CN102081767A - Poor information theory fusion-based product life characteristic information extraction method - Google Patents

Poor information theory fusion-based product life characteristic information extraction method Download PDF

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CN102081767A
CN102081767A CN2011100314912A CN201110031491A CN102081767A CN 102081767 A CN102081767 A CN 102081767A CN 2011100314912 A CN2011100314912 A CN 2011100314912A CN 201110031491 A CN201110031491 A CN 201110031491A CN 102081767 A CN102081767 A CN 102081767A
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夏新涛
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Henan University of Science and Technology
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Abstract

The invention relates to a poor information theory fusion-based product life characteristic information extraction method, which comprises the following steps of: acquiring original information of a small sample; transforming the original information into large-sample generating information by using a right self-service method and performing effective maximum likelihood processing, and acquiring maximum likelihood estimated values of large-sample content of two parameters, namely a Weibull distribution shape parameter and a scale parameter; extracting density functions of the two parameters by using a maximum entropy method; giving a confidence level, and calculating estimation intervals and expected values of the two parameters through the density functions of the shape parameter and the scale parameter respectively; and giving a failure probability, and acquiring the product life characteristic information through Weibull distribution life of the two parameters and reliability calculation thereof. The method has no requirement on completeness of the original information of the small sample, does not need priori information of the shape parameter and the scale parameter, can effectively recover total original characteristics of the product life, disclose nature of the product life information, more accurately acquire the product life characteristic information and reduce the experimental quantity of the product.

Description

Life of product characteristics information extraction method based on weary information theory fusion
Technical field
The present invention relates to a kind of life of product characteristics information extraction method, especially a kind of life characteristics information extracting method based on the rolling bearing that lacks the information theory fusion belongs to life of product and reliability assessment thereof and electric powder prediction.
Background technology
Many products, as rolling bearing, space flight and aviation bearing especially, nuclear reactor bearing and wind-power electricity generation bearing etc. not only need life-span and reliability assessment thereof, also require the fiducial interval assessment of reliability at present.This is a New Set of product accuracy life and function life-span research.Theoretically, this requirement is reasonably because according to indetermination theory, the estimated value of any parameter all has uncertainty; In addition, metrology also requires the estimated value of any parameter should follow its confidence level and fiducial interval.Draw and the calculated value of reliability is a estimated value by the life-span distribution parameter, must have indirect uncertainty.
Two parameters of Weibull are important function of many lives of product and fail-safe analysis thereof, in order to assess the fiducial interval of life-span and reliability thereof, not only will reasonably estimate the form parameter of Weibull distribution βAnd scale parameter η, the more important thing is the density function that must obtain these two parameters.
The classical theory of statistics thinks, for given Weibull distribution, βWith ηIt all is well-determined constant.But Bayesian statistics is regarded these two parameters as mutually independent random variables.Like this, βWith ηBe considered to have density function separately γ( β) and ε( η), in order to obtain γ( β) and ε( η), need parameter βWith ηThe estimated value of large sample content β j With η j ,
Figure 803776DEST_PATH_IMAGE001
For regularly truncation of small sample, adopt maximum-likelihood method, moments method or Harris method usually to obtain to determine βWith ηObviously, because experiment information and parameter information are seldom, only rely on these methods and be difficult to solve γ( β) and ε( η) evaluation problem.At present, some method, bayes method etc. for example relates to dividing value to the assessment of reliability fiducial interval, but needs βWith ηPriori.
Under condition of small sample, the effective ways that obtain a large amount of analog informations are bootstraps, but bootstrap is a kind of non-parametric estmation method, and it is continuous to use bootstrap to make up γ( β) and ε( η) time needs γ( β) and ε( η) priori, if lack γ( β) and ε( η) priori, then be difficult to reasonable implementation Interval Estimate of Parameter and fail-safe analysis.In fact, up to now, in life of product research, almost not about γ( β) and ε( η) report of priori.In addition, the sampling pattern of bootstrap requires raw information must have identical density function attribute, can not solve the have different attribute information fixed time test evaluation problem of (be non-complete information, raw information comprises out-of-service time and truncated time simultaneously).
Maximum-likelihood method is a kind of method for parameter estimation based on joint distribution, allow raw information to have different density function attributes, but maximum-likelihood method can not be carried out Interval Estimate of Parameter, can not carry out the estimation of density function of parameter.
Maximum entropy method (MEM) is one of popular approach of obtaining density function information, but maximum entropy method (MEM) needs the information of large sample content could estimate each rank square effectively, and then finds the solution Lagrange multiplier.
The maximum entropy bootstrap is a kind of new method of small sample assessment, and be applied at aspects such as the average of Frictional Moment for Rolling Bearings and maximal value estimations, see " space flight journal " 2007 the 28th the 05th phases of volume " the maximum entropy probability distribution of space flight bearing frictional torque and bootstarp infer ", but this maximum entropy bootstrap needs 3 message samples at least, and each sample must have big sample content.The density function of each sample be could at first set up so respectively, the Estimation of Mean and the interval estimation of each sample individuality obtained with maximum entropy method (MEM); Carry out subsequent treatment with bootstrap then, obtain overall Estimation of Mean and interval estimation.Therefore, the maximum entropy bootstrap can not solve the small sample evaluation problem of 3 following message samples, can not solve the seldom small sample evaluation problem of (be sample content seldom) of raw information.
Self-service maximum entropy method (MEM) is another new method of small sample assessment, and be applied at aspects such as guided missile hit rate estimations, see " equipment command technology institute journal " 03 phase in 2007 " self-service maximum entropy method (MEM) is determined prior distribution and the application in the guided missile hit probability is estimated thereof ", but this self-service maximum entropy method (MEM) needs 2 parameter information samples (i.e. 1 parameter prior imformation sample and 1 parameter field data sample) at least, also needs the density function (being binomial distribution) of 1 parameter field data.Could set up the prior distribution of parameter to parameter prior imformation sample with self-service maximum entropy method (MEM) like this, determine that with parameter field data sample it is that the posteriority of parameter distributes that the density function of parameter field data, the Bayes who obtains parameter at last distribute.In addition, self-service maximum entropy method (MEM) is that parameter information sample (being the hit rate sample) to small sample content is sampled.In order to obtain parameter information sample, need a large amount of raw information (promptly needing to carry out organizing repeatedly the MISSILE LAUNCHING test) for self-service sampling more.Therefore, self-service maximum entropy method (MEM) can not solve the small sample evaluation problem of 2 following message samples, can not solve raw information small sample evaluation problem seldom.
Especially, maximum entropy bootstrap and self-service maximum entropy method (MEM) all adopt the bootstrap sampling, can not solve evaluation problem, thereby can not have parameter estimation, parameter estimation of density function, life of product and reliability and Estimating Confidence Interval thereof in the fixed time test of non-complete information with different attribute information.
In sum,, under the condition of any prior imformation that need not the Weibull distribution parameter, how to assess the fiducial interval of life-span and reliability thereof, remain a difficult problem for the small sample fixed time test.
Summary of the invention
The purpose of this invention is to provide a kind of life of product characteristics information extraction method that merges based on weary information theory, to solve the difficult problem of the fiducial interval of assessing life of product and reliability thereof.
For achieving the above object, the life of product characteristics information extraction method step that merges based on weary information theory of the present invention is as follows:
(1) picked at random nIndividual product sample to be detected carries out fixed time test, obtains the raw information of product sample life time
Figure 242979DEST_PATH_IMAGE002
, wherein n2, have
Figure 630098DEST_PATH_IMAGE003
Individual fail message and
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Individual truncation information;
(2) choose the pseudo-sampling with equal probability pattern of putting back to of weary information theory, the raw information of the product sample life time of fixed time test is converted to large sample generation information with accurate bootstrap;
(3) at the density function formula of two parameters of Weibull
Figure 952812DEST_PATH_IMAGE005
(1) and the distribution function formula
Figure 61451DEST_PATH_IMAGE006
(2), handle large sample effectively with maximum-likelihood method and generate information, obtain form parameter βAnd scale parameter ηThe maximum likelihood estimated value of large sample content;
(4) use the maximum entropy method (MEM) that lacks information theory to handle the maximum likelihood estimated value of form parameter and scale parameter respectively, extract the density function of form parameter γ( β) and the density function of scale parameter ε( η);
(5) provide confidence level according to the failure probability of life of product and fiduciary level thereof and the requirement of confidence level, calculate the lower border value and the upper boundary values of form parameter and scale parameter respectively by the density function of form parameter and scale parameter, obtain expectation value simultaneously;
(6) provide failure probability according to the failure probability of life of product and fiduciary level thereof and the requirement of confidence level, expectation value, lower border value and upper boundary values by form parameter and scale parameter, by two parameters of Weibull life-span and Calculation of Reliability thereof, obtain expectation value, lower border value and upper boundary values information that the life of product characteristic information is life-span and reliability thereof, to realize the extraction of life of product characteristic information.
Further, puppet obtains 1 sampling information equiprobably from raw information in the described step (2), and it is put back in the raw information after as 1 self-service sample information again, samples like this rInferior, add sIndividual truncation information just obtains 1 content and is nAccurate self-service sample; This was the 1st step, repeated this sampling process BIn the step, just obtain BIndividual content is nAccurate self-service sample be that large sample generates information.
Further, hypothesis has proceeded to the in the described step (2) jGo on foot accurate self-service sampling, from formula
Figure 670287DEST_PATH_IMAGE007
(3) pseudo-equiprobability can be sampled 1 time with putting back in the out-of-service time information, obtains 1 sampling information , so sampling rInferior, obtain rIndividual sampling information about the out-of-service time is because the truncated time formula
Figure 916778DEST_PATH_IMAGE009
(4) have in s= n- rIndividual product sample truncation, then jThe accurate self-service sample of the non-complete information of individual timing truncation is
Figure 697783DEST_PATH_IMAGE010
(5)
In the formula, BThe total step number of self-service sampling of being as the criterion is accurate self-service number of samples, gets B=10000 ~ 20000 can satisfy the requirement of parameter estimation precision.
Further, two parameter maximum likelihood estimated values calculate by following two formula in the described step (3) (7)
Figure 302257DEST_PATH_IMAGE012
(8)
With formula
Figure 433024DEST_PATH_IMAGE010
(5) above-mentioned two formulas of substitution are carried out BGo on foot effective maximum likelihood and estimate, obtain BIndividual maximum likelihood estimated result.
Further, in the described step (3) effectively maximum likelihood estimate to be meant and satisfy formula
Figure 571881DEST_PATH_IMAGE013
(7) convergence is estimated, if accurate self-service sample formula
Figure 466894DEST_PATH_IMAGE014
(5) can not be from formula
Figure 716609DEST_PATH_IMAGE013
(7) obtain restraining the result in, then carry out accurate self-service sampling again, obtain new accurate self-service sample formula (5), till convergence.
Further, use symbol θUnified expression βWith η, promptly when calculating
Figure 764200DEST_PATH_IMAGE015
The time θExpression
Figure 73959DEST_PATH_IMAGE015
, work as calculating
Figure 957732DEST_PATH_IMAGE016
The time θExpression
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, use symbol ξ( θ) unified expression γ( β) and ε( η), promptly when calculating γ( β) time ξ( θ) expression γ( β), work as calculating ε( η) time ξ( θ) expression ε( η), the maximum likelihood estimated result of the large sample content of resulting two parameters of Weibull form parameters and scale parameter is in the described step (3)
Figure 913236DEST_PATH_IMAGE017
(9).
Further, maximum entropy method (MEM) is an information entropy in the described step (4)
Figure 393896DEST_PATH_IMAGE018
(10)
In the formula Be stochastic variable
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Feasible zone,
Figure 611622DEST_PATH_IMAGE021
For
Figure 263183DEST_PATH_IMAGE020
Density function;
The constraint condition of formula (10) is
Figure 121549DEST_PATH_IMAGE022
(11)
In the formula, kBe the moment of the orign number,
Figure 516758DEST_PATH_IMAGE023
Be kThe rank moment of the orign, mBe the order of high moment of the orign;
Can obtain with method of Lagrange multipliers
Figure 127868DEST_PATH_IMAGE024
Expression formula
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(12), in the formula,
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Be kIndividual Lagrange multiplier, k=0,1 ..., m, total m+ 1.
Further, establishing level of significance in the described step (5) is
Figure 727849DEST_PATH_IMAGE027
, then confidence level is
Figure 459044DEST_PATH_IMAGE028
(16)
Provide confidence level according to the failure probability of life of product and fiduciary level thereof and the requirement of confidence level p, setting parameter
Figure 717987DEST_PATH_IMAGE020
Estimation interval be (17)
In the formula,
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With
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Be respectively
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Floor value and last dividing value, and have
Figure 325555DEST_PATH_IMAGE032
(18)
(19)
Parameter
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Expectation
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For
Figure 370554DEST_PATH_IMAGE035
(20)
Parameter
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Intermediate value estimate
Figure 25975DEST_PATH_IMAGE036
For
Figure 594359DEST_PATH_IMAGE037
(21).
Further, the failure probability of establishing product in the described step (6) is q, can get the percentage probability life-span by statistics
Figure 810577DEST_PATH_IMAGE038
:
Figure 4667DEST_PATH_IMAGE039
(22)
Reliability function R( t) be
Figure 357151DEST_PATH_IMAGE040
(23)
According to formula (22) definition life expectancy For
Figure 534371DEST_PATH_IMAGE042
(24)
The life-span interval is
Figure 220567DEST_PATH_IMAGE043
, and floor value
Figure 240607DEST_PATH_IMAGE044
For
Figure 354057DEST_PATH_IMAGE045
(25)
Last dividing value
Figure 341604DEST_PATH_IMAGE046
For
Figure 565912DEST_PATH_IMAGE047
(26)
In the formula,
Figure 204573DEST_PATH_IMAGE048
With
Figure 488924DEST_PATH_IMAGE049
Be respectively
Figure 698188DEST_PATH_IMAGE050
With
Figure 726187DEST_PATH_IMAGE016
Accurate self-service likelihood maximum entropy expectation value,
Figure 720819DEST_PATH_IMAGE051
With
Figure 176071DEST_PATH_IMAGE052
Be respectively
Figure 138211DEST_PATH_IMAGE053
With Floor value,
Figure 51995DEST_PATH_IMAGE054
With
Figure 678148DEST_PATH_IMAGE055
Be respectively
Figure 862005DEST_PATH_IMAGE050
With Last dividing value;
The definition median life
Figure 201031DEST_PATH_IMAGE056
For (27)
In the formula,
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With
Figure 577151DEST_PATH_IMAGE059
Be respectively
Figure 922855DEST_PATH_IMAGE050
With
Figure 890811DEST_PATH_IMAGE016
Accurate self-service likelihood maximum entropy intermediate value estimate;
According to formula (23), the expectation reliability function
Figure 986943DEST_PATH_IMAGE060
For
Figure 760864DEST_PATH_IMAGE061
(28)
Confidence level pUnder the reliability interval be , and floor value
Figure 843537DEST_PATH_IMAGE063
For
Figure 161386DEST_PATH_IMAGE064
(29)
Last dividing value
Figure 738998DEST_PATH_IMAGE065
For
Figure 724271DEST_PATH_IMAGE066
(30)
The intermediate value reliability function
Figure 345615DEST_PATH_IMAGE067
For
Figure 150760DEST_PATH_IMAGE068
(31).
The present invention considers the complete information and the non-complete information of fixed time test, bootstrap, maximum-likelihood method and maximum entropy method (MEM) are carried out theory to be merged, played the effect of relative merits complementations, can simulate the density function of Weibull distribution parameter, obtain the fiducial interval of form parameter and dimensional parameters, and then assessment life of product, reliability and corresponding fiducial interval.The present invention does not require the integrality of small sample raw information, need not the prior imformation of form parameter and scale parameter, can recover the overall primary characteristic of life of product effectively, disclose the natural essence of life of product information, obtain the life of product characteristic information more accurately, and reduced the product experimental amount.
Description of drawings
Fig. 1 is the process flow diagram of the embodiment of the invention;
Fig. 2 is experiment one form parameter βDensity function γ( β) figure;
Fig. 3 is experiment one scale parameter ηDensity function ε( η) figure;
Fig. 4 is experiment two shape parameters βDensity function γ( β) figure;
Fig. 5 is experiment two scale parameters ηDensity function ε( η) figure;
Fig. 6 is experiment three form parameters βThe maximum likelihood estimated value of large sample content β j Figure;
Fig. 7 is experiment three scale parameters ηThe maximum likelihood estimated value of large sample content η j Figure;
Fig. 8 is experiment three form parameters βDensity function γ( β) figure;
Fig. 9 is experiment three scale parameters ηDensity function ε( η) figure;
Figure 10 is the reliability assessment and the prog chart of bearing life;
Figure 11 is the product sample number of truncation sTo the form parameter density function γ( β) influence figure;
Figure 12 is the product sample number of truncation sTo the scale parameter density function ε( η) influence figure.
Embodiment
Because experiment information and parameter information are seldom and density function the unknown of parameter, therefore the properties of product of being studied totally belong to weary infosystem.Weary information refers to that the characteristic information that research object presents is incomplete and insufficient, lacks priori.Experiment information and parameter information seldom and density function the unknown of parameter be typical case's performance of weary information.
The basic thought of weary infosystem theory is theoretical the fusion and the method fusion.Because of poor information, only be difficult to assessment with a kind of theory and method, should excavate different information characteristics with method with different theories, with the weary infosystem of full appreciation; In addition, different theories has different advantages and defective with method, by merging, can maximize favourable factors and minimize unfavourable ones, have complementary advantages, and produces new antibodies, and enhance immunity power possesses self-healing function, forms new theory and new method, solves weary information problem more perfectly.
Consider the characteristics of bootstrap, maximum-likelihood method and maximum entropy method (MEM), the present invention is theoretical and Weibull distribution based on weary infosystem, merge bootstrap, maximum-likelihood method, maximum entropy method (MEM), propose life of product characteristics information extraction method under the small sample timing truncation, be called accurate self-service likelihood maximum entropy method (MEM).With the rolling bearing is example, and as shown in Figure 1, the step of this method is as follows:
(1) picked at random nIndividual product sample to be detected carries out fixed time test, obtains the raw information of product sample life time
Figure 204166DEST_PATH_IMAGE002
, wherein n2, have
Figure 106263DEST_PATH_IMAGE003
Individual fail message and
Figure 586923DEST_PATH_IMAGE004
Individual truncation information;
(2) choose the pseudo-sampling with equal probability pattern of putting back to of weary information theory, the raw information of the product sample life time of fixed time test is converted to large sample generation information with accurate bootstrap;
(3) at the density function formula of two parameters of Weibull
Figure 957993DEST_PATH_IMAGE005
(1) and the distribution function formula
Figure 611828DEST_PATH_IMAGE006
(2), handle large sample effectively with maximum-likelihood method and generate information, obtain form parameter βAnd scale parameter ηThe maximum likelihood estimated value of large sample content;
(4) use the maximum entropy method (MEM) that lacks information theory to handle the maximum likelihood estimated value of form parameter and scale parameter respectively, extract the density function of form parameter γ( β) and the density function of scale parameter ε( η);
(5) requiring according to life of product is that the failure probability of life of product and fiduciary level thereof and the requirement of confidence level provide confidence level, calculate the lower border value and the upper boundary values of form parameter and scale parameter respectively by the density function of form parameter and scale parameter, obtain expectation value simultaneously;
(6) requiring according to life of product is that the failure probability of life of product and fiduciary level thereof and the requirement of confidence level provide failure probability, expectation value, lower border value and upper boundary values by form parameter and scale parameter, by two parameters of Weibull life-span and Calculation of Reliability thereof, obtain expectation value, lower border value and upper boundary values information that the life of product characteristic information is life-span and reliability thereof, to realize the extraction of life of product characteristic information.
Be not difficult to find out, based on small sample raw information, accurate self-service likelihood maximum entropy method (MEM) carries out theory with bootstrap, maximum-likelihood method and maximum entropy method (MEM) and merges, played the effect of relative merits complementations, can simulate the density function of Weibull distribution parameter, obtain the fiducial interval of form parameter and scale parameter, and then assessment life of product, reliability and corresponding fiducial interval.Obviously, the characteristics of accurate self-service likelihood maximum entropy method (MEM) are, when assessment, only need small sample raw information, need not any prior imformation of form parameter and scale parameter.
The present invention is also at larger samples and small sample complete information, and the non-complete information of small sample carries out timing truncation emulation and experimental study, with validity and the practicality of checking accurate self-service likelihood maximum entropy method (MEM).
1 Two-parameter Weibull Distribution
If product precision or function life-span TThe obedience stochastic variable is tTwo-parameter Weibull Distribution, its density function is
Figure 883278DEST_PATH_IMAGE069
(1)
Distribution function is
Figure 534839DEST_PATH_IMAGE006
(2)
In the formula, βBe form parameter, ηIt is scale parameter.
The raw information of 2 lives of product
The raw information of life of product refers to the working time of product, in fixed time test, comprises out-of-service time and truncated time.Out-of-service time has different attributes with truncated time.If in the raw information only arranged the out-of-service time, then be referred to as complete information; If in the raw information out-of-service time and truncated time are arranged simultaneously, then are referred to as non-complete information.Therefore, non-complete information is the set of out-of-service time and two kinds of different attribute information of truncated time.
Picked at random n2 product samples carry out fixed time test, be provided with r(
Figure 376893DEST_PATH_IMAGE003
) individual product sample inefficacy, the out-of-service time is
Figure 772103DEST_PATH_IMAGE070
(3)
Other s= n- rIndividual product sample truncation, truncated time is
Figure 399524DEST_PATH_IMAGE071
(4)
Know that by formula (3) and formula (4) fixed time test has only 1 sample content that is made of non-complete information to be n2 ensemble of communication, therefore, small sample raw information of the present invention is meant the raw information of the small sample content that has only 1 message sample.
3 accurate bootstraps
For convenient narration, use symbol θUnified expression βWith η, promptly when calculating
Figure 221987DEST_PATH_IMAGE072
The time θExpression
Figure 551337DEST_PATH_IMAGE072
, work as calculating
Figure 750237DEST_PATH_IMAGE016
The time θExpression
Figure 730700DEST_PATH_IMAGE016
, use symbol ξ( θ) unified expression βDensity function γ( β) and ηDensity function ε( η), promptly when calculating γ( β) time ξ( θ) expression γ( β), work as calculating ε( η) time ξ( θ) expression ε( η).Use the purpose of accurate bootstrap to be,, convert small sample raw information to large sample and generate information by repeatedly accurate self-service sampling; Use the purpose of maximum-likelihood method to be, generate the information acquisition parameter with large sample θThe estimated value of large sample content θ j Using the purpose of maximum entropy method (MEM) is to use parameter θThe estimated value of large sample content θ j Set up parameter θDensity function ξ( θ).
Suppose to have proceeded to jGo on foot accurate self-service sampling.Pseudo-equiprobability can be sampled 1 time with putting back to from formula (3) out-of-service time information, obtains 1 sampling information
Figure 724064DEST_PATH_IMAGE008
, so sampling rInferior, obtain rIndividual sampling information about the out-of-service time.Consider in the formula (4) s= n- rIndividual product sample truncation, then jThe accurate self-service sample of the non-complete information of individual timing truncation is
Figure 540710DEST_PATH_IMAGE010
(5)
In the formula, BThe total step number of self-service sampling of being as the criterion is accurate self-service number of samples, gets B=10000 ~ 20000 can satisfy the requirement of parameter estimation precision.Formula (5) is exactly that the large sample that converts to of the raw information with life of product generates information.
But above-mentioned pseudo-equiprobability sampling with replacement can adopt the uniformly distributed function of computer advanced language to realize.This sampling produces pseudo random number, so be referred to as pseudo-sampling with equal probability pattern.Sampling process only depends on raw information and is promptly driven by raw information.
In formula (5), truncated time is
Figure 543301DEST_PATH_IMAGE073
(6).
When above-mentioned acquisition formula (5), bootstrap has been adopted in the out-of-service time sampling, and directly used truncated time, bootstrap so this method of title is as the criterion.
4 maximum-likelihood methods
Right βWith ηCarry out maximum likelihood and estimate have
Figure 879736DEST_PATH_IMAGE074
(7)
Figure 44001DEST_PATH_IMAGE075
(8)
Can obtain respectively by formula (7) and (8) βWith ηThe maximum likelihood estimated value β j With η j
With formula (5) substitution formula (7) and formula (8), carry out BInferior effective maximum likelihood is estimated, can obtain BIndividual maximum likelihood estimated result.Here effectively maximum likelihood estimates to be meant the convergence estimation of satisfying formula (7).Maximum likelihood is estimated be difficult to convergence in some cases, for this reason, the present invention adopts effective maximum likelihood disposal route: if accurate self-service sample formula (5) can not obtain restraining the result from formula (7), then carry out accurate self-service sampling again, obtain new accurate self-service sample formula (5), till convergence.Like this, resulting BIndividual maximum likelihood estimated result is promptly about parameter θLarge sample content information be
Figure 347943DEST_PATH_IMAGE076
(9)
This is γ( β) and ε( η) maximum entropy method (MEM) make up and to have established the large information capacity basis.
5 maximum entropy method (MEM)s
Maximum entropy method (MEM) is thought, under the situation of only grasping partial information, to infer system state, should choose meet constraint condition and entropy for maximum state as a kind of rational state, this is unique adiaphorous selection, and the probability maximum that occurs of the probability distribution of entropy maximum.For this situation of having only a little information of rolling bearing life, do not have the sufficient reason selection and suppose other analytical function forms, should determine the form of the density function of Weibull distribution parameter with maximum entropy method (MEM).
According to maximum entropy method (MEM), the density function that does not have subjective prejudice most should satisfy the entropy maximum, promptly
Figure 888646DEST_PATH_IMAGE018
(10)
In the formula, HBe information entropy,
Figure 578122DEST_PATH_IMAGE077
Be stochastic variable
Figure 178868DEST_PATH_IMAGE020
Feasible zone,
Figure 704527DEST_PATH_IMAGE021
For
Figure 48921DEST_PATH_IMAGE020
Density function.
The constraint condition of formula (10) is
Figure 94368DEST_PATH_IMAGE022
(11)
In the formula, kBe the moment of the orign number,
Figure 866015DEST_PATH_IMAGE023
Be kThe rank moment of the orign, mBe the order of high moment of the orign.
Under constraint condition, regulate
Figure 816654DEST_PATH_IMAGE024
Can make the entropy maximum.This is a constrained optimization problem, can obtain with method of Lagrange multipliers Expression formula
Figure 113960DEST_PATH_IMAGE025
(12)
Formula (12) is exactly a parameter
Figure 368093DEST_PATH_IMAGE020
Density function
Figure 806027DEST_PATH_IMAGE024
Accurate self-service likelihood maximum entropy estimate.In formula (12),
Figure 554540DEST_PATH_IMAGE026
Be kIndividual Lagrange multiplier, k=0,1 ..., m, total m+ 1.First multiplier is
Figure 761531DEST_PATH_IMAGE079
(13)
Other mIndividual multiplier should satisfy condition
Figure 874980DEST_PATH_IMAGE080
(14)
By statistics, will Sort from small to large and be divided into Z-2 groups, draw histogram, obtain the zThe class mean of group
Figure 837568DEST_PATH_IMAGE082
And frequency
Figure 226961DEST_PATH_IMAGE083
, z=2,3 ..., Z-1.Again histogram is extended to ZGroup, promptly z=1,2 ..., Z, and order
Figure 245733DEST_PATH_IMAGE084
So, the kThe rank moment of the orign
Figure 969844DEST_PATH_IMAGE023
Value be
Figure 997843DEST_PATH_IMAGE085
(15)
If level of significance is , then confidence level is
Figure 696995DEST_PATH_IMAGE028
(16)
Require to provide confidence level according to life of product pIn confidence level pDown, setting parameter
Figure 331238DEST_PATH_IMAGE020
Accurate self-service likelihood maximum entropy estimation interval be (17)
In the formula,
Figure 74383DEST_PATH_IMAGE030
With
Figure 700537DEST_PATH_IMAGE031
Be respectively
Figure 133661DEST_PATH_IMAGE020
Accurate self-service likelihood maximum entropy floor value and last dividing value, and have
(18)
(19)
Defined parameters
Figure 519009DEST_PATH_IMAGE020
The expectation of accurate self-service likelihood maximum entropy For
Figure 911124DEST_PATH_IMAGE087
(20)
Defined parameters
Figure 921806DEST_PATH_IMAGE020
Accurate self-service likelihood maximum entropy intermediate value estimate
Figure 201346DEST_PATH_IMAGE036
For
Figure 297478DEST_PATH_IMAGE037
(21)
The accurate self-service likelihood maximum entropy assessment of 6 life-spans and reliability thereof
If the failure probability of product is q, can get the percentage probability life-span by statistics
Figure 71399DEST_PATH_IMAGE038
:
Figure 936587DEST_PATH_IMAGE039
(22)
Reliability function R( t) be
Figure 888493DEST_PATH_IMAGE040
(23)
Based on the self-service likelihood maximum entropy method (MEM) of standard, according to formula (22) definition life expectancy
Figure 534238DEST_PATH_IMAGE088
For
Figure 49533DEST_PATH_IMAGE089
(24)
The life-span interval is , and floor value
Figure 390571DEST_PATH_IMAGE044
For
Figure 461295DEST_PATH_IMAGE090
(25)
Last dividing value
Figure 577018DEST_PATH_IMAGE046
For
Figure 416799DEST_PATH_IMAGE047
(26)
In the formula,
Figure 710508DEST_PATH_IMAGE048
With
Figure 268528DEST_PATH_IMAGE049
Be respectively
Figure 922363DEST_PATH_IMAGE050
With
Figure 616650DEST_PATH_IMAGE016
Accurate self-service likelihood maximum entropy expectation value,
Figure 533790DEST_PATH_IMAGE051
With
Figure 625112DEST_PATH_IMAGE052
Be respectively
Figure 20321DEST_PATH_IMAGE053
With
Figure 897010DEST_PATH_IMAGE016
Floor value,
Figure 719473DEST_PATH_IMAGE054
With
Figure 799555DEST_PATH_IMAGE055
Be respectively
Figure 998456DEST_PATH_IMAGE050
With
Figure 729651DEST_PATH_IMAGE016
Last dividing value.
The definition median life
Figure 723015DEST_PATH_IMAGE056
For
Figure 477344DEST_PATH_IMAGE057
(27)
In the formula,
Figure 525941DEST_PATH_IMAGE058
With
Figure 49326DEST_PATH_IMAGE091
Be respectively With
Figure 783113DEST_PATH_IMAGE092
Accurate self-service likelihood maximum entropy intermediate value estimate.
According to formula (23), define accurate self-service likelihood maximum entropy expectation reliability function
Figure 871285DEST_PATH_IMAGE060
For
Figure 514756DEST_PATH_IMAGE061
(28)
The definition confidence level pUnder the reliability interval be
Figure 849923DEST_PATH_IMAGE093
, and floor value
Figure 641161DEST_PATH_IMAGE063
For
Figure 719976DEST_PATH_IMAGE064
?(29)
Last dividing value
Figure 529538DEST_PATH_IMAGE065
For
Figure 301185DEST_PATH_IMAGE094
(30)
Define accurate self-service likelihood maximum entropy intermediate value reliability function
Figure 314140DEST_PATH_IMAGE067
For
Figure 196645DEST_PATH_IMAGE095
(31)。
7 specifically experiments
In order to check method of the present invention, that is the correctness of accurate self-service likelihood maximum entropy method (MEM) and carry out effect comparison with existing common method, carry out 3 kinds of dissimilar concrete experiments below.
7.1 the complete information case of larger samples
Experiment 1: this is the assessment case of the complete information of a larger samples, checking the parameter estimation effect of accurate self-service likelihood maximum entropy method (MEM), and compares with moments method, maximum-likelihood method and Harris method.Consider Weibull distribution, get n= r=50, the setup parameter true value is respectively β=2.5 Hes η=200, obtain the out-of-service time with the statistical simulation method:
40?49?59?70?85?93?96?99?105?111?115?116?116?118?123?128?130?131?132?135?136?139?146?154?157?162?169?170?188?191?199?205?207?210?215?222?234?253?264?279?281?287?316?319?319?321?326?344?386?392
Relevant CALCULATION OF PARAMETERS result is as shown in table 1, and wherein, because in fact the Harris method has adopted the maximum-likelihood method estimated parameter, so its estimated result is identical with maximum-likelihood method.As can be seen, accurate self-service likelihood maximum entropy method (MEM), maximum-likelihood method (Harris method) and moments method are right ηThe relative error of estimating is no more than 5%, and effect is all fine; Right βEstimation effect, the relative error of accurate self-service likelihood maximum entropy method (MEM) is less than 10%, and the relative error of maximum-likelihood method (Harris method) and moments method is greater than 10%.
Relevant Calculation for life result is as shown in table 2.As can be seen, the life value that 3 kinds of methods calculate all is lower than the life-span true value, and is relatively conservative.But comparatively speaking, accurate self-service likelihood maximum entropy method (MEM) is to the evaluated error minimum in life-span, and effect is best; Maximum-likelihood method (Harris method) is taken second place, and moments method is the poorest.
Especially, shown in Fig. 2, Fig. 3 and table 1, accurate self-service likelihood maximum entropy method (MEM) can also obtain the estimation of density function of parameter, thereby can implement intermediate value and estimate to estimate with interval.Obviously, accurate self-service likelihood maximum entropy method (MEM) is more excellent, more perfect than the maximum-likelihood method (Harris method) and the estimation effect of moments method.
Figure 362179DEST_PATH_IMAGE096
Figure 304727DEST_PATH_IMAGE097
7.2 small sample complete information case
Experiment 2: this is the assessment case of a small sample complete information.When carrying out small sample rolling bearing life experimental evaluation, the most frequently used method is the Harris method.For ease of comparative analysis, present case quote Harris the small sample out-of-service time ( n= r=10):
14.01?15.38?20.94?29.44?31.15?36.72?40.32?48.61?56.42?56.97
Fig. 4 and Fig. 5 be respectively accurate self-service likelihood maximum entropy method (MEM) obtain about parameter βWith ηDensity function.The relevant calculation result of accurate self-service likelihood maximum entropy method (MEM) and Harris method is shown in table 3 and table 4.
As can be seen, accurate self-service likelihood maximum entropy method (MEM) and Harris method are to parameter βWith ηThe estimated value difference, thereby inevitable variant to the estimated result of bearing life intermediate value and expectation value.As previously mentioned, because the Harris method adopts the maximum-likelihood method estimated parameter, so its estimation effect is too conservative, and it is good to be not so good as accurate self-service likelihood maximum entropy method (MEM).
Must be pointed out that the Harris method can not get parms ηIntermediate value estimate and intervally estimate, can not get parms βWith ηDistribution function, thereby be difficult to reasonably assess the confidence level and the fiducial interval of bearing life and reliability thereof.Very strict bearing life assessment is crucial to reliability requirement for this.On this meaning, accurate self-service likelihood maximum entropy method (MEM) more has superiority than Harris method.
Figure 804978DEST_PATH_IMAGE098
7.3 the non-complete information case of small sample
Experiment 3: this is the detailed evaluation and prediction case of a non-complete information of small sample, by present case, also will narrate concrete operations step of the present invention and computation process details.
7.3.1 acquisition raw information
1. select the regularly bearing unit of truncation durability test of small sample
Certain type gyro machine rotor bearing is installed on the bearing life testing machine, is carried out small sample fixed time test accuracy life, drop into altogether n=8 bearing units.
2. determine truncated time
Truncated time is taken as 4000h.
3. carry out the truncation test
In process of the test, have r=5 element failures, the resulting out-of-service time (unit: h) be T F=(1313,2288,2472,2506,3382) are formula (3).
4. obtain truncation unit number
During to truncated time, off-test.Total s=3 unit truncation, resulting truncated time (unit: h) be T C=(4000,4000,4000) are formula (4).
5. acquisition raw information
Raw information is by the out-of-service time T FAnd truncated time T CTwo parts constitute, i.e. (1313,2288,2472,2506,3382,4000,4000,4000).
Generate information 7.3.2 raw information is converted to large sample with accurate bootstrap
(1) producing interval with the random function in the computer advanced language is the even distribution pseudo random number of [0,1] u, will uLinear mapping to interval [1, r]=[1,5] and round (round and still be [1,5] between the back zone), obtain 1 integer sequence number i
(2) with the out-of-service time T FMiddle subscript sequence number is iInformation note, as 1 sampling information;
(3) repeat (1) ~ (2) totally 5 times, obtain 5 sampling informations;
(4) again with truncated time T CIn information regard 3 sampling informations as, so just obtain n= s+ rIt is 8 accurate self-service sample that=8 sampling informations promptly obtain 1 content;
(5) repeat (1) ~ (4) altogether B=10000 times, obtain 10000 content and be 8 accurate self-service sample, Here it is has converted raw information to large sample and has generated information T j Be formula (5), here j=1,2 ..., 10000.
7.3.3 obtain the large sample content estimated value of form parameter and scale parameter with maximum-likelihood method
Large sample is generated information T j Substitution formula (7) and formula (8) calculate 10000 form parameters respectively βAnd scale parameter ηLarge sample content estimated value β j With η j , will β j With η j Rank order from small to large, the result sees Fig. 6 and Fig. 7 respectively.
7.3.4 extract the density function of form parameter and scale parameter with maximum entropy method (MEM)
(1) establishes θ j = β j With θ= β, getting, the order of high moment of the orign is m=5, according to the histogram principle, after will sorting respectively β j Be divided into 10 groups, obtain the class mean of each group θ z And frequency Ξ z , use again formula (15) calculate about β j Each rank moment of the orign m k , k=1,2 ..., 5, use formula (12) ~ formula (14) to obtain form parameter then βDensity function γ( β)= ξ( θ), as shown in Figure 8;
(2) establish θ j = η j With θ= η, getting, the order of high moment of the orign is m=5, according to the histogram principle, after will sorting respectively η j Be divided into 10 groups, obtain the class mean of each group θ z And frequency Ξ z , with formula (15) calculate about η j Each rank moment of the orign m k , k=1,2 ..., 5, use formula (12) ~ formula (14) to obtain scale parameter then ηDensity function ε( η)= ξ( θ), as shown in Figure 9.
7.3.5 calculate the estimation interval and the expectation value of form parameter and scale parameter
(1) gets confidence level respectively p=90%, 95% and 99%;
(2) establish θ= βWith ξ( θ)= γ( β), calculate form parameter by formula (17) ~ formula (20) βEstimation interval [ β L, β U] and expectation value β Mean, the result is as shown in table 5;
(3) establish θ= ηWith ξ( θ)= ε( η), calculate scale parameter by formula (17) ~ formula (20) ηEstimation interval [ η L, η U] and expectation value η Mean, the result is as shown in table 5.
Figure 225595DEST_PATH_IMAGE099
7.3.6 extract the life of product characteristic information
(1) gets crash rate respectively q=10%, 5% and 1%;
(2) the principal character information of extraction life of product: calculate life expectancy by formula (24) L Mean q , by formula (25) and formula (26) calculate the life-span interval [ L L q , L U q ], calculate the reliability of life expectancy by formula (28) R Mean( L Mean q ), by formula (29) ~ formula (30) calculate life expectancy the reliability interval [ R L( L Mean q ), R U( L Mean q )], the extraction of all characteristic informations the results are shown in Table 5 and Figure 10.
7.3.7 the assessment of life of product feature and deduction
Gyro manufacturer takes up with 10 these profile shafts of cover again joins 5 certain type high speed gyro motors, the longevity test of switching on, and each gyro machine accumulative total electrical working time reaches 1023h as a result, and all satisfies performance requirement.Like this, can infer that by table 5 and Figure 10 the reliability that this profile shaft holds life expectancy is at least R Mean(1023)=96.8%, and in confidence level pReliability interval under=96.8% be [ R L(1023), R U(1023)]=[90.08,99.20].Therefore, this profile shaft holds the least reliability of accuracy life and is higher than 90%.
By above-mentioned 3 experiments as can be known, the advantage of multiple mathematical theory has been merged in the present invention, integrality to small sample raw information does not require, need not the prior imformation of form parameter and scale parameter, can recover the overall primary characteristic of life of product effectively, disclose the natural essence of life of product information, therefore, obtain the life of product characteristic information more accurately than additive method.
7.4 the density function feature of Weibull distribution parameter
By Fig. 2～Fig. 5 and Fig. 8 ~ Fig. 9 as can be seen, in 3 kinds of dissimilar cases, form parameter βDensity function γ( β) all have unimodal left avertence feature, a scale parameter ηDensity function ε( η) all have a unimodal near symmetrical feature.
In order to further investigate this problem, in the non-complete information case of small sample, change the product sample number of truncation sSize, look at γ( β) and ε( η) what has change.As Figure 11 and shown in Figure 12, along with sIncrease, γ( β) and ε( η) width diminish, highly increase; γ( β) summit be moved to the left, and ε( η) summit move right.This shows that the density function of Weibull distribution parameter and the information number of truncation have substantial connection.It can also be seen that, γ( β) unimodal left avertence feature and ε( η) unimodal near symmetrical feature not with sChange and change.Consider that again Fig. 2～Fig. 5 and Fig. 8 ~ Fig. 9 can find, γ( β) unimodal left avertence feature and ε( η) unimodal near symmetrical feature and quantity of information what and information integrity irrelevant.Therefore, different with the classical theory of statistics, accurate self-service likelihood maximum entropy method (MEM) thinks that for Weibull distribution, form parameter and scale parameter all belong to stochastic variable, and their density function has unimodal left avertence attitude feature and unimodal near symmetrical feature respectively.
In sum, based on weary infosystem theory, bootstrap, maximum-likelihood method and maximum entropy method (MEM) are carried out theory merge, propose accurate self-service likelihood maximum entropy method (MEM), can solve the regularly small sample evaluation problem of truncation Weibull distribution parameter density function of rolling bearing life effectively.Based on this, obtain intermediate value, expectation value and the interval estimation of parameter, and then propose the appraisal procedure of life-span and reliability fiducial interval thereof.
The characteristics of accurate self-service likelihood maximum entropy method (MEM) are when assessment life-span and reliability fiducial interval thereof, only by small sample raw information, to need not any prior imformation of boolean's distribution parameter.
Compare with moments method, maximum-likelihood method and Harris method, accurate self-service likelihood maximum entropy method (MEM) is not only to the evaluated error minimum in parameter and life-span, and density function, intermediate value and interval that can also estimated parameter.Therefore, accurate self-service likelihood maximum entropy method (MEM) is more excellent, more perfect than the estimation effect of moments method, maximum-likelihood method and Harris method.
Accurate self-service likelihood maximum entropy method (MEM) can be assessed the fiducial interval of bearing life and reliability thereof, thereby aspect reliability assessment, accurate self-service likelihood maximum entropy method (MEM) more has superiority than moments method, maximum-likelihood method and Harris method.
The density function of form parameter has unimodal left avertence feature, and the density function of scale parameter has unimodal near symmetrical feature.These distribution characteristicss of Weibull distribution parameter and experiment information amount and information integrity are irrelevant.
The inventive method will lack the infosystem theory first and be applied in the extraction of life of product characteristic information, merged the advantage of multiple mathematical theory, integrality to small sample raw information does not require, need not the prior imformation of form parameter and scale parameter, can recover the overall primary characteristic of life of product effectively, disclose the natural essence of life of product information, obtain the life of product characteristic information more accurately, and reduced the product experimental amount.

Claims (9)

1. based on the life of product characteristics information extraction method of weary information theory fusion, it is characterized in that the step of this method is as follows:
(1) picked at random nIndividual product sample to be detected carries out fixed time test, obtains the raw information of product sample life time
Figure 2011100314912100001DEST_PATH_IMAGE002
, wherein n2, have
Figure 2011100314912100001DEST_PATH_IMAGE004
Individual fail message and
Figure 2011100314912100001DEST_PATH_IMAGE006
Individual truncation information;
(2) choose the pseudo-sampling with equal probability pattern of putting back to of weary information theory, the raw information of the product sample life time of fixed time test is converted to large sample generation information with accurate bootstrap;
(3) at the density function formula of two parameters of Weibull
Figure 2011100314912100001DEST_PATH_IMAGE008
(1) and the distribution function formula
Figure 2011100314912100001DEST_PATH_IMAGE010
(2), handle large sample effectively with maximum-likelihood method and generate information, obtain form parameter βAnd scale parameter ηThe maximum likelihood estimated value of large sample content;
(4) use the maximum entropy method (MEM) that lacks information theory to handle the maximum likelihood estimated value of form parameter and scale parameter respectively, extract the density function of form parameter γ( β) and the density function of scale parameter ε( η);
(5) provide confidence level according to the failure probability of life of product and fiduciary level thereof and the requirement of confidence level, calculate the lower border value and the upper boundary values of form parameter and scale parameter respectively by the density function of form parameter and scale parameter, obtain expectation value simultaneously;
(6) provide failure probability according to the failure probability of life of product and fiduciary level thereof and the requirement of confidence level, expectation value, lower border value and upper boundary values by form parameter and scale parameter, by two parameters of Weibull life-span and Calculation of Reliability thereof, obtain expectation value, lower border value and upper boundary values information that the life of product characteristic information is life-span and reliability thereof, to realize the extraction of life of product characteristic information.
2. the life of product characteristics information extraction method that merges based on weary information theory according to claim 1, it is characterized in that: puppet obtains 1 sampling information equiprobably in the described step (2) from raw information, it is put back in the raw information after as 1 self-service sample information again, like this sampling rInferior, add sIndividual truncation information just obtains 1 content and is nAccurate self-service sample; This was the 1st step, repeated this sampling process BIn the step, just obtain BIndividual content is nAccurate self-service sample be that large sample generates information.
3. the life of product characteristics information extraction method that merges based on weary information theory according to claim 2 is characterized in that: hypothesis has proceeded to the in the described step (2) jGo on foot accurate self-service sampling, from formula
Figure 2011100314912100001DEST_PATH_IMAGE012
(3) pseudo-equiprobability can be sampled 1 time with putting back in the out-of-service time information, obtains 1 sampling information , so sampling rInferior, obtain rIndividual sampling information about the out-of-service time is because the truncated time formula
Figure 2011100314912100001DEST_PATH_IMAGE016
(4) have in s= n- rIndividual product sample truncation, then jThe accurate self-service sample of the non-complete information of individual timing truncation is
Figure 2011100314912100001DEST_PATH_IMAGE018
(5)
In the formula, BThe total step number of self-service sampling of being as the criterion is accurate self-service number of samples, gets B=10000 ~ 20000 can satisfy the requirement of parameter estimation precision.
4. the life of product characteristics information extraction method that merges based on weary information theory according to claim 3, it is characterized in that: two parameter maximum likelihood estimated values calculate by following two formula in the described step (3) (7)
Figure 2011100314912100001DEST_PATH_IMAGE022
(8)
With formula
Figure DEST_PATH_IMAGE018A
(5) above-mentioned two formulas of substitution are carried out BGo on foot effective maximum likelihood and estimate, obtain BIndividual maximum likelihood estimated result.
5. the life of product characteristics information extraction method that merges based on weary information theory according to claim 4 is characterized in that: effective maximum likelihood estimation is meant and satisfies formula in the described step (3)
Figure 2011100314912100001DEST_PATH_IMAGE024
(7) convergence is estimated, if accurate self-service sample formula
Figure DEST_PATH_IMAGE018AA
(5) can not be from formula
Figure DEST_PATH_IMAGE024A
(7) obtain restraining the result in, then carry out accurate self-service sampling again, obtain new accurate self-service sample formula (5), till convergence.
6. the life of product characteristics information extraction method that merges based on weary information theory according to claim 5 is characterized in that: use symbol θUnified expression βWith η, promptly when calculating
Figure 2011100314912100001DEST_PATH_IMAGE026
The time θExpression
Figure DEST_PATH_IMAGE026A
, work as calculating
Figure 2011100314912100001DEST_PATH_IMAGE028
The time θExpression
Figure DEST_PATH_IMAGE028A
, use symbol ξ( θ) unified expression γ( β) and ε( η), promptly when calculating γ( β) time ξ( θ) expression γ( β), work as calculating ε( η) time ξ( θ) expression ε( η), the maximum likelihood estimated result of the large sample content of resulting two parameters of Weibull form parameters and scale parameter is in the described step (3)
Figure 2011100314912100001DEST_PATH_IMAGE030
(9).
7. the life of product characteristics information extraction method that merges based on weary information theory according to claim 6, it is characterized in that: maximum entropy method (MEM) is an information entropy in the described step (4)
Figure 2011100314912100001DEST_PATH_IMAGE032
(10)
In the formula
Figure 2011100314912100001DEST_PATH_IMAGE034
Be stochastic variable
Figure 2011100314912100001DEST_PATH_IMAGE036
Feasible zone,
Figure 2011100314912100001DEST_PATH_IMAGE038
For
Figure DEST_PATH_IMAGE036A
Density function;
The constraint condition of formula (10) is
Figure 2011100314912100001DEST_PATH_IMAGE040
(11)
In the formula, kBe the moment of the orign number,
Figure 2011100314912100001DEST_PATH_IMAGE042
Be kThe rank moment of the orign, mBe the order of high moment of the orign;
Can obtain with method of Lagrange multipliers Expression formula
Figure 2011100314912100001DEST_PATH_IMAGE044
(12), in the formula,
Figure 2011100314912100001DEST_PATH_IMAGE046
Be kIndividual Lagrange multiplier, k=0,1 ..., m, total m+ 1.
8. the life of product characteristics information extraction method that merges based on weary information theory according to claim 7, it is characterized in that: establishing level of significance in the described step (5) is
Figure 2011100314912100001DEST_PATH_IMAGE048
, then confidence level is
Figure 2011100314912100001DEST_PATH_IMAGE050
(16)
Provide confidence level according to the failure probability of life of product and fiduciary level thereof and the requirement of confidence level p, setting parameter
Figure DEST_PATH_IMAGE036AA
Estimation interval be (17)
In the formula,
Figure 2011100314912100001DEST_PATH_IMAGE054
With
Figure DEST_PATH_IMAGE056
Be respectively
Figure DEST_PATH_IMAGE036AAA
Floor value and last dividing value, and have
Figure DEST_PATH_IMAGE058
(18)
Figure DEST_PATH_IMAGE060
(19)
Parameter Expectation
Figure DEST_PATH_IMAGE062
For
Figure DEST_PATH_IMAGE064
(20)
Parameter
Figure DEST_PATH_IMAGE036AAAAA
Intermediate value estimate
Figure DEST_PATH_IMAGE066
For
Figure DEST_PATH_IMAGE068
(21).
9. the life of product characteristics information extraction method that merges based on weary information theory according to claim 8, it is characterized in that: the failure probability of establishing product in the described step (6) is q, can get the percentage probability life-span by statistics :
Figure DEST_PATH_IMAGE072
(22)
Reliability function R( t) be
Figure DEST_PATH_IMAGE074
(23)
According to formula (22) definition life expectancy
Figure DEST_PATH_IMAGE076
For
Figure DEST_PATH_IMAGE078
(24)
The life-span interval is
Figure DEST_PATH_IMAGE080
, and floor value
Figure DEST_PATH_IMAGE082
For
Figure DEST_PATH_IMAGE084
(25)
Last dividing value
Figure DEST_PATH_IMAGE086
For (26)
In the formula,
Figure DEST_PATH_IMAGE090
With
Figure DEST_PATH_IMAGE092
Be respectively
Figure DEST_PATH_IMAGE094
With
Figure DEST_PATH_IMAGE028AA
Accurate self-service likelihood maximum entropy expectation value, With Be respectively
Figure DEST_PATH_IMAGE094A
With
Figure DEST_PATH_IMAGE028AAA
Floor value,
Figure DEST_PATH_IMAGE100
With Be respectively With
Figure DEST_PATH_IMAGE028AAAA
Last dividing value;
The definition median life
Figure DEST_PATH_IMAGE104
For
Figure DEST_PATH_IMAGE106
(27)
In the formula,
Figure DEST_PATH_IMAGE108
With
Figure DEST_PATH_IMAGE110
Be respectively
Figure DEST_PATH_IMAGE094AAA
With
Figure DEST_PATH_IMAGE028AAAAA
Accurate self-service likelihood maximum entropy intermediate value estimate;
According to formula (23), the expectation reliability function For
Figure DEST_PATH_IMAGE114
(28)
Confidence level pUnder the reliability interval be
Figure DEST_PATH_IMAGE116
, and floor value
Figure DEST_PATH_IMAGE118
For
Figure DEST_PATH_IMAGE120
(29)
Last dividing value For
Figure DEST_PATH_IMAGE124
(30)
The intermediate value reliability function
Figure DEST_PATH_IMAGE126
For
Figure DEST_PATH_IMAGE128
(31).
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Application publication date: 20110601