CN101620045B - Method for evaluating reliability of stepping stress quickened degradation experiment based on time sequence - Google Patents

Abstract

The invention discloses a method for evaluating the reliability of a stepping stress quickened degradation experiment based on a time sequence. The method utilizes a correlation coefficient stationary sequence analytical method to describe non-equidistant stochastic information in stepping stress quickened degradation experiment data, establishes a regressive-non-equidistant autoregression degradation model on the basis of obtaining the respective advantages of a deterministic regression model and a stochastic correlation coefficient stationary sequence autoregression model, combines a grey theory to predict a product degradation trend and forms a function and a curve for predicting the life and evaluating the reliability of the stepping stress quickened degradation experiment based on the time sequence. The method obviously decreases the sample quantity of a product and shortens the time of the degradation experiment, thereby saving a large amount of expenses and resources, enhancing the fitting accuracy of the degradation model and enhancing the reliability of a reliability evaluating result.

Description

Based on seasonal effect in time series stepstress accelerated degradation test reliability estimation method

Technical field

The present invention relates to a kind of accelerated degradation test life prediction and reliability estimation method, belong to accelerated test assessment technology field.

Background technology

For the long-life product of high reliability, often be difficult to observe in a short time product failure, become a kind of effective way based on properties of product degraded data analytic product reliability.In order to be difficult to obtain fail data at these, but product that can the obtained performance degraded data carries out reliability assessment, and the method for accelerated degradation test is arisen at the historic moment.

Degradation failure is quickened in research, and key will be set up the time dependent acceleration degradation model of properties of product degenerative character parameter.Quickening degradation model generally is divided into based on degradation mechanism with based on two kinds of models of statistics.The former need have deep research and a large amount of experimental basis to the degradation mechanism of product self, and is confined to the identical product of degradation mechanism, is not suitable for the popularization that engineering is used.The latter uses statistical model to describe degraded data, and is more suitable in engineering.

In accelerated degradation test reliability assessment in the past, quicken a certain regression model of the many employings of degradation model and represent based on statistics.This method can reflect the determinacy variation tendency that the performance characteristic parameter is degenerated, but the feature of performance degradation can not be described comprehensively, because the performance degradation rule of product is not only to comprise the determinacy part that rule changes, if will carry out Accurate Analysis, also need its irregular random partial is carried out deep research to it.

In addition, existing almost only effective at the accelerated degradation test of constant stress level based on the accelerated degradation test reliability estimation method of statistics.Yet the constant stress accelerated degradation test needs the test sample of some usually, thereby is unsuitable for newly the development and the product of cost costliness carry out the assessment of reliable life.And the stepstress accelerated degradation test has faster relatively expedite product degradation failure, and only needs the still less advantage of test sample number, thereby has higher practical value at the accelerated degradation test reliability estimation method of stepstress.

A branch of the former coefficient reason of time series analysis statistics subject, it is the important mathematical tool of research stochastic process.In fields such as engineering, for having randomness, and passing in time and have the information of certain statistical law, can not or be difficult to use the general analytic method of determining and describe its process, time series analysis then utilizes modern statistics and information data treatment technology, fully randomness rule and feature in the mined information are the strong instruments that solves this practical problems.Because stepstress accelerated degradation test data are time series, therefore can adopt Time series analysis method that it is analyzed.In the existing relevant stepstress accelerated degradation test reliability estimation method document, do not see the report of applied time series analysis method as yet at home and abroad.

Time Series Method need solve following problem in stepstress accelerated degradation test Application in Reliability Estimation:

1, adopt the test period in the Step-Stress Accelerated Life Testing based to amount to method, stepstress accelerated degradation test data can be converted into product test period under the normal stress level in the test period under each stress level, the degraded data after amounting to is the time series of non-equidistance.Yet traditional Time series analysis method only is applicable to the analysis of constant stress accelerated degradation test intermediate reach seasonal effect in time series, therefore needs research to be applicable to non-equidistance seasonal effect in time series analytical approach in the stepstress accelerated degradation test.

2, in the accelerated degradation test in the past, seldom consider the influence of testing equipment to the properties of product degenerative process, comprise the influence of determinacy and randomness, how deterministic analytic model is combined with the non-equidistance time series models of describing stochastic process, and the built-up pattern parameter estimated, remain further research.

3, in order to improve time series models accuracy of predicting and length, on the basis of retention time series model self advantage, how to adopt the combination with it of existing prediction theory, be the practical problems during engineering is used.

4, how will be applied to the stepstress accelerated degradation test based on the acceleration degradation model that Time series analysis method is set up, formation is applicable to the stepstress level, based on statistics, accelerated degradation test reliability estimation method with a high credibility, be a difficult point in the present field.

Summary of the invention

The objective of the invention is to describe randomness information in the accelerated degradation test statistics in order to solve traditional acceleration degradation model, and existing accelerated degradation test reliability estimation method is difficult to be applied to the problem of the accelerated degradation test reliability assessment of stepstress level, take time series analysis modeling technique means, reach technique effect according to stepstress accelerated degradation test data reasonable prediction life of product and reliability assessment.

The present invention utilizes the correlation coefficient stationary series analytical approach to describe non-equidistance random information in the stepstress accelerated degradation test data, set up recurrences-non-equidistance autoregression acceleration degradation model on the basis of advantage separately at the correlation coefficient stationary series autoregressive model that draws deterministic regression model and randomness, in conjunction with gray theory prediction product degradation trend, form method based on accelerated degradation test life prediction of seasonal effect in time series stepstress and reliability assessment.

The stepstress accelerated degradation test hypothesis that the present invention proposes:

The performance degradation process of supposing 1 product has monotonicity, i.e. the degeneration of performance generation is irreversible;

The failure mechanism of supposing product under 2 each stress level is constant;

The residual life of supposing 3 products only depends on totally inefficacy part and stress level at that time at that time, and irrelevant with the accumulation mode;

Suppose not have any inefficacy that causes by degenerating in 4 tests, promptly the performance degradation of product does not pass through failure threshold.

For ease of explanation, the present invention also needs following hypothesis to explain: suppose n product carried out the stepstress accelerated degradation test total k stress level S in the test _i, i=1,2 ..., k.Sampling interval under each stress level is Δ t, and the number of samples to properties of product under each stress level is m _i, total number of samples under all stress levels $m=\underset{i=1}{\overset{k}{\mathrm{\Σ}}}{m}_i,$ Then the test period of each stress level is τ _i=Δ tm _i, the performance degradation data of each product under all stress levels are y _l, l=1,2 ... m.

Based on above-mentioned hypothesis, reliability estimation method provided by the invention mainly comprises following five big steps:

Step 1, acquisition test data and data pre-service.

The original degraded data that is collected by testing equipment is difficult to directly it be carried out data analysis usually, the influence that causes according to one's analysis for fear of excessive degeneration value logarithm, improve the fitting precision that quickens degradation model, and the criterion of unified degradation failure, the original degraded data of tackling each product is done the pre-service of value just respectively.

Step 2, sampling interval are amounted to.

In the stepstress accelerated degradation test, stress is stepped in whole test, and the properties of product degradation ratio changes with the variation of stress, is to set up to quicken degradation model, at first will obtain same degradation ratio and be the degraded data under the same stress.In Step-Stress Accelerated Life Testing based, what adopt is that test period is amounted to method, promptly under the constant condition of the accumulation amount of degradation that keeps degraded data under each stress, test period to degraded data under each stress is amounted to, thereby convert out the test period that is experienced under the stress of product when it lost efficacy, be the life-span of product under this stress.

And for the stepstress accelerated degradation test, do not observe product failure in the middle of the test, therefore need to adopt the method for amounting to of this test period to obtain each stress level and convert test period under the normal stress level, thereby obtain the sampling interval of degraded data under the normal stress level, could set up the degraded data under the normal stress level like this and quicken degradation model, thus the prediction life-span of product under normal stress.Because the different stress levels sampling interval difference of degraded data under the normal stress level down, therefore the degraded data of being converted into the normal stress level by each stress level will no longer be the time series of equidistantly sampling.

To each product carry out respectively specifically to amount to step as follows:

1, respectively according to the accumulation amount of degradation and the corresponding test period thereof of degraded data under each stress level, determines the performance degradation rate under each stress level.

2, the degradation ratio under each stress level is corresponding with stress level.Each stress level and degradation ratio are set up regression model, just can obtain the relational model of degradation ratio and stress level.

3, by counter stress-degradation ratio relational model extrapolation, can obtain the degradation ratio of product under normal stress.

4, according to the accumulated damage equivalence principle, under the constant condition of the accumulation amount of degradation that keeps degraded data, test period under the different stress levels of stepstress accelerated degradation test is converted test period under the normal stress level by the ratio of degradation ratio, in like manner, the sampling interval under all proof stress levels can be converted sampling interval under the normal stress level by the ratio of degradation ratio.

Step 3, sequence acceleration Time Created degradation model.

The data of stepstress accelerated degradation test have characteristics such as non-stationary trend, cyclical variation and randomness variation, utilize Time Series Method to set up when quickening degradation model, must analyze item by item degraded data.If be illustrated in l=1 with Y (l), 2 ..., the performance degradation stochastic variable of m observation, according to the Cramer decomposition theorem as can be known, any one time series { Y (l) } can be decomposed into determinacy part and the steadily stack of random partial

Y(l)＝T(l)+S(l)+R(l)，l＝1，2，…m。(1)

In the formula: T (l) is a trend term, and S (l) is a periodic term, and R (l) is the residual error item.Trend term and periodic term are the determinacy part in the nonstationary time series, adopt regression model to represent usually.And the residual error item is steady random partial, adopts autoregressive model to represent.

As follows to the concrete modeling procedure that each product carries out respectively:

1, to through pretreated degraded data, sets up the trend term model.Because of trend term is relevant with stress level, so should adopt linear regression model (LRM) to set up the trend term model to the degraded data under each stress level respectively

T _i(l)＝slp _i·l+y _0i，l＝1，2，…m _i，i＝1，2，…k。(2)

In the formula: slp _iBe the degradation ratio under each stress level, y _0iBe degraded data initial value under each stress level.

2, remove trend term and the degraded data of removing behind the trend term set up the periodic term model.With the temperature stress test is example, is subjected to the influence of the temperature control characteristic of testing equipment, and temperature can be cyclic fluctuation up and down in a certain setting value usually.And for some product, for example can there be the phenomenon of temperature drift usually in electronic product, so the fluctuation up and down of temperature will inevitably be reflected on some performance output valve of electronic product.Owing to cyclic fluctuation is caused by testing equipment, and irrelevant with stress level, therefore can adopt the periodic regression modelling periodic term model of diving to the degraded data under all stress together

In the formula: q is the angular frequency number, A _jBe the amplitude of j angular frequency,

ω _jBe j angular frequency,

Be j phasing degree.

3, remove periodic term and the residual error item under each stress is converted into residual error item under the normal stress.The residual error item is relevant with stress level, for the degraded data under the normal stress is predicted, should be folded to sampling interval under the normal stress according to different stress, the equidistant sampling residual error item under each stress is converted into non-equidistance sampling residual error item under the normal stress.

4, the residual error item under the normal stress is set up residual error item model.Each stress level is folded to the equally spaced time series of residual error item right and wrong under the normal stress level, and traditional Time series analysis method only is applicable to the time series of equidistant sampling, and therefore traditional Time Series Method is not suitable for the analysis of these data.Because the residual error item is the related coefficient stationary process, " the non-equidistance correlation coefficient stationary series analytical approach " according to horse dogface, Fu Huimin were published in the aviation power journal in 2003 can adopt non-equidistance correlation coefficient stationary series analytical approach, with t _lRepresent l=1,2 ... m observation time corresponding under normal stress is set up non-equidistance seasonal effect in time series autoregressive model, thereby is obtained the residual error item model of product under the normal stress level

$R\left(l\right)=\underset{j=1}{\overset{p}{\mathrm{\Σ}}}{\mathrm{\η}}_j\left({\mathrm{\τ}}_l\right)R(l-j)+\mathrm{\ϵ}\left({\mathrm{\τ}}_l\right),l=\mathrm{1,2},\·\·\·m.---\left(4\right)$

In the formula: p is the exponent number of autoregressive model, η _jBe the function of autoregressive coefficient,

τ _l＝(τ _l1，τ _l2，…，τ _lp)′，τ _lj＝τ _i-τ _l-j(j＝1，2，…，p)，

ε (τ _l) be separate and obedience N[0, σ ₀ ²φ (α, τ _l)] white noise.

5, the latent periodic model of associating residual error item and the autoregression of periodic term modelling non-equidistance.Owing to have correlativity between residual error item R (l) and the stochastic variable Y (l),, should after determining every structure of models, all parametric joints be found the solution in order to improve the model fitting precision.The present invention proposes the latent periodic regression model substitution residual error item non-equidistance autoregressive model of periodic term is carried out interative computation, sets up the latent periodic model of non-equidistance autoregression, obtains the production life cycle item and residual error item conjunctive model under normal stress.

The latent periodic model of non-equidistance autoregression is

In the formula: A _j ^*,

The parameter of making corresponding conversion when diving periodic model for the periodic regression model substitution non-equidistance autoregression of diving.Concrete transform method can be with reference to the content of " applied time series analysis " write by He Shuyuan, the BJ University Press publishes middle chapter 7 about the latent periodic model of equidistant autoregression.

6, set up trend term model under the normal stress.According to the degradation ratio slp under the normal stress ₀And degraded data initial value y ₀, can obtain the trend term model under the normal stress

T ^*(l)＝slp ₀·l+y ₀，l＝1，2，…，m， (6)

7, combination recurrence-non-equidistance autoregressive model forms final acceleration degradation model.With trend term model under the normal stress and the latent direct addition of periodic model of non-equidistance autoregression, set up recurrence-non-equidistance autoregressive model, obtain the final acceleration degradation model of product degradation data under the normal stress level

Y ^*(l)＝T ^*(l)+X(l)，l＝1，2，…，m。(7)

Step 4, quicken degradation model in conjunction with gray theory prediction.

Time series analysis method is the randomness information in the mining data fully, yet this model exists the assurance of precision of prediction to depend on the limitation of large sample amount.If adopt conventional DIRECT FORECASTING METHOD to carry out the time series models prediction, then precision of prediction is difficult to guarantee, thereby has reduced reliability assessment result's confidence level.Gray theory be research minority according to probabilistic theory, promptly study minority according to probabilistic background under, the decision-making of the foundation of the processing of data, analysis of phenomenon, model, the prediction of development trend, things and the control of system and the assessment of state.Adopt the gray theory prediction to quicken degradation model, can on the basis of retention time sequence modeling advantage, overcome the limitation of time series models prediction well, improve the precision of prediction that quickens degradation model.

System's gray prediction nesting in the gray theory is the system's grey forecasting model at certain structure, and GM (1,1) model is embedded GM, and (1, N) model solution is to obtain the predicted value of each behavior variable.Quicken degradation model for time series, if according to raw data directly to future anticipation, its precision is often undesirable.Based on system's gray prediction nesting, each step predicted value is added raw data, and leave out first raw data, again degradation model is quickened in match again, model is revised constantly, by that analogy, till predicting desired step number, this nested prediction is usually than direct precision of prediction height.

Because the trend term model under the normal stress is to be got according to the degradation ratio that stress-the degradation ratio relation directly is extrapolated under the normal stress by the degradation ratio under each stress, therefore need not revise this model again, need only adopt system's gray prediction nesting to predict to the latent periodic model of the non-equidistance autoregression of setting up by residual error item under periodic term and the normal stress, again with normal stress under the direct predicted value addition of trend term model, thereby obtain final acceleration degradation model predicted value.

The concrete prediction steps that each product is carried out respectively is as follows:

1, set prediction step number h, make u=1,2 ..., h represents the number of times predicted.

2, establish x _l, l=1,2 ..., m represents one group of degraded data behind the removal trend term to be predicted, and it is adopted the periodic regression modelling periodic term model of diving.

In the formula: q _u, A _Uj, ω _Uj,

J=1,2 ..., q _uFor carrying out the model parameter that the u time when prediction revise.

3, remove periodic term and obtain the residual error item.Equidistant residual error item under each stress is converted into non-equidistance residual error item under the normal stress.

4, adopt the non-equidistance autoregressive model to set up residual error item model to the residual error item under the normal stress.

$R\left(l\right)=\underset{j=1}{\overset{p_u}{\mathrm{\Σ}}}{\mathrm{\η}}_{\mathrm{uj}}\left({\mathrm{\τ}}_{\mathrm{ul}}\right)R(l-j)+{\mathrm{\ϵ}}_u\left({\mathrm{\τ}}_{\mathrm{ul}}\right),l=\mathrm{1,2},\·\·\·,m.---\left(9\right)$

In the formula: τ _Ul, p _u, η _Uj, ε _u, l=1,2 ..., m, j=1,2 ..., p _uFor carrying out the model parameter that the u time when prediction revise.

5, associating residual error item and periodic term model, and the average sample spacing of getting residual error item under the normal stress

$\stackrel{\‾}{\mathrm{\Δt}}=\underset{i=1}{\overset{k}{\mathrm{\Σ}}}{\mathrm{\Δt}}_i,---\left(10\right)$

As the prediction spacing of model, correspondingly get τ _u=(Δ t, Δ t2 ..., Δ tp _u) ', sets up the predictor formula of the latent periodic model of non-equidistance autoregression.

In the formula: τ _u, A _Uj ^*,

J=1,2 ..., q _uFor carrying out the model parameter that the u time when prediction revise.

6, adopt dive the best of periodic model of non-equidistance autoregression not have inclined to one side predictor formula and carry out one-step prediction, the no inclined to one side predictor formula of described the best is as follows:

Obtain a new predicted value x of this model according to formula (12) _M+1, so far finish once prediction.

7, remove first data x of this group degraded data ₁, the one-step prediction value x of the latent periodic model of adding non-equidistance autoregression _M+1, obtain one group of degraded data x behind the new removal trend term _l, l=2,3 ..., m+1.

8, with x _l, l=2,3 ..., m+1 is data to be predicted, repeats above-mentioned steps 2～6.

9, repeat h-1 above-mentioned steps 7,8, can dope x one by one _M+u, u=1,2 ..., h.

10, the trend term model formation (6) under the normal stress is carried out the directly prediction of h step, obtain predicted value t _M+u ^*, u=1,2 ..., h.

11, with two kinds of model prediction results added, returned-the non-equidistance autoregressive model promptly quickens h step predicted value of degradation model

$y_{m+u}^{*}=t_{m+u}^{*}+x_{m+u},u=\mathrm{1,2},\·\·\·,h.$

Step 5, predicated error analysis and reliability assessment.

For accelerated degradation test, properties of product degenerate in the time of can not satisfying the performance index requirement of stipulating in the product specification, think that promptly inefficacy has taken place product.Its life-span is exactly that properties of product are from meeting the requirements to undesirable time.Set up acceleration degradation model method for what the present invention proposed based on time series, the inefficacy of properties of product is equivalent to degenerative process seasonal effect in time series value and passes through a certain setting failure threshold first, and the life-span of product is exactly the time of passing through failure threshold first.Because in accelerated degradation test, usually do not observe the properties of product amount of degradation and pass through failure threshold, can only pass through failure threshold first according to the predicted value of quickening degradation model and judge inefficacy, therefore this pairing life-span of inefficacy that obtains by prediction is not the true lifetime of product, and is referred to as the pseudo-life-span.The present invention is directed to the acceleration degradation model that the stepstress accelerated degradation test is set up, can directly obtain the pseudo-life-span under the normal stress level, so the distribution function in pseudo-life-span is exactly the unreliable degree function of product among the present invention.

In addition, for the reliability assessment of accelerated degradation test, the precision of prediction that quickens degradation model has determined the accuracy in pseudo-life-span, thereby has determined reliability assessment result's confidence level, so the present invention has carried out error analysis to pseudo-pairing predicted value of life-span.

As follows to the concrete appraisal procedure that all products carry out:

1, sets the degradation failure criterion of product according to the engineering actual conditions, provide failure threshold.

2, according to the step predicted value y of the acceleration degradation model h under each product normal stress _M+u ^*, u=1,2 ..., h determines that the predicted value of each product passes through the prediction step number u of failure threshold first _LifeAnd the time under the corresponding normal stress

$t_{\mathrm{life}}=\underset{i=1}{\overset{k}{\mathrm{\Σ}}}{\mathrm{\τ}}_{0i}+u_{\mathrm{life}}\·\stackrel{\‾}{\mathrm{\Δt}},---\left(13\right)$

Be the pseudo-life-span of each product under normal stress.

3, to the predicted value of the pseudo-life-span correspondence of each product

Carry out error analysis.Promptly return-the non-equidistance autoregressive model for quickening degradation model, its error is to be that the non-equidistance autoregressive model produces by the residual error item, and then the square error formula of its gray theory prediction is

${\mathrm{\&sigma;}}_{m+\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="H2009100891378E00031.png,H2009100891378E00041.png"\; state="\{\{state\}\}">u</figure-callout>}}^2=\underset{j=1}{\overset{\mathrm{<figure-callout\; id="2"\; label="u\; G\; uj"\; filenames="H2009100891378E00031.png,H2009100891378E00041.png"\; state="\{\{state\}\}">u</figure-callout>}}{\mathrm{\&Sigma;}}}{G}_{\mathrm{uj}}^{}{}_2^{}{\mathrm{\&sigma;}}_{(u-j+1)\mathrm{\&epsiv;}}^2,u=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,h,---\left(14\right)$

In the formula: G _Uj, j=1,2 ..., u is the Green function of non-equidistance autoregressive model u step prediction;

σ _{(u-j+1) ε} ², j=1,2 ..., u is the white noise variance of non-equidistance autoregressive model u-j+1 step prediction.Each pseudo-life prediction step number u then _LifePairing square error is institute and asks.

4,, determine the pseudo-life-span distribution function structure and the parameter estimation of product in conjunction with the pseudo-life-span under all product normal stresses.

5, according to the pseudo-life-span distribution function of product, provide Reliability Function and curve, thereby provide the reliability assessment result of stepstress accelerated degradation test.

The invention has the advantages that:

(1) the present invention existing accelerated degradation test reliability estimation method almost only at the effective situation of constant stress level under, complete feasible, the accelerated degradation test reliability estimation method that is applicable to stepstress of one cover is proposed, this method has realized the reliability assessment to stepstress accelerated degradation test data, compare with the existing method of the accelerated degradation test reliability assessment of constant stress that only is applicable to, the time that has significantly reduced the sample size of product and shortened degradation experiment, thus a large amount of funds and resource saved.

(2) the present invention adopts Time Series Method that stepstress accelerated degradation test data are set up first and quickens degradation model, because time series models are a kind of statistical models, describes the acceleration degraded data with it, is based on the acceleration degradation model of statistics.Carry out the method for stepstress accelerated degradation test reliability assessment compares with setting up the acceleration degradation model based on degradation mechanism, this method does not need that the physical chemistry degradation mechanism of product self is had deep research and a large amount of experimental basis, be not limited to the identical product of assessment degradation mechanism, thereby be more suitable for the popularization that engineering is used.

(3) the present invention proposes the method that the non-equidistance correlation coefficient stationary series analytical approach in the Time Series Method is applied to non-equidistance degraded data in the stepstress accelerated degradation test is set up the acceleration degradation model, successfully solved the problem that the existing acceleration degradation model of setting up at constant stress accelerated test data can't be applied to stepstress accelerated degradation test data.

(4) the present invention proposes recurrence-non-equidistance autoregressive model, and with its acceleration degradation model as stepstress accelerated degradation test data.This model combines regression model can accurately hold the advantage that performance determinacy variation tendency and autoregressive model can fully excavate randomness rule and feature in the random information, remedied tradition and only can not describe the shortcoming of properties of product degenerative character comprehensively quickening method that degraded data sets up regression model, improved the fitting precision that quickens degradation model, and the fitting precision that quickens degradation model has determined the accuracy of life prediction, thereby has improved reliability assessment result's confidence level;

(5) the present invention is except the performance degradation process of analytic product self, also proposed to consider the influence of testing equipment to the properties of product degenerative process, the cyclical variation process that is caused by testing equipment is set up the method for the periodic regression model of diving, only consider that with tradition the modeling method that the product self performance is degenerated compares, the acceleration degradation model that this method is set up is more comprehensive to the description of degenerative process, thereby has improved reliability assessment result's confidence level.

(6) the present invention proposes the latent periodic model of non-equidistance autoregression, and use it for description periodic term in the degraded data under the normal stress and residual error item.This model is not only with the two the simple addition of model, but proposed to consider that periodic term is with the correlativity of residual error item in the stepstress accelerated degradation test data, interative computation with the periodic regression model substitution non-equidistance autoregressive model of diving, and two kinds of model parameters are united the method for finding the solution, only the modeling method of every simple addition of model is compared with tradition, improved and quickened the fitting precision of degradation model, thereby improved reliability assessment result's confidence level;

(7) the present invention proposes and adopt in the gray theory system's gray prediction nesting that product is quickened degradation model to carry out forecast method, compare with traditional DIRECT FORECASTING METHOD, this method can be predicted according to probabilistic information at minority, can describe on the basis of this advantage of properties of product degenerative character at the retention time series model comprehensively, not only overcome based on seasonal effect in time series and quickened the limitation that the degradation model precision of prediction depends on the large sample amount, and improved and quickened the precision of prediction of degradation model, thereby improved reliability assessment result's confidence level.

(8) the present invention proposes the gray theory predicated error analytical approach of a kind of recurrence-non-equidistance autoregressive model, quantized to quicken the precision of prediction of degradation model, thereby can estimate the confidence level that adopts the inventive method to obtain the reliability assessment result objectively.

Description of drawings

Fig. 1 is the process flow diagram of reliability estimation method of the present invention;

Fig. 2 is original degraded data;

Fig. 3 is the trend term under the first pretreated degraded data of value and each stress level;

Fig. 4 is stress-degradation ratio relation;

Fig. 5 is degraded data and the periodic term behind the removal trend term;

Fig. 6 is that the latent periodic model of non-equidistance autoregression predicts the outcome;

Fig. 7 is recurrence-non-equidistance autoregressive model prediction result;

Fig. 8 is the original degraded data of embodiment 1;

Fig. 9 is the embodiment 1 pretreated degraded data of value just;

Figure 10 is embodiment 1 stress-degradation ratio relation;

Figure 11 is the degraded data after embodiment 1 removes trend term;

Figure 12 is degraded data and the prediction under embodiment 1 normal stress;

Figure 13 is embodiment 1 a fiduciary level curve.

Among the figure: 1-is the pretreated degraded data of value just, trend term under each stress level of 2-, 3-removes the degraded data behind the trend term, the 4-periodic term, the latent periodic model of 5-non-equidistance autoregression predicts the outcome, degraded data under the 6-normal stress level, 7-recurrence-non-equidistance autoregressive model prediction result, trend term under the 8-normal stress level, degraded data under the 9-embodiment 1 normal stress level, 10-embodiment 1 recurrence-non-equidistance autoregressive model prediction result, the trend term under the 11-embodiment 1 normal stress level.

Embodiment

The present invention is described in further detail below in conjunction with drawings and Examples 1.

The present invention is a kind of based on seasonal effect in time series stepstress accelerated degradation test reliability estimation method, at first carries out following hypothesis before method is carried out:

The performance degradation process of supposing 1 product has monotonicity, i.e. the degeneration of performance generation is irreversible;

The failure mechanism of supposing product under 2 each stress level is constant;

The residual life of supposing 3 products only depends on totally inefficacy part and stress level at that time at that time, and irrelevant with the accumulation mode;

Suppose not have any inefficacy that causes by degenerating in 4 tests, promptly the performance degradation of product does not pass through failure threshold.

And hypothesis is carried out the stepstress accelerated degradation test to n product, total k stress level S in the test _i, i=1,2 ..., k.Sampling interval under each stress level is Δ t, and the number of samples to properties of product under each stress level is m _i, total number of samples under all stress levels $m=\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{m}_i,$ Then the test period of each stress level is τ _i=Δ tm _i, the performance degradation data of each product under all stress levels are y _l, l=1,2 ... m.

The concrete grammar implementing procedure is realized as shown in Figure 1 as follows:

Step 1, acquisition test data are carried out the data pre-service.

Gather original degraded data by testing equipment, as shown in Figure 2, it is done the pre-service of value just.

As follows to the concrete grammar that each product carries out respectively:

If the original degraded data of stepstress accelerated degradation test is y _l, l=1,2 ... m.Then be through the first pretreated degraded data of value

$y_l^{\&prime;}=\frac{y_l}{y_1},l=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,m---\left(15\right)$

As the curve among Fig. 31, the initial value of pretreated degraded data all is 1.For ease of explanation, below still use y through pretreated degraded data _lExpression.

Step 2, sampling interval are amounted to.

As follows to the concrete grammar that each product carries out respectively:

1, for pretreated degraded data y _l, l=1,2 ..., mi calculates the accumulation amount of degradation under each stress level respectively

${\mathrm{\&Delta;y}}_i=y_{m_i}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="H2009100891378E00052.png,H2009100891378E00061.png"\; state="\{\{state\}\}">y</figure-callout>}}_1,i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,k,---\left(16\right)$

And corresponding test period τ _i, determine the performance degradation rate slp under each stress level _i, being curve 2 slopes as shown in Figure 3, the degradation ratio computing formula is as follows

${\mathrm{slp}}_i=\frac{{\mathrm{\&Delta;y}}_i}{{\mathrm{\&tau;}}_i},i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,k.---\left(17\right)$

2, with the degradation ratio slp under each stress level _iWith stress level S _iCorresponding.To each stress level S _iWith corresponding degradation ratio slp _iSet up regression model, to obtain deterioration velocity slp _iWith stress level S _iRelation.

With Arrhenius temperature acceleration model is example, as shown in Figure 4.This moment stress S _iBe absolute temperature, promptly

${\mathrm{slp}}_i=\mathrm{Aexp}\left(\frac{E}{{\mathrm{KS}}_i}\right),i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,k---\left(18\right)$

In the formula: K is a Boltzmann constant;

E is an activation energy, and is relevant with material;

A is a constant.

3, by counter stress-degradation ratio relational model extrapolation, can obtain product at normal stress S ₀Under degradation ratio slp ₀, just as the slope of the curve among Fig. 78.

4, according to the accumulated damage equivalence principle, promptly product is at stress S _iUnder the τ that works _iThe accumulation amount of degradation of time equals this product at stress level S _jUnder the τ that works _jThe accumulation amount of degradation of time, wherein i ≠ j can obtain the different stress level step-by-step test time τ of stepstress accelerated degradation test _iConvert the test period τ under the normal stress level _0iThereby, obtain all proof stress levels and convert sampling interval Δ t under the normal stress level _i, as shown in Figure 7.

Because

slp ₀·τ _0i＝Δy _i，i＝1，2，…k， (19)

Can get by formula (17)

slp ₀·τ _0i＝slp _i·τ _i， (20)

So the test period of converting under the normal stress is

${\mathrm{\&tau;}}_{0i}=\frac{{\mathrm{slp}}_i}{{\mathrm{slp}}_0}\&CenterDot;{\mathrm{\&tau;}}_i.---\left(21\right)$

In like manner, the sampling interval of converting under the normal stress level is

${\mathrm{\&Delta;t}}_i=\frac{{\mathrm{\&tau;}}_{0i}}{m_i}=\frac{{\mathrm{slp}}_i}{{\mathrm{slp}}_0}\&CenterDot;\frac{{\mathrm{\&tau;}}_i}{m_i}=\frac{{\mathrm{slp}}_i}{{\mathrm{slp}}_0}\&CenterDot;\mathrm{\&Delta;t}.---\left(22\right)$

Step 3, sequence acceleration Time Created degradation model.

As follows to the concrete grammar that each product carries out respectively:

1, to one group of pretreated degraded data y of process _l, l=1,2 ..., m sets up the trend term model.Adopt linear regression model (LRM) formula (2) to set up the trend term model to the degraded data under each stress level respectively, make t _l, l=1,2 ..., m _i, i=1,2 ..., k represents the sample value of this model, then the set of the trend term under all stress levels is

t _l＝{t ₁，t ₂，…，t _k}，l＝1，2，…，m，

Curve 2 as shown in Figure 3.

2, remove trend term t _lAnd the degraded data of removing behind the trend term set up the periodic term model.Data behind the removal trend term are

y _l-t _l，l＝1，2，…，m，

Adopt the periodic regression model formation (3) of diving to set up the periodic term model to it, make s _l, l=1,2 ..., m represents the sample value of this model, is periodic term, curve 4 as shown in Figure 5.

3, remove periodic term s _lAnd the equidistant sampling residual error item under each stress is converted into non-equidistance sampling residual error item under the normal stress.The residual error item is expressed as

r _l＝y _l-t _l-s _l，l＝1，2，…，m。

4, the sampling of the non-equidistance under normal stress residual error item is set up non-equidistance seasonal effect in time series autoregressive model, thereby obtain the residual error item model formation (4) of product under the normal stress level.

5, the latent periodic model formula (5) of associating residual error item and the autoregression of periodic term modelling non-equidistance makes x _l, l=1,2 ..., m represents the sample value of this model, is periodic term and residual error item under the normal stress, curve 3 as shown in Figure 6.

6, set up trend term model under the normal stress.According to the degradation ratio slp under the normal stress ₀And degraded data initial value y ₀, obtain the trend term model formation (6) under the normal stress, make t _l ^*, l=1,2 ..., m represents this model sample value, is the trend term under the normal stress, curve 8 as shown in Figure 7.

7, combination recurrence-non-equidistance autoregressive model.With trend term model under the normal stress and the latent direct addition of periodic model of non-equidistance autoregression, set up recurrence-non-equidistance autoregressive model formula (7), obtain the final acceleration degradation model of product degradation data under the normal stress level, make y _l ^*, l=1,2 ..., m represents this model sample value, then the degraded data under the normal stress is expressed as

$y_l^{*}=t_l^{*}+x_l,l=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,m.$

Curve 6 as shown in Figure 7.

Step 4, quicken degradation model and predicated error analysis in conjunction with gray theory prediction.

As follows to the concrete grammar that each product carries out respectively:

1, set prediction step number h, make u=1,2 ..., h represents the number of times predicted.

2, make x _l, l=1,2 ..., m represents one group of degraded data behind the removal trend term to be predicted, and it is adopted the periodic regression modelling periodic term model formation (8) of diving, and makes s _l, l=1,2 ..., m represents the sample value of this model, as shown in Figure 5.

3, remove periodic term and obtain the residual error item

r _l＝x _l-s _l，l＝1，2，…，m，

And the equidistant residual error item under each stress is converted into non-equidistance residual error item under the normal stress.

4, to the residual error item r under the normal stress _lAdopt the non-equidistance autoregressive model to set up residual error item model formation (9).

5, associating residual error item and periodic term model, and get the prediction spacing of the average sample separation delta t of residual error item under the normal stress as model, set up the predictor formula (10) of the latent periodic model of non-equidistance autoregression.

6, adopt dive the best of periodic model of non-equidistance autoregression not have inclined to one side predictor formula (11) and carry out one-step prediction, obtain a new predicted value x of this model _M+1, so far finish once prediction.

7, remove first data x of this group degraded data ₁, the one-step prediction value x of the latent periodic model of adding non-equidistance autoregression _M+1, obtain one group of degraded data x behind the new removal trend term _l, l=2,3 ..., m+1.

8, with x _l, l=2,3 ..., m+1 is data to be predicted, repeats above-mentioned steps 2～6.

9, repeat h-1 above-mentioned steps 7,8, dope x one by one _M+u, u=1,2 ..., h, as shown in Figure 6 the curve 5 of predicting the outcome that obtains.

10, the trend term model formation (6) under the normal stress is carried out the directly prediction of h step, obtain predicted value t _M+u ^*, u=1,2 ..., h, curve 8 as shown in Figure 7.

11, with two kinds of model prediction results added, returned-the non-equidistance autoregressive model promptly quickens h step predicted value of degradation model

$y_{m+u}^{*}=t_{m+u}^{*}+x_{m+u},u=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,h,$

Curve 7 as shown in Figure 7.

Step 5, predicated error analysis and reliability assessment.

As follows to the concrete grammar that all products carry out:

1, according to the engineering actual conditions, a certain percent value of setting the properties of product initial value is a failure threshold.

2, according to the step predicted value y of the acceleration degradation model h under the normal stress _M+u ^*, u=1,2 ..., h determines that the predicted value of each product passes through the prediction step number u of failure threshold first _LifeAnd the pseudo-life-span t under the normal stress level _Life

3, to the predicted value of the pseudo-life-span correspondence of each product

Carry out error analysis.According to the square error formula (14) that quickens the prediction of degradation model gray theory, the then pseudo-life prediction step number u of each product _LifePairing square error is institute and asks.

4, in conjunction with the pseudo-life-span t under all product normal stress levels _Life, determine the pseudo-life-span distribution function structure and the parameter estimation of product.Can distribute with reference to the life-span commonly used in the accelerated test, comprise exponential distribution, Weibull distribution, lognormal distribution etc.Its parameter can adopt maximum likelihood method to estimate.

5, obtain pseudo-life-span t under the normal stress level for the present invention _Life, its distribution function is exactly the unreliable degree function of product, is converted into Reliability Function, thereby provides the reliability assessment result of stepstress accelerated degradation test.

Embodiment 1:

With certain type electronic product stepstress accelerated degradation test is example, specifically introduces based on seasonal effect in time series stepstress accelerated degradation test reliability estimation method.

Step 1, acquisition test data are carried out the data pre-service.

10 these products are carried out the stepstress accelerated degradation test of 4 temperature stress levels, and test interval is 1 hour, and original degraded data as shown in Figure 8.Test parameters is as shown in table 1.

Table 1 stepstress accelerated degradation test parameter

Temperature	Number of samples
		55℃	1320
70℃	5460
		85℃	900
95℃	1320

Each product is carried out respectively:

To the pre-service of value at the beginning of the original degraded data work that directly collects by testing equipment, adopt formula (15) to obtain through the pretreated degraded data of first value, as shown in Figure 9.

Step 2, sampling interval are amounted to.

Each product is carried out respectively:

1, obtains the accumulation amount of degradation of degraded data under each stress level and corresponding test period thereof according to formula (16), thereby determine performance degradation rate under each stress level according to formula (17).

2, the degradation ratio under each stress level is corresponding with stress level.Each stress level and corresponding degradation ratio are set up Arrhenius temperature acceleration model formula (18), obtain degradation ratio and stress level relation, as shown in figure 10.

3, by the extrapolation of counter stress-degradation ratio relational model, obtain the degradation ratio of product under 25 ℃ of normal stress levels, the slope of curve 11 is degradation ratio as shown in figure 12.

4,, obtain each stress level and convert sampling interval under 25 ℃ of the normal stress levels, curve 9 as shown in figure 12 according to accumulated damage equivalence principle formula (22).

Step 3, sequence acceleration Time Created degradation model.

Each product is carried out respectively:

1, adopt linear regression model (LRM) formula (2) to set up the trend term model through pretreated degraded data down to each stress level respectively.

2, remove trend term and adopt the periodic regression model formation (3) of diving to set up the periodic term model to the degraded data of removing behind the trend term, as shown in figure 11 be the degraded data of removing trend term.

3, remove periodic term and the equidistant sampling residual error item each stress under is converted into non-equidistance under the normal stress residual error item of sampling.

4, the residual error item under the normal stress is set up non-equidistance seasonal effect in time series autoregressive model, thereby obtain the residual error item model formation (4) of product under the normal stress level.

5, the latent periodic model formula (5) of associating residual error item and the autoregression of periodic term modelling non-equidistance.

6, set up trend term model under the normal stress.According to degradation ratio under the normal stress and degraded data initial value, obtain the trend term model formation (6) under the normal stress, trend term curve 11 as shown in figure 12.

7, combination recurrence-non-equidistance autoregressive model.With trend term model under the normal stress and the latent direct addition of periodic model of non-equidistance autoregression, set up recurrence-non-equidistance autoregressive model formula (7), obtain the final acceleration degradation model of product degradation data under the normal stress level, curve 9 as shown in figure 12.

Step 4, quicken degradation model and predicated error analysis in conjunction with gray theory prediction.

Each product is carried out respectively:

1, setting the prediction step number was 5000 steps.

2, to the degraded data behind one group of removal trend term to be predicted, the periodic regression modelling periodic term model formation (8) of adopt diving, as shown in figure 11 be the degraded data of removing trend term.

3, remove periodic term and obtain the residual error item, the equidistant residual error item under each stress is converted into non-equidistance residual error item under the normal stress.

4, adopt the non-equidistance autoregressive model to set up residual error item model formation (9) to the residual error item under the normal stress.

5, associating residual error item and periodic term model, and 100 hours prediction spacings as model of average sample spacing of getting residual error item under the normal stress set up the dive predictor formula (10) of periodic model of non-equidistance autoregression.

6, adopt dive the best of periodic model of non-equidistance autoregression not have inclined to one side predictor formula (11) and carry out one-step prediction, obtain a new predicted value of this model.

7, remove first data of this group degraded data, add the one-step prediction value of the latent periodic model of non-equidistance autoregression, obtain one group of degraded data behind the new removal trend term.

8, be data to be predicted with this group new data, repeat above-mentioned steps 2～6.

9, repeat 4999 above-mentioned steps 7,8, can obtain the predicted value in 5000 steps one by one.

10, the trend term model formation (6) under the normal stress is predicted directly that obtain 5000 step predicted values, curve 11 as shown in figure 12.

11, two kinds of model prediction results added are returned-the non-equidistance autoregressive model promptly quicken degradation model 5000 the step predicted values, curve 10 as shown in figure 12.

Step 5, predicated error analysis and reliability assessment.

All products are carried out:

1, according to the engineering actual conditions, set this type electronic product the performance initial value 97% for failure threshold.

2, according to 5000 step of the acceleration degradation model under normal stress predicted value, determine the pseudo-life-span of each product under normal stress.

3, the predicted value of the pseudo-life-span correspondence of each product is carried out error analysis.According to the square error formula (14) that quickens the prediction of degradation model gray theory, then the pairing square error of pseudo-life prediction step number of each product is institute and asks, and is as shown in table 2.

Pseudo-lifetime data of table 2 and predicated error

Model	Pseudo-lifetime data (hour)	Square error
			Recurrence-non-equidistance autoregressive model	1569948，991970，1025120，1345578，1564491，1390313，1621545，1024051，928334，1053697	0.0087，0.0087，0.0094，0.0090，0.0091 0.0091，0.0096，0.0094，0.0088，0.0092
Pure regression model	1654080，1011170，1042520，1366178，1578591，1409213，1639145，1043451，950334，1066597	0.0838，0.0834，0.0872，0.0845，0.0856， 0.0871，0.0895，0.0884，0.0841，0.0869

4, in conjunction with the pseudo-life-span under all product normal stress levels, determine that the pseudo-life-span distribution function of this type electronic product is a lognormal distribution, its distribution function is

F(t)＝Φ{[1nt-μ]/σ}，t＝1，2，…。

5, obtain pseudo-life-span under the normal stress level for the present invention, its distribution function is exactly the unreliable degree function of product, and then its Reliability Function is

R(t)＝1-Φ{[1nt-μ]/σ}，t＝1，2，…，

Concrete parameter is as shown in table 3, the fiduciary level curve as shown in figure 13, thereby provide the reliability assessment result of stepstress accelerated degradation test.

Table 3 reliability assessment result

Model	Recurrence-non-equidistance autoregressive model	Pure regression model
			The pseudo-life-span distribution average of lognormality	14.0231	14.0593
The pseudo-life-span distribution variance of lognormality	0.051	0.2692
			Pseudo-life-span average (hour)	1251504	1276127

Provide result below, with comparing that the present invention proposes based on the seasonal effect in time series method based on the stepstress accelerated degradation test reliability estimation method of pure regression model.

Stepstress accelerated degradation test with this type electronic product is an example equally, and each test parameters is with table 1.

The acceleration degradation model that this method is set up as shown in figure 12, the pseudo-lifetime data and the predicated error of its calculating see Table 2, reliability assessment the results are shown in Table 3, the fiduciary level curve is as shown in figure 13.

Can obviously find out from table 2, the pseudo-life-span that obtains based on recurrence-non-equidistance autoregressive model method the square error of corresponding degraded data predicted value significantly less than result based on pure regression model method, it is more accurate to illustrate that the resultant life prediction result of method of the present invention compares based on the method for pure regression model, and therefore the resultant result of stepstress accelerated degradation test reliability estimation method of the present invention compares based on the methods and results of pure regression model more credible.

Claims (3)

1. based on seasonal effect in time series stepstress accelerated degradation test reliability estimation method,

Suppose 1: the performance degradation process of product has monotonicity, i.e. the degeneration of performance generation is irreversible;

Suppose 2: the failure mechanism of product is constant under each stress level;

Suppose 3: the residual life of product only depends on totally inefficacy part and stress level at that time at that time, and irrelevant with the accumulation mode;

Suppose 4: do not have any inefficacy that causes by degenerating in the test, promptly the performance degradation of product does not pass through critical value;

And hypothesis is carried out the stepstress accelerated degradation test to n product, total k stress level S in the test _i, i=1,2 ..., k, the sampling interval under each stress level is Δ t, and the number of samples to properties of product under each stress level is m _i, total number of samples under all stress levels

Then the test period of each stress level is τ _i=Δ tm _i, the performance degradation data of each product under all stress levels are y _l, l=1,2 ... m;

It is characterized in that the concrete grammar step is as follows:

Step 1, acquisition test data are carried out the data pre-service;

Step 2, sampling interval are amounted to;

Step 3, Time Created the sequence degradation model;

If be illustrated in l=1 with Y (l), 2 ..., the performance degradation stochastic variable of m observation, according to the Cramer decomposition theorem as can be known, any one time series { Y (l) } can be decomposed into determinacy part and the steadily stack of random partial

Y(l)＝T(l)+S(l)+R(l)，l＝1，2，…m (1)

In the formula: T (l) is a trend term, and S (l) is a periodic term, and R (l) is the residual error item, and trend term and periodic term are the determinacy part in the nonstationary time series, adopt regression model to represent usually, and the residual error item is steady random partial, adopts autoregressive model to represent;

As follows to the concrete modeling procedure that each product carries out respectively:

(1), adopt linear regression model (LRM) to set up the trend term model to the degraded data under each stress level respectively to the pretreated degraded data of process step 1:

T _i(l)＝slp _i·l+y _0i (2)

In the formula: slp _iBe the degradation ratio under each stress level, y _0iBe the degraded data initial value, l=1,2 ... m _i, i=1,2 ... k;

(2) remove the trend term in the degraded data and the degraded data of removing behind the trend term adopted the periodic regression modelling periodic term model of diving:

In the formula: q is the angular frequency number, A _jBe the amplitude of j angular frequency, ω _jBe j angular frequency,

Be j phasing degree, l=1,2 ... m;

(3) remove periodic term and the equidistant sampling residual error item each stress under is converted into non-equidistance under the normal stress residual error item of sampling;

(4) the residual error item under the normal stress is set up residual error item model;

Adopt non-equidistance correlation coefficient stationary series analytical approach, represent the l time observation time corresponding under normal stress, set up non-equidistance seasonal effect in time series autoregressive model, thereby obtain the residual error item model of product under the normal stress level with tl:

$R\left(l\right)=\underset{j=1}{\overset{p}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_j\left({\mathrm{\&tau;}}_l\right)R(l-j)+\mathrm{\&epsiv;}\left({\mathrm{\&tau;}}_l\right)---\left(4\right)$

In the formula: p is the exponent number of autoregressive model, η _jBe the function of autoregressive coefficient, τ _l=(τ _L1, τ _L2..., τ _Lp) ', τ _Lj=t _l-t _L-j(j=1,2 ..., p), ε (τ _l) be separate and obedience

White noise, l=1,2 ... m;

(5) the latent periodic model of residual error item in the joint step (4) and the periodic term modelling non-equidistance autoregression in the step (2);

The latent periodic regression model substitution residual error item non-equidistance autoregressive model of periodic term is carried out interative computation, set up the latent periodic model of non-equidistance autoregression, obtain the production life cycle item and residual error item model under normal stress;

The latent periodic model of non-equidistance autoregression is

In the formula:

The parameter of making corresponding conversion when diving periodic model for the periodic regression model substitution non-equidistance autoregression of diving, l=1,2 ... m;

(6) set up trend term model under the normal stress;

According to the degradation ratio slp under the normal stress ₀And degraded data initial value y ₀, obtain the trend term model under the normal stress:

T ^*(l)＝slp ₀·l+y ₀ (6)

In the formula: l=1,2 ... m;

(7) combination recurrence-non-equidistance autoregressive model forms final degradation model;

With trend term model under the normal stress and the latent direct addition of periodic model of non-equidistance autoregression, set up recurrence-non-equidistance autoregressive model, obtain the final degradation model of product degradation data under the normal stress level:

Y ^*(l)＝T ^*(l)+X(l) (7)

In the formula: l=1,2 ... m;

Step 4, in conjunction with gray theory prediction degradation model;

The concrete prediction steps that each product is carried out respectively is as follows:

(1) set prediction step number h, make u=1,2 ..., h represents the number of times predicted;

(2) establish x _l, l=1,2 ..., m represents one group of degraded data behind the removal trend term to be predicted, and it is adopted the periodic regression modelling periodic term model of diving:

In the formula; q _u, A _Uj, ω _Uj,

J=1,2 ..., q _u, for carrying out the model parameter l=1 that the u time when prediction revise, 2 ..., m;

(3) remove periodic term and obtain the residual error item, the equidistant residual error item under each stress is converted into non-equidistance residual error item under the normal stress;

(4) adopt the non-equidistance autoregressive model to set up residual error item model to the residual error item under the normal stress

$R\left(l\right)=\underset{j=1}{\overset{p_u}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_{\mathrm{uj}}\left({\mathrm{\&tau;}}_{\mathrm{ul}}\right)R(l-j)+{\mathrm{\&epsiv;}}_u\left({\mathrm{\&tau;}}_{\mathrm{ul}}\right)---\left(9\right)$

In the formula: τ _Ul, p _u, η _Uj, ε _u, j=1,2 ..., p _uFor carrying out the model parameter that the u time when prediction revise, l=1,2 ..., m;

(5) unite residual error item and periodic term model, and get the average sample spacing of residual error data under the normal stress

$\stackrel{\&OverBar;}{\mathrm{\&Delta;t}}=\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;t}}_i,---\left(10\right)$

As the prediction spacing of model, correspondingly get Set up the predictor formula of the latent periodic model of non-equidistance autoregression

In the formula: τ _u,

J=1,2 ..., q _uFor carrying out the model parameter that the u time when prediction revise, Δ t _iBe the sampling interval under the normal stress level;

(6) adopt dive the best of periodic model of non-equidistance autoregression not have inclined to one side predictor formula and carry out one-step prediction, the no inclined to one side predictor formula of described the best is as follows:

Obtain a new predicted value x of this model according to formula (12) _M+1, so far finish once prediction;

(7) remove first data x of this group degraded data ₁, the one-step prediction value x of the latent periodic model of adding non-equidistance autoregression _M+1, obtain one group of degraded data x behind the new removal trend term _l, l=2,3 ..., m+1;

(8) with x _l, l=2,3 ..., m+1 is data to be predicted, repeats above-mentioned steps 2～6;

(9) repeat h-1 above-mentioned steps 7,8, can dope x one by one _M+u, u=1,2 ..., h;

(10) the trend term model formation (6) under the normal stress is carried out the directly prediction of h step, obtain predicted value

U=1,2 ..., h;

(11) with two kinds of model prediction results added, returned-the non-equidistance autoregressive model is the h step predicted value of degradation model:

$y_{m+u}^{*}=t_{m+u}^{*}+x_{m+u},$ u＝1，2，…，h；

Step 5, predicated error analysis and reliability assessment;

As follows to the concrete appraisal procedure that all products carry out:

(1) according to the degradation failure criterion of engineering actual conditions setting product, provides failure threshold;

(2) according to the step predicted value of the acceleration degradation model h under each product normal stress

U=1,2 ..., h determines that the predicted value of each product passes through the prediction step number u of failure threshold first _LifeAnd the time under the corresponding normal stress

$t_{\mathrm{life}}=\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&tau;}}_{0i}+u_{\mathrm{life}}\&CenterDot;\stackrel{\&OverBar;}{\mathrm{\&Delta;t}},---\left(13\right)$

Be the pseudo-life-span of each product under normal stress; Wherein, τ _0iBe the test period under the normal stress level;

(3) to the predicted value of the pseudo-life-span correspondence of each product

Carry out error analysis, formula is

${\mathrm{\&sigma;}}_{m+u}^2=\underset{j=1}{\overset{u}{\mathrm{\&Sigma;}}}{G}_{\mathrm{uj}}^2{\mathrm{\&sigma;}}_{(u-j+1)\mathrm{\&epsiv;}}^2,---\left(14\right)$

In the formula: G _Uj, j=1,2 ..., u is the Green function of non-equidistance autoregressive model u step prediction;

J=1,2 ..., u is the white noise variance of non-equidistance autoregressive model u-j+1 step prediction;

u＝1，2，…，h；

Each pseudo-life prediction step number u then _LifePairing square error is institute and asks;

(4), determine the pseudo-life-span distribution function structure and the parameter estimation of product in conjunction with the pseudo-life-span under all product normal stresses;

(5) according to the pseudo-life-span distribution function of product, provide Reliability Function and curve, thereby provide the reliability assessment result of stepstress accelerated degradation test.

2. according to claim 1 based on seasonal effect in time series stepstress accelerated degradation test reliability estimation method, it is characterized in that: it is as follows that the sampling interval described in the step 2 is amounted to step:

(1) respectively according to the accumulation amount of degradation and the corresponding working time thereof of degraded data under each stress level, determines the performance degradation rate under each stress level;

(2) degradation ratio under the different stress levels is corresponding with stress level, each stress level and degradation ratio are set up regression model, obtain the relational model of degradation ratio and stress level;

(3) by counter stress-degradation ratio relational model extrapolation, obtain the degradation ratio of product under normal stress;

(4) according to the accumulated damage equivalence principle, under the constant condition of the accumulation amount of degradation that keeps degraded data, working time under the different stress levels of stepstress accelerated degradation test is converted working time under the normal stress level by the ratio of degradation ratio, in like manner, the sampling interval under all proof stress levels is converted sampling interval under the normal stress level by the ratio of degradation ratio.

3. according to claim 1 based on seasonal effect in time series stepstress accelerated degradation test reliability estimation method, it is characterized in that: the described data pre-service of step 1 specifically realizes by the following method:

The original degraded data of each product being established the stepstress accelerated degradation test respectively is y _l, l=1,2 ... m then through the first pretreated degraded data of value is

$y_l^{\&prime;}=\frac{y_l}{y_1}---\left(15\right)$

With the initial value unification of pretreated degraded data is 1.

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