CN102779208A - Sequential accelerated degradation test optimal design method based on relative entropy - Google Patents

Abstract

The invention discloses a sequential accelerated degradation test optimal design method based on relative entropy, comprising the specific steps of: step 1, utilizing a Bayesian theory to establish an accelerated degradation test optimal design method based on the relative entropy; step 2, establishing a sequential truncation judging method; and step 3, carrying out sequential accelerated degradation test based on the relative entropy. According to the method disclosed by the invention, a 'sequential design' is introduced to an optimal design of the accelerated degradation test for the first time and the sequential accelerated degradation test optimal design method is provided. With the adoption of the 'sequential design', not only can prior information before the test be sufficiently utilized, but also performance degradation information obtained in the test is gradually utilized, and a test design error caused by that the deviation between the prior information and a product real condition is larger is reduced, so that compared with a partial optimal design and Bayesian optimal design of the accelerated degradation test, the sequential accelerated degradation test optimal design method based on the relative entropy has the greater advantage.

Description

Sequential accelerated degradation test Optimization Design based on relative entropy

Technical field

The present invention is a kind of sequential accelerated degradation test Optimization Design based on relative entropy, belongs to the accelerated degradation test technical field, is used to solve the technical matters in reliability and systems engineering field.

Background technology

In current reliability engineering both at home and abroad; The life-span of long-life high reliability products such as Aero-Space electronic product, photovoltaic and reliability assessment have become a research difficult problem; Accelerated life test (Accelerated Life Testing; ALT) and accelerated degradation test (Accelerated Degradation Testing ADT) is the gordian technique that solves this difficult problem at present.Wherein ADT has overcome the shortcoming of ALT record product fail data, becomes the fail-test hot research fields.

The accelerated degradation test technology at first must be faced with the design problem of testing program in practical applications; Promptly how under limited time and expense scientific and reasonable arrangement proof stress level, receive test variablees such as sample basis, test period; To obtain the most effectively performance degradation information, make life of product and reliability assessment the most accurate.If can solve this problem, the testing program of adopt optimizing not only can obtain product life-span and reliability assessment result accurately, for product development and user provide correct decision-making foundation; Can also improve test efficiency greatly, the test resource is fully used, reduce the development cost of product.Therefore study the accelerated degradation test optimal design and have important theory value and practical applications value.

The ADT optimal design is according to the different design of ADT local optimum, ADT Bayes optimal design and the sequential ADT optimal design of being divided into of the utilization of prior imformation and Test Information.

For most of products, all can there be a large amount of prior imformations usually.Such as, the historical test data of like product or the actual data of using; Product is from the performance test of before ADT, carrying out, environmental test etc.The accelerated degradation test optimal design depends on the information of model and model unknown parameter, according to different priori conditions, can obtain different designs.When model (degradation model and acceleration model) is known,, claim that this estimated value is the assumed value (or default) of parameter as if the estimated value that can provide model parameter according to prior imformation and engineering experience etc.On this basis, select for use different optimization criterions to obtain the optimum test scheme, be called classics or local accelerated degradation test optimal design (Locally optimal design), be also referred to as traditional accelerated degradation test optimal design.When the estimated value deviation of parameter was big, then the scheme that obtains of local optimum design was not an optimal case.Bayes's test design is to utilize prior imformation to confirm prior distribution but not the concrete value of model parameter, based on this, sets up Bayes and optimizes criterion and obtain optimal case, thereby avoided the shortcoming of local optimum design.Bayes's optimization is called average (average) again and optimizes or overall (population) optimization.Bayes's optimal design is to be the basis with the optimizing decision theory under the condition of uncertainty to a certain extent, is a kind of decision theory, is important branch of statistical theory.Its objective is the expected utility of maximization test.The default of parameter and prior distribution are the important difference of local optimum design and Bayes's optimal design.

In the test of traditional reliability statistics, classifying by the truncation mode to be divided into fixed time test, fixed failure number test and sequential truncation test.Fixed time test is meant n sample is made an experiment, the truncated time of regulation test in advance t ₀, to moment t ₀All test specimens stop test, utilize test figure assessment reliability of products characteristic quantity; Fixed failure number test is meant n sample is made an experiment, the number of faults r of the truncation of regulation test in advance, and test proceeds to and occurs till r the fault, utilizes test figure assessment reliability of products characteristic quantity; Sequential truncation test is by the acceptance of drafting in advance, rejection and truncated time line; At duration of test; To observed continuously by trial product, and the criterion that reception, rejection or the continuation of correlation test time of accumulation and number of faults and regulation are tested done a kind of test of comparison.In these three kinds of tests, all be to have formulated scheme before the test with fixed failure number test regularly, clear and definite sample size and the truncated time or the truncation number of faults of test enforcement no longer adjusted scheme in the process of the test.We might as well claim be designed to " Static Design " of this type testing program.Along with carrying out of test, we should constantly sum up the information that from test, obtains, and in conjunction with Mathematical Statistics Analysis, progressively understand the understanding product, therefore, constantly adjust testing program, and " limit test, limit arrangement " is only rational test design method.Sequential truncation test is exactly to adopt this thinking; Before test, do not stipulate concrete given the test agent number and truncated time; But in process of the test, the criterion comparison of Test Information such as the accumulation test period that progressively utilize to obtain and number of faults and regulation, and then make the judgement of next step test; Therefore can early obtain conclusion (of pressure testing), thereby reduce experimentation cost.Claim that this conceptual design is " dynamic design ".

In the accelerated test conceptual design, be generally and reduce life of product and reliability extrapolation error under the normal stress, can select 3 or 3 above stress levels to implement test.In implementation process,, can progressively respectively be quickened the True Data that properties of product are degenerated under the stress level along with the carrying out of test.Therefore, also can according to the utilization of Test Information, the testing program design be divided into " Static Design " and " dynamic design " according to aforesaid way.Design of ADT local optimum and ADT Bayes optimal design all belong to " Static Design ", and sequential ADT optimal design then belongs to " dynamic design ".

The existing big quantity research of ADT local optimum design:

To CSADT; H-F Yu etc. studied that degradation ratio obeys respectively that Weibull reciprocal distributes and the lognormal distribution situation under the Optimization Design (list of references [1]: Yu H.F.Designing an accelerated degradation experiment with a reciprocal Weibull degradation rate.Journal of statistical planning and inference of accelerated degradation test; 2006,136:282-297).Ying Shi; Luis A Escobar and W Q Meeker has studied the optimal design of the degradation experiment that accelerates the failure; To minimize the progressive variance of specifying out-of-service time quantile maximum likelihood to estimate is optimization aim, and (general equivalence theorem GET) verifies the optimality (list of references [2]: Ying Shi of different schemes to adopt general equivalence theorem; Luis A Escobar; William Q Meeker.Accelerated Destructive Degradation Test Planning.Technometrics, 2009,51 (1): 1-13).Yashun Wang etc. are based on the melange effect model; Dividing the position estimated mean-square with minimize production Life Distribution p is target, is constraint with testing expenses, adopts Monte Carlo emulation; Studied the optimal design (list of references [3]: Yashun Wang of ADT; Chunhua Zhang, Xun Chen, Yongqiang Mo.Simulation-based optimal design for accelerated degradation tests.ICRMS.July 2009:1302-1306).To SSADT; S-T Tseng etc. based on the Gamma process study optimization of SSADT (list of references [4]: Tseng S.T.; Narayanaswamy Balakrishnan; Chih-Chun Tsai.Optimal Step-Stress Accelerated Degradation Test Plan for Gamma Degradation Processes.IEEE Transactions on Reliability, 2009,58 (4): 611-618).Xiaoyang Li is a target with the progressive variance that minimizes competition reliability model p branch position life estimation under the normal stress; Studied the optimization method (list of references [9]: Xiaoyang Li of considering the SSADT under the competition inefficacy mechanism; Tongmin Jiang.Optimal Design for Step-Stress Accelerated Degradation Testing with Competing Failure Modes Proceedings Annual Reliability and Maintainability Symposium, 2009:64-68).

The research of ADT Bayes optimal design mainly contains:

In " Bayesian reliability " book, author Michael S.Hamada, Alyson G.Wilson etc. are based on mixing utility models; Distributing as the priori of melange effect model parameter with normal distribution and contrary gamma, is target with the Reliability Estimation precision, and test sample amount and Performance Detection number of times have been carried out optimal design (list of references [10]: Michael S.Hamada; Alyson G.Wilson, C.Shane Reese, Harry F.Martz.Bayesian Reliability.Springer Science+Business Media; LLC; New York, USA, 2008:330-331).Kathryn Chaloner and Isabella Verdinelli; Provided the summary (list of references [11]: Chaloner of Bayes's test design; K, Verdinelli, I..Bayesian Experimental Design:A Review.Statistical Science; 1995,10:273-304; ).Bayes's test design generally is difficult to obtain the mathematic(al) representation that posteriority distributes, two kinds of methods of therefore having derived and having addressed this problem: a kind of emulation, a kind of large sample theory that is based on of being based on.More existing scholars of ALT design based on bayesian theory make research.People such as Alaattin Erkanli have studied Bayes's design (list of references [12]: AlaattinErkanli of the constant ALT of different utility functions from the angle of decision theory; R.Soyer.Simulation-based designs for accelerated life tests.Journal of Statistical Planning and Inference; 2000,90:335-348).People such as Gladys D.C.Barriga propose under index-Weibull Life Distribution and the Arrhenius model, based on the bayes method of the ALT of Markov chain Monte Carlo (MCMC).Yao Zhang and W.Q.Meeker utilizes the approximate bayesian criterion that obtains of large sample; Having studied the Bayes that logarithm position yardstick distributes with the CSALT of the linear acceleration model of location parameter designs; And propose to seek optimal case with general equivalence theorem (GET); To solve nonlinear problem (list of references [13]: Yao Zhang and William Q.Meeker.Bayesian Methods for Planning Accelerated Life Tests.Technometrics; 2006,48 (1): 49-60).

The research of sequential accelerated life test design mainly contains:

Xiao Liu and L-C Tang etc. have proposed sequential CSALT method for designing, and according to this method, the test of at first implementing high stress is with the snatch fail data; Then; Based on Bayesian inference method, utilize these fail datas, set up the prior imformation under the low stress level.Divide the prognosis of the progressive posterior variance of position estimation to test expectation through life characteristics under the minimize production normal stress level; Make test design be able to optimize (list of references [13]: Liu X; And Tang LC.A Sequential Constant-stress Accelerated Life Testing Scheme and Its Bayesian Inference.Quality and Reliability Engineering International; 2009,25 (1): 91-109; List of references [14]: Tang LC, and Liu X Planning Sequential Constant-Stress Accelerated Life Tests With Step Wise Loaded Auxiliary Acceleration Factor.Journal of Statistical Planning and Inference.140 (2010) 1968 – 1985).Further, in their 2010 paper, Bayes's Optimization Design is proposed to sequential CSALT.With the Squared Error Loss minimum serves as to optimize criterion, based on emulation and combine curved surface fitting method to provide Optimization result.Through contrast; Proof Bayes method for designing can obviously improve the robustness of test design; Reduce uncertain (list of references [15]: Tang LC; And Liu X.Planning and Inference of a Sequential Accelerated Life Test.Journal of Quality Technology, 2010,42 (1): 103-118).

But, up to the present also do not have sequential accelerated degradation test Study on Optimal Design Method.

Application number: 201210048774.2, title: based on the accelerated degradation test Optimization Design of bayesian theory, the applying date: 2012-02-28 wherein discloses, and adopting relative entropy is the stepstress accelerated degradation test method of optimization aim.

Summary of the invention

The objective of the invention is the prior imformation of all only utilizing product for the local optimum design that solves present accelerated degradation test and Bayes's optimal design; Be " Static Design ", do not consider the performance degradation information in the process of the test, and this information reflect the problem of the degradation characteristics of product own exactly most; A kind of sequential accelerated degradation test Optimization Design based on relative entropy has been proposed; This method is optimization aim to the maximum with the relative entropy that test obtains, and adopts the stress of " step falls " to apply mode, not only utilizes the prior imformation before testing; Utilize the Test Information in each step simultaneously, constantly the follow-up test scheme is optimized design.Through making full use of the information of per step test, can make the own characteristics of product that more meet, more practice thrift the scheme of experimentation cost.

A kind of accelerated degradation test Optimization Design of the present invention based on relative entropy, concrete steps are:

Step 1, utilize bayesian theory, set up accelerated degradation test Optimization Design based on relative entropy;

Step 2, set up sequential truncation decision method;

Step 3, carry out sequential accelerated degradation test based on relative entropy;

The advantage and the good effect of the inventive method are:

(1) the inventive method is incorporated into " sequential design " in the optimal design of accelerated degradation test first, proposes sequential accelerated degradation test Optimization Design.Adopt " sequential design "; Not only made full use of the prior imformation before the test; Also progressively utilized obtained performance degradation information in the test; Reduced the test design error that causes when prior imformation and product truth exist than large deviation, therefore local optimum design and the Bayes's optimal design than accelerated degradation test all has very big superiority;

(2) the inventive method is set up the optimization criterion of sequential trials designs based on relative entropy, and this criterion is target to the maximum with test expectation information gain, has considered obtainable quantity of information in the test comprehensively, the Optimization result that the obtains engineering reality of more fitting;

(3) set up sequential trials design cycle and sequential truncation decision rule, for the engineering construction of the design of accelerated degradation test from now on provides foundation, sequential trials have the test period of saving and sample size, and the advantage that can comparatively fast enter a judgement.

Description of drawings

Fig. 1 is a sequential trials design cycle synoptic diagram of the present invention;

Fig. 2 is the process flow diagram of the sequential accelerated degradation test Optimization Design based on relative entropy of the present invention;

Fig. 3 is the sequential ADT design flow diagram of three levels of the step 2 of Optimization Design of the present invention;

Fig. 4 is the sequential ADT initial trial of three levels scheme optimization result in the embodiment of the invention;

Fig. 5 is the testing program Optimization result of sequential ADT stages 2 of three levels in the embodiment of the invention;

Fig. 6 is that the sequential ADT of three levels implements S in the embodiment of the invention ₂Back parameter beta ₁With τ prior distribution and posteriority changes in distribution situation;

Fig. 7 is that the sequential ADT of three levels implements S in the embodiment of the invention ₁Back parameter beta ₁With τ prior distribution and posteriority changes in distribution situation

Embodiment

To combine accompanying drawing and embodiment that technical scheme of the present invention is done further to specify below.

The present invention is incorporated into " sequential trials " in the accelerated degradation test design, in conjunction with bayesian theory, studies sequential accelerated degradation test optimal design.Sequential trials design cycle synoptic diagram is seen shown in Figure 1, utilizes the product prior imformation, confirms the preliminary test scheme; The stress level number of clearly implementing (representing) with K; After every enforcement one step test (promptly whenever carrying out test under the stress level) obtains True Data, utilize the prior imformation and true test figure of product, progressively follow-up test is optimized design; Make the testing program that more meets the characteristics of product own, until having implemented this K step test.Sequential trials have the test period of saving and sample size, and the advantage that can comparatively fast enter a judgement.The present invention is incorporated into " sequential design " in the accelerated degradation test optimal design; Foundation provides the concrete steps of sequential accelerated degradation test Optimization Design based on the sequential accelerated degradation test optimal design flow process framework and the sequential truncation decision method of relative entropy.

A kind of accelerated degradation test Optimization Design that the present invention proposes based on relative entropy, as shown in Figure 2, comprise following step:

Step 1, utilize bayesian theory, set up accelerated degradation test Optimization Design based on relative entropy.

(1) confirms properties of product degradation model and acceleration model, and then provide the prior distribution of model parameter based on historical data.

The properties of product degradation model is used the more three kinds of models of melange effect model, gamma process and Brownian movement (Wiener process) that mainly comprise.Acceleration model commonly used has Allan Nice (Arrhenius) model, contrary power rate model, Aileen (Eyring) model etc.; Its form all can be represented the log-linear form: wherein;

is a certain known function of stress s; For example; As far as Allan Nice (Arrhenius) model;

s=T, T is an absolute temperature; As far as contrary power rate model,

s can represent voltage, electric current, power etc.; D (s) is the performance degradation rate; A, b are constant.According to product own characteristic, sensitive stress and performance parameter degenerate case etc., confirm properties of product degradation model and acceleration model, and then the probability density function of definite degradation model and log-likelihood function etc.

According to product historical data, like product information, performance degradation amount distribution situation is in conjunction with the theoretical prior distribution of confirming unknown parameter in properties of product degradation model and the acceleration model of Bayes's conjugation prior distribution.

(2) make up testing program set D

Constitute or other condition by experimentation cost, confirm the value of sample size n, test period t and monitoring interval of delta t, by test period and the total monitoring number of times m=t/ Δ t of monitoring interval calculation.

To stepstress accelerated degradation test (SSADT),, confirm that in conjunction with the testing expenses constraint under the situation of total sample size n and overall measurement number of times m, the decision variable that needs to optimize is the horizontal S of each proof stress in confirmed test stress level scope _kWith the measurement number of times m under each stress level _k

Make S represent the proof stress vector, comprise K element among the S, K is a positive integer, S=(S ₁..., S _k..., S _K), stress level of each element representation, S _kRepresent k stress level, for example, 3 stress levels are arranged in the test, then S comprises 3 elements, S=(S ₁, S ₂, S ₃).Make M represent to measure time number vector, then M=(m ₁..., m _k..., m _K), m _kRepresent k stress level S _kUnder the measurement number of times (k=1 ..., K).

Scheme set D=S * M constitutes (S by the value space of proof stress and the value space of measurement number of times _k, m _k) be a certain scheme (or claiming design) η in the design space.

Stress level all is a continuous variable with measuring number of times, can mark off numerous scheme.Because objective function needs the bulk sampling simulation calculation, usually need move several days or even a few week.The present invention adopts the surface fitting scheme to avoid this problem for this reason.Along stress level and measurement number of times direction; Five equilibrium value in its span; The design space is comprised that boundary demarcation is that limited scheme formed the testing program set; To each computation schemes objective function in the testing program set, utilize surface fitting to find out the maximum zone of objective function, and then find out optimal case.

The present invention adopts the three-dimension curved surface fitting technique, and two independents variable are stress S ₁Compare m with the measurement number of times ₁, dependent variable is an objective function.Also being about to the optimization aim function representation is stress S ₁With measurement number of times m ₁Function.Stress S under other proof stress levels _k(1<k≤K) with measure number of times m _k(1<k≤K) provides or is expressed as S according to actual conditions ₁, m ₁Function.

To two stress levels (being also referred to as 2 designs), the decision variable of testing program is: stress level S ₁, S ₂, detect number of times m ₁, m ₂Generally get S ₂=S _Max, S ₂Under measurement number of times m ₂=m-m ₁

The situation of counter stress number of levels K>=3, the decision variable of testing program is: stress level S ₁, S ₂..., S _K, detect number of times m ₁, m ₂..., m _KMake S _k=S _Max, make the horizontal S of intermediate stress _k(1＜k＜K) is expressed as S ₁And S _kFunction, for example to temperature stress, can adopt inverse uniformly-spaced; Can adopt logarithm uniformly-spaced to electric stress.Equally, to measuring number of times, middle m _k(the available m of 1＜k＜K) ₁Function representation, then

To constant stress accelerated degradation test (CSADT), need consider that also sample size is distributed n under each stress level _kSolution is following:

Count sample dispensing n under K and each stress according to total sample number n and stress level _kConstraint condition: n _k>=n,

Provide several kinds of different sample dispensing, to every kind of sample dispensing, can handle according to the mode of above-mentioned SSADT, obtain the optimal case under every kind of sample dispensing, compare, the maximum scheme of select target value is an optimal case.

(3) foundation is based on the optimization aim of relative entropy

In theory of probability and information theory, relative entropy (relative entropy) is called KL divergence (Kullback – Leibler divergence), information entropy (information entropy) again.In Bayesian statistical theory, relative entropy is the tolerance of distance between prior distribution and posteriority distribute.Say from Shannon (Shannon) information viewpoint, relative entropy also represented the information gain that obtains through test (information gain, IG).Therefore, as utility function, (Expected Information Gain, EIG) maximum turns to optimization aim with the expectation information gain that obtains in the test with relative entropy in the present invention.

According to the research of Lindley, the information I that comprises in the prior distribution ₀For:

I ₀＝∫p(θ)log?p(θ)dθ＝E _θ?log?p(θ) （1）

Wherein, p (θ) representation model parameter prior distribution probability density function; E _θExpression is about the expectation of θ.

The gross information content I that from posteriority distributes, obtains ₁(x) be:

I ₁(x)＝∫p(θ|x)log?p(θ|x)dθ （2）

Wherein, p (θ | x) representation model parameter posteriority distribution probability density function.

The Lindley information that definition obtains from testing program η in its research is:

I(η,x,p(θ))＝I ₁(x)-I ₀ （3）

The testing program design should provide before data x obtains, and therefore need hope that then the expectation of I (η, x, p (θ)) is to the information peek term of sample space:

I(η,p(θ))＝E _x[I ₁(x)-I ₀] （4）

Wherein, E _xExpression hopes the information of sample space peek term, I (η, p (θ)) be also referred to as the expectation information gain (Expected Information Gain, EIG).Based on bayesian theory, the expectation of Test Information can be expressed as:

$\mathrm{EIG}\left(\mathrm{\&eta;}\right)=I(\mathrm{\&eta;},p\left(\mathrm{\&theta;}\right))=E_x{E}_{\mathrm{\&theta;}}\left[\log \frac{p\left(x\right|\mathrm{\&theta;})}{p\left(x\right)}\right]---\left(5\right)$

In the formula, p (x) is marginal likelihood function, also is the standard constant, and is as follows:

p(x)＝∫p(x|θ)p(θ)dθ （6）

Wherein, the likelihood function under p (x| θ) the expression parameter θ known conditions.

Therefore, the optimization aim based on relative entropy is:

$\underset{\mathrm{\&eta;}\&Element;\&Element;D}{\max }\mathrm{EIG}\left(\mathrm{\&eta;}\right)---\left(7\right)$

(4),, utilize software WinBUGS14 calculation optimization target based on Monte Carlo Markov chain (MCMC) method to each testing program.

In most cases, be difficult to obtain posterior analytical expression, so numerical simulation calculating is solution commonly used in the bayesian theory.

Expectation information gain (EIG) also can be write:

$\mathrm{EIG}\left(\mathrm{\&eta;}\right)=E_x{E}_{\mathrm{\&theta;}}\left[\log \frac{p\left(x\right|\mathrm{\&theta;})}{p\left(x\right)}\right]=E_x{E}_{\mathrm{\&theta;}}\left[\log p\left(x\right|\mathrm{\&theta;})\right]-E_x\left[\log p\left(x\right)\right]---\left(8\right)$

Because following formula is difficult to have the demonstration expression formula usually, therefore general employing Monte Carlo (Monte Carlo) emulation mode is calculated.At first, the p in the formula (8) (x| θ) is a likelihood function, can directly adopt Monte Carlo emulation mode to calculate E at parameter space and sample space _xE _θ[log p (x| θ)], computing formula is following:

$E_x{E}_{\mathrm{\&theta;}}\left[\log p\left(x\right|\mathrm{\&theta;})\right]=\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HDA00001785950000021.png"\; state="\{\{state\}\}">R</figure-callout>}}_2}\&CenterDot;{\&Sum;}_{h=1}^{{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HDA00001785950000021.png"\; state="\{\{state\}\}">R</figure-callout>}}_2}\log p\left(x_h\right|{\mathrm{\&theta;}}_h)---\left(9\right)$

R in the formula (9) ₂Be simulation times, get bigger integer usually, generally be chosen as 100.

Secondly, for marginal likelihood function p (x), the inventive method adopts the Laplace-Metropolis algorithm to estimate that computing formula is following:

$p\left(x\right)\&ap;{\left(2\mathrm{\&pi;}\right)}^{d/2}{\left|{\mathrm{\&Sigma;}}_{\mathrm{\&theta;}}\right|}^{1/2}p\left(x\right|\stackrel{\&OverBar;}{\mathrm{\&theta;}})p\left(\stackrel{\&OverBar;}{\mathrm{\&theta;}}\right)---\left(10\right)$

$\stackrel{\&OverBar;}{\mathrm{\&theta;}}=\frac{1}{R_{\mathrm{ML}}}\&CenterDot;\underset{g=1}{\overset{R_{\mathrm{ML}}}{\mathrm{\&Sigma;}}}{\mathrm{\&theta;}}_g\mathrm{and}{\mathrm{\&Sigma;}}_{\mathrm{\&theta;}}=\frac{1}{R_{\mathrm{ML}}-1}\&CenterDot;\underset{g=1}{\overset{R_{\mathrm{ML}}}{\mathrm{\&Sigma;}}}({\mathrm{\&theta;}}_g-\stackrel{\&OverBar;}{\mathrm{\&theta;}}){({\mathrm{\&theta;}}_g-\stackrel{\&OverBar;}{\mathrm{\&theta;}})}^T---\left(11\right)$

Wherein, π is a circular constant, π ≈ 3.14; D is the dimension of model parameter vector; R _MLBe the number of the performance degradation increment x of emulation, be sample size n and total product that detects number of times m, i.e. R _ML=n * m; θ _gBe based on the parameter Posterior Mean that MCMC adopts g the performance degradation increment that software WinBUGS14 calculates,

Be the θ that obtains by N emulation degeneration incremental data _gAverage; Σ _θIt is the posterior variance-covariance matrix of parameter.

Based on above-mentioned formula (8) ~ (11), following based on the solution procedure of the optimization aim of relative entropy:

Substep 1. (is total to R from the testing program set _DIndividual scheme) gets scheme η in _r, r=1 ..., R _D

2. couples of scheme η of substep _r, R is extracted in emulation from its corresponding prior distribution ₂Subparameter θ _Rh, h representes simulation times, h=1 in this step ..., R ₂, and utilize the parameter θ of each emulation _Rh, from sampling distribution f (x| θ _Rh, η _r) the middle degeneration properties of product incremental data x that generates _Rh

The degeneration incremental data x that substep 3. generates according to emulation _Rh, numerical procedure η _rLog-likelihood function log p (x _Rh| θ _Rh, η _r), and according to formula (9) numerical procedure η _rE _xE _θ[logp (x _Rh| θ _Rh, η _r)]:

$E_x{E}_{\mathrm{\&theta;}}\left[\log p\left(x_{\mathrm{rh}}\right|{\mathrm{\&theta;}}_{\mathrm{rh}},{\mathrm{\&eta;}}_r)\right]=\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HDA00001785950000021.png"\; state="\{\{state\}\}">R</figure-callout>}}_2}\&CenterDot;{\&Sum;}_{h=1}^{{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HDA00001785950000021.png"\; state="\{\{state\}\}">R</figure-callout>}}_2}\log p\left(x_{\mathrm{rh}}\right|{\mathrm{\&theta;}}_{\mathrm{rh}},{\mathrm{\&eta;}}_r);$

4. couples of scheme η of substep _r, parameter θ is extracted in emulation from its corresponding prior distribution _r, to θ _rEmulation R ₃Inferior degeneration incremental data x _Rh, h representes simulation times, h=1 in this step ..., R ₃The degeneration incremental data that obtains based on each emulation is in conjunction with the parameter Posterior Mean θ of MCMC method and g performance degradation incremental data of WinBUGS14 computed in software _g, and then obtain θ according to formula (11) _gAverage

With the posterior variance-covariance matrix Σ of parameter _θ:

$\stackrel{\&OverBar;}{\mathrm{\&theta;}}=\frac{1}{R_{\mathrm{ML}}}\&CenterDot;\underset{g=1}{\overset{R_{\mathrm{ML}}}{\mathrm{\&Sigma;}}}{\mathrm{\&theta;}}_g\mathrm{and}{\mathrm{\&Sigma;}}_{\mathrm{\&theta;}}=\frac{1}{R_{\mathrm{ML}}-1}\&CenterDot;\underset{g=1}{\overset{R_{\mathrm{ML}}}{\mathrm{\&Sigma;}}}({\mathrm{\&theta;}}_g-\stackrel{\&OverBar;}{\mathrm{\&theta;}}){({\mathrm{\&theta;}}_g-\stackrel{\&OverBar;}{\mathrm{\&theta;}})}^T;$

Substep 5. calculates marginal likelihood function p (x according to formula (10) _Rh):

$p\left(x_{\mathrm{rh}}\right)\&ap;{\left(2\mathrm{\&pi;}\right)}^{d/2}{\left|{\mathrm{\&Sigma;}}_{\mathrm{\&theta;}}\right|}^{1/2}p\left(x_{\mathrm{rh}}\right|\stackrel{\&OverBar;}{\mathrm{\&theta;}})p\left(\stackrel{\&OverBar;}{\mathrm{\&theta;}}\right);$

And then obtain

wherein; Likelihood function under

representation model parameter Posterior Mean

known conditions,

representation model parameter Posterior Mean

is brought the probable value that prior distribution obtains into.

Substep 6. calculates EIG according to formula (8), promptly obtains scheme η _rThe expectation information gain;

$\mathrm{EIG}\left({\mathrm{\&eta;}}_r\right)=E_x{E}_{\mathrm{\&theta;}}\left[\log \frac{p\left(x\right|\mathrm{\&theta;})}{p\left(x\right)}\right]=E_x{E}_{\mathrm{\&theta;}}\left[\log p\left(x_{\mathrm{rh}}\right|{\mathrm{\&theta;}}_{\mathrm{rh}},{\mathrm{\&eta;}}_r)\right]-E_x\left[\log p\left(x\right)\right];$

Substep 7. is got back to substep 1, to each scheme iteron step 2 ~ 6 in the scheme set; Obtain all scheme optimization targets in the scheme set.

(5) utilize curved surface fitting method, relatively the desired value of all schemes finds optimal case.

In some cases, the definite structure of curved surface regression model is difficult to confirm.Therefore the present invention selects for use nonparametric technique that the desired value of all schemes is carried out regression fit, for example, and the nuclear smoothing method.Be the regression fit effect of more different models, the inventive method returns with parametric polynomial respectively and the local weighted recurrence of nonparametric is loose, and (Locally Weighted Scatterplot Smoothing is LOWESS) to the match of data march face for the some smoothing method.

It is several to (η that all scheme desired values that obtain based on (4) constitute _r, EIG (η _r)) carry out fitting surface, find the maximum corresponding scheme of desired value to be the optimum test scheme.

Step 2, set up sequential truncation decision method.

Sequential truncation decision method is specially:

(1) (promptly judging based on actual information gain and the contrast of expectation information gain) judged in contrast based on relative entropy.

At first, implementing a certain stress level S _kBefore the following test, calculate this stress level S _kThe expectation information gain of following test is designated as EIG (S _k), numerical procedure is seen (4) in the step 1.

Then, implement S _kFollowing test, (be designated as y, performance degradation incremental data x is the poor of adjacent twice performance degradation data to the some performance degradation data of every acquisition, x _j=y _J+1-y _j), calculate actual information gain (IG), be designated as IG (S _k).IG(S _k)＝I ₁(x)-I ₀。Its calculation procedure and EIG (S _k) calculation procedure similar, different is, and finding the solution of expectation information gain EIG need be hoped the information peek term of parameter space θ and sample space x, so will from prior distribution, extract parameter θ R ₂Inferior, and to the parameter θ of each extraction _h, repeatedly emulation generates the performance degradation incremental data and gets expectation from the sampling distribution of sample x.And to the actual information gain, the performance degradation incremental data is the actual tests data, need in sample space, not get expectation, directly brings the actual tests data computation into and gets final product.

At last, relatively actual information gain and expectation information gain.When the actual information that obtains gains greater than expectation information gain, i.e. IG (S _k)>EIG (S _k) time, can truncation, stop this step test.

The relative entropy reflection is the prior distribution of parameter before and after the test and the distance between the posteriority distribution among the present invention.The distance that distributes when parameter prior distribution and posteriority is bigger, and when differing far away, test acquired information gain meeting is bigger for parameter priori and its actual conditions model parameter value of actual rule (the reflection properties of product degenerate); And the distance of working as parameter prior distribution and posteriority distribution is less; Parameter priori and its actual value differ hour, and the parameter prior distribution can reflect its truth, then implement test after; Information entropy by real data obtains is less; Along with the progress of test, the situation that the actual information gain is difficult to surpass the expectation information gain possibly appear, and therefore the posteriority changes in distribution situation of observed parameter judges whether to stop test simultaneously.

(2) judge according to the posteriority changes in distribution of parameter.

Implement S _kFollowing test, the some performance degradation data of every acquisition, the posteriority distribution p of calculating parameter (θ | x).The posteriority distribution p (θ | x) utilize software WinBUGS14 to obtain based on the Markov monte carlo method.

When parameter posteriority changes in distribution less, when tending towards stability, can truncation, stop this step test.

The purpose of implementing accelerated degradation test is through acquisition properties of product degraded data, in conjunction with prior imformation, provides the estimation of model parameter, and then life-span and reliability under the assessment product normal stress.When the parameter posteriority is estimated to have tended towards stability, then need not continue test; If the some degraded datas of every increase, parameters calculated posteriority estimated difference then need continue test apart from bigger.

(3) the monitoring number of times of implementing to stipulate in the preceding scheme with test is truncation.

Consider under each step stress and all need obtain certain performance degradation information; Can not " stop " go on foot under the test after for a long time at certain; Therefore, when actual information gain and parameter posteriority distribute can not satisfy the truncation requirement time, should be truncation with the monitoring number of times of stipulating in the preceding scheme of this step test enforcement.

To sum up, the truncation decision rule can be expressed as: implement test, based on actual tests data and prior information, calculate actual information gain and the distribution of parameter posteriority, relatively actual information gain and expectation information gain, and the variation of parameter posteriority distribution.If the actual information gain can stop test greater than the expectation information gain; If the actual information gain is difficult to surpass the expectation information gain, judge with parameter posteriority changes in distribution that then posteriority distributes to tend towards stability and then can stop test; All can not meet the demands if information gain and posteriority distribute, the monitoring number of times of stipulating in the scheme before then implementing with this step test is truncation.

Step 3, carry out sequential accelerated degradation test based on relative entropy.

If the stress level number is K, sequential ADT conceptual design can be divided into K stage, is specially:

Stage 1:

(1) confirms the prior distribution of model parameter according to prior imformation, be designated as p _K(θ);

(2) constitute the confirmed test resource according to testing expenses: always monitor number of times m and sample size n;

(3) according to the accelerated degradation test Optimization Design of step 1 based on relative entropy, the initial scheme before confirmed test is implemented is designated as η _K

(4) implement high stress S _KUnder test, obtain the performance degradation incremental data x under this stress level _K

Stage 2:

(1) confirms that according to the Test Information under the prior imformation of model parameter and the high stress SK experience stage posteriority of 1 back parameter distributes, and is designated as p _K(θ | x _K), and then the prior distribution of parameter of definite stages 2, be designated as p _K-1(θ);

(2) combine remaining test resource,, obtain the follow-up test scheme, be designated as η according to the accelerated degradation test Optimization Design of step 1 based on relative entropy _K-1

(3) implement stress level S _K-1Under test, the some performance degradation data of every collection judge whether to continue under this stress level test according to truncation decision rule in the step 2.

Stage 3:

(1) according to stress level S _KAnd S _K-1Under Test Information, confirm that experience stages 2 posteriority of back parameters distributes, and is designated as p _K-1(θ | x _K-1), and then the prior distribution of parameter of definite stages 3, be designated as p _K-2(θ);

(2) combine remaining test resource,, obtain the follow-up test scheme, be designated as η according to the accelerated degradation test Optimization Design of step 1 based on relative entropy _K-2

(3) implement stress level S _K-2Under test, the some performance degradation data of every collection judge whether to continue under this stress level test according to truncation decision rule in the step 2.

So go on

Stage K:

(1) according to before all stress levels Test Information down, confirm the posteriority distribution of parameter behind the experience stage K-1, be designated as p ₂(θ | x ₂), and then the prior distribution of definite stage K parameter, be designated as p _K-2(θ);

(2) combine remaining test resource,, obtain the testing program of last stress level, be designated as η according to the accelerated degradation test Optimization Design of step 1 based on relative entropy ₁

(3) implement stress level S ₁Under test, the some performance degradation data of every collection judge whether to continue under this stress level test according to truncation decision rule in the step 2.

Above-mentioned x _kRepresent performance of products degeneration incremental data under k the stress level, k=1 ... K.

Be example with 3 levels below, be discussed in more detail sequential ADT conceptual design flow process, as shown in Figure 3, totally 3 stages.Corresponding stress level of each stage; Confirm the parameter prior distribution according to the information before this stage; For example the parameter posteriority that obtains according to model parameter prior distribution and the 1st stage of the 2nd stage distributes, and confirm should stage parameter prior distribution, and the parameter posteriority that the 3rd stage obtained according to preceding two stages distributes, and confirm should stage parameter prior distribution, utilizes Bayes's Optimization Design; Formulate the optimum test scheme based on relative entropy, implement step test.Along with the continuous acquisition of test figure, the prior distribution of parameter is constantly revised, and the optimal case that therefore obtains is constantly adjustment also, is specially:

Stage 1: high stress S ₃Following test

(1) at first according to like product information; The mathematical model of definite properties of product degenerative process such as historical test data; Derivation is based on the probability density function of this model and log-likelihood function etc., and then confirms parameter prior distribution p according to the prior imformation of model parameter _K(θ).

(2) according to testing expenses formation and the definite test resource that can participate in implementing of experimentation cost constraint: always monitor number of times m and sample size n etc.;

(3) according to the accelerated degradation test Optimization Design of step 1 based on relative entropy, the initial scheme before confirmed test is implemented:

η ₃={(S ₁,S ₂,S ₃),(m ₁,m ₂,m ₃),(n ₁,n ₂,n ₃)}

Wherein, η ₃Expression initial trial scheme;

S ₁, S ₂, S ₃Three stress levels confirming in the expression initial scheme, and S ₁＜S ₂＜S ₃General S ₃=S _Max

m ₁, m ₂, m ₃Represent three stress level S ₁, S ₂, S ₃Following performance of products monitoring number of times; m ₁>=m ₂>=m ₃,

n ₁, n ₂, n ₃Represent three stress level S ₁, S ₂, S ₃Under tried sample size.To constant stress ADT, n ₁, n ₂, n ₃Can be identical can be different, and To stepstress ADT, all samples are progressively applied each stress level, so n ₁=n ₂=n ₃=n.

(4) implement high stress S ₃Stress S is gathered in following test ₃Following performance of products degraded data, every collection one piece of data calculates the information gain that test obtained (IG) and the parameter posteriority that have carried out, and carries out comparative analysis.For making the enough quantity of information of acquisition under each stress level, test period is generally shorter under the high stress, does not therefore do truncation in advance, promptly implements to test under the high stress according to initial scheme, and its actual monitoring number of times is used m _3rExpression, then m _3r=m ₃

Stage 2: the horizontal S of intermediate stress ₂Following test

(1) according to the prior imformation and the S of model parameter ₃Following parameter posteriority distribution p ₃(θ | x ₃), (x ₃Be S ₃The performance degradation incremental data of following acquisition; Subscript 3 index stress levels, p ₃(θ | x ₃) expression acquisition S ₃Under Test Information after the posteriority of parameter θ distribute) confirm stress level S ₂The prior distribution p of following parameter ₂(θ) (subscript 2 index stress levels, p ₂(θ) stress level S is implemented in expression ₂The prior distribution of preceding parameter θ);

(2) combine remaining test resource,, obtain follow-up S according to the accelerated degradation test Optimization Design of step 1 based on relative entropy ₁, S ₂Following testing program:

η ₂={(S ₁′,S ₂′),(m ₁′,m ₂′),(n ₁′,n ₂′)}

Wherein, η ₂Expression is through S ₃After the following test, S ₁, S ₂Following testing program is by initial trial scheme η ₃In { (S ₁, S ₂), (m ₁, m ₂), (n ₁, n ₂) be adjusted into { (S ₁', S ₂'), (m ₁', m ₂'), (n ₁', n ₂'), promptly respectively test the corresponding adjustment of variable, stress level S ₁→ S ₁', S ₂→ S ₂'; Monitoring number of times m ₁→ m ₁', m ₂→ m ' ₂Sample size n ₁→ n ₁', n ₂→ n ' ₂

(3) implement the 2nd stress level S ₂' under test; The some performance degradation data of every collection; Calculate the information gain (IG) and the parameter posteriority that are obtained, judge, if satisfy the truncation decision rule according to truncation decision rule in the step 2; Then stop under this stress level test, according to the Test Information optimization that obtains and implement next step test; If do not satisfy the truncation decision rule, then continuing should force level test down.When satisfying the truncation decision rule, make S ₂' following actual monitoring frequency table is shown m _2r

Stage 3: stress level S ₁Following test

(1) the posteriority distribution p of parameter under two stress levels having implemented of basis ₃(θ | x ₃) and p ₂(θ | x ₂), x ₂Expression stress level S ₂' under the performance degradation incremental data, confirm stress level S ₁The prior distribution p of following parameter ₁(θ);

(2) combine remaining test resource,, confirm S according to the accelerated degradation test Optimization Design of step 1 based on relative entropy ₁Following testing program η ₁={ S ₁", m ₁", n ₁" }, S ₁" be to scheme η ₂Adjustment, m ₁", n ₁" be remaining monitoring number of times and sample size.

(3) implement S ₁" under test, the some performance degradation data of every collection are calculated information gain (IG) and the parameter posteriority obtained, judge according to truncation decision rule in the step 2, if satisfy the truncation decision rule, then stop to test; If do not satisfy the truncation decision rule, then continuing should force level test down.When satisfying the truncation decision rule, make S ₁" the actual monitoring frequency table is shown m down _1r

Optimal design when being k=3 in the embodiment of the invention.

Embodiment:

To certain super-radiance light emitting diode (super luminescent diode SLD), implements the sequential accelerated degradation test optimal design of stepstress based on relative entropy, and step is following:

Step 1, utilize bayesian theory, set up accelerated degradation test Optimization Design based on relative entropy.

Step 1 is the basis of step 2 of the present invention and step 3, is the practical implementation of example description of step one to optimize the initial trial scheme according to the method for step 1 in the step 3 below.

(1) confirms properties of product degradation model and acceleration model, and then provide the prior distribution of model parameter based on historical data.Provide model and hypothesis according to the design information that receives trial product SLD, historical data, like product information etc., as follows,

1. suppose:

A1: degradation trend is dull irreversible;

A2: degradation failure mechanism does not change with stress;

A3: at the horizontal S of normal stress ₀With k acceleration stress level S ₁<s ₂<<s _KDown, performance degradation process Y _kObey Brownian Motion with Drift, coefficient of deviation d (S _k)>0, coefficient of diffusion σ _k＞0, k=1 ..., K:

Y _k(t)＝σ _kB(t)+d(S _k)·t+y ₀ （12）

In the formula, Y _k(t) be the performance of products parameter, reflection properties of product degenerative process; y ₀Be the starting point of Brownian Motion with Drift, promptly properties of product are at initial time t ₀Initial value; B (t) is the standard Brownian movement, and B (τ) ~ N (0, t); T representes the performance degradation time.

A4: coefficient of diffusion σ _kDo not change with stress level, that is, and σ ₀=σ ₁=...=σ _K=σ.

A5: coefficient of deviation d (S _k) also can be described as degradation ratio, be the function of stress S, also be coefficient of deviation is an acceleration model

Wherein, A and B be the representation model undetermined parameter respectively, S _kBe k and quicken stress level,

Be stress S _kThe function of certain form is for example when acceleration stress is absolute temperature

2. model:

1) be to quicken stress with the temperature, acceleration model is elected the ArrheniuS model as:

carried out parametrization:

In the formula, S ₀, S _MaxRepresent that respectively product does not change normal applied stress level and the highest acceleration stress level that test can apply under the failure mechanism prerequisite, generally gets S _K=S _MaxThe horizontal υ of proof stress after the parametrization is at υ ₀=1 and υ _MaxBetween=0.

Degradation ratio is expressed as with υ:

ln?d _k＝β ₀+β ₁υ _k （16）

Can know β according to formula (15) and (16) ₀=ln d _Max, β ₀The logarithm of representing properties of product degradation ratio under the high stress; β ₀+ β ₁The logarithm of properties of product degradation ratio under the expression normal stress level; β ₁Logarithm poor of representing performance degradation rate under the most heavily stressed and the normal stress, β ₁=(ln d _k-β ₀)/υ _k=lnd ₀-β ₀d ₀Properties of product degradation ratio under the expression normal stress level; d _kExpression stress level S _kFollowing properties of product degradation ratio; υ _kExpression stress level S _kForm after the parametrization.A lot of important acceleration models comprise that Arrhenius model, contrary power rate model are through all being expressed as the form of formula (16) after the suitable parametrization.To the Arrhenius model, can know β ₀=A+B/S _Max,

Therefore, the unknown parameter of model also can be written as: θ={ β ₀, β ₁, τ }, τ=1/ σ ²

2) degradation model is a Brownian Motion with Drift:

Y _k(t)＝σ _kB(t)+d(S _k)·t+y ₀ （17）

Wherein, σ _kThe expression coefficient of diffusion, d (S _k) the expression coefficient of deviation, also can be described as degradation ratio, be the function of stress S, also be coefficient of deviation is an acceleration model, make initial value y ₀=0.Model unknown parameter vector θ={ A, B, τ } then, τ=1/ σ ²Also can be written as: θ={ β ₀, β ₁, τ }, τ=1/ σ ²

3) likelihood function:

If n product implemented the SSADT of K level.Do not have in the hypothesis test because the inefficacy that performance degradation causes.Among the SSADT, the performance monitoring number of times is m under k the stress level _k, then the accumulation of SSADT monitoring number of times does

The properties of product monitoring time is spaced apart Δ t, then the test period t under k stress level _k=m _kΔ t, total testing time are t=m Δ t.To receive the j Measuring Time of trial product be t to i under k the stress level _Kij(i=1 ..., n, k=1 ..., K, j=1 ... M _k), the performance number that monitors is y _KijBrownian movement is a Gaussian process, so the performance degradation increment x on the detection time interval of delta t is independent and to obey average be d (s) Δ t, and variance is σ ²The normal distribution of Δ t, i.e. x～N (d (s) Δ t, σ ²Δ t).The probability density function of independent increment x does,

$f\left(x\right|\mathrm{\&theta;})=\frac{1}{\sqrt{2\mathrm{\&pi;}{\mathrm{\&tau;}}^{-1}\mathrm{\&Delta;t}}}\exp \{-\frac{[x-\exp (A+B/S_k)\&CenterDot;\mathrm{\&Delta;t}{]}^2}{2{\mathrm{\&tau;}}^{-1}\&CenterDot;\mathrm{\&Delta;t}}\}---\left(18\right)$

According to accumulated damage hypothesis and formula (18), the likelihood function of n all degeneration increments of sample is following under K stress level SSADT:

$p\left(x\right|\mathrm{\&theta;})=\underset{k=1}{\overset{K}{\mathrm{\&Pi;}}}\underset{i=1}{\overset{n}{\mathrm{\&Pi;}}}\underset{j=1}{\overset{m_{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="HDA00001785950000021.png"\; state="\{\{state\}\}">k</figure-callout>}}}{\mathrm{\&Pi;}}}\frac{}{\sqrt{2\mathrm{\&pi;}{\mathrm{\&tau;}}^{-1}\mathrm{\&Delta;t}}}\exp \{-\frac{{[x_{\mathrm{kij}}-\exp (A+B/S_k)\&CenterDot;\mathrm{\&Delta;t}]}^2}{2{\mathrm{\&tau;}}^{-1}\mathrm{\&Delta;t}}\}---\left(19\right)$

x _KijRepresent i under k the stress level j performance degradation incremental data that receives trial product, its log-likelihood function does,

$\ln p\left(x\right|\mathrm{\&theta;})\&Proportional;-\frac{1}{2}\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{m_k}{\mathrm{\&Sigma;}}}\left\{\right[\ln \left(2\mathrm{\&pi;\&Delta;t}\right)-\ln \left(\mathrm{\&tau;}\right)]+\frac{{[x_{\mathrm{kij}}-\exp (A+B/S_k)\&CenterDot;\mathrm{\&Delta;t}]}^2}{{\mathrm{\&tau;}}^{-1}\mathrm{\&Delta;t}}\}---\left(20\right)$

Then, provide the prior distribution of model parameter based on historical data:

According to degeneration increment x～N (d (S) Δ t, σ ²Δ t) and the conjugation prior distribution theoretical, can suppose parameter beta ₀And β ₁Obeying average respectively is μ ₀, μ ₁, variance is σ ₀, σ ₁Normal distribution, σ ²τ reciprocal to obey scale parameter be a, form parameter is that the gamma of b distributes.Also be; τ ~ Γ (a, b).

According to the design information of like product information, product, the test etc. of knowing the real situation, confirm that the prior distribution of model parameter is:

$p\left({\mathrm{\&beta;}}_0\right)~N({\mathrm{\&mu;}}_0,{\mathrm{\&sigma;}}_0^2)=N(-\mathrm{3.7,1})---\left(21\right)$

$p\left({\mathrm{\&beta;}}_1\right)~N({\mathrm{\&mu;}}_1,{\mathrm{\&sigma;}}_1^2)=N(-\mathrm{5.6,1})---\left(22\right)$

p(τ)~Γ(a,b)＝Γ(1,100) （23）

Can know E (ln d according to formula (16) from the angle of degradation ratio _k)=E (β ₀)+E (β ₁) υ _k

(2) make up testing program set D

According to definite sample size n=3 such as testing expenses formations, always monitor number of times m=120, monitoring interval of delta t=1.

Divide stress vector and measure time number vector in conjunction with engineering experience.At first, confirm stress level S ₁The value space, make S ₁=[60,65,70,75,80,85,90] ℃, S ₃=110 ℃, S ₂Get S ₁With S ₃The absolute temperature inverse uniformly-spaced, promptly

Corresponding S ₁Each element obtain S ₂≈ [83,86,89,92,94,97,100] ℃.Secondly, confirm monitoring number of times m ₁The value space, make m ₁For guaranteeing all to obtain comparatively sufficient performance degradation data under each stress level, generally there is m=[40,50,60,70] ₁>=m ₂>=m ₃, we make the horizontal S of intermediate stress ₂Monitor down number of times for monitoring number of times under height two levels and half the, i.e. m ₂=0.5 (m ₁+ m ₃), m then ₂=40, m ₃=80-m ₁Constitute testing program set D (S in the value space of monitoring number of times down according to each stress level and each stress level ₁Get 7 values, m ₁Therefore get 4 values, totally 28 schemes).

(3) foundation is based on the optimization aim of relative entropy

The optimization aim of setting up based on relative entropy according to (3) in the embodiment step 1 is:

$\underset{\mathrm{\&eta;}\&Element;D}{\max }\mathrm{EIG}\left(\mathrm{\&eta;}\right)$

$=\underset{\mathrm{\&eta;}\&Element;D}{\max }\left(E_x{\text{E}}_{\mathrm{\&theta;}}\right[\log p\left(x\right|\mathrm{\&theta;})]-E_x[\log p\left(x\right)\left]\right)$

(4),, utilize software WinBUGS14 calculation optimization target based on Monte Carlo Markov chain (MCMC) method to each testing program.

To each testing program among the scheme set D, based on the MCMC method, utilize WinBUGS14 software, according to the concrete calculation procedure calculation optimization target in the step 1 in the embodiment (4).

(5) utilize curved surface fitting method, relatively the desired value of all schemes finds optimal case.

According to the method for step 1 in the embodiment (5), constitute several to (η to all scheme optimization desired values that obtain _r, EIG (η _r)), adopt polynomial expression (secondary) recurrence and local weighted (secondary) to return the some smoothing method (LOWESS) that looses and carry out fitting surface.

The result that is optimized is as shown in Figure 4, and the tentative programme before the test is as shown in table 1.

The sequential ADT initial trial of table 1 three levels scheme

S 1,S 2,S 3(℃)	m 1,m 2,m 3	EIG
			65,86,110	50,40,30	≈200

Step 2, set up sequential truncation decision method.

Three angles of monitoring number of times of stipulating the scheme before conversion that distributes from actual information gain and the contrast of expectation information gain, parameter posteriority according to step 2 in the practical implementation method and test are implemented are set up the sequential truncation decision rule of this instance.

(1) (promptly judging based on actual information gain and the contrast of expectation information gain) judged in contrast based on relative entropy.

At first, implementing a certain stress level S _kBefore the following test, calculate this stress level S _kThe expectation information gain EIG (S of following test _k).Then, implement S _kFollowing test, the some performance degradation incremental datas of every acquisition are calculated actual information gain IG (S _k).At last, relatively actual information gain and expectation information gain.When the actual information that obtains gains greater than expectation information gain, i.e. IG (S _k)>EIG (S _k) time, can truncation, stop this step test.

(2) judge according to the posteriority changes in distribution of parameter.

If the actual information gain is difficult to surpass the situation of expectation information gain, then the posteriority changes in distribution situation of observed parameter judges whether to stop test simultaneously.

Implement S _kFollowing test, the some performance degradation data of every acquisition, the posteriority distribution p of calculating parameter (θ | x).When parameter posteriority changes in distribution less, when tending towards stability, can truncation, stop this step test.

(3) the monitoring number of times of implementing to stipulate in the preceding scheme with test is truncation.

Consider under each step stress and all need obtain certain performance degradation information; Can not " stop " go on foot under the test after for a long time at certain; Therefore, when actual information gain and parameter posteriority distribute can not satisfy the truncation requirement time, should be truncation with the monitoring number of times of stipulating in the preceding scheme of this step test enforcement.

Step 3, set up the sequential trials design cycle.

The temperature stress number of levels is 3 among the sequential SSADT of this instance, i.e. K=3.Establish the sequential trials design cycle of this instance according to the design cycle of step 3 in the practical implementation method.

Stage 1:

(1) confirms the prior distribution of model parameter according to prior imformation, be designated as p _K(θ), see embodiment step 1 Chinese style (21) ~ (23);

(2) constitute the confirmed test resource according to testing expenses: always monitor number of times m=120 and sample size n=3, monitoring interval of delta t=1; Receive the working stress S of trial product SLD ₀Be 25 ℃, minimum acceleration stress S _MinBe 60 ℃, working limit S _MaxIt is 110 ℃.

(3) according to the accelerated degradation test Optimization Design of step 1 based on relative entropy, the initial scheme before confirmed test is implemented is designated as η _K, see table 1 in the embodiment step 1;

(4) according to initial trial scheme η ₃, implement high stress S ₃Under test, obtain the performance degradation incremental data x under this stress level ₃Suppose the performance degradation incremental data x of " actual tests " ₂Be by parameter

${\stackrel{\&OverBar;}{\mathrm{\&beta;}}}_1=-4.32,$ $\stackrel{\&OverBar;}{\mathrm{\&sigma;}}=0.05$ Emulation generates.

Calculate S ₃Under expectation information gain EIG (S ₃)=57.03.Performance degradation incremental data x ₃Actual information gain be IG (S ₃)=112.87.Adopt the MCMC method, utilize the WinBugs computed in software to obtain model parameter posteriority distribution p (θ ₃| x ₃):

p(β ₀|x ₃)～N(-2.17,0.051 ²) （24）

p(τ|x ₃)～Γ(44.836,7.5) （25）

Stage 2:

(1) with parameter beta ₀With τ at S ₃Under posteriority distribute and β ₁Prior distribution as follow-up S ₁And S ₂The parameter prior distribution p of following test ₂(θ).

(2) utilize surplus resources to confirm S ₁And S ₂Under scheme set D ₂, the result that is optimized is as shown in Figure 5, optimal case:

η ₂={(S ₁′,S ₂′),(m ₁′,m ₂′)}={(70,89),(50,40)}。

Implement S ₂' test down, its expectation information gain EIG (S ₂')=64.6.

So S ₂The performance degradation incremental data x of ' following " actual tests " ₂～N (0.0489,0.05 ²).Every collection plurality of data calculates actual information gain and parameter posteriority, sees table 2 and table 3.

The sequential ADTS of table 2 three levels ₂The actual information gain of ' different down monitoring number of times

m 2	20	25	30	35	40
						IG	32.7	40.8	47.9	54.9	65.5

The sequential ADTS of table 3 liang level ₂The parameter posteriority of ' different down monitoring number of times distributes

Because m ₂=35 o'clock, the actual information gain did not reach expectation information gain and parameter beta yet ₁Posterior Mean still have than great fluctuation process, therefore should not shift to an earlier date truncation, according to implement in the original plan should the step test, m ₂Actual information gain in=40 o'clock reaches the expectation information gain, and parameter beta ₀The posteriority change that distributes less, β ₁See shown in Figure 6 with the posteriority changes in distribution of τ.Stop this step test, get into next stage.

(3) implement S ₁" under test, the some performance degradation data of every collection are calculated information gain (IG) and the parameter posteriority obtained, judge according to truncation decision rule in the step 2, if satisfy the truncation decision rule, then stop to test; If do not satisfy the truncation decision rule, then continuing should force level test down.When satisfying the truncation decision rule, make S ₁" the actual monitoring frequency table is shown m down _1r

Stage 3:

(1) with S ₂' following parameter beta ₀, β ₁Distribute as S with the posteriority of τ ₁The parameter prior distribution p of ' test ₁(θ).

(2) to S ₁Span [S _Min, S ₂) carry out discretize and obtain several S ₁Value, confirm monitoring number of times m in conjunction with remaining test resource ₁"=50, constitute testing program set D ₁, calculate each S respectively ₁Corresponding expectation information gain (EIG) is got the maximum corresponding S of EIG ₁Be optimum solution.Promptly find the solution

${\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="HDA00001785950000021.png,HDA00001785950000031.png"\; state="\{\{state\}\}">S</figure-callout>}}_1^{\&prime;}=\underset{S_{}}{\mathrm{<figure-callout\; id="1"\; label="arg\; max\; S"\; filenames="HDA00001785950000021.png,HDA00001785950000031.png"\; state="\{\{state\}\}">arg</figure-callout>}\max }\mathrm{EIG}\left(S_1\right)---\left(26\right)$

Obtain S ₁"=68 ℃, EIG (S ₁")=76.5.

(3) implement S ₁" test down, So S ₁" descend the performance degradation incremental data x of " actual tests " ₁～N (0.0182,0.05 ²).Every collection plurality of data calculates the actual information gain and posteriority distributes, and the result sees table 4 and table 5.The posteriority changes in distribution of parameter is seen shown in Figure 7.Though m ₁=40 o'clock, the actual information gain did not reach the expectation information gain, but parameter beta ₁Tend towards stability with the posteriority distribution of τ, can stop test.

The sequential ADTS of table 4 three levels ₂The actual information gain of ' different down monitoring number of times

m 1	25	30	35	40
					IG	33.5	39.1	45.8	56.1

The sequential ADTS of table 5 liang level ₂The parameter posteriority of ' different down monitoring number of times distributes

If the test resource is sufficient, also can continue to test to m ₁=50.In this instance, m ₁=50 o'clock, actual information gain IG=68.4, the parameter posteriority is distributed as β ₁~ N (4.504,0.403), τ～Γ (172.861,2.114) is with m ₁Changed little at=40 o'clock.

Sequential Bayes's experimental design is not only utilized the parameter prior distribution, and along with the carrying out of test, constantly utilizes actual tests data adjustment testing program, can stop test in advance based on the test actual conditions, saves time and tests resource.

Claims (4)

1. the accelerated degradation test Optimization Design based on relative entropy is characterized in that, comprises following step:

Step 1, utilize bayesian theory, set up accelerated degradation test Optimization Design based on relative entropy;

(1) confirms properties of product degradation model and acceleration model, and then provide the prior distribution of model parameter based on historical data;

(2) make up testing program set D;

Confirm the value of sample size n, test period t and monitoring interval of delta t, by test period and the total monitoring number of times m=t/ Δ t of monitoring interval calculation;

1),, confirms that in conjunction with the testing expenses constraint under the situation of total sample size n and overall measurement number of times m, the decision variable that needs to optimize is the horizontal S of each proof stress in confirmed test stress level scope to stepstress accelerated degradation test (SSADT) _kWith the measurement number of times m under each stress level _k

Make S represent the proof stress vector, comprise K element among the S, K is a positive integer, S=(S ₁..., S _k..., S _K), stress level of each element representation, S _kRepresent k stress level, make M represent to measure time number vector, then M=(m ₁..., m _k..., m _K), m _kRepresent k stress level S _kUnder the measurement number of times, k=1 ..., K;

Scheme set D=S * M constitutes (S by the value space of proof stress and the value space of measurement number of times _k, m _k) be a certain scheme η in the design space;

Adopt the surface fitting scheme; Along stress level and measurement number of times direction; Five equilibrium value in its span comprises that with the design space boundary demarcation is that limited scheme formed the testing program set, to each computation schemes objective function in the testing program set; Utilize surface fitting to find out the maximum zone of objective function, and then find out optimal case;

2), need consider that also sample size is distributed n under each stress level to constant stress accelerated degradation test (CSADT) _kConcrete grammar is following:

Count sample dispensing n under K and each stress according to total sample number n and stress level _kConstraint condition: n _k>=n,

Provide several kinds of different sample dispensing, to every kind of sample dispensing, can handle according to the mode of above-mentioned SSADT, obtain the optimal case under every kind of sample dispensing, compare, the maximum scheme of select target value is an optimal case;

(3) foundation is based on the optimization aim of relative entropy

The present invention as utility function, turns to optimization aim with the expectation information gain maximum that obtains in the test with relative entropy;

The information I that comprises in the prior distribution ₀For:

I ₀＝∫p(θ)logp(θ)dθ＝E _θlogp(θ) （1）

Wherein, p (θ) representation model parameter prior distribution probability density function; E _θExpression is about the expectation of θ;

The gross information content I that from posteriority distributes, obtains ₁(x) be:

I ₁(x)＝∫p(θ|x)logp(θ|x)dθ （2）

Wherein, p (θ | x) representation model parameter posteriority distribution probability density function;

The information that definition obtains from testing program η is:

I(η,x,p(θ))＝I ₁(x)-I ₀ （3）

The testing program design should provide before data x obtains, and therefore need hope that then the expectation of I (η, x, p (θ)) is to the information peek term of sample space:

I(η,p(θ))＝E _x[I ₁(x)-I ₀] （4）

Wherein, E _xExpression hopes that to the information peek term of sample space I (η, p (θ)) is also referred to as the expectation information gain; Based on bayesian theory, the expectation of Test Information is expressed as:

In the formula, p (x) is marginal likelihood function, also is the standard constant, and is as follows:

p(x)＝∫p(x|θ)p(θ)dθ （6）

Wherein, the likelihood function under p (x| θ) the expression parameter θ known conditions;

Therefore, the optimization aim based on relative entropy is:

(4),, utilize software WinBUGS14 calculation optimization target based on Monte Carlo Markov chain (MCMC) method to each testing program;

(5) utilize curved surface fitting method, relatively the desired value of all schemes finds optimal case;

Select for use nonparametric technique that the desired value of all schemes is carried out regression fit, loose the some smoothing method to the match of data march face with parametric polynomial recurrence and the local weighted recurrence of nonparametric respectively;

It is several to (η that all scheme desired values that obtain based on (4) constitute _r, EIG (η _r)) carry out fitting surface, find the maximum corresponding scheme of desired value to be the optimum test scheme;

Step 2, set up sequential truncation decision method;

Sequential truncation decision method is specially:

(1) contrast is judged based on relative entropy;

At first, implementing a certain stress level S _kBefore the following test, calculate this stress level S _kThe expectation information gain of following test is designated as EIG (S _k);

Then, implement S _kFollowing test, the some performance degradation data of every acquisition are designated as y, and performance degradation incremental data x is the poor of adjacent twice performance degradation data, x _i=y _J+1-y _j, calculate actual information gain IG, be designated as IG (S _k); IG (S _k)=I ₁(x)-I ₀

At last, relatively actual information gain and expectation information gain; When the actual information that obtains gains greater than expectation information gain, i.e. IG (S _k)>EIG (S _k) time, can truncation, stop this step test;

(2) judge according to the posteriority changes in distribution of parameter;

Implement S _kFollowing test, the some performance degradation data of every acquisition, the posteriority distribution p of calculating parameter (θ | x); The posteriority distribution p (θ | x) utilize software WinBUGS 14 to obtain based on the Markov monte carlo method;

Less when parameter posteriority changes in distribution, when tending towards stability, truncation stops this step test;

(3) the monitoring number of times of implementing to stipulate in the preceding scheme with test is truncation;

When actual information gain and parameter posteriority distribute can not satisfy the truncation requirement time, should be truncation with the monitoring number of times of stipulating in the preceding scheme of this step test enforcement;

To sum up, the truncation decision method can be expressed as: implement test, according to actual tests data and prior imformation, calculate actual information gain and the distribution of parameter posteriority, relatively actual information gain and expectation information gain, and the variation of parameter posteriority distribution; If the actual information gain can stop test greater than the expectation information gain; If the actual information gain is difficult to surpass the expectation information gain, judge with parameter posteriority changes in distribution that then posteriority distributes to tend towards stability and then can stop test; All can not meet the demands if information gain and posteriority distribute, the monitoring number of times of stipulating in the scheme before then implementing with this step test is truncation;

Step 3, carry out sequential accelerated degradation test based on relative entropy;

If the stress level number is K, be divided into K stage, be specially:

Stage 1:

(1) confirms the prior distribution of model parameter according to prior imformation, be designated as p _K(θ);

(2) constitute the confirmed test resource according to testing expenses: always monitor number of times m and sample size n;

(3) according to the accelerated degradation test Optimization Design of step 1 based on relative entropy, the initial scheme before confirmed test is implemented is designated as η _K

(4) implement high stress S _KUnder test, obtain the performance degradation incremental data x under this stress level _K

Stage 2:

(1) according to the prior imformation and the high stress S of model parameter _KUnder Test Information confirm that experience stage 1 posteriority of back parameter distributes, and is designated as p _K(θ | x _K), and then the prior distribution of parameter of definite stages 2, be designated as p _K-1(θ);

(2) combine remaining test resource,, obtain the follow-up test scheme, be designated as η according to the accelerated degradation test Optimization Design of step 1 based on relative entropy _K-1

(3) implement stress level S _K-1Under test, the some performance degradation data of every collection judge whether to continue under this stress level test according to truncation decision rule in the step 2;

Stage 3:

(1) according to stress level S _KAnd S _K-1Under Test Information, confirm that experience stages 2 posteriority of back parameters distributes, and is designated as p _K-1(θ | x _K-1), and then the prior distribution of parameter of definite stages 3, be designated as p _K-2(θ);

(2) combine remaining test resource,, obtain the follow-up test scheme, be designated as η according to the accelerated degradation test Optimization Design of step 1 based on relative entropy _K-2

(3) implement stress level S _K-2Under test, the some performance degradation data of every collection judge whether to continue under this stress level test according to truncation decision rule in the step 2;

So go on

Stage K:

(1) according to before all stress levels Test Information down, confirm the posteriority distribution of parameter behind the experience stage K-1, be designated as p ₂(θ | x ₂), and then the prior distribution of definite stage K parameter, be designated as p _K-2(θ);

(2) combine remaining test resource,, obtain the testing program of last stress level, be designated as η according to the accelerated degradation test Optimization Design of step 1 based on relative entropy ₁

(3) implement stress level S ₁Under test, the some performance degradation data of every collection judge whether to continue under this stress level test according to truncation decision rule in the step 2;

Above-mentioned x _kRepresent performance of products degeneration incremental data under k the stress level, k=1 ... K.

2. a kind of accelerated degradation test Optimization Design according to claim 1 based on relative entropy; It is characterized in that; Degradation model is melange effect model, gamma process model or Brownian movement model in the described step 1 (1), and described acceleration model is Allan Nice model, contrary power rate model or Aileen's model.

3. a kind of accelerated degradation test Optimization Design based on relative entropy according to claim 1 is characterized in that, in the described step 1 (2), and described three-dimension curved surface fitting technique, two independents variable are that stress S1 compares m with the measurement number of times ₁, be stress S with the optimization aim function representation ₁With measurement number of times m ₁Function; Stress S under other proof stress levels _kWith measurement number of times m _kProvide or be expressed as S according to actual conditions ₁, m ₁Function, 1＜k≤K;

To two stress levels, the decision variable of testing program is: stress level S ₁, S ₂, detect number of times m ₁, m ₂Get S ₂=S _Max, S ₂Under measurement number of times m ₂=m-m ₁

The situation of counter stress number of levels K>=3, the decision variable of testing program is: stress level S ₁, S ₂..., S _K, detect number of times m ₁, m ₂..., m _KMake S _k=S _Max, make the horizontal S of intermediate stress _kBe expressed as S ₁And S _kFunction, 1＜k＜K, same, to measuring number of times, middle m _kUse m ₁Function representation, then

1＜k＜K.

4. a kind of accelerated degradation test Optimization Design based on relative entropy according to claim 1 is characterized in that, in the described step 1 (4); To each testing program; Based on Monte Carlo Markov chain method, utilize software WinBUGS14 calculation optimization target, be specially:

The writing of expectation information gain:

At first, the p in the formula (8) (x| θ) is a likelihood function, directly adopts Monte Carlo emulation mode to calculate E at parameter space and sample space _xE _θ[log p (x| θ)], computing formula is following:

R in the formula (9) ₂Be simulation times, get bigger integer usually, generally be chosen as 100, p (x _h| θ _h) be likelihood function, wherein θ _hThe expression parameter that emulation is extracted from the parameter prior distribution, x _hExpression is from sampling distribution f (x| θ _h, η) the middle degeneration properties of product incremental data that generates;

Secondly, for marginal likelihood function p (x), adopt the Laplace-Metropolis algorithm to estimate that computing formula is following:

Wherein, π is a circular constant, π ≈ 3.14; D is the dimension of model parameter vector; R _MLBe the number of the performance degradation increment x of emulation, be sample size n and total product that detects number of times m, i.e. R _ML=n * m; θ _gBe based on the parameter Posterior Mean that MCMC adopts g the performance degradation increment that software WinBUGS14 calculates, Be the θ that obtains by N emulation degeneration incremental data _gAverage; Σ _θIt is the posterior variance-covariance matrix of parameter;

Based on above-mentioned formula (8) ~ (11), following based on the solution procedure of the optimization aim of relative entropy:

Substep 1. is total to R from the testing program set _DGet scheme η in the individual scheme _r, r=1 ..., R _D

2. couples of scheme η of substep _r, R is extracted in emulation from its corresponding prior distribution ₂Subparameter θ _Rh, h representes simulation times, h=1 in this step ..., R ₂, and utilize the parameter θ of each emulation _Rh, from sampling distribution f (x| θ _Rh, η _r) the middle degeneration properties of product incremental data x that generates _Rh

The degeneration incremental data x that substep 3. generates according to emulation _Rh, numerical procedure η _rLog-likelihood function logp (x _Rh| θ _Rh, η _r), and according to formula (9) numerical procedure η _rE _xE _θ[logp (x _Rh| θ _Rh, η _r)]:

4. couples of scheme η of substep _r, parameter θ is extracted in emulation from its corresponding prior distribution _r, to θ _rEmulation R ₃Inferior degeneration incremental data x _Rh, h representes simulation times, h=1 in this step ..., R ₃The degeneration incremental data that obtains based on each emulation is in conjunction with the parameter Posterior Mean θ of MCMC method and g performance degradation incremental data of WinBUGS14 computed in software _g, and then obtain θ according to formula (11) _gAverage

With the posterior variance-covariance matrix Σ of parameter _θ:

Substep 5. calculates marginal likelihood function p (x according to formula (10) _Rh):

And then obtain wherein; Likelihood function under

representation model parameter Posterior Mean

known conditions,

representation model parameter Posterior Mean

is brought the probable value that prior distribution obtains into;

Substep 6. calculates EIG according to formula (8), promptly obtains the expectation information gain of scheme η r;

Get back to substep 1, to each scheme iteron step 2 ~ 6 in the scheme set; Obtain all scheme optimization targets in the scheme set.

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