CN109543234B - Component life distribution parameter estimation method based on random SEM algorithm - Google Patents
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Abstract
The invention belongs to the technical field of digital computing or data processing equipment or methods specially applicable to specific applications, and discloses a component life distribution parameter estimation method based on a random SEM algorithm; solving a survival function of the service life of the system by using a cumulative distribution function of the service life of the components and a system signature vector, and writing a maximum likelihood function of the service life of the system; estimating the component parameters using a random SEM algorithm: s, combining a sequence system signature and a conditional probability vector to obtain a life estimation sample, and M, bringing the estimation sample into a likelihood function and maximizing an update parameter; carrying out Monte Carlo simulation experiments under two conditions of a small sample and a medium-to-large sample of the system and generating four indexes of deviation and mean square error of point estimation, coverage rate and expected width of confidence interval; compared with the prior algorithm, the SEM algorithm estimation is found to be better and is used for processing the situation of estimating the service life distribution parameters of the component of the II-type bilateral deleting system.
Description
Technical Field
The invention belongs to the technical field of digital computing or data processing equipment or methods specially suitable for specific applications, and particularly relates to a component life distribution parameter estimation method based on a random SEM algorithm.
Background
Currently, the current state of the art commonly used in the industry is as follows: in 1985, the concept of signature and its basic calculation was first proposed by samanigo in 1985, from which signature vectors began to be tried into the methods and tools of system reliability research; later in 1999, kocha et al developed that the life distribution function of the system could be expressed by a mixture of signature vectors and component life distribution functions, further establishing a relationship between the vectors and the system; in addition, boland in 2001, provided a system-way-set-number-based associated system Signature calculation method. Through the above work, the concept of system Signature is widely and deeply applied by people to various branches of system reliability research. However, at the same time, because of the limitation of the existing signature-based method (the calculated system life structure function is often complex), a lot of time is consumed in the solving process, and estimation and calculation of parameters are not facilitated, so that the method becomes a hot spot and a difficult problem in parameter estimation of the life distribution of the system components. Meanwhile, a key link for predicting the service life of the system is to estimate unknown parameters and determine whether the prediction of the service life of the system is accurate or not. To date scholars have explored several algorithms for parameter estimation for the system component lifetime distribution, such as: an optimal linear unbiased estimation algorithm aiming at single-side deletion of a system, an optimal linear unbiased estimation algorithm, a moment estimation algorithm, a Bayesian algorithm, a maximum likelihood estimation algorithm, a regression estimation algorithm and the like. In the prior art, the distribution and linear inference of the service life of the component are discussed in a parameter estimation method based on optimal linear unbiased estimation on the service life distribution of the system component, and the algorithm derives the optimal linear unbiased estimation of parameters in a general scale parameter group and a position-scale parameter group and gives an accurate calculation formula of variance and covariance. However, this method only considers one component life distribution parameter, and does not take more parameter factors into consideration, which makes it difficult to estimate for systems with uncertain component life distribution families. The calculation method of the second parameter estimation method based on moment estimation on the service life distribution of the system component in the prior art is simpler in principle, and compared with other algorithms, the calculation method has no complicated calculation process, but the method can only obtain more accurate estimation when the sample capacity is larger. In practical applications, the deviation of the estimation result is larger for the case of a smaller number of samples. In the third prior art, the method solves the parameter estimation of the non-monotonic failure rate distribution condition based on the Bayesian estimation on the parameter estimation method of the service life distribution of the system component, but the method depends on the selection of a loss function to a great extent, if the loss function does not well describe the system, the method has great influence on the superiority of the estimated quantity, and the technology is difficult to obtain an estimation result with higher precision under the asymmetric loss condition. The fourth prior art, namely a parameter estimation method for service life distribution of a system component based on maximum likelihood, solves the problem of deleting data by using order statistics, but theory deviation is long, and a numerical method is needed to solve, so that the convergence of a calculation result can be ensured, and the situation that the result is not converged often occurs. The method has the advantages of simplicity and easy understanding, no additional calculation process is needed, an ideal result is given, meanwhile, the operation time is saved, and the method can be used as an initial value of other methods to carry in calculation. However, the method also needs numerical solution, can not completely ensure the convergence of calculation, and when the proportion of missing data is high, the obtained result is often unstable.
In summary, the problems of the prior art are: the existing parameter estimation method for the service life of the system component has the problem of second-order bias which is caused by the fact that the numerical problem of huge calculation amount is needed or the calculation process is complicated and the convergence of the numerical method solution cannot be guaranteed, so that the calculation process has higher time complexity and higher hardware requirements, and finally, the calculation time is long and the precision is not high, and the method is not beneficial to being applied to projects with higher requirements on time and precision; in addition, in the prior art, most systems aiming at single-side deletion are adopted, and the situation of double-side deletion of type II is considered to be less, so that no suitable method aiming at double-side deletion systems exists at present, and a plurality of double-side deletion systems exist in the engineering fields of aerospace, construction and the like, and the development of partial fields is slow due to the lack of research and the blank of technology. The application range of the original method has a certain limitation, and the method is particularly embodied in a model which can only be used for unilateral deletion with low time requirements and low precision requirements in engineering.
Difficulty and meaning for solving the technical problems: in the parameter estimation of the system component lifetime distribution, if the system component lifetime is unknown, the calculation is difficult by the conventional method, and if the conventional component lifetime distribution is brought in, the accuracy of the estimation is reduced. If the situation that the result of the calculation value solution is lengthy and the second-order partial derivative cannot be converged occurs, the prior art is difficult to execute, so that the application range of the signature method is narrower. In addition, in the case of II-type bilateral deletion, the difficulty in estimating the parameters of the component is increased due to the deletion of bilateral data, and the accuracy of estimation is reduced. The superiority of parameter estimation of component life distribution can affect the accuracy of prediction of system life; only if a wide application range is found, the parameter distribution of the service life of the component can be accurately estimated, and the service life of the system can be accurately predicted.
Disclosure of Invention
Aiming at the problems existing in the prior art, the invention provides a component life distribution parameter estimation method based on a random SEM algorithm.
The invention is realized in such a way that a component life distribution parameter estimation method based on a random SEM algorithm comprises the following steps:
(1) Solving a survival function of the service life of the system by using a cumulative distribution function of the service life of the components and a system signature vector;
(2) Estimating the parameters of the component by using a random SEM algorithm, S step, combining a sequence system signal and a conditional probability vector to obtain a life estimation sample, and M step, bringing the estimation sample into a likelihood function and maximizing an update parameter;
(3) And carrying out Monte Carlo simulation experiments under two conditions of a small sample and a medium-to-large sample of the system, and generating four indexes of deviation and mean square error of point estimation, coverage rate and expected width of a confidence interval.
Further, the component life distribution parameter estimation method based on the random SEM algorithm specifically comprises the following steps:
step one, calculating a system signature vector S according to a system structure function Γ (s 1 ,s 2 ...s n ) Order signature vector
Omega ith system failures are arranged, and the ith component of the signature vector of the system is obtained as follows:
event a is that the kth order system failure is caused by the ith order component failure; generating m system lives for a determined system, observing which component fails to cause the system to fail, and repeating the process for N times; the order system signature vector may be approximated as:
step two, using the cumulative distribution function F of the component life X And System signature vector S Γ Survival function representing system lifetime
For n independent co-distributed component systems, X represents a random variable of component life and T represents a random variable of system life; survival function of System lifetime->
Expressed as:
step three, establishing a random SEM algorithm to estimate the component parameters; s, combining a sequence system signature and a conditional probability vector to obtain a life estimation sample; m, introducing the estimated sample into a likelihood function and maximizing an update parameter; repeating the step S and the step M, and iterating to obtain a component parameter estimated value;
fourth, construct confidence intervals, bilateral 100 (1- α)% confidence intervals for component distribution parameters μ and σ, may be constructed as follows:
further, the third step specifically includes:
(1) The right truncated density formula and the left truncated density formula can be written according to the conditional density of the service life of the sequential parts, and are respectively as follows:
(2) For the r-l+1 systems from the first to the r failure in the m systems;
(3) Maximizing the likelihood function relative to θ yields θ (h+1) The next cycle is performed:
(4) Repeating the steps (2) and (3) to obtain theta (h) H=1, 2, …, B, discarding the first few iteration values, and averaging the remaining iteration values to obtain an estimate of θ
Further, the component life for the r-l+1 th system from the first to the r-th failures in the m systems is generated by:
(1) Kth system (based on system lifetime ordering), based on order system signature, with probability quality function
λ=1, 2, …, n produces a discrete random variable Λ, this realization being denoted λ;
(2) Conditional distribution of truncated density from left, let θ=θ (h) Generating lambda-1 followersMachine variable:
(3) Conditional distribution of right truncated density, let θ=θ (h) Generating n-lambda random variables:
(4) Finally, a pseudo-complete sample of the system k is obtained
(5) Wherein the method comprises the steps of
Repeating steps (1) - (3) for k=l, …, r to obtain pseudo-complete samples (r-l+1) n for each of the first l-1 of the m systems, and the last m-r systems.
Further, the obtaining pseudo-complete samples (r-l+1) n for each of the first l-1 of the m systems, and the last m-r systems, the component life is generated by:
(1) Based on conditional probability vector p (j:m) As a probability mass function
δ=0, 1, …, n-1 yields a discrete random variable Δ, known as τ=t r:m ；
(2) Conditional distribution of truncated density from left to truncated point t k ＝t r:m ,θ＝θ (h) Generating delta random variables
(3) From right to intercept density conditional distribution to intercept point t k ＝t r:m ,θ＝θ (h) Generating n-delta random variables
(4) Obtaining the j thPseudo-complete sample of tail-biting system (part life)
(5) Repeating the steps (1) - (4) to obtain an estimated sample
l-1,r+1,r+2,…，m
Another object of the present invention is to provide a type II double-sided deletion system to which the random SEM algorithm-based component lifetime distribution parameter estimation method is applied.
Another object of the present invention is to provide a vehicle engineering component life distribution parameter estimation system to which the random SEM algorithm-based component life distribution parameter estimation method is applied.
Another object of the present invention is to provide a material engineering component life distribution parameter estimation system to which the random SEM algorithm-based component life distribution parameter estimation method is applied.
Another object of the present invention is to provide an electronic engineering component life distribution parameter estimation system to which the random SEM algorithm-based component life distribution parameter estimation method is applied.
It is another object of the present invention to provide a computer to which the random SEM algorithm-based component lifetime distribution parameter estimation method is applied.
In summary, the invention has the advantages and positive effects that: the invention considers the situation that the component life distribution parameter estimation is subjected to II-type bilateral deletion, is indispensable in the actual component life distribution parameter estimation, and has wider application range compared with other deletion situations. The method uses three estimation methods based on maximum likelihood estimation, regression estimation and random SEM algorithm, and uses order statistics to reduce the influence of deleted data; the three estimation methods complement each other, so that the parameter estimation of the component can achieve better effects on point estimation and interval estimation. The method can effectively solve the problem that the service life distribution parameters of the system components are estimated by bilateral deletion of type II. Compared with the prior art, the method has the advantages of higher estimation accuracy and wider application range.
Drawings
FIG. 1 is a flowchart of a method for estimating component lifetime distribution parameters based on a random SEM algorithm according to an embodiment of the present invention.
Fig. 2 is a graph of four data obtained by simulation of the system 1 according to the embodiment of the present invention by using a random SEM method when the system number m=10 and the deletion rate q=0% -50%.
Fig. 3 is a graph of four data obtained by simulation of the system 2 according to the embodiment of the present invention by using a random SEM method when the system number m=10 and the deletion rate q=0% -50%.
Fig. 4 is a graph of four data obtained by simulation of the system 3 according to the embodiment of the present invention by using a random SEM method when the system number m=10 and the deletion rate q=0% -50%.
Fig. 5 is a graph of four data obtained by simulation of the system 1 according to the embodiment of the present invention by using a random SEM method when the system number m=50 and the deletion rate q=0% -90%.
Fig. 6 is a graph of four data obtained by simulation of the system 2 according to the embodiment of the present invention using a random SEM method when the system number m=50 and the deletion rate q=0% -90%.
Fig. 7 is a graph of four data obtained by simulation of the system 3 according to the embodiment of the present invention by using a random SEM method when the system number m=50 and the deletion rate q=0% -90%.
Fig. 8 is a four-item data graph of the system 2 according to the embodiment of the present invention, which is simulated by using the MLE method when the system number m=10 and the deletion rate q=0% -50%.
Fig. 9 is a four-item data graph of the system 2 according to the embodiment of the present invention, which is simulated by using the MLE method when the system number m=50 and the deletion rate q=0% -90%.
Fig. 10 is a graph of four data simulated by REG method when the system number m=10 and the deletion rate q=0% -50% in the system 2 according to the embodiment of the present invention.
Fig. 11 is a graph of four data simulated by REG method when the system number m=50 and the deletion rate q=0% -90% in the system 2 according to the embodiment of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the following examples in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention.
The parameter estimation method aiming at the service life of the existing system component is a second-order bias method which is complicated in calculation process or by a numerical method requiring huge calculation amount and can not ensure the convergence of the numerical method; in the prior art, the problem that the situation of type II bilateral deletion is less and the application range is limited to a certain extent is considered. The invention adopts a cumulative distribution function of the service life of the component and a system signature vector; a Monte Carlo simulation experiment under two conditions of a small sample and a medium-to-large sample is adopted; in particular to a method for estimating component life distribution parameters of a type II double-side deleting system based on a random SEM algorithm.
The principle of application of the invention is described in detail below with reference to the accompanying drawings.
As shown in fig. 1, the method for estimating the component life distribution parameters based on the random SEM algorithm according to the embodiment of the present invention includes the following steps:
s101: solving a survival function of the service life of the system by using a cumulative distribution function of the service life of the components and a system signature vector;
s102: estimating the parameters of the component by using a random SEM algorithm, S step, combining a sequence system signal and a conditional probability vector to obtain a life estimation sample, and M step, bringing the estimation sample into a likelihood function and maximizing an update parameter;
s103: and carrying out Monte Carlo simulation experiments under two conditions of a small sample and a medium-to-large sample of the system, and generating four indexes of deviation and mean square error of point estimation, coverage rate and expected width of a confidence interval.
The component life distribution parameter estimation method based on the random SEM algorithm provided by the embodiment of the invention specifically comprises the following steps:
step one, calculating a system signature vector S according to a system structure function Γ (s 1 ,s 2 ...s n ) Order signature vector
Assuming omega is arranged for the ith system failure, the ith component of the signature vector of the system is obtained as follows:
assuming event a is that the kth order system failure is caused by the ith order component failure. M system lives are generated for a given system and the failure of which component is observed to cause the system to fail, and the process is repeated N times. The order system signature vector may be approximated as:
step two, using the cumulative distribution function F of the component life X And System signature vector S Γ Survival function representing system lifetime
For n independent co-distributed component systems, X represents a random variable of component life and T represents a random variable of system life; survival function of System lifetime->
Expressed as:
step three, establishing a random SEM algorithm to estimate the component parameters; s, combining a sequence system signature and a conditional probability vector to obtain a life estimation sample; and M, introducing the estimated samples into a likelihood function and maximizing an update parameter. Repeating the step S and the step M, and iterating to obtain a component parameter estimated value;
and step four, constructing a confidence interval. The bilateral 100 (1- α)% confidence interval for the component distribution parameters μ and σ may be constructed as follows:
in a preferred embodiment of the present invention, the third step specifically includes:
(1) The right truncated density formula (6) and the left truncated density formula (7) can be written according to the conditional density of the service life of the sequential parts, and are respectively as follows:
(2) For the r-l+ 1 systems from the first to the r failure in the m systems, the component life is generated by:
ikth system (based on system life ordering), based on order system signature, with probability mass function
λ=1, 2, …, n produces a discrete random variable Λ, which is denoted as λ.
II conditional distribution of left cut-off density, let θ=θ (h) Generating lambda-1 random variables:
III conditional distribution of right cut-off density, let θ=θ (h) Generating n-lambda random variables:
IV finally obtaining a pseudo-complete sample of system k
V wherein
Repeating step (I-III) for k=l, …, r to obtain pseudo-complete samples (r-l+1) n
For each of the first l-1 of the m systems, and the last m-r systems, the component life is generated by:
i is based on a conditional probability vector p (j:m) As a probability mass function
δ=0, 1, …, n-1 yields a discrete random variable Δ, known as τ=t r:m ；
II conditional distribution of truncated density from left to break point t k ＝t r:m ,θ＝θ (h) Generating delta random variables
III from the right truncated Density Condition to truncated Point t k ＝t r:m ,θ＝θ (h) Generating n-delta random variables
IV obtaining a (part life) pseudo-complete sample of the jth tail-biting system
V repeating steps (I-IV) to obtain an estimated sample
j＝1,2,…，l-1,r+1,r+2,…，m；
(3) Maximizing the likelihood function relative to θ yields θ (h+1) To perform the next cycle:
(4) Repeating the steps (2) and (3) to obtain theta (h) H=1, 2, …, B, discarding the first few iteration values, and averaging the remaining iteration values to obtain an estimate of θ
The principle of application of the invention is further described below with reference to the accompanying drawings.
The method for estimating the service life distribution parameters of the system component under the II-type bilateral deleting system based on the signature vector provided by the embodiment of the invention comprises the following steps:
(1) Derivation of system lifetime distribution function and likelihood function
The random variable of the component lifetime is denoted by X, and the random variable of the system lifetime is denoted by T. The cumulative distribution function (cdf), probability density function (pdf), survival function (sf) of X are expressed as
T cumulative distribution function (cdf), probability density function (pdf), survival function (sf) are denoted as +.>
For n independent co-distributed component systems, the cumulative distribution function of component life represents the survival function of system life:
wherein s is i I element, s of the design vector for an n-component system i ＝Pr(T＝X i：n )，
Because the failure sequence of n independent co-distributed components has n-! One possible setting, each with a specific order of component failure. Then assuming omega of the ith system failure arrangement, the ith component of the system signature vector is finally obtained as s i ＝ω/n！。
By combining densities of the order statistics (X (l), X (2), …, X (r)), a survival function of the system lifetime is derived using the assumed cumulative lifetime distribution function (weibull distribution) of the individual components of the system:
let θ= (μσ)' be the parameter vector. The likelihood function of theta based on the life data of the II-type bilateral deletion system is obtained through the survival function and is as follows:
to estimate θ, the present invention maximizes (11), or equivalently equation (12):
(2) Derivation of conditional density formulas for order signature and conditional order components
Consider the life X of n components in the kth system (m total systems) k1 ，X k2 ，...，X kn K=1, 2,..m, corresponding order component life X k，1：n ＜X k，2：n ＜...＜X k，n：n Intuitively, a system that fails first out of m systems is more likely to be caused by critical component failure. If the arrangement of the system life is given, the life is T a：m And T b：m The system signature vectors of (a < b) should be different. More precisely, balakrishnan et al will kth order System lifetime T k：m Is expressed as a systematic signature vector of (a)
Wherein:
the above equation is a vector that represents the probability that the failure of the kth system is due to its ith order component failure. Order the
Representing the number of failed systems caused by the failure of the ith sequence component,
then->
Can be expressed as:
wherein the method comprises the steps of
Indicating the conditional probability that the ith order component failure causes the kth system failure, known +.>
The system failure is caused by the failure of its jth order component. If the system life is independently distributed (iid), then->
Can be expressed as:
(3) Conditional density formula for sequence components
Assuming that the lambda-th component failure in the kth system causes system failure, denoted as t k ＝x k,λ:n Then the conditional distribution of the remaining n-1 components is a random variable of the right truncated or left truncated distribution. In particular, t is known to be k ＝x k,λ:n Front lambda-1 sequence part life x k,1:n ,x k,2:n ,…,x k,(λ-1):n The conditional density of (2) is a right cut-off density:
t is also known as k ＝x k,λ:n Last n-lambda sequential component life x k,(λ+1):n ,x k,(λ+2):n ,…,x k,n:n The conditional density of (2) is a left cut-off density:
conditional probability vector:
(4) Principle of estimation method based on random SEM
(i) For the r-l+ 1 systems from the first to the r failure in the m systems, the component life is generated by:
ikth system (based on system life ordering), based on order system signature, with probability mass function
λ=1, 2, …, n produces a discrete random variable Λ, which is denoted as λ.
II conditional distribution of left cut-off density, let θ=θ (h) Generating lambda-1 random variables:
III conditional distribution of right cut-off density, let θ=θ (h) Generating n-lambda random variables:
IV finally obtaining a pseudo-complete sample of system k
V wherein
Repeating step (I-III) for k=l, …, r to obtain pseudo-complete samples (r-l+1) n
For each of the first l-1 of the m systems, and the last m-r systems, the component life is generated by:
i is based on a conditional probability vector p (j:m) As a probability mass function
δ=0, 1, …, n-1 yields a discrete random variable Δ, known as τ=t r:m ；
II conditional distribution of truncated density from left to break point t k ＝t r:m ,θ＝θ (h) Generating delta random variables
III from the right truncated Density Condition to truncated Point t k ＝t r:m ,θ＝θ (h) Generating n-delta random variables
IV obtaining a (part life) pseudo-complete sample of the jth tail-biting system
V repeating steps (I-IV) to obtain an estimated sample
j＝1,2,…，l-1,r+1,r+2,…，m；
(ii) Maximizing the likelihood function relative to θ yields θ (h+1) To perform the next cycle:
(iii) Repeating the steps (i) and (ii) to obtain theta (h) H=1, 2, …, B, discarding the first few iteration values, and averaging the remaining iteration values to obtain an estimate of θ
Description of tracking problem, i.e., assuming m systems with n parts placed in life test, signature of each system is (s 1 ,s 2 ...s n ). In the life test, starting from the first system failure experiment, the experiment is terminated when the r-th system fails, wherein l and r are 1 < l < r < m determined by the experimenter. The observed failure time of the sequence system is T l:m ＜T l+1:m ＜…＜T r:m And obtaining a type II bilateral deletion system life sample. The invention is mainly used for researching the estimation of parameters of the service life distribution of the component by observing service life data of the system, so that all service life test data are assumed to be obtained.
How to make accurate parameter estimation of the component life distribution of the system is the primary task to be solved. Inaccuracy in parameter estimation of component life distribution is mainly due to the fact that data loss occurs in the observed samples. In an actual life test, the failure time of all systems is often not recorded completely, and the failure time of the systems is still not failed beyond the life test time, so that the right tail of the observed data is deleted, and the failure time of the first few systems is too short to record or lose, so that the left tail of the observed data is deleted. These missing data may render inaccurate parameter estimation of the component life distribution. Therefore, in the parameter estimation of the service life distribution of the system component, the influence of different deletion conditions needs to be considered, so that the accurate estimation of the parameter of the service life distribution of the component is ensured. In addition, the amount of the deleted data also affects the parameter estimation of the component life distribution, and the more the deleted data is, the more inaccurate the estimation is. On the other hand, the influence of the algorithm itself is considered, the accuracy of the point estimation and the interval estimation generated by different algorithms is different, and the estimation deviation is different.
The principle of application of the invention is further described below in connection with specific embodiments.
Assuming that in life test, the observed failure time of the serial system is T l:m ＜T l+1:m ＜…＜T r:m If the data are directly used as complete data to carry out parameter estimation, the influence of the data deleted on the two sides on the component parameters is ignored. It is therefore necessary to supplement the observed test data to obtain a pseudo-complete sample.
The method for estimating the component life distribution parameters of the II type bilateral deleting system based on the random SEM algorithm provided by the embodiment of the invention specifically comprises the following steps:
step one, calculating a system signature vector S according to a system structure function Γ (s 1 ,s 2 ...s n ) Order signature vector
Assuming omega is arranged for the ith system failure, the ith component of the signature vector of the system is obtained as follows:
assuming event a is that the kth order system failure is caused by the ith order component failure. M system lives are generated for a given system and the failure of which component is observed to cause the system to fail, and the process is repeated N times. The order system signature vector may be approximated as:
step two, using the cumulative distribution function F of the component life X And System signature vector S Γ Survival function representing system lifetime
For n independent co-distributed component systems, X represents the random variable of component life and T represents the random variable of system life. Survival function of System lifetime->
The method comprises the following steps:
step three, a random SEM algorithm is established to estimate the component parameters: s, combining the order system signature and the conditional probability vector to obtain a life estimation sample, and M, bringing the estimation sample into a likelihood function and maximizing an update parameter. Repeating the step S and the step M, and iterating to obtain the component parameter estimated value.
And step four, constructing a confidence interval. The bilateral 100 (1- α)% confidence interval for the component distribution parameters μ and σ can be constructed as follows
Further, the third step specifically includes:
(1) The right truncated density formula (6) and the left truncated density formula (7) can be written according to the conditional density of the service life of the sequential parts
(2) For the r-l+ 1 systems from the first to the r failure in the m systems, the component life is generated by:
ikth system (based on system life ordering), based on order system signature, with probability mass function
λ=1, 2, …, n produces a discrete random variable Λ, which is denoted as λ.
II conditional distribution of left cut-off density, let θ=θ (h) Generating lambda-1 random variables:
III conditional distribution of right cut-off density, let θ=θ (h) Generating n-lambda random variables:
IV finally obtaining a pseudo-complete sample of system k
V wherein
Repeating step (I-III) for k=l, …, r to obtain pseudo-complete samples (r-l+1) n
For each of the first l-1 of the m systems, and the last m-r systems, the component life is generated by:
i is based on a conditional probability vector p (j:m) As a probability mass function
δ=0, 1, …, n-1 yields a discrete random variable Δ, known as τ=t r:m ；
II conditional distribution of truncated density from left to break point t k ＝t r:m ,θ＝θ (h) Generating delta random variables
III from the right truncated Density Condition to truncated Point t k ＝t r:m ,θ＝θ (h) Generating n-delta random variables
IV obtaining a (part life) pseudo-complete sample of the jth tail-biting system
V repeating steps (I-IV) to obtain an estimated sample
j＝1,2,…，l-1,r+1,r+2,…，m；/>
(3) Maximizing the likelihood function relative to θ yields θ (h+1) To perform the next cycle:
(4) Repeating the steps (2) and (3) to obtain theta (h) H=1, 2, …, B sequence, discard the first fewIterative values, and averaging from the remaining iterative values to obtain an estimate of θ
The application effect of the invention is described in detail below in connection with a simulation experiment based on Monte Carlo:
1. simulation conditions
Three systems with different signature and system structure functions are considered in simulation research:
System 2 is a 4-component parallel-serial system with system signature of (1/4, 1/2, 0) and structure function of ψ (X) =min { X 1 ,max{X 2 ,X 3 ,X 4 }}。
System 3 is a 4-component hybrid parallel system with system signature of (0, 1/2, 1/4) and structure function of ψ (X) =max { X 1 ,min{X 2 ,X 3 ,X 4 }}。
The Weibull distribution is a widely used lifetime distribution in survivability modeling due to its flexibility. Let Y be a random variable subject to Weibull distribution, then the cumulative distribution function of X is:
since the random variable x= lnY obeys the minimum extremum distribution, the minimum extremum distribution belongs to the family of position scale distributions. Therefore, the logarithm of the service life is more convenient to calculate. The cumulative distribution function of the random variable x= lnY is:
assuming that the component life distribution of the three systems obeys the Weibull distribution, the logarithmic life function obeys the minimum value distribution, the position parameter mu=1 and the scale parameter sigma=0.5 are set for simulation study.
The experiment of Monte Carlo size samples is carried out on the component life through parameter estimation by adopting a random SEM algorithm, the simulation process is repeated 10000 times, the deviation and the mean square error of the point estimation, the coverage rate and the expected width of the confidence interval are given based on 10000 times of simulation, and the results are calculated by the following formulas:
estimating deviation:
mean square error:
coverage rate:
confidence interval average width:
ideal results: the smaller the estimated bias, mean square error, confidence interval average width, the better, and the larger the coverage ratio, the better.
2. Small sample simulation
For small sample simulation, the invention considers the situation that m=10 systems exist in one life test experiment, and obtains a type II double-side deleted sample data from the beginning of the record of the failure of the first system to the stopping of the failure of the r system. Where l and r are determined by the erasure rate q, as in equation (27), equation (28):
in small sample simulation, because the data are less, when the deletion rate is more than 50%, only a few data can be utilized, the deviation of parameter estimation can be greatly increased, and reasonable estimation accuracy cannot be ensured. Therefore, in small samples, only cases where the erasure rate is 50% -0% are considered. Calculating the estimated deviation, the mean square error, the coverage rate of the confidence interval and the expected width of the small sample. The results of the operation of systems 1-3 are shown in fig. 2-4.
3. Medium to large sample emulation
For medium to large sample simulation, the invention considers the situation that m=50 systems exist in one life test experiment, and similar to a small sample, the invention also obtains a type II double-side deleted sample data from the beginning of the failure record of the first system to the stopping of the failure of the r system. Where l and r are the same as the erasure rate q as shown in equations (27) (28). The estimated bias, mean square error, coverage of confidence interval and expected width of the medium to large samples are calculated. The results of the operation of systems 1-3 are shown in fig. 5-7.
Analysis of the above estimation results generated by random SEM algorithm, and comparison of the simulation result graph results can find that the results of the systems 1,2, 3 expressed in random SEM simulation are similar, and considering this fact, the system 2 (4-component parallel-serial system, system signature is (1/4, 1/2, 0), and the structure function is ψ (X) =min { X ] 1 ,max{X 2 ,X 3 ,X 4 Analysis was performed as a representative and compared with the prior art using a method based on maximum likelihood and a regression-based estimation method.
1. For small sample analysis
In terms of estimated deviation, the invention discovers that the estimated deviation obtained by a random SEM algorithm tends to zero, and the deviation of SEM to mu and sigma is negative; in terms of mean square error, the random SEM algorithm can control the mean square error to be within 0.05, while the traditional method can only control the deviation to be within 0.8. The random algorithm is significantly better for μ and σ estimation than the previous method. In terms of coverage rate, a random SEM algorithm can ensure reasonable coverage rate; the confidence interval estimation for μmay be within 0.8 and the confidence interval estimation for σ may be within 0.5 over the average width length of the confidence interval.
2. For large sample analysis
In terms of estimated deviation, the estimated deviation of a large sample also tends to be zero, positive and negative deviations are consistent with the appearance of a small sample, and the estimated deviation is still that the random SEM is negative to mu and sigma deviations; in the aspect of the mean square error, the mean square error of a large sample is obviously smaller than that of a small sample, and the mean square error can be controlled within the range of 0.005; in terms of coverage, a large sample can obtain higher coverage; the estimation of the large sample is also less accurate over the average width of the confidence interval, the confidence interval of mu is controlled to be 0.30, and the confidence interval of sigma can be controlled to be within 0.45.
3. Overall analysis
In the simulation, increasing the erasure rate deteriorates the performance of both the point estimation and the section estimation, specifically, the variance estimation MSE increases, the coverage CR decreases, and the confidence section WCI becomes longer as the erasure rate increases.
By comparing the small sample with the large sample, the estimated deviation gradually approaches zero when the small sample and the large sample are compared, and the positive and negative deviations are consistent in the large sample and the small sample, and are negative. The estimation effect of large samples can be found to be much better than that of small samples in terms of mean square error, coverage and confidence interval. The number of data samples plays a great role in estimation, and the more the number of samples, the more accurate the estimation. The coverage rate is too small, and when the deletion rate is larger than 50%, the coverage rate cannot reach a reasonable range, so that the deletion rate of the required data cannot be too large when the parameter estimation is performed on the small sample.
The same simulation was performed by the conventional estimation method, and the simulation results based on Maximum Likelihood (MLE) are shown in fig. 8 and 9, and the simulation results based on Regression (REG) are shown in fig. 10 and 11, which are compared with the random SEM algorithm. The estimation accuracy of the random SEM algorithm on point estimation is higher than that of the prior maximum likelihood estimation and regression-based estimation. The performance of the section estimation is not higher than that of the conventional method. The random SEM algorithm is therefore more suitable for point estimation for type II double-sided deletion systems.
The foregoing description of the preferred embodiments of the invention is not intended to be limiting, but rather is intended to cover all modifications, equivalents, and alternatives falling within the spirit and principles of the invention.
Claims (9)
1. The component life distribution parameter estimation method based on the random SEM algorithm is characterized by comprising the following steps of:
(1) Solving a survival function of the service life of the system by using a cumulative distribution function of the service life of the components and a system signature vector;
(2) Estimating the parameters of the component by using a random SEM algorithm, S step, combining a sequence system signal and a conditional probability vector to obtain a life estimation sample, and M step, bringing the estimation sample into a likelihood function and maximizing an update parameter;
(3) Carrying out Monte Carlo simulation experiments under two conditions of a small sample and a medium-to-large sample of the system and generating four indexes of deviation and mean square error of point estimation, coverage rate and expected width of a confidence interval;
the component life distribution parameter estimation method based on the random SEM algorithm specifically comprises the following steps:
step one, calculating a system signature vector S according to a system structure function Γ (s 1 ,s 2 …s n ) Order signature vector
Omega ith system failures are arranged, and the ith component of the signature vector of the system is obtained as follows:
event a is that the kth order system failure is caused by the ith order component failure; generating m system lives for a determined system, observing which component fails to cause the system to fail, and repeating the process for N times; the order system signature vector may be approximated as:
step two, using the cumulative distribution function F of the component life X And System signature vector S Γ Survival function representing system lifetime
For n independent co-distributed component systems, X represents a random variable of component life and T represents a random variable of system life; survival function of System lifetime->
Expressed as:
step three, establishing a random SEM algorithm to estimate the component parameters; s, combining a sequence system signature and a conditional probability vector to obtain a life estimation sample; m, introducing the estimated sample into a likelihood function and maximizing an update parameter; repeating the step S and the step M, and iterating to obtain a component parameter estimated value;
fourth, construct confidence intervals, bilateral 100 (1- α)% confidence intervals for component distribution parameters μ and σ, may be constructed as follows:
2. the method for estimating component lifetime distribution parameters based on random SEM algorithm according to claim 1, wherein said step three specifically comprises:
(1) The right truncated density formula and the left truncated density formula can be written according to the conditional density of the service life of the sequential parts, and are respectively as follows:
(2) For the r-l+1 systems from the first to the r failure in the m systems;
(3) Maximizing the likelihood function relative to θ yields θ (h+1) The next cycle is performed:
(4) Repeating the steps (2) and (3) to obtain theta (h) H=1, 2, …, B, discarding the first few iteration values, and averaging the remaining iteration values to obtain an estimate of θ
3. The method for estimating component lifetime distribution parameters based on random SEM algorithm according to claim 2, wherein said component lifetime is generated for the r-l+1 systems from the i-th to the r-th failures of the m systems by:
(1) The kth system is based on system life ordering, based on order system signature, with probability quality function
Generating a discrete random variable Λ, representing this implementation as λ;
(2) Conditional distribution of truncated density from left, let θ=θ (h) Generating lambda-1 random variables:
(3) Conditional distribution of right truncated density, let θ=θ (h) Generating n-lambda random variables:
(4) Finally, a pseudo-complete sample of the system k is obtained
(5) Wherein the method comprises the steps of
Repeating steps (1) - (3) for k=l, …, r to obtain pseudo-complete samples (r-l+1) n, for each of the first l-1 and the last m-r of the m systems.
4. A method for estimating component lifetime distribution parameters based on random SEM algorithms according to claim 3, wherein said obtaining pseudo-complete samples (r-l+1) n pairs of the first l-1 of m systems, and the last m-r systems, component lifetime is generated by:
(1) Based on conditional probability vector p (j:m) As a probability mass function
Generating a discrete random variable delta, known as τ=t r:m ；
(2) Conditional distribution of truncated density from left to truncated point t k ＝t r:m ,θ＝θ (h) Generating delta random variables
(3) From right to intercept density conditional distribution to intercept point t k ＝t r:m ,θ＝θ (h) Generating n-delta random variables
(4) Obtaining a pseudo-complete sample of component life of a jth tail-biting system
(5) Repeating the steps (1) - (4) to obtain an estimated sample
5. A type II double-sided deletion system applying the random SEM algorithm-based component lifetime distribution parameter estimation method of any one of claims 1 to 4.
6. A vehicle engineering component life distribution parameter estimation system applying the random SEM algorithm-based component life distribution parameter estimation method according to any one of claims 1 to 4.
7. A material engineering component life distribution parameter estimation system applying the random SEM algorithm-based component life distribution parameter estimation method according to any one of claims 1 to 4.
8. An electronic engineering component life distribution parameter estimation system applying the random SEM algorithm-based component life distribution parameter estimation method according to any one of claims 1 to 4.
9. A computer applying the random SEM algorithm-based component lifetime distribution parameter estimation method of any one of claims 1 to 4.
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