. 1(a), where X is2And X3Is a parent node, X1As child nodes, directed edge X2→X1，X3→X1A set E is formed, and the set E is a Bayesian network connection relation;
the Bayesian network can be directly provided or derived from a system fault tree, and FIG. 1(a) can describe the top event as X in FIG. 1(b)1The bottom event is X2，X3Fault tree (X)2And X3The relationship is either an or gate/and gate/not gate).
In the step (2), the normal fuzzy random prior distribution of the system reliability is assumed according to the subjective and objective information, wherein the 'objective and subjective information' is based on limited known data and then the fuzzy random prior distribution of the system is determined according to subjective judgment;
the fuzzy random prior distribution refers to distribution based on fuzzy random variables, wherein the distribution is random variables, but the distribution parameters are uncertain variables of fuzzy numbers; i.e. shaped as
Wherein f (-) is a random probability density function, and the parameters of the distribution
Is a fuzzy number; take a fuzzy random normal distribution as an example: is provided with
And
fuzzy mean and fuzzy standard deviation of fuzzy random variable respectively, then the fuzzy random normal distribution can be expressed as
The "normal fuzzy random prior distribution assuming system reliability according to the main and objective information" in step (2) is implemented as follows:
determining a triangular fuzzy number of a mean value and a standard deviation of fuzzy random distribution of the system reliability according to the existing data (test times T and success times N); maximum possible value of standard deviation triangular ambiguity number is available
Upper bound of the table is available
Lower bound retrievable
Maximum possible value of mean triangular blur number can be taken
Upper bound of the table is available
Lower bound retrievable
Then, the expert or experienced engineering technician modifies the parameters properly to obtain the assumed prior distribution of the system reliability.
Wherein, in the step (3), "sampling is performed according to the upper (lower) boundary envelope of the prior distribution, and the maximum likelihood estimation is adopted to fit the equivalent Beta distribution parameter of the envelope, so as to obtain the upper (lower) boundary envelope equivalent Beta distribution expression", the specific method is as follows:
"envelope" refers to the envelope consisting of the upper and lower boundaries of a fuzzy randomly distributed CDF, the envelope constructed and sampled as follows:
is provided with
And
fuzzy mean and fuzzy standard deviation of fuzzy random variable respectively; all membership functions are assumed to be triangular fuzzy numbers; thus, the fuzzy mean and standard deviation can be expressed as
And
where superscripts L, M and U are lower bound, median, and upper bound, respectively; the fuzzy random distribution CDF upper bound envelope and the sample points are composed of two parts: in that
To the left side of
For distribution sampling, in
To the right side of
For distribution sampling, an upper bound set of sampling points is defined as
Correspondingly, the lower bound envelope and sample points of the fuzzy random CDF are
To the left side of
For distribution sampling, in
To the right side of
For distribution sampling, a lower bound set of sampling points is definedX α＝{x 1,x 2，…,x n}; in particular, for compliance with x to N (mu)M,σM) The distribution of (a) is called a nominal distribution; figure 2 shows the shape of
And
schematic of the envelope and nominal distribution of the fuzzy random variables of (1);
"Beta distribution" means
The probability density function is a function represented by:
wherein (. cndot.) is a gamma distribution,
for variables x shaped as having a f (x; α) probability density function, called obedience parametersα Beta distribution (Chinese translation: Beta distribution);
maximum likelihood estimation α refers to the use of a set of sample points
And Beta distribution probability density function constructing likelihood function:
and solving the following equation set to obtain alpha and Beta, and then obtaining the equivalent Beta distribution expression
Wherein, the step (4) of establishing the system likelihood function expression according to the existing data is implemented as follows:
according to each node XiN (E) existing data P (1, 2)iH), constructing a likelihood function p (eh) in the bayesian network:
wherein E ═ E (E)1,E2,...,En)。
Wherein, in the step (5), "based on the bayesian network model, the likelihood function is multiplied by the equivalent Beta distribution expression of the envelope to obtain an upper bound (lower bound) expression of the posterior distribution of the system reliability", which is specifically performed as follows:
"Bayesian network model-based" means: solving the model description according to the Bayesian network in the step (1),
"multiplying the likelihood function by the equivalent Beta distribution expression of the envelope" means: beta distribution equivalent expression of prior distribution envelope obtained in step (3)
And the Bayesian network likelihood function expression constructed based on the current data obtained in the step (4)
Multiplication, upper bound (lower bound) expression of the system reliability posterior distribution
Wherein E ═ E (E)1,E2,...,En)。
Wherein, in the step (6), "Metropolis-Hastings sampling is performed on the part with the normalization constant being cut off according to the upper bound (lower bound) expression of the system posterior distribution, and the statistics of the system edge sample points is performed on the result to obtain the approximate upper bound (lower bound) probability density function PDF of the system reliability posterior distribution; ", the concrete method is as follows:
"truncation of normalization constant" means to truncate the denominator part of the P (HE) obtained in step (5) and retain the molecular expression; then carrying out Metropolis-Hastings sampling on the molecular expression;
"Metropolis-Hastings sampling" means that sampling is performed as follows:
setting a molecular expression obtained after the normalization constant is omitted as pi (x), and selecting a sampling frequency N;
1) from the proposed distribution density function x-N (x)i0.1) extracting the sample point xi+1Wherein x isiInitial value x1＝0.5，σ1＝0.5；
2) Calculating the transition probability P (x)i；xi+1)＝P(xi|x～N(xi+1,0.1))，P(xi+1；xi)＝P(xi+1|x～N(xi,0.1))
3) Calculating an acceptance probability αi
4) From uniform distributionGet uiU (0,1) if Ui＜αiThen σi+1＝xi+1(ii) a Otherwise σi+1＝xi(ii) a Repeating the steps 1) to 3) for N times, and obtaining an N-dimensional sample point set; for the sample point set, sample points in the 'system' dimension are extracted, namely upper bound (lower bound) sample points { sigma } of the posterior distribution of the systemi},i＝1,2,...,N；
"obtaining an approximate upper bound (lower bound) probability density function PDF of a posterior distribution of system reliability" refers to:
according to sample point { sigma i1,2, N, in the interval [0,1 ═ N]The frequency histogram is plotted up to approximate the upper (lower) probability density function PDF of the posterior distribution: will be interval [0,1]Is uniformly divided into m portions { [ dx { [ dX ]0,dx1],(dx1,dx2],...,(dxm-1,dxm]Where dx is0＝0,dxm＝1,
i is 0, 1.. times.m, and { σ is calculatedi1,2, N, the number of samples in each interval { σ }Number,i1,2, m, then the histogram is
Wherein, the step (7) of obtaining the upper bound (lower bound) cumulative probability density function CDF of the system reliability posterior distribution means that the frequency cumulative calculation is performed on the interval [0,1] according to the frequency histogram obtained in the step (6), and the upper bound (lower bound) approximate CDF histogram of the system reliability posterior distribution is drawn according to the following formula
The reliability analysis method of the present invention is illustrated in the flow chart of fig. 3.
(III) advantages and Effect of the invention
The structural system reliability analysis method provided by the invention combines the characteristics of the Bayesian network analysis method and the fuzzy random variables, provides a Bayesian network system reliability calculation framework based on fuzzy random parameters, and has the advantages and effects that:
(1) the Bayesian network solving process under inherent and cognitive mixed uncertainty is constructed, and the reliability analysis of a structural system under incomplete data can be guided;
(2) based on the characteristics of fuzzy random variables, the upper and lower boundaries of the distribution of the system reliability are obtained according to the Bayesian network, and the posterior reference of the system reliability can be provided for engineering/scientific research personnel;
(3) the method is convenient and quick to calculate, can fully utilize the existing data to calculate the reliability of the system, can be implemented under the probability of unknown conditions, and can solve the problems of large numerical calculation amount and low confidence coefficient 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 figures, the numbers, symbols, and symbols 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 the system reliability analysis of 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 case 1 system reliability equivalent prior Beta distribution.
FIG. 7 is the acceptance rate (upper bound) 1000 times before sampling of case 1 Metropolis-Hastings.
FIG. 8 is the acceptance rate (lower bound) 1000 times before sampling of case 1 Metropolis-Hastings.
Fig. 9a and 9b are posterior distribution frequency histograms for the case 1 system.
Fig. 10 is the cumulative density of posterior distribution frequencies for the case 1 system.
FIG. 11 is a fault tree for "turbine system faults".
FIG. 12 is a Bayesian network of "turbine system failure".
Fig. 13 is a case 2 system reliability prior distribution.
Fig. 14 is a case 2 system reliability equivalent prior Beta distribution.
FIG. 15 is the acceptance rate (upper bound) 1000 times before sampling of case 2 Metropolis-Hastings.
FIG. 16 is the acceptance rate (lower bound) 1000 times before sampling of case 2 Metropolis-Hastings.
Fig. 17a and 17b are posterior distribution frequency histograms for the case 2 system.
Fig. 18 is the cumulative density of posterior distribution frequencies for the case 2 system.
The numbers, symbols and codes in the figures are explained as follows:
in FIG. 1, X1，X2And X3Is a Bayesian network node; is a fault tree AND gate;
in FIG. 2, CDF is the cumulative distribution function;
in FIG. 5, CDF is the 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, and S represents the system of case 1;
in FIG. 10, CDF is the cumulative distribution function, S represents the system for case 1;
in FIG. 13, CDF is the cumulative distribution function;
in fig. 14, PDF is a probability density function, and Beta distribution is Beta distribution;
17a and 17b, PDF is a probability density function, S represents the system of case 2;
in fig. 18, CDF is the cumulative distribution function, and S represents the system of case 2.
The symbols and codes referred to in the present 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 sampling-Halastine sampling
Detailed Description
The technical scheme of the invention is explained in detail by combining the calculation examples and the attached drawings.
Case 1 (numerical case):
suppose a Bayesian network is shown in FIG. 4, with 3 nodes, node S representing the system and node C1And C2Representing a 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 data1,C2S). Assume that the known data are as shown in table 1.
TABLE 1 known data
| Bayesian network node names | Known data (number of successes/total number of tests) |
| Subsystem C1 | 11/14 |
| Subsystem C2 | 37/41 |
| System S | 8/11 |
The invention relates to a structural system reliability analysis method based on fuzzy random uncertainty Bayesian network, as shown in FIG. 3, the implementation steps are as follows:
step (1), establishing a Bayesian network of a system object; according to the description, a Bayesian network is established, as shown in FIG. 4 in a Bayesian network schematic diagram 1.
Step (2), according to the main and objective information, the normal fuzzy random prior distribution of the system reliability is assumed;
according to the known data system data of 8/11, the prior distribution of the system is assumed to be in accordance with the normal fuzzy random distribution, and the mean value and the standard deviation of the distribution parameters are both assumed to be triangular fuzzy numbers, which are shown in table 2, and as shown in fig. 5 of the normal fuzzy random cumulative probability diagram 2.
TABLE 2 systematic fuzzy random prior distribution
Sampling according to the upper (lower) boundary envelope of the prior distribution, and fitting the equivalent Beta distribution parameters of the envelope by adopting maximum likelihood estimation to obtain an upper (lower) boundary envelope equivalent Beta distribution expression;
the fuzzy random variable constructed envelope generated in step (2) is sampled 1000 times according to the method explained in step (3) as shown in fig. 5, and the sample point is marked as xi( i 1, 2.., 1000), the likelihood function in the likelihood estimation is
Wherein f is Beta function expression, then α can be obtained by solving the following equation system, and then the equivalent Beta distribution expression is obtained
The expression parameters α -2.2510, β -1.4075 of upper boundary envelope equivalent Beta distribution expression, α -1.4969, β -1.5324 of lower boundary envelope equivalent Beta distribution expression are obtained by solution
Even the expression corresponding thereto. The upper (lower) envelope equivalent Beta distribution is shown in FIG. 6.
Step (4), establishing a system likelihood function expression according to the existing data;
according to the data in table 1, assuming that all nodes are binomial distributed, the system likelihood function expression is:
L(p1,p2,pS)＝p1 11(1-p1)3p2 37(1-p2)4pS 8(1-pS)3
multiplying the likelihood function and the equivalent Beta distribution expression of the envelope based on a 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), carrying out Metropolis-Hastings sampling on the part with the normalization constant omitted according to an upper bound (lower bound) expression of the posterior distribution of the system, and carrying out system edge sample point statistics on the result to obtain an approximate upper bound (lower bound) probability density function PDF of the posterior distribution of the system reliability; (ii) a
The expression after the normalization constant is cut off in the step (5) is as follows:
Metropolis-Hastings sampling is carried out, and the sampling time N is 10000 times. For clarity, the acceptance rate of the first 1000 extractions is shown in fig. 7 (upper bound), fig. 8 (lower bound).
Dividing the [0,1] interval into 50 parts, carrying out system edge frequency statistics on Metropolis-Hastings sampling results to finally obtain a fitting posterior distribution PDF frequency table shown in a table 3, wherein the corresponding posterior distribution PDF histogram is shown in a table 9a and a table 9b,
TABLE 3 system posterior distribution PDF frequency table
Step (7), obtaining an upper bound (lower bound) accumulative 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 chart shown in a table 4, wherein a corresponding posterior distribution CDF histogram is shown in a table 10,
TABLE 4 posterior distribution CDF frequency table of system
Case 2 (engineering case):
the aeroengine turbine system fault tree is shown in fig. 11, and the top event is defined as a "turbine system fault" and is composed of 2 fault events which are connected in series, namely a turbine disk fault and a turbine blade fault, wherein 2 possible fault modes exist on the turbine disk, and the occurrence of each fault mode has a corresponding fault reason. Therefore, in order to more accurately find the fundamental events reflecting the rotor system failure, all failure causes that cause the failure mode to occur are defined as the bottom events of the failure tree. Wherein, all child nodes are in a serial relation. The known data are shown in table 5.
TABLE 5 known data
The invention relates to a structural system reliability analysis method based on fuzzy random uncertainty Bayesian network, as shown in FIG. 3, the implementation steps are as follows:
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 in a Bayesian network schematic diagram 1, and each node is marked with known data.
Step (2), according to the main and objective information, the normal fuzzy random prior distribution of the system reliability is assumed;
according to the known data system data of 194/195, the prior distribution of the system is assumed to be in accordance with the normal fuzzy random distribution, and the mean value and the standard deviation of the distribution parameters are both assumed to be triangular fuzzy numbers, which are shown in table 6, as shown in fig. 13 of a normal fuzzy random cumulative probability diagram 2.
TABLE 6 systematic fuzzy random prior distribution
Sampling according to the upper (lower) boundary envelope of the prior distribution, and fitting the equivalent Beta distribution parameters of the envelope by adopting maximum likelihood estimation to obtain an upper (lower) boundary envelope equivalent Beta distribution expression;
the fuzzy random variable constructed envelope generated in step (2) is sampled 1000 times according to the method explained in step (3) as shown in fig. 13, and the sample point is marked as xi( i 1, 2.., 1000), the likelihood function in the likelihood estimation is
Wherein f is Beta distribution function expression, then α can be obtained by solving the following equation system, and the equivalent Beta distribution expression is obtained
The expression parameters α -56.8116, β -1.8496 of upper boundary envelope equivalent Beta distribution expression, α -25.2457, β -1.2317 of lower boundary envelope equivalent Beta distribution expression are obtained by solution
Even the expression corresponding thereto. The upper (lower) envelope equivalent Beta distribution is shown in FIG. 14.
Step (4), establishing a system likelihood function expression according to the existing data;
according to the data in table 1, assuming that all nodes are binomial distributed, the system likelihood function expression is:
multiplying the likelihood function and the equivalent Beta distribution expression of the envelope based on a 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), carrying out Metropolis-Hastings sampling on the part with the normalization constant omitted according to an upper bound (lower bound) expression of the posterior distribution of the system, and carrying out system edge sample point statistics on the result to obtain an approximate upper bound (lower bound) probability density function PDF of the posterior distribution of the system reliability;
the expression after the normalization constant is cut off in the step (5) is as follows:
Metropolis-Hastings sampling is carried out, and the sampling time N is 10000 times. For clarity, the acceptance rate of the first 1000 extractions is shown in fig. 15 (upper bound), fig. 16 (lower bound).
Dividing the [0,1] interval into 25 parts, carrying out system edge frequency statistics on Metropolis-Hastings sampling results to finally obtain a fitting posterior distribution PDF frequency table shown in a table 7, wherein the corresponding posterior distribution PDF histogram is shown in a figure 17a and a figure 17b,
TABLE 7 system posterior distribution PDF frequency table
Step (7), obtaining an upper bound (lower bound) accumulative 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 chart shown in a table 8, wherein a corresponding posterior distribution CDF histogram is shown in a table 18,
TABLE 8 posterior distribution CDF frequency table of system
Claims (8)
1. 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, is characterized in that: the implementation steps are as follows:
step (1), establishing a Bayesian network of a system object;
step (2), according to the main and objective information, the normal fuzzy random prior distribution of the system reliability is assumed;
sampling according to the upper and lower boundary envelopes of the prior distribution, and fitting the equivalent Beta distribution parameters of the envelopes by adopting maximum likelihood estimation to obtain an upper and lower boundary envelope equivalent Beta distribution expression;
step (4), establishing a system likelihood function expression according to the existing data;
multiplying the likelihood function and the equivalent Beta distribution expression of the envelope based on a Bayesian network model to obtain an upper bound expression and a lower bound expression of the posterior distribution of the system reliability;
step (6), carrying out Metropolis-Hastings sampling on the part with the normalization constant omitted according to the upper bound expression and the lower bound expression of the posterior distribution of the system, and carrying out system edge sample point statistics on the result to obtain approximate upper bound probability density function PDF and lower bound probability density function PDF of the posterior distribution of the system reliability;
and (7) obtaining the upper bound and the lower bound of the posterior distribution of the system reliability and accumulating the probability density function CDF.
2. The structural system reliability analysis method based on uncertainty bayesian network according to claim 1, namely a structural system reliability analysis method based on fuzzy random uncertainty bayesian network, characterized in that:
the "bayesian network for establishing system objects" described in step (1) is implemented by:
the Bayesian network is formed by directed acyclic graphs formed by connecting directed edges of event relations; directed acyclic graphs are defined as G ═<V,E>(ii) a Where the set of nodes is defined as V ═ X1,X2,...,XnAn oriented edge E between the nodes represents the connection relation between events and also corresponds to a causal relation; x connected by directed edgesi→XjIn, XiIs a parent node, and XjIs a child node;
Xican use parent (X) in the parent node seti) Or pa (X)i) To represent;
the directed graph G expresses the 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 descendant node can be determined, the joint probability distribution of all the nodes can be obtained by the following formula;
given the assumption S and the evidence set E ═ E1,E2,...,EnThe bayes theorem can be expressed as:
wherein,
p (S): represents the probability that H is true, also known as the prior probability;
P(S|Ei): the conditional probability that E is true for a given evidence is denoted as H, also called a posterior probability;
P(Ei| S): representing evidence E given that assumption H is trueiConditional probability of occurrence, also known as likelihood probability;
(1) prior probability: the probability of occurrence of various events is determined according to expert data and objective facts;
(2) posterior probability: the probability of occurrence of the event A, B and the probability of occurrence of A when B occurs are known, and the probability of occurrence of B under the condition that A is determined to occur is obtained by combining a Bayesian formula;
(3) the total probability formula: if (A)1,A2,...,An) A set of events that is E, and satisfies: p (A)i) If the total probability formula is more than or equal to 0, the total probability formula is as follows:
the bayesian network can be given directly and derived from the system fault tree.
3. The structural system reliability analysis method based on uncertainty bayesian network according to claim 1, namely a structural system reliability analysis method based on fuzzy random uncertainty bayesian network, characterized in that:
in the step (2), the normal fuzzy random prior distribution of the system reliability is assumed according to the subjective and objective information, wherein the 'according to the subjective and objective information' is that the fuzzy random prior distribution of the system is determined based on limited known data and according to subjective judgment;
the fuzzy random prior distribution refers to distribution based on fuzzy random variables, wherein the distribution is random variables, but the distribution parameters are uncertain variables of fuzzy numbers; i.e. shaped as
Wherein f (-) is a random probability density function, and the parameters of the distribution
Is a fuzzy number; take a fuzzy random normal distribution as an example: is provided with
And
fuzzy mean and fuzzy standard deviation of fuzzy random variable respectively, then the fuzzy random normal distribution can be expressed as
The "normal fuzzy random prior distribution assuming system reliability according to the main and objective information" in step (2) is implemented as follows:
according to the existing data: determining the mean value of fuzzy random distribution of the system reliability and the triangular fuzzy number of standard deviation according to the test times T and the success times N; maximum possible value of standard deviation triangular ambiguity number
Upper boundary extraction
Lower boundary extraction
The maximum possible value of the mean triangular blur number can be taken
Upper bound of energy can get
Lower bound of energy can get
Then, experts and experienced engineering technicians modify the parameters appropriately to obtain the assumed prior distribution of the system reliability.
4. The structural system reliability analysis method based on uncertainty bayesian network according to claim 1, namely a structural system reliability analysis method based on fuzzy random uncertainty bayesian network, characterized in that:
in the step (3), "sampling is performed according to the upper and lower envelopes of the prior distribution, and the maximum likelihood estimation is adopted to fit the equivalent Beta distribution parameters of the envelopes, so as to obtain the equivalent Beta distribution expressions of the upper and lower envelopes", which specifically comprises the following steps:
"envelope" refers to the envelope consisting of the upper and lower boundaries of a fuzzy randomly distributed CDF, the envelope constructed and sampled as follows:
is provided with
And
fuzzy mean and fuzzy standard deviation of fuzzy random variable respectively; all membership functions are assumed to be triangular fuzzy numbers; thus, the fuzzy mean and standard deviation can be expressed as
And
where superscripts L, M and U are lower bound, median, and upper bound, respectively; the fuzzy random distribution CDF upper bound envelope and the sample points are composed of two parts: in that
To the left side of
For distribution sampling, in
To the right side of
For distribution sampling, an upper bound set of sampling points is defined as
Corresponding, fuzzy random scoreThe CDF lower bound envelope and sample points are
To the left side of
For distribution sampling, in
To the right side of
For distribution sampling, a lower bound set of sampling points is definedX α＝{x 1,x 2，...,x n}; in particular, for compliance with x to N (mu)M,σM) The distribution of (a) is called a nominal distribution;
"Beta distribution" means
The probability density function is a function represented by:
wherein (. cndot.) is a gamma distribution,
for a variable x shaped as a probability density function with f (x; α), we call the Beta distribution with compliance parameter α;
maximum likelihood estimation α refers to the use of a set of sample points
And Beta distribution probability density function constructing likelihood function:
solving the following equation system to obtain alpha, Beta and then obtaining the equivalent Beta distribution expression
5. The structural system reliability analysis method based on uncertainty bayesian network according to claim 1, namely a structural system reliability analysis method based on fuzzy random uncertainty bayesian network, characterized in that:
the "establishing a system likelihood function expression based on the existing data" in step (4) is implemented as follows:
according to each node XiN existing data P (E), i 1,2iH), constructing a likelihood function P (E | H) in the bayesian network:
wherein E ═ E1,E2,...,En。
6. The structural system reliability analysis method based on uncertainty bayesian network according to claim 1, namely a structural system reliability analysis method based on fuzzy random uncertainty bayesian network, characterized in that:
in the step (5), "based on the bayesian network model, the likelihood function is multiplied by the equivalent Beta distribution expression of the envelope to obtain the upper and lower boundary expressions of the posterior distribution of the system reliability", which is specifically performed as follows:
"Bayesian network model-based" means: solving the model description according to the Bayesian network in the step (1),
"multiplying the likelihood function by the equivalent Beta distribution expression of the envelope" means: beta distribution equivalent expression of prior distribution envelope obtained in step (3)
And the Bayesian network likelihood function expression constructed based on the current data obtained in the step (4)
Multiplication, upper and lower bound expressions of posterior distribution of system reliability
Wherein E ═ E1,E2,...,En。
7. The structural system reliability analysis method based on uncertainty bayesian network according to claim 1, namely a structural system reliability analysis method based on fuzzy random uncertainty bayesian network, characterized in that:
in the step (6), "Metropolis-Hastings sampling is performed on the part with the normalization constant omitted according to the upper and lower boundary expressions of the posterior distribution of the system, and system edge sample point statistics is performed on the result to obtain approximate upper and lower boundary probability density functions PDF of the posterior distribution of the system reliability; ", the concrete method is as follows:
"truncation of normalization constant" means to truncate the denominator part of P (H | E) obtained in step (5) and retain the numerator expression; then carrying out Metropolis-Hastings sampling on the molecular expression;
"Metropolis-Hastings sampling" means that sampling is performed as follows:
setting a molecular expression obtained after the normalization constant is omitted as pi (x), and selecting a sampling frequency N;
1) from the proposed distribution density function x-N (x)i0.1) extracting the sample point xi+1Wherein x isiInitial value x1＝0.5，σ1＝0.5；
2) Calculating the transition probability P (x)i；xi+1)＝P(xi|x～N(xi+1,0.1))，
P(xi+1；xi)＝P(xi+1|x～N(xi,0.1))
3) Calculating an acceptance probability αi
4) Extracting u from the uniform distributioniU (0,1) if Ui＜αiThen σi+1＝xi+1(ii) a Otherwise σi+1＝xi(ii) a Repeating the steps 1) to 3) for N times to obtain an N-dimensional sample point set; for the sample point set, sample points of 'system' dimension are extracted, namely upper and lower bound sample points { sigma over the posterior distribution of the systemi},i＝1,2,...,N；
"obtaining the upper and lower bound probability density functions PDF of the approximate system reliability posterior distribution" means:
according to sample point { sigmai1,2, N, in the interval [0,1 ═ N]And drawing a frequency histogram to approximate the upper and lower bound probability density functions PDF of the posterior distribution: will be interval [0,1]Is uniformly divided into m portions { [ dx { [ dX ]0,dx1],(dx1,dx2],...,(dxm-1,dxm]}，
Wherein dx is0＝0,dxm＝1,
Calculation of { σi1,2, N, the number of samples in each interval { σ }Number,i1,2, m, then the histogram is
8. The structural system reliability analysis method based on uncertainty bayesian network according to claim 1, namely a structural system reliability analysis method based on fuzzy random uncertainty bayesian network, characterized in that:
the step (7) of obtaining the upper bound and the lower bound cumulative probability density function CDF of the posterior distribution of the system reliability means that the frequency histogram obtained in the step (6) is subjected to frequency cumulative calculation on the interval [0,1], and the upper bound and the lower bound approximate CDF histogram of the posterior distribution of the system reliability are drawn according to the following formula
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