CN106650204A - Product failure behavior coupling modeling and reliability evaluation method - Google Patents
Product failure behavior coupling modeling and reliability evaluation method Download PDFInfo
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Abstract
A product failure behavior coupling modeling and reliability evaluation method, namely, a product failure behavior coupling modeling and reliability evaluation method based on Vine-Copula and accelerated degradation testing, includes the following steps: 1, degradation trajectory modeling; 2, failure mechanism consistency testing; 3, distribution model making of performance parameters under normal stress with the extrapolation method; 4, failure behavior coupling modeling based on Vine-Copula; 5, reliability evaluation. Through the steps, product failure behavior coupling modeling is completed, a model with joint distribution of reliability degree is established, the problem of solving the reliability function with immediate integration is solved, the product multivariate coupling relationship is clearly described, and the method for reliability evaluation of products with multi-performance parameter correlated characteristics is provided.
Description
Technical field
The present invention proposes a kind of product bug behavior Coupling method and reliability estimation method, more particularly to a kind of base
The product bug behavior Coupling method and reliability estimation method of (Vine-Copula) and accelerated degradation test are drawn in rattan-Kapp,
Belong to the failure behavior modeling in reliability engineering field.
Background technology
The safety of the Grand Equipments such as space flight, aviation, nuclear equipment, electric power networks is on active service for the national economic development and national defence are built
If it is all significant, but its service condition complexity, bad environments, it is easily caused serious accident generation, therefore correct understanding
Failure, assess and predict its reliability for ensure equipment safety operation, increasing economic efficiency makes great sense.Due to system
The raising of complexity and going deep into for failure understanding, for the research of reliability theory, progressively from towards " during failure
Between " turn to towards " failure process ".On the one hand research based on failure process occurred from the angle research failure of microcosmic
Mechanism, i.e. physics of failure.On the other hand it is that the generation development of failure and biography in systems are studied from macroscopic perspective
Broadcast, i.e. the behavior of failure.Product bug behavior Coupling method of the invention based on Vine-Copula and accelerated degradation test and can
It is a kind of to the product bug behavior modeling with polytomy variable and reliability estimation method by property appraisal procedure.By to existing
Looking into for technology is new, Vine-Copula and accelerated degradation test are also not based on both at home and abroad carry out product failure behavior coupling building
Research in terms of mould and reliability assessment.
The content of the invention
The characteristics of for high reliability long life complex product there is failure mechanism to couple, is building macroscopical failure behavior model
Carry out the aspect of reliability assessment, it is proposed that a kind of to produce with polytomy variable based on Vine-Copula and accelerated degradation test
Product failure behavior is modeled and reliability estimation method.On the one hand the method can clearly describe product polytomy variable coupled relation,
On the other hand carry out in time and cost Reliability Assessment be able to bearing.
The present invention before product bug behavior Coupling method and Reliability assessment is carried out, the system that need to clearly constitute product
Failure behavior method is described, select the token state of descriptive system failure behavior.Due to the failure between the part for constituting system
Mechanism influences each other can be shown in the state of part, and a kind of available method is state vector Z (t) for adopting part for table
Levy, the failure behavior of descriptive system, failure behavior coupling model is built, so as to the reliability of estimating system.Wherein, Z (t)=(D
(t), X (t), F (t))T, D (t) is components damage state variable, and X (t) is unit response variable, F (t) be part bear should
Power variable.System is said, the mechanism competition or coupled problem that either components interior is present between each part can use each
Faulted condition variable in the state vector of part represents its degeneration damage status to build product bug behavior coupling model；And
From actual angle, faulted condition variable reacts typically from the monitoring parameter of product, therefore the present invention is from accelerated degradation test
Under the performance parameter test data of each level start with, be modeled and analysis, be embodied as flow process and see accompanying drawing 1.
The present invention employs the following technical solutions realization, a kind of product bug behavior Coupling method of the invention and reliability
Appraisal procedure, i.e., a kind of product bug behavior Coupling method and reliability based on Vine-Copula and accelerated degradation test is commented
Estimate method, its step is as follows：
Step one：Degradation path is modeled
The degradation path model of product is represented by amount of degradation with time and the function of suffered environmental stress；Entering row degradation
When track models, need
1. degraded data feature is analyzed, degradation path model Δ y is set upt=f (t, S), Δ ytThe amount of degradation of t is represented,
S represents suffered environmental stress horizontal size；
2. likelihood function is set upfiRepresent the probability density function of i-th amount of degradation test point, μ,
σ represents degradation path model parameter；
3. degraded data is substituted into, the maximum problem of likelihood function is solved using Nelder-Mead simplex method,
Obtain the parameter Estimation of each performance parameter degradation path model under each stress level；
Step 2：Failure mechanism consistency check
It is the validity of guarantee test in accelerated degradation test, the same performance parameter under different accelerated stress levels
Failure mechanism consistency check should be met.What is existed between model parameter and failure mechanism are constant in Degradation path modeling contacts foundation
Degradation path modeling method different and there is difference, so that the Degradation path for obeying Brownian Motion with Drift is modeled as an example：
1. its acceleration mechanism consistency check formula is determinedWherein μiAnd σiRespectively stress level Si
Under degradation path model coefficient of deviation and diffusion coefficient；
2. hypothesis testing statistic is builtTs of the T from the free degree for n+m-2 is distributed；Wherein：X obeys average
For μ1Variance is σ2Normal distribution, Y obey average be μ2Variance is σ2Normal distribution, and X1, X2... XnIt is from overall X
Sample, Y1,Y2,…YmThe sample of overall Y is taken from, n and m is respectively sample size size；
3. region of rejection is determined.Given level of significance α, substitutes into data and calculates | T |, looks into t distribution tables and understands：t1-α/2(n+m-
2) size, if | T | >=t1-α/2(n+m-2), then assay falls in region of rejection, failure mechanism of the parameter under different stress
There is no uniformity.Conversely, then meeting consistency check；
Step 3：The distributed model of performance parameter under extrapolation normal stress
Relation between life of product feature and accelerated stress level can be stated by acceleration model, using acceleration model
The degradation model and edge distribution of each variable under normal stress can be derived by；In the present invention, using simplified many stress Aileens
Acceleration model：T is characterized the life-span, and T and RH represents respectively temperature and humidity variables, and A, B and C are to wait to seek ginseng
Number；
1. accelerated factor is determined
Make AFi,0=t0/tiFor the horizontal S of accelerated stressiRelative to the horizontal S of normal stress0Accelerated factor, tiAnd t0Respectively
Represent product in the horizontal S of accelerated stressiS horizontal with normal stress0Under, when performance parameter reaches required during identical amount of degradation
Between；
2. acceleration model parameter is solved
A. the relational equation of accelerated factor and life-span, coefficient of deviation, diffusion coefficient between different stress is built
Wherein AF represents accelerated factor, and L represents life-span, μiAnd σiRepresent in accelerated stress SiThe drift system of lower performance parameter Degradation path
Number and diffusion coefficient；Subscript h represents under higher stress level that subscript l is represented under relatively low stress level；
B. to performance parameter p, in k1And k2Under stress level, sum of squares of deviations function is constructed
The minimum problem of sum of squares of deviations function is solved using Nelder-Mead simplex method, is obtained in acceleration model most
Excellent model parameter B, C；
3. degradation path model parameter under normal stress is estimated
After trying to achieve the parameter in acceleration model, degradation path model parameter under normal stress is entered using least square method
Row estimation, obtains sum of squares of deviations functionWithUsing simple
Shape method solve make sum of squares of deviations minimum corresponding to parameter value, so as to obtain normal stress under degradation path model parameter μp0With
σp0Point estimation；Degradation path model is under final normal stress：yp0(t)=μp0t+σp0B(t)；
Step 4：Product bug behavior Coupling method based on Vine-Copula
Copula functions are a class " connection " functions, and connection is edge distribution and Joint Distribution.The reason of Vine-Copula
It is conditional probability by basis, by the way that Joint Distribution is resolved into into the form that multiple binary copula and corresponding conditional probability are even taken advantage of
It is modeled, the product bug behavior Coupling method main flow based on Vine-Copula is as follows：
1. performance parameter correlation analysis
Before the failure behavior Coupling method for carrying out product, for the polynary performance parameter degradation of product, phase is first carried out
The analysis of closing property；With Kendall ' s τ coefficients as relativity measurement index, the absolute value of index is got over closer to 1 expression correlation
By force, and Kendall ' s τ coefficients represent nonlinear correlation degree；
2. Joint Distribution is decomposed into into the form that suitable conditional probability is multiplied by vine graph models, using copula letters
Number builds multiple correlation relational model；Specifically,
A. it is based on the modeling method of D-Vine
1) first, in ground floor Vine graph models, each variable presses correlation as the node in tree graph, between node
It is attached with short-term, for example, a kind of four-dimensional its ground floor relation of D-vine is as shown in Figure 2.With cijRepresent connecting node i
With the probability density of the copula functions of node j, then cij=cpq, p, q are respectively the variable that node i and node j are included.
2) second layer tree graph in Vine figures is set up on the basis of ground floor tree graph, is made with the connection in ground floor tree graph
For the node in second layer tree graph, adjacent line is attached in the unit of the formation of the second layer with line in ground floor, is built
The tree graph of the second layer, such as accompanying drawing 3 be with accompanying drawing 2 (1,2), (2,3), (3,4) set up second layer tree graph for unit.Still with cijTable
Show the probability density of connecting node i and the copula functions of node j, there is cij=cpq|k, { p, k }, { q, k } be respectively node i and
The variable that node j is included, wherein k are the total variable of two nodes.
3) third layer tree graph is also to be set up on the basis of preceding layer tree graph using similar method, by that analogy, when
Only it is left a pair of dependency relations in tree graph, i.e., is exactly the tree graph of last layer when only remaining a line.In the same manner, third layer tree
Scheme have c including the probability density of the copula functions between all connecting nodes including last layer of tree graphij=cpq|k,
{ p, k }, { q, k } are the variable that node i and node j are included, and wherein k is the total variable of two nodes.
4) the corresponding copula function probabilities density of each bar line in Vine figures is multiplied, you can obtain scheming based on Vine
Copula functions decomposition texture .c=∏ cij.Production reliability Joint Distribution probability density function is represented byxiRepresent i-th performance parameter of product, riRepresent the corresponding reliability probability density of i-th parameter
Function, is so far set up based on Vine-Copula coupling models.
B. Vine-Copula coupling models are solved
Solve Vine-Copula coupling models, you can by spending Joint Distribution probability density, first to determine reliability joint point
Cloth probability density r (x1,...,xn) in each copula probability density function form and parameter, i.e. cijForm and parameter.
1) c in ground floor tree graph in Vine models is determinedij:Substitute into i-th, j-th node and correspond to performance parameter (p, q)
Amount of degradation data.Likelihood function is set up according to alternative copula functional formsUsing
Nelder-Mead simplex method solves the maximum and its corresponding copula function parameters value of likelihood function, is then based on AIC
Criterion chooses copula functional forms the most suitable and corresponding parameter.Wherein, j represents sample number in likelihood function L, altogether
M sample, i represents test point sequence number, common n test point, Rp(t) and RqT () represents respectively the reliability of pth and q performance parameters
Degree distribution function.
2) c in second layer tree graph in Vine models is determinedij:Make that { p, k }, { q, k } are respectively node i and node j is included
Variable, then have cij=cpq|k, likelihood function is set up according to alternative copula functional forms
The maximum and its corresponding copula function parameters value of likelihood function, Ran Houji are solved using Nelder-Mead simplex method
Copula functional forms the most suitable and corresponding parameter are chosen in AIC criterion.Wherein, j represents that sample is compiled in likelihood function L
Number, common m sample, i represents test point sequence number, common n test point, Rp|k(t) and Rq|kT () represents respectively reliability Joint Distribution
Conditional probability expression formula, its computing formula follows：
I. when k is one-dimensional variable, i.e., an identical variable is only included between two nodes, now Rp|k(t) and Rq|k(t)
Value according toCalculate.Wherein R is reliability distribution function,Represent the copula
Function pair RjDerivation.
Ii. when k is vectorial more than 1 of dimension, i.e., multiple identical variables, now R are included between two nodesp|k(t) and
Rq|kThe value of (t) according toCalculate.Wherein v is conditional vector, v=(v1,
v2,...,vd), vjIt is any one element in v, v-jRepresent that v removes vjThe vector for obtaining.
Determine the c in second layer tree graph in Vine modelsijAfterwards, by that analogy, the c of other layer of tree graph is calculatedij, until really
Determine in Vine figures till all optimum copula functional forms.
3) all c are determinedijFunctional form after, to selected cijParameter estimated again together.Set up likelihood
Function L=r (x1,...,xn), the maximum of likelihood function is solved using Nelder-Mead simplex method method, correspondence can be obtained
Optimized parameter result.By all cijIn the breakdown of the parameter back substitution Vine-Copula of function, that is, obtain based on Vine-
The multiple correlation relational model that copula sets up.
Step 5：Reliability assessment
Complete that the reliability connection of product can be obtained after product bug behavior Coupling method using Vine-Copula functions
Close distribution probability density function.It is various yet with the density function complex structure of Copula functions, while each performance parameter
Reliability probability density function expression formula is also sufficiently complex, causes the analytic expression of probability density function of reliability Joint Distribution not
It is only long and extremely complex, it is difficult to the method for passing through direct integral solves Reliability Function.For non-critical dullness degenerate case
Under production reliability Joint Distribution accurate solution problem, present invention employs conditional probability decomposition method solved.
1. the conditional probability expression formula of the reliability Joint Distribution of binary variable is calculated
Wherein R is reliability distribution function,Represent copula function pairs RjDerivation.
2. the conditional probability expression formula of polytomy variable reliability Joint Distribution is calculated
Wherein v is conditional vector, v=(v1,v2,...,vd), vjIt is any one element in v, v-jRepresent that v removes vj
The vector for obtaining, v-j=(v1,v2,...,vj-1,vj+1,...,vd)。
3. polynary reliability Joint Distribution expression formula is calculated
Wherein K represents the performance parameter number for occurring to degenerate, akIt is the code name of performance parameter, v is conditional vector, vjIt is bar
An element in part vector v, v-jRepresent in conditional vector v and remove vjThe vector for obtaining,For vjThe corresponding time,Be to
Amount v-jThe time arrow that the middle element corresponding time is constituted.
Wherein, " the choosing copula functional forms the most suitable based on AIC criterion " described in step 4 refers to：With
Copula corresponding to AIC minimum of a values is considered selection most widely suited in alternative copula forms.The calculating formula of AIC is AIC
=-2lnL+2p, wherein L are maximum likelihood function values, and p is model parameter number, and M is sample size, and corresponding value is product
Observation station number.
Wherein, in step 5, for ease of the solution of reliability Joint Distribution, reliability Joint Distribution is decomposed into into condition
Every binary Copula function that the principle of probability is so that in final decomposition result is used in Vine-Copula
And the binary Copula function established, described " method that conditional probability is decomposed ", some principles can be followed：
1) first reliability edge distribution is (i.e. in breakdown), for D-vine preferably selects ground floor two ends any
End node；
2) Section 2 in breakdown, i.e.,For the homonymy end node that D-vine preferably selects the second layer；
3) by that analogy, for D-vine preferably selects corresponding end node.
By above step, the failure behavior Coupling method to product is completed, establish the mould of reliability Joint Distribution
Type, solves the difficult problem that immediate integration solves Reliability Function, clearly describes product polytomy variable coupled relation, there is provided
The method that Reliability Assessment is carried out in time and cost can be born.
It is an advantage of the current invention that：
For polytomy variable high reliability long life product, based on accelerated degradation test data and Brownian Motion with Drift, adding
On the basis of fast mechanism consistency check, structure has obtained the variable edge distribution under normal stress level.Based on layering thought,
Using Vine-Copula methods, the decoupling problem of multivariate joint probability distribution is described with binary Copula function, i.e., to potential portion
Part correlation mechanism has carried out decoupling modeling, it is to avoid numerical simulation Solve problems of the polynary Copula functions on higher-dimension.Together
When, in terms of reliability calculating, because the modeling method of current Copula functions is generally all moving back for collocation strictly monotone degeneration
Changing model carries out life appraisal, and the method for solving based on conditional probability proposed by the present invention causes what is degenerated for non-critical is dull
Situation also can be solved.Finally, contrast multiple normal distribution modeling method, overcome only obey between degradation parameter it is linear or uncorrelated
Relation, show the Life estimating that Vine-Copula can be under effective process nonlinear correlation.
Description of the drawings
Fig. 1 the method for the invention implementing procedure figures.
Fig. 2 four-dimension D-vine ground floor tree graphs.
Fig. 3 four-dimension D-vine second layer tree graphs.
The D-vine models of Fig. 4 intelligent electric meter elementary errors.
The joint reliability figure of four elementary errors of Fig. 5 intelligent electric meters.
Sequence number, code name are described as follows in figure：
Numeral represents respectively the code of performance parameter in Fig. 2 and Fig. 3.BE1, BE5, BE9, BE10 represent intelligent electricity in Fig. 4
Four elementary errors of table, CE represents the cumulative errors of intelligent electric meter.BE1BE5 represents the company of connecting node BE1 and node BE5
Line, sign performance parameter BE1 is related to BE5, and as the node of the second layer in vine figures.BE5BE9 and BE9BE10 is characterized and contained
Justice is ibid.BE1BE9 | BE5 represents the line of connecting node BE1BE5 and node BE5BE9, characterize performance parameter BE5 both with BE1
Correlation, but it is related to BE9, and BE1BE9 | nodes of the BE5 as third layer in vine figures.BE5BE10 | BE9 implications are ibid.
BE1BE10 | BE5BE9 is represented and represented connecting node BE1BE9 | BE5 and node BE5BE10 | the lines of BE9, the more than performance of sign
Parameter BE5 is related to BE1, BE10, and has performance parameter BE9 related to BE1, BE10.
Specific embodiment
Below in conjunction with drawings and Examples, the present invention is described in further detail.The present invention is defeated by analysis product
Going out the correlation between parameter carries out Reliability assessment building coupling model, intelligent electric meter is chosen as application, to intelligence
The output performance parameter of ammeter enters row degradation modeling, and carries out the reliable life assessment of ammeter.Intelligent electric meter is that typically have
Many performance parameter electronic products, can characterize the performance parameter of intelligent electric meter degenerate state has cumulative errors 1, elementary error 4
, it is respectively cumulative errors CE and elementary error BE1, BE5, BE9, BE10.The Constant Acceleration degradation experiment of intelligent electric meter is selected altogether
Taking two kinds of accelerated stress (temperature and humidity) has carried out the test of three stress levels, and 8 samples of each stress level specifically should
Power combination is shown in Table 1.Understand that the output performance parameter of intelligent electric meter possesses degradation characteristics by accelerated degradation test data, it is moved back
Change trend is gradually increasing, and has clear and definite failure threshold according to design objective, and under test conditions, temperature and humidity is to promote it
The raw accelerated stress degenerated.
A kind of product bug behavior Coupling method of the present invention and reliability estimation method, i.e., it is a kind of to be based on Vine-Copula
With the product bug behavior Coupling method and reliability estimation method of accelerated degradation test, as shown in Figure 1, its specific implementation step
It is as follows：
Step one：Degradation path is modeled
Test data of 5 critical performance parameters of intelligent electric meter under each stress level is obtained by accelerated degradation test.
The characteristics of according to intelligent electric meter various performance parameters Degradation path, performance parameter Degradation path is built using Brownian Motion with Drift
Mould.
1. the Degradation path of performance parameter obeys model Δ yt=μ t+ σ B (t), wherein t be the time, Δ ytFor t performance
The amount of degradation of parameter, μ is coefficient of deviation, and σ is diffusion coefficient, and B (t) is standard Brownian movement.
2. likelihood function is set upWherein j represents jth sample, and common n is individual
Sample, i represents i-th test point, common mjIndividual test point, tjiThe time of jth sample i ＆ lt test is represented,By the property surveyed
The amount of degradation of energy parameter, μ is coefficient of deviation, and σ is diffusion coefficient.
3. the maximum problem of likelihood function is solved using Nelder-Mead simplex method, obtain each stress water
The parameter estimation result of each performance parameter degradation path model is as shown in table 2 under flat.
Step 2：Failure mechanism consistency check
For Degradation path obeys Brownian Motion with Drift model, by taking performance parameter CE of intelligent electric meter as an example, its failure machine
Reason consistency check can basis
1. each sample for the parameter under each accelerated stress, using Nelder-Mead simplex method to likelihood
FunctionMaximum problem solved, obtain degradation model parameter valueWithWherein j represents jth sample, and i represents i-th test point, common mjIndividual test point, tjiRepresent the test of jth sample i ＆ lt
Time,By survey performance parameter amount of degradation, μ is coefficient of deviation, and σ is diffusion coefficient.
2. hypothesis testing statistic is builtT obeys the free degree for the t distributions of n+m-2.OrderFor the test samples under three accelerated stress, the test samples X under three stress can be obtained1, X2, X3, as shown in table 3.
Then one by one to the test samples X under wantonly 2 stressi、XjCarry out T inspections.
3. significance 0.05 is given, t distribution tables is looked into and is understood that region of rejection is | T | >=t0.975(14)=2.1448, order inspection
Test sample Xi、XjReplace respectively in T statisticsWithAnd bring data calculating into, contrast t0.975(14) value understands have | T
| ＜ t0.975(14), i.e. X1With X2With X3Between be not significantly different from, failure mechanism of the parameter under three stress has consistent
Property, assay is shown in Table 4.
The repeatable step 1 of the failure mechanism consistency check of other performance parameters, 2 and 3, until all parameters are by consistent
Property inspection.
Step 3：The distributed model of performance parameter under extrapolation normal stress
In the case where each stress level is completed after each performance parameter Degradation path modeling, need the degeneration mould under high stress level
Shape parameter is extrapolated under normal stress.
1. accelerated factor is determined
According to accelerated test condition, many stress Aileen acceleration models are chosen：Order
For the horizontal S of accelerated stresshRelative to stress level SlAccelerated factor.
2. acceleration model parameter is solved
To performance parameter p, in k1And k2Under stress level, sum of squares of deviations function is constructed
The minimum problem of sum of squares of deviations function is solved using Nelder-Mead simplex method, correspondence acceleration side can be obtained
Optimal model parameters B, C in journey are as shown in table 5.
3. degradation path model parameter under normal stress is estimated
After trying to achieve the parameter in acceleration model, using least square method to (20 DEG C of temperature under normal stress；Relative humidity
45%) degradation path model parameter estimated, obtains sum of squares of deviations functionWith
Solution is optimized using simplex method, so as to obtain normal stress under degradation path model parameter μp0And σp0Point estimation, such as
Shown in table 6.
By above step, finally giving degradation path model under normal stress is：yp0(t)=μp0t+σp0B(t)。
Step 4：Product bug behavior Coupling method based on Vine-Copula
1. correlation analysis
Keep constant it is assumed that can only select under one of accelerated stress level based on dependency relation under each accelerated stress
Test data carry out correlation analysis, herein select the 3rd group of stress level under test data.Using relativity measurement index
The correlation that Kendall ' s τ coefficients are calculated between sample critical performance parameters is shown in Table 7, it is concluded that as follows：
1) there is no correlation between cumulative errors CE and elementary error BE
2) elementary error BE1, BE5, BE9, BE10 are present compared with strong nonlinearity correlation.
2. Joint Distribution is decomposed into into the form that suitable conditional probability is multiplied by vine graph models, using copula structures
Build multiple correlation relational model.
A. it is based on the modeling method of D-Vine
Understand that the correlation between four elementary errors of intelligent electric meter is more close by analysis, four elementary errors can be set up
D-vine models between BE, as shown in Figure 4.
1) first, in ground floor Vine graph models, successively to four elementary errors BE1, BE5 of intelligent electric meter, BE9,
BE10 is encoded to 1,2,3,4, using four elementary errors as the node in tree graph, is carried out with short-term by correlation between node
Connection, as shown in Figure 2.With cijThe probability density of connecting node i and the copula functions of node j is represented, there is cij={ c12,c23,
c34}
2) second layer tree graph in Vine figures is set up on the basis of ground floor tree graph, with the connection in ground floor tree graph
(1,2), (2,3), (3,4) as the node in second layer tree graph, it is attached with line, the tree graph of the second layer is built, with Fig. 3
It is shown.Still with cijThe probability density of connecting node i and the copula functions of node j is represented, there is cij={ c13|2,c24|3}
3) the third layer tree graph in Vine figures is set up on the basis of second layer tree graph, with the connection in second layer tree graph
(13 | 2), (24 | 3) as the node in third layer tree graph, it is attached with line.Still with cijRepresent connecting node i and node j
Copula functions probability density, have cij={ c14|23}
Can obtain four elementary error reliability Joint Distribution probability density functions of intelligent electric meter according to D-vine models is：
r(x1,x2,x3,x4)=c12·c23·c34·c13|2·c24|3·c14|23·r1·r2·r3·r4
Wherein, variable x1,x2,x3,x4Elementary error BE1, BE5, BE9, BE10 are represented respectively, and c is correspondence copula functions
Probability density, riRepresent the reliability distribution probability density function of i-th performance parameter.
B. Vine-Copula coupling models are solved
Determine that the reliability joint distribution function of intelligent electric meter needs to determine formula r (x first1,x2,x3,x4) in each
The form and parameter of Copula functions.Alternatively collection is Copula functions in the present invention：Gaussian Copula, Frank
Copula, Ali-Mikhail-Haq Copula, Clayton Copula.
1) c in ground floor tree graph in Vine models is determinedij：cij={ c12,c23,c34, according to alternative copula letters
Number form formula solves likelihood function using Nelder-Mead simplex methodMaximum, obtain
The parameter Estimation and maximum likelihood function value of copula functions, is then based on AIC criterion and chooses copula letters the most suitable
Number form formula and corresponding parameter.Wherein, (k1, k2)=(1 | 2,3 | 2), according to result of calculation (being shown in Table 8), it may be determined that c12、
c23、c34Optimal selection be Frank Copula.
2) complete Vine figure ground floors copula functions it is preferred after, determine in Vine models in second layer tree graph
cij:cij={ c13|2,c24|3Likelihood function is set up according to alternative copula functional forms
The maximum and its corresponding copula function parameters value of likelihood function, Ran Houji are solved using Nelder-Mead simplex method
Copula functional forms the most suitable and corresponding parameter are chosen in AIC criterion.Wherein, (p | k, q | k)={ (1 | 2,3 | 2),
(2 | 3,4 | 3) }, Jing determines c13|2Optimal selection be Frank Copula, c24|3Optimal selection be Clayton Copula.
By that analogy, to c14|23Carry out functional form selection, it is known that, c14|23Optimal selection be Frank Copula, so far complete
To formula r (x1,x2,x3,x4) each copula functional forms it is preferred, obtain result
r(x1,x2,x3,x4)=cFrank(α12,R1,R2)·cFrank(α23,R2,R3)·cFrank(α34,R3,R4)
·cFrank(α13|2,R1|2R3|2)·cClayton(α24|3,R2|3,R4|3)
·cFrank(α14|23,R1|23,R4|23)·r1·r2·r3·r4
Wherein, R1|2, R2|3, R1|23Formula can be expressed according to the conditional probability of polytomy variable reliability Joint Distribution to try to achieve.
3) after determining all of copula functional forms, the parameter of selected copula functions is estimated again together
Meter, sets up likelihood function L=r (x1,x2,x3,x4), employ Nelder-Mead simplex method method and solve maximum likelihood letter
Number, is obtained optimized parameter and the results are shown in Table 9, by the breakdown of the parameter back substitution vine-copula of all copula functions, i.e.,
The multiple correlation relational model set up based on Vine-Copula is obtained.
Step 5：Reliability Solution
1. the conditional probability expression formula of the reliability Joint Distribution of binary variable is calculated
2. the conditional probability expression formula of polytomy variable reliability Joint Distribution is calculated
3., according to polynary reliability Joint Distribution expression formula, the reliability Joint Distribution of four elementary error couplings is obtained
For：
RBEs(t)=R1234(t1,t2,t3,t4)
=R4(t4)·R3|4(t3|t4)·R2|34(t2|t3,t4)·R1|234(t1|t2,t3,t4)
Wherein R3|4, R2|34, R1|234Value for can be according to Rhj(xh|xj) and Rx|v(x | v) calculate.
Final its four elementary error reliability joint distribution function figures are as shown in Figure 5.Accelerated by analyzing intelligent electric meter
Degradation experiment data modeling result understands, when there is correlation between many performance parameters, if to degenerate do not consider correlation when modeling
Property is only modeled with series model, and the Reliability Function curve and life appraisal result for obtaining can seem conservative.By contrast,
Considering the modeling method of dependency relation can to a certain extent provide assessment result be well grounded but radical not too much.Cause
, when there is correlation between performance parameter, the effect of consideration dependency relation in degeneration modeling process is a need for for this.
The related problem of performance parameter is characterized in for product high reliability long life feature and failure mechanism coupling, this
It is bright have studied based on Vine-Copula and accelerated degradation test comment with polytomy variable product bug behavior modeling and reliability
Estimate method.Modeling method based on Vine-Copula is by Vine-Copula flexible and changeable structure, the phase to Various Complex
The Vine models that closing implementations has each self application are available for analyzing, setting up correlation models, and no matter performance parameter is moved back
It is linearly or nonlinearly relation between change amount, can be suitable for well, solves in polytomy variable reliability model variable two-by-two
The problem that dependency relation is characterized.
It is related to all forms in the present invention as follows：
The each performance parameter degradation path model parameter estimation result of table 1
Degradation path model parameter Estimation under each performance parameter accelerated stress of table 2
The each products C E estimates of parameters of table 3 and test samples
Table 4CE parameter consistency assays
The each performance parameter acceleration model parameter estimation result of table 5
Each performance parameter degradation path model parameter under the normal stress of table 6
Correlativity calculation result between the performance parameter of table 7
The each copula functional forms of table 8 and parameter Estimation
Table 9Copula function preferred results
Claims (3)
1. a kind of product bug behavior Coupling method and reliability estimation method, i.e., a kind of to be moved back based on Vine-Copula and acceleration
Change the product bug behavior Coupling method and reliability estimation method of test, it is characterised in that：Its step is as follows：
Step one：Degradation path is modeled
The degradation path model of product is expressed as amount of degradation with time and the function of suffered environmental stress；Build Degradation path is carried out
During mould, need
1. degraded data feature is analyzed, degradation path model Δ y is set upt=f (t, S), Δ ytRepresent the amount of degradation of t, S tables
Show suffered environmental stress horizontal size；
2. likelihood function is set upfiThe probability density function of i-th amount of degradation test point is represented, μ, σ is represented
Degradation path model parameter；
3. degraded data is substituted into, the maximum problem of likelihood function is solved using Nelder-Mead simplex method, obtained
The parameter Estimation of each performance parameter degradation path model under each stress level；
Step 2：Failure mechanism consistency check
It is the validity of guarantee test in accelerated degradation test, the same performance parameter under different accelerated stress levels should be accorded with
Close failure mechanism consistency check；What is existed between model parameter and failure mechanism are constant in Degradation path modeling contacts according to degeneration
Track modeling method different and there is difference, so that the Degradation path for obeying Brownian Motion with Drift is modeled as an example：
1. its acceleration mechanism consistency check formula is determinedWherein μiAnd σiRespectively stress level SiUnder
The coefficient of deviation and diffusion coefficient of degradation path model；
2. hypothesis testing statistic is builtTs of the T from the free degree for n+m-2 is distributed；Wherein：It is μ that X obeys average1Side
Difference is σ2Normal distribution, Y obey average be μ2Variance is σ2Normal distribution, and X1, X2... XnIt is the sample from overall X,
Y1,Y2,…YmThe sample of overall Y is taken from, n and m is respectively sample size size；
3. region of rejection is determined；Given level of significance α, substitutes into data and calculates | T |, looks into t distribution tables and knows：
t1-α/2(n+m-2) size, if | T | >=t1-α/2(n+m-2), then assay falls in region of rejection, and parameter is answered different
Failure mechanism under power does not have uniformity, conversely, then meeting consistency check；
Step 3：The distributed model of performance parameter under extrapolation normal stress
Relation between life of product feature and accelerated stress level is stated by acceleration model, is derived using acceleration model
The degradation model and edge distribution of each variable under normal stress；In the present invention, using simplified many stress Aileen acceleration models：T is characterized the life-span, and T and RH represents respectively temperature and humidity variables, and A, B and C are parameter to be asked；
1. accelerated factor is determined
Make AFi,0=t0/tiFor the horizontal S of accelerated stressiRelative to the horizontal S of normal stress0Accelerated factor, tiAnd t0Represent respectively
Product is in the horizontal S of accelerated stressiS horizontal with normal stress0Under, performance parameter reaches the time required during identical amount of degradation；
2. acceleration model parameter is solved
A. the relational equation of accelerated factor and life-span, coefficient of deviation, diffusion coefficient between different stress is built
Wherein AF represents accelerated factor, and L represents life-span, μiAnd σiRepresent in accelerated stress SiThe drift system of lower performance parameter Degradation path
Number and diffusion coefficient；Subscript h represents high stress level, and subscript l represents low stress level；
B. to performance parameter p, in k1And k2Under stress level, sum of squares of deviations function is constructed
The minimum problem of sum of squares of deviations function is solved using Nelder-Mead simplex method, is obtained in acceleration model most
Excellent model parameter B, C；
3. degradation path model parameter under normal stress is estimated
After trying to achieve the parameter in acceleration model, degradation path model parameter under normal stress is estimated using least square method
Meter, obtains sum of squares of deviations functionWithUsing simplex method
Solution make sum of squares of deviations minimum corresponding to parameter value, so as to obtain normal stress under degradation path model parameter μp0And σp0
Point estimation；Degradation path model is under final normal stress：yp0(t)=μp0t+σp0B(t)；
Step 4：Product bug behavior Coupling method based on Vine-Copula
Copula functions are a class " connection " functions, and connection is edge distribution and Joint Distribution；The theoretical base of Vine-Copula
Plinth is conditional probability, is carried out by the way that Joint Distribution is resolved into into the form that multiple binary copula and corresponding conditional probability even take advantage of
Modeling, the product bug behavior Coupling method main flow based on Vine-Copula is as follows：
1. performance parameter correlation analysis
Before the failure behavior Coupling method for carrying out product, for the polynary performance parameter degradation of product, correlation point is first carried out
Analysis；With Kendall ' s τ coefficients as relativity measurement index, the absolute value of index represents that correlation is stronger closer to 1,
Kendall ' s τ coefficients represent nonlinear correlation degree；
2. Joint Distribution is decomposed into into the form that suitable conditional probability is multiplied by vine graph models, using copula function structures
Build multiple correlation relational model；Specifically,
A. it is based on the modeling method of D-Vine
1) first, in ground floor Vine graph models, each variable as the node in tree graph, by correlation with short between node
Line is attached；With cijThe probability density of connecting node i and the copula functions of node j is represented, then cij=cpq, p, q are respectively
The variable that node i and node j are included；
2) second layer tree graph in Vine figures is set up on the basis of ground floor tree graph, using the connection in ground floor tree graph as the
Node in two-layer tree figure, adjacent line is attached in the unit of the formation of the second layer with line in ground floor, builds second
The tree graph of layer；Still with cijThe probability density of connecting node i and the copula functions of node j is represented, there is cij=cpq|k, { p, k },
{ q, k } is respectively the variable that node i and node j are included, and wherein k is the total variable of two nodes；
3) third layer tree graph is also to be set up on the basis of preceding layer tree graph using similar method, by that analogy, works as tree graph
In be only left a pair of dependency relations, i.e., be exactly the tree graph of last layer when only remaining a line；In the same manner, third layer tree graph is arrived
There is c including the probability density of the copula functions between all connecting nodes including last layer of tree graphij=cpq|k, p,
K }, { q, k } be the variable that includes of node i and node j, wherein k is the total variable of two nodes；
4) the corresponding copula function probabilities density of each bar line in Vine figures is multiplied, that is, is obtained based on Vine figures
The decomposition texture .c=∏ c of copula functionsij；Production reliability Joint Distribution probability density function is expressed asxiRepresent i-th performance parameter of product, riRepresent the corresponding reliability probability density of i-th parameter
Function, is so far set up based on Vine-Copula coupling models；
B. Vine-Copula coupling models are solved
Solve Vine-Copula coupling models, you can by spending Joint Distribution probability density, first to determine that reliability Joint Distribution is general
Rate density r (x1,...,xn) in each copula probability density function form and parameter, i.e. cijForm and parameter；
1) c in ground floor tree graph in Vine models is determinedij:Substitute into i-th, j-th node and correspond to the degeneration of performance parameter (p, q)
Amount data；Likelihood function is set up according to alternative copula functional formsUsing Nelder-
Mead simplex methods solve the maximum and its corresponding copula function parameters value of likelihood function, are then based on AIC criterion choosing
Take copula functional forms the most suitable and corresponding parameter；Wherein, j represents sample number, common m sample in likelihood function L
This, i represents test point sequence number, common n test point, Rp(t) and RqT () represents respectively the reliability point of pth and q performance parameters
Cloth function；
2) c in second layer tree graph in Vine models is determinedij:{ p, k }, { q, k } is made to be respectively the change that node i and node j are included
Amount, then have cij=cpq|k, likelihood function is set up according to alternative copula functional forms
The maximum and its corresponding copula function parameters value of likelihood function, Ran Houji are solved using Nelder-Mead simplex method
Copula functional forms the most suitable and corresponding parameter are chosen in AIC criterion, wherein, j represents that sample is compiled in likelihood function L
Number, common m sample, i represents test point sequence number, common n test point, Rp|k(t) and Rq|kT () represents respectively reliability Joint Distribution
Conditional probability expression formula, its computing formula follows：
I. when k is one-dimensional variable, i.e., an identical variable is only included between two nodes, now Rp|k(t) and Rq|kT the value of () is pressed
According toCalculate；Wherein R is reliability distribution function,Represent the copula functions
To RjDerivation；
Ii. when k is vectorial more than 1 of dimension, i.e., multiple identical variables, now R are included between two nodesp|k(t) and Rq|k(t)
Value according toCalculate；Wherein v is conditional vector, v=(v1,v2,…,
vd), vjIt is any one element in v, v-jRepresent that v removes vjThe vector for obtaining；
Determine the c in second layer tree graph in Vine modelsijAfterwards, by that analogy, the c of other layer of tree graph is calculatedij, until determining Vine
In figure till all optimum copula functional forms；
3) all c are determinedijFunctional form after, to selected cijParameter estimated again together；Set up likelihood function L
=r (x1,...,xn), the maximum of likelihood function is solved using Nelder-Mead simplex method method, obtain corresponding optimum
Parametric results；By all cijIn the breakdown of the parameter back substitution Vine-Copula of function, that is, obtain based on Vine-copula
The multiple correlation relational model of foundation；
Step 5：Reliability assessment
The reliability Joint Distribution for completing to be obtained after product bug behavior Coupling method product using Vine-Copula functions is general
Rate density function；It is various yet with the density function complex structure of Copula functions, while the reliability of each performance parameter is general
Rate density function expression formula is also sufficiently complex, cause the analytic expression of the probability density function of reliability Joint Distribution not only long but also
It is extremely complex, it is difficult to the method for passing through direct integral solves Reliability Function；For the product under non-critical dullness degenerate case
The accurate solution problem of reliability Joint Distribution, the method that present invention employs conditional probability decomposition is solved；
1. the conditional probability expression formula of the reliability Joint Distribution of binary variable is calculated
Wherein R is reliability distribution function,Represent copula function pairs RjDerivation；
2. the conditional probability expression formula of polytomy variable reliability Joint Distribution is calculated
Wherein v is conditional vector, v=(v1,v2,…,vd), vjIt is any one element in v, v-jRepresent that v removes vjObtain
Vector, v-j=(v1,v2,…,vj-1,vj+1,…,vd)；
3. polynary reliability Joint Distribution expression formula is calculated
Wherein K represents the performance parameter number for occurring to degenerate, akIt is the code name of performance parameter, v is conditional vector, vjCondition to
An element in amount v, v-jRepresent in conditional vector v and remove vjThe vector for obtaining,For vjThe corresponding time,For vector v-j
The time arrow that the middle element corresponding time is constituted；
By above step, the failure behavior Coupling method to product is completed, establish the model of reliability Joint Distribution, solved
Immediate integration of having determined solves a difficult problem for Reliability Function, clearly describes product polytomy variable coupled relation, there is provided can hold
Method by Reliability Assessment is carried out in time and cost.
2. a kind of product bug behavior Coupling method according to claim 1 and reliability estimation method, i.e. one kind are based on
The product bug behavior Coupling method and reliability estimation method of Vine-Copula and accelerated degradation test, it is characterised in that：
" choosing copula functional forms the most suitable based on AIC criterion " described in step 4 refers to：With corresponding to AIC minimum of a values
Copula be considered selection most widely suited in alternative copula forms；The calculating formula of AIC is AIC=-2lnL+2p, wherein L
It is maximum likelihood function value, p is model parameter number, and M is sample size, corresponding value is the observation station number of product.
3. a kind of product bug behavior Coupling method according to claim 1 and reliability estimation method, i.e. one kind are based on
The product bug behavior Coupling method and reliability estimation method of Vine-Copula and accelerated degradation test, it is characterised in that：
It is to make by the principle that reliability Joint Distribution is decomposed into conditional probability for ease of the solution of reliability Joint Distribution in step 5
The every binary Copula function obtained in final decomposition result is all as far as possible the binary for having used in Vine-Copula and having established
Copula functions, described " method that conditional probability is decomposed ", it then follows following some principles：
1) first reliability edge distribution is (i.e. in breakdown), for D-vine preferably selects ground floor two ends arbitrarily to hold
Node；
2) Section 2 in breakdown, i.e.,For the homonymy end node that D-vine preferably selects the second layer；
3) by that analogy, for D-vine preferably selects corresponding end node.
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