CN102779208B - 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

Based on the sequential accelerated degradation test Optimization Design of relative entropy

Technical field

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

Background technology

In Now Domestic external reliability engineering, the life-span of the 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) be the gordian technique that solves at present this difficult problem.Wherein ADT has overcome the shortcoming of ALT record product fail data, becomes the study hotspot in fail-test field.

First accelerated degradation test technology must be faced with the design problem of testing program in engineering application, the i.e. test variable such as scientific and reasonable arrangement proof stress level, tested sample, test period under limited time and expense how, to obtain the most effective performance degradation information, make life of product and reliability assessment the most accurate.If can solve this problem, adopt the testing program of 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 greatly improve test efficiency, make to test resource and be fully used, reduce the development cost of product.Therefore study accelerated degradation test optimal design and there is important theory value and engineering using value.

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 products, conventionally all can there is a large amount of prior imformations.Such as, the historical test data of like product or actual usage data; Product is from performance test, the environmental test etc. of carrying out before ADT.Accelerated degradation test optimal design depends on the information of model and unknown-model parameter, according to different priori conditions, can obtain different designs.In the time that model (degradation model and acceleration model) is known, if can provide the estimated value of model parameter according to prior imformation and engineering experience etc., claim that this estimated value is the assumed value (or default) of parameter.On this basis, select different Optimality Criterias to obtain optimal test scheme, be called classics or local accelerated degradation test optimal design (Locally optimal design), also referred to as traditional accelerated degradation test optimal design.In the time that the estimated value deviation of parameter is larger, the scheme that local optimum design obtains is not optimal case.Bayes's test design is to utilize prior imformation to determine the prior distribution of model parameter but not concrete value based on this, is set up Bayes's Optimality Criteria and obtained optimal case, thereby has avoided the shortcoming of local optimum design.Bayes's optimization is called again average (average) and optimizes or overall (population) optimization.Bayes's optimal design is taking the optimizing decision theory under condition of uncertainty as basis to a certain extent, is a kind of decision theory, is important branch of statistical theory.Its objective is the expected utility that maximizes test.The default of parameter and prior distribution are the important difference of local optimum design and Bayes's optimal design.

In traditional reliability statistics test, classify and can be divided into fixed time test, fixed failure number test and sequential Censored Test by truncation mode.Fixed time test refers to be tested n sample, the truncated time of regulation test in advance t ₀, to moment t ₀all test specimens stop test, utilize the characteristic quantities of test figure assessment product; Fixed failure number test refers to be tested n sample, and the number of faults r of the truncation of regulation test in advance, till test proceeds to and occurs r fault, utilizes the characteristic quantities of test figure assessment product; Sequential Censored 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 one test that the criterion of the reception of the correlation test time of accumulation and number of faults and regulation, rejection or continuation test is compared.In these three kinds of tests, timing and fixed failure number test are all to make scheme before test, and sample size and truncated time or the truncation number of faults of clear and definite test enforcement, no longer adjust scheme in process of the test.We might as well claim be designed to " Static Design " of this class testing program.Along with carrying out of test, we should constantly sum up the information obtaining from test, in conjunction with Mathematical Statistics Analysis, progressively understand understanding product, therefore, constantly adjust testing program, and " limit test, limit arranges " is only rational test design method.Sequential Censored Test adopts this thinking exactly, before test, do not specify concrete given the test agent number and truncated time, but in process of the test, progressively utilize the Test Information such as accumulation test period and number of faults of acquisition and the criterion comparison of regulation, and then make the judgement of next step test, therefore can early obtain conclusion (of pressure testing), thereby reduce experimentation cost.Claim this conceptual design for " dynamic design ".

In accelerated test conceptual design, be generally and reduce life of product and reliability extrapolation error under normal stress, can select more than 3 or 3 stress level to implement test.In implementation process, along with the carrying out of test, can progressively obtain the True Data that under each acceleration stress level, properties of product are degenerated.Therefore, also can in the manner described above, according to the utilization of Test Information, plan design be divided into " Static Design " and " dynamic design ".The design of ADT local optimum and ADT Bayes optimal design all belong to " Static Design ", and sequential ADT optimal design belongs to " dynamic design ".

The existing large quantity research of ADT local optimum design:

To CSADT, H-F Yu etc. studied that degradation ratio obeys respectively that Weibull reciprocal distributes and 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, specify the progressive variance of out-of-service time quantile maximum likelihood estimation as optimization aim to minimize, adopt general equivalence theorem (general equivalence theorem, GET) verify the optimality (list of references [2]: Ying Shi of different schemes, Luis A Escobar, William Q Meeker.Accelerated Destructive Degradation Test Planning.Technometrics, 2009,51 (1): 1-13).Yashun Wang etc. are based on melange effect model, divide the square error of position estimation as target taking minimum production life-span distribution p, taking testing expenses as constraint, adopt Monte Carlo emulation, study 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 Gamma process study the 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 competes reliability model p under normal stress and divides the progressive variance of a life estimation as target to minimize, study the optimization method (list of references [9]: Xiaoyang Li of considering the SSADT under 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, distribute as the priori of melange effect model parameter with normal distribution with against gamma, taking reliability estimated accuracy as target, test sample amount and Performance Detection number of times are optimized to 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, provide 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 is generally difficult to obtain the mathematic(al) representation that posteriority distributes, and has therefore derived two kinds of methods that address this problem: one is based on emulation, and one is based on large sample theory.More existing scholars of ALT design based on bayesian theory make research.The people such as Alaattin Erkanli have studied the Bayesian 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).The people such as Gladys D.C.Barriga propose under the distribution of index-Weibull life-span and Arrhenius model, the bayes method of the ALT based on Markov Chain Monte Carlo (MCMC).Yao Zhang and W.Q.Meeker utilizes the approximate bayesian criterion that obtains of large sample, study the Bayesian Design of the CSALT of the linear acceleration model of the distribution of logarithm Location Scale and location parameter, and propose to find 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, according to the method, first implement the test of high stress level to obtain rapidly fail data, then, based on Bayesian inference method, utilize these fail datas, set up the prior imformation under low stress level.Divide the pre-posteriority of the progressive posterior variance of position estimation to expect by life characteristics under minimum production normal stress level, make test design optimized (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, for sequential CSALT, Bayes's Optimization Design is proposed.Taking Squared Error Loss minimum as Optimality Criteria, provide optimum results based on emulation and in conjunction with curved surface fitting method.By contrast, prove that Bayesian Design method 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 there is no the research of sequential accelerated degradation test Optimization Design.

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

Summary of the invention

The object of the invention is the design of local optimum in order to solve current accelerated degradation test and Bayes's optimal design and all only utilize the prior imformation of product, " Static Design ", do not consider the performance degradation information in process of the test, and this information reflects 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, the relative entropy that the method obtains with test is optimization aim to the maximum, adopt the stress of " step is fallen " to apply mode, not only utilize the prior imformation before test, utilize the Test Information of each step simultaneously, constantly follow-up test scheme is optimized to design.By making full use of the information of every step test, can make the scheme that more meets the feature of product own, more saves experimentation cost.

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

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

Step 2, set up sequential truncation decision method;

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

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 take full advantage of the prior imformation before test, also progressively utilize in test and obtained performance degradation information, reduced the test design error causing when prior imformation and product truth exist relatively large deviation, therefore than the local optimum design of accelerated degradation test and Bayes's optimal design all tool have an enormous advantage;

(2) the inventive method is set up the Optimality Criteria of sequential trials designs based on relative entropy, and this criterion expects that with test information gain is target to the maximum, has considered obtainable quantity of information in test comprehensively, the optimum results the obtaining engineering reality of more fitting;

(3) set up sequential trials design cycle and sequential truncation decision rule, for the engineering construction of accelerated degradation test design 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.

Brief description of the drawings

Fig. 1 is sequential trials design cycle schematic 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 present invention;

Fig. 5 is three testing program optimum results of sequential ADT stages 2 of level in the embodiment of the present invention;

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

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

Embodiment

Below in conjunction with drawings and Examples, technical scheme of the present invention is described in further detail.

The present invention is incorporated into " sequential trials " in accelerated degradation test design, in conjunction with bayesian theory, studies sequential accelerated degradation test optimal design.Sequential trials design cycle schematic diagram as shown in Figure 1, utilize product prior imformation, determine preliminary test scheme, the stress level number (representing with K) of clearly implementing, every enforcement one step test (often carrying out testing under a stress level) obtains after True Data, utilizes prior imformation and the true test figure of product, progressively follow-up test is optimized to design, make the testing program that more meets the feature of product own, until 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 accelerated degradation test optimal design, set up sequential accelerated degradation test optimal design flow process framework and sequential truncation decision method based on relative entropy, provide the concrete steps of sequential accelerated degradation test Optimization Design.

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

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

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

What the application of properties of product degradation model was more mainly comprises three kinds of models of melange effect model, gamma process and Brownian movement (Wiener process).Conventional acceleration model has Arrhenius (Arrhenius) model, contrary power rate model, Aileen (Eyring) model etc., and its form all can represent log-linear form: wherein, the a certain known function of stress s, for example, for Arrhenius (Arrhenius) model, s=T, T is absolute temperature; For contrary power rate model, s can represent voltage, electric current, power etc.; D (s) is performance degradation rate; A, b are constant.According to product own characteristic, sensitive stress and performance parameter degenerate case etc., determine properties of product degradation model and acceleration model, and then probability density function and the log-likelihood function etc. of definite degradation model.

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

(2) build testing program set D

Formed or other condition by experimentation cost, determine 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), in definite proof stress horizontal extent, determine that in conjunction with testing expenses constraint, in the situation of total sample size n and overall measurement number of times m, needing the decision variable of optimizing is the horizontal S of each proof stress _kwith the measurement number of times m under each stress level _k.

Make S represent proof stress vector, S comprises K element, and K is positive integer, S=(S ₁..., S _k..., S _k), stress level of each element representation, S _krepresent k stress level, for example, have 3 stress levels in test, S comprises 3 elements, S=(S ₁, S ₂, S ₃).Make M represent to measure Vector of degree, M=(m ₁..., m _k..., m _k), m _krepresent k stress level S _kunder measurement number of times (k=1 ..., K).

Scheme set D=S × M, is made up of (S 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 design space.

Stress level and measurement number of times are all continuous variables, can mark off numerous scheme.Because objective function needs bulk sampling simulation calculation, usually need to move several days or even a few week.The present invention adopts surface fitting scheme to avoid this problem for this reason.Along stress level and measurement number of times direction, decile value in its span, design space is comprised to boundary demarcation is the set of limited scheme composition testing program, to each scheme calculating target function in testing program set, utilize surface fitting to find out the region of objective function maximum, and then find out optimal case.

The present invention adopts three-dimension curved surface fitting technique, and two independents variable are stress S ₁compare m with measurement number of times ₁, dependent variable is objective function.Also be stress S by optimization aim function representation ₁with measurement number of times m ₁function.Stress S under other proof stress levels _k(1<k≤K) and measurement 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 (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 ₂.Generally 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 _k.Make 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(1 < k < K) available m ₁function representation,

To constant stress accelerated degradation test (CSADT), also need to consider that under each stress level, sample size is distributed n _k.Solution is as follows:

Count sample distribution n under K and each stress according to total sample number n and stress level _kconstraint condition: n _k>=n, provide several different samples and distribute, every kind of sample is distributed, can be according to the mode processing of above-mentioned SSADT, obtain every kind of optimal case under sample distribution, then compare, the scheme of select target value maximum is optimal case.

(3) set up the optimization aim based on relative entropy

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

According to the research of Lindley, the information I comprising in prior distribution ₀for:

I ₀＝∫p(θ)log p(θ)dθ＝E _θ log p(θ) （1）

Wherein, p (θ) represents model parameter prior distribution probability density function; E _θrepresent the expectation about θ.

The gross information content I obtaining from posteriority distributes ₁(x) be:

I ₁(x)＝∫p(θ|x)log p(θ|x)dθ （2）

Wherein, p (θ | x) represent 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）

Plan design should provide before data x obtains, and therefore needed the information of sample space to get mathematical expectation, and the expectation of I (η, x, p (θ)) is:

I(η,p(θ))＝E _x[I ₁(x)-I ₀] （4）

Wherein, E _xrepresent the information of sample space to get mathematical expectation, I (η, p (θ)) is also referred to as expecting 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 formula, p (x) is marginal likelihood function, is also standard constant, as follows:

p(x)＝∫p(x|θ)p(θ)dθ （6）

Wherein, p (x| θ) represents the likelihood function under 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), to each testing program, based on Monte Carlo Markov chain (MCMC) method, utilize software WinBUGS14 calculation optimization target.

In most cases, be difficult to obtain posterior analytical expression, therefore numerical simulation calculating is solution conventional in 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 above formula is difficult to have Explicit Expression formula conventionally, therefore generally adopt Monte Carlo (Monte Carlo) emulation mode to calculate.First, the p (x| θ) in formula (8) is 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 as follows:

$E_x{E}_{\mathrm{\&theta;}}\left[\log p\left(x\right|\mathrm{\&theta;})\right]=\frac{1}{R_2}\&CenterDot;{\&Sum;}_{h=1}^{R_2}\log p\left(x_h\right|{\mathrm{\&theta;}}_h)---\left(9\right)$

R in formula (9) ₂be simulation times, conventionally get larger integer, be generally chosen as 100.

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

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

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

Wherein, π is circular constant, π ≈ 3.14; D is the dimension of model parameter vector; R _mLbeing the number of the performance degradation increment x of emulation, is sample size n and total product that detects number of times m, i.e. R _mL=n × m; θ _gthe parameter Posterior Mean that adopts g the performance degradation increment that software WinBUGS14 calculates based on MCMC, the θ being obtained by N emulation degeneration incremental data _gaverage; Σ _θit is the posterior variance-covariance matrix of parameter.

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

Sub-step 1. is from testing program set (R altogether _dindividual scheme) in get scheme η _r, r=1 ..., R _d.

Sub-step 2. is to scheme η _r, from its corresponding prior distribution, R is extracted in emulation ₂subparameter θ _rh, h represents 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 sub-step 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)]:

$E_x{E}_{\mathrm{\&theta;}}\left[\log p\left(x_{\mathrm{rh}}\right|{\mathrm{\&theta;}}_{\mathrm{rh}},{\mathrm{\&eta;}}_r)\right]=\frac{1}{R_2}\&CenterDot;{\&Sum;}_{h=1}^{R_2}\log p\left(x_{\mathrm{rh}}\right|{\mathrm{\&theta;}}_{\mathrm{rh}},{\mathrm{\&eta;}}_r);$

Sub-step 4. is to scheme η _r, from its corresponding prior distribution, parameter θ is extracted in emulation _r, for θ _remulation R ₃inferior degeneration incremental data x _rh, h represents simulation times, h=1 in this step ..., R ₃; The degeneration incremental data obtaining based on each emulation, calculates the parameter Posterior Mean θ of g performance degradation incremental data in conjunction with MCMC method and WinBUGS14 software _g, and then obtain θ according to formula (11) _gaverage with the posterior variance-covariance matrix Σ of parameter _θ:

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

Sub-step 5. is calculated 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, represent model parameter Posterior Mean likelihood function under known conditions, represent model parameter Posterior Mean bring the probable value that prior distribution obtains into.

Sub-step 6. is calculated EIG according to formula (8), obtains scheme η _rexpectation information gain;

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

Sub-step 7. is got back to sub-step 1, to each scheme iteron step 2 ~ 6 in scheme set; Obtain the optimization aim of all schemes in scheme collection.

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

In some cases, the precise structure of Response surface regression model is difficult to determine.Therefore the present invention selects nonparametric technique to carry out regression fit to the desired value of all schemes, for example, and core smoothing method.For the regression fit effect of more different models, the inventive method returns with parametric polynomial respectively and the local weighted recurrence of nonparametric is fallen apart some smoothing method (Locally Weighted Scatterplot Smoothing, LOWESS) to the matching of data march face.

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

Step 2, set up sequential truncation decision method.

Sequential truncation decision method is specially:

(1) based on relative entropy, (judging with expecting information gain contrast according to actual information gain) judged in contrast.

First, implementing a certain stress level S _kbefore lower test, calculate this stress level S _kthe expectation information gain of lower test, is designated as EIG (S _k), numerical procedure is shown in (4) in step 1.

Then, implement S _klower test, (be designated as y, performance degradation incremental data x is the poor of adjacent twice Performance Degradation Data to the some Performance Degradation Datas 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 expects solving and need to getting mathematical expectation to the information of parameter space θ and sample space x of information gain EIG, therefore will from prior distribution, extract parameter θ R ₂inferior, and parameter θ to each extraction _h, from the sampling distribution of sample x, Multi simulation running generates performance degradation incremental data and gets expectation.And to actual information gain, performance degradation incremental data is actual tests data, need to not get expectation in sample space, directly brings actual tests data into and calculate.

Finally, relatively actual information gain and expectation information gain.When the actual information gain obtaining is greater than expectation information gain, i.e. IG (S _k) >EIG (S _k) time, can truncation, stop this step test.

In the present invention, relative entropy reflection is the distance between the prior distribution of parameter before and after test and posteriority distribute.The distance distributing when parameter prior distribution and posteriority is larger, and when parameter priori and its actual conditions model parameter value of actual rule (the reflection properties of product degenerate) differ far away, test acquired information gain meeting is larger; And the distance of working as parameter prior distribution and posteriority distribution is less, parameter priori and its actual value differ hour, parameter prior distribution can reflect its truth, implement after test, the information entropy being obtained by real data is less, along with the progress of test, may occur that actual information gain is difficult to exceed the situation of expecting information gain, 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 _klower test, the some Performance Degradation Datas of every acquisition, the posteriority distribution p of calculating parameter (θ | x).Posteriority distribution p (θ | x) utilize software WinBUGS14 to obtain based on Markov monte carlo method.

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

The object of implementing accelerated degradation test is by obtaining properties of product degraded data, in conjunction with prior imformation, providing the estimation of model parameter, and then assesses life-span and the reliability under product normal stress.In the time that parameter posteriority is estimated to have tended towards stability, do not need to continue test; If the some degraded datas of every increase, the parameter posteriority estimated difference of calculating, apart from larger, needs to continue test.

(3) to test the monitoring number of times of implementing to specify in front scheme as truncation.

Consider under each step stress and all need to obtain certain performance degradation information, can not " stop " walk under test at certain after for a long time, therefore,, when actual information gain and parameter posteriority distribute can not meet truncation requirement time, the monitoring number of times that should implement to specify in front scheme taking this step test is as truncation.

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

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

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

Stage 1:

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

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

(3) the accelerated degradation test Optimization Design based on relative entropy according to step 1, determines the initial scheme before test is implemented, and is designated as η _k;

(4) implement high stress level S _kunder test, obtain the performance degradation incremental data x under this stress level _k.

Stage 2:

(1) determine that according to the Test Information under the prior imformation of model parameter and high stress level SK the posteriority of rear parameter of experience stage 1 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), in conjunction with remaining test resource, the accelerated degradation test Optimization Design according to step 1 based on relative entropy, obtains follow-up test scheme, is designated as η _k-1;

(3) implement stress level S _k-1under test, the some Performance Degradation Datas of every collection, judge whether to continue to test under this stress level according to truncation decision rule in step 2.

Stage 3:

(1) according to stress level S _kand S _k-1under Test Information, determine rear parameter of experience stages 2 posteriority distribute, be 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), in conjunction with remaining test resource, the accelerated degradation test Optimization Design according to step 1 based on relative entropy, obtains follow-up test scheme, is designated as η _k-2.

(3) implement stress level S _k-2under test, the some Performance Degradation Datas of every collection, judge whether to continue to test under this stress level according to truncation decision rule in step 2.

So go on

Stage K:

(1) according to before Test Information under all stress levels, determine that the posteriority of parameter distributes after 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), in conjunction with remaining test resource, the accelerated degradation test Optimization Design according to step 1 based on relative entropy, obtains the testing program of last stress level, is designated as η ₁.

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

Above-mentioned x _krepresent the performance degradation incremental data of product under k stress level, k=1 ... K.

Taking 3 levels as example, elaborate sequential ADT conceptual design flow process below, as shown in Figure 3, totally 3 stages.The corresponding stress level of every one-phase, determine parameter prior distribution according to the information before this stage, the parameter posteriority that for example the 2nd stage obtained according to model parameter prior distribution and the 1st stage distributes and determines this stage parameter prior distribution, the parameter posteriority that the 3rd stage obtained according to the first two stage distributes and determines this stage parameter prior distribution, utilize Bayes's Optimization Design, formulate optimal test scheme based on relative entropy, implement a step test.Along with the continuous acquisition of test figure, the prior distribution of parameter is constantly revised, and the optimal case therefore obtaining is also constantly adjusted, and is specially:

Stage 1: high stress level S ₃lower test

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

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

(3) the accelerated degradation test Optimization Design based on relative entropy according to step 1, determine the initial scheme before 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 ₃represent initial trial scheme;

S ₁, S ₂, S ₃represent three stress levels determining in initial scheme, and S ₁< S ₂< S ₃; General S ₃=S _max;

M ₁, m ₂, m ₃represent three stress level S ₁, S ₂, S ₃the performance monitoring number of times of lower product; m ₁>=m ₂>=m ₃,

N ₁, n ₂, n ₃represent three stress level S ₁, S ₂, S ₃under tested 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 to each stress level, therefore n ₁=n ₂=n ₃=n.

(4) implement high stress level S ₃lower test, gathers stress S ₃the Performance Degradation Data of lower product, every collection one piece of data, information gain (IG) and parameter posteriority that the test that calculating has been carried out obtains, and be analyzed.For making to obtain enough quantity of information under each stress level, under high stress level, test period is generally shorter, does not therefore do truncation in advance, implements to test under high stress level its actual monitoring number of times m according to initial scheme _3rrepresent m _3r=m ₃.

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

(1) according to the prior imformation of model parameter and S ₃lower parameter posteriority distribution p ₃(θ | x ₃), (x ₃for S ₃the performance degradation incremental data of lower acquisition; Subscript 3 index stress levels, p ₃(θ | x ₃) expression acquisition S ₃under Test Information after the posteriority of parameter θ distribute) determine stress level S ₂the prior distribution p of lower parameter ₂(θ) (subscript 2 index stress levels, p ₂(θ) represent to implement stress level S ₂the prior distribution of front parameter θ);

(2), in conjunction with remaining test resource, the accelerated degradation test Optimization Design according to step 1 based on relative entropy, obtains follow-up S ₁, S ₂lower testing program:

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

Wherein, η ₂represent through S ₃after lower test, S ₁, S ₂lower 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'), 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 ' ₂; Sample size n ₁→ n ₁', n ₂→ n ' ₂.

(3) implement the 2nd stress level S ₂' under test, the some Performance Degradation Datas of every collection, calculate the information gain (IG) and the parameter posteriority that obtain, according to truncation decision rule judgement in step 2, if meet truncation decision rule, stop testing under this stress level, according to obtain Test Information optimization and implement next step test; If do not meet truncation decision rule, continue under force level, to test.In the time meeting truncation decision rule, make S ₂' lower actual monitoring frequency table is shown m _2r.

Stage 3: stress level S ₁lower test

(1) according to the posteriority distribution p of parameter under two stress levels having implemented ₃(θ | x ₃) and p ₂(θ | x ₂), x ₂represent stress level S ₂' under performance degradation incremental data, determine stress level S ₁the prior distribution p of lower parameter ₁(θ);

(2), in conjunction with remaining test resource, the accelerated degradation test Optimization Design according to step 1 based on relative entropy, determines S ₁lower 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 Datas of every collection, calculate the information gain (IG) and the parameter posteriority that obtain, according to truncation decision rule judgement in step 2, if meet truncation decision rule, stop testing; If do not meet truncation decision rule, continue under force level, to test.In the time meeting truncation decision rule, make S ₁" lower actual monitoring frequency table is shown m _1r.

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

embodiment:

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

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

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

(1) determine 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 design information, historical data, like product information of being subject to trial product SLD etc., as follows,

1. hypothesis:

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 _kunder, 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 formula, Y _k(t) be the performance parameter of product, reflection properties of product degenerative process; y ₀for the starting point of Brownian Motion with Drift, properties of product are at initial time t ₀initial value; B (t) is standard Brownian movement, and B (τ) ~ N (0, t); T represents 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, be also coefficient of deviation is acceleration model

Wherein, A and B represent respectively model undetermined parameter, S _kbe k and accelerate stress level, stress S _kthe function of certain form, for example, in the time that acceleration stress is absolute temperature

2. model:

1) taking temperature as accelerating stress, acceleration model is elected ArrheniuS model as:

Right carry out parametrization:

In 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 failure mechanism prerequisite, generally gets S _k=S _max.The horizontal υ of proof stress after 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）

Known according to formula (15) and (16), β ₀=ln d _max, β ₀represent the logarithm of properties of product degradation ratio under high stress level; β ₀+ β ₁represent the logarithm of properties of product degradation ratio under normal stress level; β ₁represent logarithm poor of performance degradation rate under the most heavily stressed and normal stress, β ₁=(ln d _k-β ₀)/υ _k=lnd ₀-β ₀; d ₀represent properties of product degradation ratio under normal stress level; d _krepresent stress level S _klower properties of product degradation ratio; υ _krepresent stress level S _kform after parametrization.A lot of important acceleration models comprise that Arrhenius model, contrary power rate model all can be expressed as the form of formula (16) after suitable parametrization.To Arrhenius model, known 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 Brownian Motion with Drift:

Y _k(t)＝σ _kB(t)+d(S _k)·t+y ₀ （17）

Wherein, σ _krepresent coefficient of diffusion, d (S _k) represent coefficient of deviation, also can be described as degradation ratio, be the function of stress S, be also coefficient of deviation is acceleration model, make initial value y ₀=0.Unknown-model parameter vector θ={ A, B, τ }, τ=1/ σ ².Also 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.In hypothesis test, do not have because the inefficacy that performance degradation causes.In SSADT, under k stress level, performance monitoring number of times is m _k, the accumulation of SSADT monitoring number of times is properties of product monitoring time is spaced apart Δ t, the test period t under k stress level _k=m _kΔ t, total testing time is t=m Δ t.Under k stress level, to be subject to the j Measuring Time of trial product be t to i _kij(i=1 ..., n, k=1 ..., K, j=1 ... m _k), the performance number monitoring is y _kij.Brownian movement is Gaussian process, and therefore the independence of the performance degradation increment x in interval of delta t detection time and obedience average are 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 is,

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

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

$p\left(x\right|\mathrm{\&theta;})=\underset{k=1}{\overset{K}{\mathrm{\&Pi;}}}\underset{i=1}{\overset{n}{\mathrm{\&Pi;}}}\underset{j=1}{\overset{m_k}{\mathrm{\&Pi;}}}\frac{}{\sqrt{2\mathrm{\&pi;}{\mathrm{\&tau;}}^{-1}\mathrm{\&Delta;t}}}\exp \{-\frac{{[x_{\mathrm{kij}}-\exp (A+B/S_k)\&CenterDot;\mathrm{\&Delta;t}]}^2}{2{\mathrm{\&tau;}}^{-1}\mathrm{\&Delta;t}}\}---\left(19\right)$

X _kijrepresent i under k stress level j performance degradation incremental data that is subject to trial product, its log-likelihood function is,

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

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 conjugate prior distribution theory, can be supposed parameter beta ₀and β ₁obeying respectively average is μ ₀, μ ₁, variance is σ ₀, σ ₁normal distribution, σ ²τ reciprocal to obey scale parameter be a, the gamma that form parameter is b distributes.Also, τ ~ Γ (a, b).

According to the design information of like product information, product, the test etc. of knowing the real situation, determine 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）

The known E of angle (ln d according to formula (16) from degradation ratio _k)=E (β ₀)+E (β ₁) υ _k.

(2) build 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.

Incorporation engineering experience is divided stress vector and is measured Vector of degree.First, determine stress level S ₁value space, make S ₁=[60,65,70,75,80,85,90] DEG C, S ₃=110 DEG C, S ₂get S ₁with S ₃absolute temperature inverse uniformly-spaced, corresponding S ₁each element obtain S ₂≈ [83,86,89,92,94,97,100] DEG C.Secondly, determine monitoring number of times m ₁value space, make m ₁, for ensureing 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 ₂lower monitoring number of times be height monitor under two levels number of times and half, i.e. m ₂=0.5 (m ₁+ m ₃), m ₂=40, m ₃=80-m ₁.Form testing program set D(S according to the value space of monitoring number of times under each stress level and each stress level ₁get 7 values, m ₁get 4 values, therefore totally 28 schemes).

(3) set up the optimization aim based on relative entropy

The optimization aim of setting up based on relative entropy according to (3) in 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), to each testing program, based on Monte Carlo Markov chain (MCMC) method, utilize software WinBUGS14 calculation optimization target.

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

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

According to the method for step 1 in embodiment (5), form several to (η to the optimization target values of all schemes that obtain _r, EIG (η _r)), adopt polynomial expression (secondary) recurrence and local weighted (secondary) to return a loose some smoothing method (LOWESS) and carry out fitting surface.

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

The sequential ADT initial trial of table 1 three level 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.

Set up the sequential truncation decision rule of this example from actual information gain with expecting three angles of monitoring number of times that specify in scheme conversion that information gain contrast, parameter posteriority distribute and test are implemented according to step 2 in specific implementation method.

(1) based on relative entropy, (judging with expecting information gain contrast according to actual information gain) judged in contrast.

First, implementing a certain stress level S _kbefore lower test, calculate this stress level S _kthe expectation information gain EIG (S of lower test _k).Then, implement S _klower test, the some performance degradation incremental datas of every acquisition, calculate actual information gain IG (S _k).Finally, relatively actual information gain and expectation information gain.When the actual information gain obtaining is 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.

Be difficult to exceed if there is actual information gain the situation of expecting information gain, the posteriority changes in distribution situation of observed parameter judges whether to stop test simultaneously.

Implement S _klower test, the some Performance Degradation Datas of every acquisition, the posteriority distribution p of calculating parameter (θ | x).When parameter posteriority changes in distribution less, while tending towards stability, can truncation, stop this step test.

(3) to test the monitoring number of times of implementing to specify in front scheme as truncation.

Consider under each step stress and all need to obtain certain performance degradation information, can not " stop " walk under test at certain after for a long time, therefore,, when actual information gain and parameter posteriority distribute can not meet truncation requirement time, the monitoring number of times that should implement to specify in front scheme taking this step test is as truncation.

Step 3, set up sequential trials design cycle.

In the sequential SSADT of this example, temperature stress number of levels is 3, i.e. K=3.Establish the sequential trials design cycle of this example according to the design cycle of step 3 in specific implementation method.

Stage 1:

(1) determine 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) form and determine test resource according to testing expenses: always monitor number of times m=120 and sample size n=3, monitoring interval of delta t=1; Be subject to the working stress S of trial product SLD ₀be 25 DEG C, minimum acceleration stress S _minbe 60 DEG C, working limit S _maxit is 110 DEG C.

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

(4) according to initial trial scheme η ₃, implement high stress level S ₃under test, obtain the performance degradation incremental data x under this stress level ₃.Suppose the performance degradation incremental data x of " actual tests " ₂by 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 MCMC method, utilize WinBugs software to calculate 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 lower test ₂(θ).

(2) utilize surplus resources to determine S ₁and S ₂under scheme set D ₂, be optimized result as shown in Figure 5, optimal case:

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

Implement S ₂' lower test, it expects information gain EIG (S ₂')=64.6. therefore S ₂' under the performance degradation incremental data x of " actual tests " ₂～N (0.0489,0.05 ²).The some data of every collection, calculate actual information gain and parameter posteriority, in table 2 and table 3.

The sequential ADTS of table 2 three level ₂the actual information gain of ' lower different 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 ' lower different monitoring number of times distributes

Due to m ₂=35 o'clock, actual information gain does not reach yet expected information gain and parameter beta ₁posterior Mean still have larger fluctuation, therefore should not shift to an earlier date truncation, according to implementing in the original plan this step test, m ₂actual information gain in=40 o'clock reaches expectation information gain, and parameter beta ₀posteriority distribute variation less, β ₁with the posteriority changes in distribution of τ as shown in Figure 6.Stop this step test, enter next stage.

(3) implement S ₁" under test, the some Performance Degradation Datas of every collection, calculate the information gain (IG) and the parameter posteriority that obtain, according to truncation decision rule judgement in step 2, if meet truncation decision rule, stop testing; If do not meet truncation decision rule, continue under force level, to test.In the time meeting truncation decision rule, make S ₁" lower actual monitoring frequency table is shown m _1r.

Stage 3:

(1) with S ₂' lower 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, determine monitoring number of times m in conjunction with remaining test resource ₁"=50, form testing program set D ₁, calculate respectively each S ₁corresponding expectation information gain (EIG), gets the maximum corresponding S of EIG ₁for optimum solution.Solve

$S_1^{\&prime;}=\underset{S_1\&Element;[S_{\min },S_{\max })}{\arg \max }\mathrm{EIG}\left(S_1\right)---\left(26\right)$

Obtain S ₁"=68 DEG C, EIG (S ₁")=76.5.

(3) implement S ₁" lower test, therefore S ₁" under the performance degradation incremental data x of " actual tests " ₁～N (0.0182,0.05 ²).The some data of every collection are calculated actual information gain and posteriority distribution, the results are shown in Table 4 and table 5.The posteriority changes in distribution of parameter as shown in Figure 7.Although m ₁=40 o'clock, actual information gain did not reach expectation information gain, but parameter beta ₁distribute and tend towards stability with the posteriority of τ, can stop test.

The sequential ADTS of table 4 three level ₂the actual information gain of ' lower different 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 ' lower different monitoring number of times distributes

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

Sequential Bayes's test design is not only utilized parameter prior distribution, and along with carrying out of testing, constantly utilizes actual tests data to adjust testing program, can stop in advance test according to 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 the accelerated degradation test Optimization Design based on relative entropy;

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

(2) build testing program set D;

Determine 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), to stepstress accelerated degradation test, in definite proof stress horizontal extent, determine that in conjunction with testing expenses constraint, in the situation of total sample size n and overall measurement number of times m, needing the decision variable of optimizing is the horizontal S of each proof stress _kwith the measurement number of times m under each stress level _k;

Make S represent proof stress vector, S comprises K element, and K is 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 Vector of degree, M=(m ₁..., m _k..., m _k), m _krepresent k stress level S _kunder measurement number of times, k=1 ..., K;

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

Adopt surface fitting scheme, along stress level and measurement number of times direction, decile value in its span, design space is comprised to boundary demarcation is the set of limited scheme composition testing program, to each scheme calculating target function in testing program set, utilize surface fitting to find out the region of objective function maximum, and then find out optimal case;

2), to constant stress accelerated degradation test, also need to consider that under each stress level, sample size is distributed n _k; Concrete grammar is as follows:

Count sample distribution n under K and each stress according to total sample number n and stress level _kconstraint condition: n _k≤ n, provide several different samples and distribute, every kind of sample is distributed, according to the mode processing of stepstress accelerated degradation test, obtain every kind of optimal case under sample distribution, then compare, the scheme of select target value maximum is optimal case;

(3) set up the optimization aim based on relative entropy

Using relative entropy as utility function, turn to optimization aim with the expectation information gain maximum obtaining in testing;

The information I comprising in prior distribution ₀for:

I ₀＝∫p(θ)logp(θ)dθ＝E _θlogp(θ) (1)

Wherein, p (θ) represents model parameter prior distribution probability density function; E _θrepresent the expectation about θ;

The gross information content I obtaining from posteriority distributes ₁(x) be:

I ₁(x)＝∫p(θ|x)logp(θ|x)dθ (2)

Wherein, p (θ | x) represent 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)

Plan design should provide before data x obtains, and therefore needed the information of sample space to get mathematical expectation, and the expectation of I (η, x, p (θ)) is:

I(η,p(θ))＝E _x[I ₁(x)-I ₀] (4)

Wherein, E _xrepresent the information of sample space to get mathematical expectation, I (η, p (θ)) is also referred to as expecting information gain; Based on bayesian theory, the Expectation-based Representation for Concepts of Test Information is:

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

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

p(x)＝∫p(x|θ)p(θ)dθ (6)

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

Therefore, the optimization aim based on relative entropy is:

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

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

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

Select nonparametric technique to carry out regression fit to the desired value of all schemes, fall apart some smoothing method to the matching 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) form _r, EIG (η _r)) carry out fitting surface, find the maximum corresponding scheme of desired value to be optimal test scheme;

Step 2, set up sequential truncation decision method;

Sequential truncation decision method is specially:

(1) based on relative entropy, contrast is judged;

First, implementing a certain stress level S _kbefore lower test, calculate this stress level S _kthe expectation information gain of lower test, is designated as EIG (S _k);

Then, implement S _klower test, the some Performance Degradation Datas of every acquisition, are designated as y, and performance degradation incremental data x is the poor of adjacent twice Performance Degradation Data, 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;

Finally, relatively actual information gain and expectation information gain; When the actual information gain obtaining is greater than expectation information gain, i.e. IG (S _k) >EIG (S _k) time, truncation, stops this step test;

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

Implement S _klower test, the some Performance Degradation Datas of every acquisition, the posteriority distribution p of calculating parameter (θ | x); Posteriority distribution p (θ | x) utilize software WinBUGS14 to obtain based on Markov chain Monte-Carlo method;

When parameter posteriority changes in distribution is less, while tending towards stability, truncation, stops this step test;

(3) to test the monitoring number of times of implementing to specify in front scheme as truncation;

When actual information gain and parameter posteriority distribute can not meet truncation requirement time, the monitoring number of times that should implement to specify in front scheme taking this step test is as truncation;

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

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

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

Stage 1:

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

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

(3) the accelerated degradation test Optimization Design based on relative entropy according to step 1, determines the initial scheme before test is implemented, and is designated as η _k;

(4) implement high stress level S _kunder test, obtain the performance degradation incremental data x under this stress level _k;

Stage 2:

(1) according to the prior imformation of model parameter and high stress level S _kunder Test Information determine rear parameter of experience stage 1 posteriority distribute, be 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), in conjunction with remaining test resource, the accelerated degradation test Optimization Design according to step 1 based on relative entropy, obtains follow-up test scheme, is designated as η _k-1;

(3) implement stress level S _k-1under test, the some Performance Degradation Datas of every collection, judge whether to continue to test under this stress level according to truncation decision rule in step 2;

Stage 3:

(1) according to stress level S _kand S _k-1under Test Information, determine rear parameter of experience stages 2 posteriority distribute, be 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), in conjunction with remaining test resource, the accelerated degradation test Optimization Design according to step 1 based on relative entropy, obtains follow-up test scheme, is designated as η _k-2;

(3) implement stress level S _k-2under test, the some Performance Degradation Datas of every collection, judge whether to continue to test under this stress level according to truncation decision rule in step 2;

So go on

Stage K:

(1) according to before Test Information under all stress levels, determine that the posteriority of parameter distributes after 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 ₁(θ);

(2), in conjunction with remaining test resource, the accelerated degradation test Optimization Design according to step 1 based on relative entropy, obtains the testing program of last stress level, is designated as η ₁;

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

Above-mentioned x _krepresent the performance degradation incremental data of product under k stress level, k=1 ... K.

2. a kind of accelerated degradation test Optimization Design based on relative entropy according to claim 1, it is characterized in that, in described step 1 (1), degradation model is melange effect model, gamma process model or Brownian Motion Model, and described acceleration model is Arrhenius relationship, 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 described step 1 (2), and described surface fitting scheme, two independents variable are stress S ₁with measurement number of times m ₁, be stress S by 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 ₂; 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 _k; Make 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, 1<k<K.

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

Expect information gain writing:

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

First, the p (x| θ) in formula (8) is likelihood function, directly adopts Monte Carlo emulation mode to calculate E at parameter space and sample space _xe _θ[logp (x| θ)], computing formula is as follows:

$E_x{E}_{\mathrm{\&theta;}}\left[\log p\left(x\right|\mathrm{\&theta;})\right]=\frac{1}{R_2}\&CenterDot;{\mathrm{\&Sigma;}}_{h=1}^{R_2}\log p\left(x_h\right|{\mathrm{\&theta;}}_h)---\left(9\right)$

R in formula (9) ₂be simulation times, get 100, p (x _h| θ _h) be likelihood function, wherein θ _hrepresent the parameter that emulation is extracted from parameter prior distribution, x _hrepresent from sampling distribution f (x| θ _h, η) and the middle degeneration properties of product incremental data that generates;

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

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

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

Wherein, π is circular constant, π ≈ 3.14; D is the dimension of model parameter vector; R _mLbeing the number of the performance degradation increment x of emulation, is sample size n and total product that detects number of times m, i.e. R _mL=n × m; θ _gbe the parameter Posterior Mean that adopts g the performance degradation increment that software WinBUGS14 calculates based on Markov chain Monte-Carlo, θ is the θ being obtained by N emulation degeneration incremental data _gaverage; Σ _θθ _gvariance-covariance matrix;

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

Sub-step 1. is from testing program set R altogether _din individual scheme, get scheme η _r, r=1 ..., R _d;

Sub-step 2. is to scheme η _r, from its corresponding prior distribution, R is extracted in emulation ₂subparameter θ _rh, h represents 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 sub-step 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)]:

$E_x{E}_{\mathrm{\&theta;}}\left[\log p\left(x_{\mathrm{rh}}\right|{\mathrm{\&theta;}}_{\mathrm{rh}},{\mathrm{\&eta;}}_r)\right]=\frac{1}{R_2}\&CenterDot;{\mathrm{\&Sigma;}}_{h=1}^{R_2}\log p\left(x_{\mathrm{rh}}\right|{\mathrm{\&theta;}}_{\mathrm{rh}},{\mathrm{\&eta;}}_r);$

Sub-step 4. is to scheme η _r, from its corresponding prior distribution, parameter θ is extracted in emulation _r, for θ _remulation R ₃inferior degeneration incremental data x _rh, h represents simulation times, h=1 in this step ..., R ₃; The degeneration incremental data obtaining based on each emulation, calculates the parameter Posterior Mean θ of g performance degradation incremental data in conjunction with Markov chain Monte Carlo and WinBUGS14 software _g, and then obtain θ according to formula (11) _gaverage θ and θ _gvariance-covariance matrix Σ _θ:

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

Sub-step 5. is calculated marginal likelihood function p (xrh) according to formula (10):

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

And then obtain $E_x\left[\log p\left(x\right)\right]=\frac{1}{R_3}\&CenterDot;{\mathrm{\&Sigma;}}_{h=1}^{R_3}\log p\left(x_{\mathrm{rh}}\right);$ Wherein, represent model parameter Posterior Mean likelihood function under known conditions, represent model parameter Posterior Mean bring the probable value that prior distribution obtains into;

Sub-step 6. is calculated EIG according to formula (8), obtains scheme η _rexpectation information gain;

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

Get back to sub-step 1, to each scheme iteron step 2～6 in scheme set; Obtain the optimization aim of all schemes in scheme collection.