A remaining life prediction method considering prior credibility

Abstract

The present application relates to the technical field of reliability engineering, and particularly relates to a residual life prediction method considering prior credibility, comprising the following steps: S1, constructing conjugate prior distribution: adopting conjugate prior distribution to model degradation model parameters, and determining hyperparameter values of the prior distribution; S2, posterior weighting representation: representing posterior parameters of the degradation model as a weighted combination form of field observation information and prior information; S3, credibility calibration weight: calibrating the weight of the prior information in the posterior distribution; S4, posterior distribution updating: updating the posterior distribution of the parameters in the degradation model based on the calibrated weight, reflecting the actual characteristics of the individual degradation behavior, and using the updated posterior distribution to perform residual life prediction; the present application introduces a prior credibility factor to perform weighted updating on the degradation model parameters, and improves the life prediction accuracy and modeling robustness under the condition that the prior information is not completely reliable.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of reliability engineering, and in particular to a residual life prediction method considering prior credibility. BACKGROUND

[0002] With the rapid development of science and technology and the increasingly complex structure of industrial products, the degradation behavior of products in service is becoming increasingly diversified, which puts forward higher requirements for reliability modeling and life prediction. In order to improve the prediction accuracy, traditional reliability modeling methods usually rely on a large amount of prior information, such as accelerated degradation data, historical degradation data of the same type of product, etc., and take these prior information as the basis for modeling.

[0003] However, in actual application, the prior information may be inaccurate, incomplete or inconsistent in data source, which is easy to cause distortion of the prior model. In addition, the classical Bayesian modeling method does not consider the credibility of the prior information when fusing the prior and observed information. Regardless of the quality of the prior data, the influence weight of the posterior distribution is fixed. In the scenario where the prior information is unreliable, it is easy to cause prediction deviation and reduce the ability of the model to characterize the real degradation process. SUMMARY

[0004] The present application provides a residual life prediction method considering prior credibility, which weights and calibrates the prior information by introducing a credibility factor, and dynamically adjusts the posterior parameter distribution combined with the field observation data, so as to more accurately reflect the actual degradation behavior of the product and improve the accuracy and engineering applicability of reliability analysis and life prediction.

[0005] A residual life prediction method considering prior credibility, comprising the following steps:

[0006] S1, constructing a conjugate prior distribution: for the degradation difference between product individuals, a conjugate prior distribution is used to model the degradation model parameters, and the hyperparameter value of the prior distribution is determined;

[0007] S2, posterior weighted representation: the posterior parameters of the degradation model are represented as a weighted combination of field observation information and prior information;

[0008] S3, credibility calibration weight: introducing a prior credibility coefficient, the weight of the prior information in the posterior distribution is calibrated;

[0009] S4, posterior distribution updating: based on the calibrated weight, the posterior distribution of the parameters in the degradation model is updated to reflect the actual characteristics of the individual degradation behavior, and the updated posterior distribution is used for residual life prediction.

[0010] Optionally, the constructing a conjugate prior distribution in S1 comprises:

[0011] S11, determine the prior distribution form: the same batch of products due to manufacturing, storage factors produce differences, through the gamma distribution to describe the difference between individual degradation parameters, set the prior distribution as:

[0012] ;

[0013] wherein, is a shape parameter, is a drift parameter, is a scale parameter in the gamma process, is a gamma function;

[0014] S12, define the prior probability density function: defined by the gamma distribution, the prior probability density function is expressed as:

[0015] ;

[0016] wherein, is a prior distribution function;

[0017] S13, establish the full likelihood function: the full likelihood function is constructed to estimate the hyperparameters of the prior distribution, which is expressed as:

[0018] ;

[0019] wherein, L is the set target likelihood function, is the prior hyperparameter value to be estimated, is the hyperparameter value in each iteration process, is the measurement interval between the th sample and the previous sample, is the increment of the th product performance measurement value under the th stress level, N is the set N th accelerated stress level, n is the set th sample under each stress level, is a gamma function;

[0020] S14, take the logarithm of the likelihood function: take the logarithm of the likelihood function to obtain the log-likelihood function;

[0021] S15, derive the log-likelihood function: derive ;

[0022] S16, give the maximum likelihood estimate value of the hyperparameter: the maximum likelihood estimate value of the hyperparameter is calculated by derivation;

[0023] S17, establish an iterative update condition: A set of initial values is set and iterative updating is performed, and the iteration is stopped when the difference between the new iteration value and the last iteration value reaches a predetermined precision, and the last iteration value is set as the hyperparameter value of the prior distribution. A set of initial values is set and iterative updating is performed, and the iteration is stopped when the difference between the new iteration value and the last iteration value reaches a predetermined precision, and the last iteration value is set as the hyperparameter value of the prior distribution.

[0024] Optionally, the iterative updating condition is expressed as:

[0025] .

[0026] Optionally, the posterior weighting in S2 includes:

[0027] S21, defining the field observation data and the measurement interval: setting the field measured data and the measurement interval time, wherein the field measured data is , the measurement interval time is , the measurement interval and the increment are defined, and are expressed as:

[0028] ;

[0029] ;

[0030] wherein, and are field data;

[0031] S22, Bayesian derivation and parameter updating of the posterior distribution: the field observation data is combined with the gamma distribution by the Bayesian derivation method to derive the posterior distribution, which is expressed as:

[0032]

[0033] wherein, is the posterior distribution function, and are hyperparameters in the posterior distribution, is prior information;

[0034] S23, updating the posterior distribution parameters: since the prior and the posterior have conjugate property, the posterior distribution parameter updating is expressed as:

[0035] ;

[0036] .

[0037] Optionally, the credibility calibration weight in S3 includes:

[0038] S31, defining the weighted composition of the posterior parameters: in the posterior distribution parameter updating, the parameters and are obtained by weighted superposition of prior information and field measured data, wherein, in the calculation of is field measured data, is prior hyperparameter estimation, in the calculation of is field measured data, is prior hyperparameter estimation;

[0039] S32, introduce credibility factor and reconstruct weighting: introduce credibility factor B, suppress unreliable prior information by adjusting prior weight, obtain calibration update of posterior parameter, expressed as:

[0040] ;

[0041] .

[0042] Optionally, the posterior distribution update in the S4 comprises:

[0043] S41, calculate posterior expectation as distribution parameter: let degradation parameter be , combine credibility calibrated posterior parameter and , calculate posterior expectation, expressed as:

[0044] ;

[0045] wherein, is the expected value of the hypothetical distribution;

[0046] S42, construct probability density function of degradation process: substitute the expected value of into the probability density function , construct probability density function under product degradation state;

[0047] S43, update reliability function and life expression: based on the calibrated posterior parameter, derive and update the reliability function, life distribution function and expected life function of the product, to realize quantitative prediction of the remaining life.

[0048] Optionally, the probability density function under the product degradation state is expressed as:

[0049] ;

[0050] wherein, is the probability density function after introducing prior credibility.

[0051] Optionally, the update of reliability function and life expression in the S43 comprises:

[0052] reliability function update: ;

[0053] Wherein, D is the failure threshold of the product, is the reliability function after introducing the prior credibility;

[0054] Lifetime distribution function update: ;

[0055] Wherein, is the cumulative failure distribution function after introducing the prior credibility;

[0056] Expected lifetime function update: ;

[0057] Wherein, is the product lifetime expression of BS distribution after introducing the prior credibility.

[0058] The beneficial effects of the present application are:

[0059] The present application, by constructing conjugate prior distribution and introducing credibility factor to calibrate prior information, can dynamically adjust the influence degree of prior information source quality, overcome the defect that prior reliability cannot be distinguished in traditional Bayesian modeling, improve the robustness and flexibility of degradation model parameter estimation, and is suitable for product heterogeneity, prior incomplete reliability reliability analysis scene.

[0060] The present application, by fusing the field observation data and the prior information weighted by credibility, updating the posterior distribution parameters of the degradation model, further deriving the reliability function, lifetime distribution function and lifetime expectation expression, can realize the quantitative prediction of the remaining life of the product, effectively improve the prediction accuracy, and provide important decision basis for product maintenance strategy formulation and life cycle management. BRIEF DESCRIPTION OF DRAWINGS

[0061] In order to more clearly illustrate the technical solutions in the present application or prior art, the following will briefly introduce the drawings needed to be used in the embodiments or prior art description. Obviously, the drawings in the following description are only a part of the present application, and other drawings can also be obtained by those skilled in the art without creative labor.

[0062] Figure 1 The modeling method flowchart of the embodiment of the present application;

[0063] Figure 2 The carbon film resistance degradation data schematic diagram under 83 DEG C accelerated stress of the embodiment of the present application;

[0064] Figure 3 The carbon film resistance degradation data schematic diagram under 133 DEG C accelerated stress of the embodiment of the present application;

[0065] Figure 4 The carbon film resistance degradation data under 173℃ accelerated stress for the embodiment of the present application is shown in the diagram;

[0066] Figure 5 The residual life diagram of the embodiment of the present application without considering prior credibility is shown in the diagram;

[0067] Figure 6 The residual life diagram of the embodiment of the present application considering prior credibility is shown in the diagram. DETAILED DESCRIPTION

[0068] The present application will be described in detail below with reference to the accompanying drawings and specific embodiments. For some known technologies, other alternative ways can also be implemented by those skilled in the art; and the accompanying drawings are only used to more specifically describe the embodiments, and are not intended to specifically limit the present application.

[0069] As shown in the diagram, a residual life prediction method considering prior credibility includes the following steps: Figures 1-6

[0070] 1. Construct the conjugate prior distribution of the product. Due to the influence of manufacturing process, storage conditions and other factors, there will be slight differences in the degradation of different individuals in the same batch of products, which often introduces random parameters to describe the differences between individuals. The conjugate prior distribution has good statistical properties in Bayesian statistical inference, so it is assumed that the scale parameter (α) in the Gamma process obeys the Gamma distribution, i.e.

[0071]

[0072]

[0073]

[0074]

[0075]

[0076] In order to find the hyperparameter value of the prior distribution, the full likelihood function is established:

[0077]

[0078] N ​​​​​​​​​​​​​​​​the set of the N accelerated stress levels, n the set of the samples for each stress level, L the set of the target likelihood function, the set of the prior hyperparameter values to be estimated, the set of the measurement interval between the the set of the product performance measurement increment for the the set of the gamma function.

[0079]

[0080] ;

[0081] where, the set of the gamma function.

[0082]

[0083] ;

[0084] where, the set of the reciprocal of the gamma function multiplied by the derivative of the gamma function.

[0085] the set of the

[0086] maximum likelihood estimate of the hyperparameter is:

[0087] ;

[0088] ;

[0089] a set of initial values is set, and the iteration is updated according to the above, and it is set that the iteration is stopped when the difference between the new iteration value and the last value reaches a certain precision, and the last iteration value is set as the hyperparameter value of the prior distribution. 2. Solution of posterior distribution parameter updating method. Assuming is the measured data on site,

[0090] is the measurement interval time, , , the posterior probability density can be obtained by Bayes formula:

[0091] ​​

[0092] wherein, is a posterior distribution function, and is a hyperparameter in the posterior distribution, and is field data, α is a shape parameter.

[0093] The prior probability density is:

[0094] ;

[0095] wherein, is a prior distribution function, and is a hyperparameter in the prior distribution.

[0096] Since the conjugate prior distribution and the posterior distribution have no formal change in distribution but only change in parameters, the updating method of the posterior distribution parameters can be obtained as:

[0097] ;

[0098] ;

[0099] wherein, and are hyperparameters in the posterior distribution.

[0100] 3. The weighted sum form of the posterior information and the prior information, through the updating method of the posterior parameters, it can be seen that in the updating formula of the parameter , there are two parts of weighted addition to obtain, wherein is the prior hyperparameter estimation value, is the field information; similarly, in the updating formula of the parameter , it is also obtained by two parts of weighted addition, wherein is the prior hyperparameter estimation value, is the field information. The advantage of arranging the posterior distribution into the weighted sum of the field information and the prior information is that the posterior information can be changed accordingly with the change of the credibility of the prior information.

[0101] In the classical Bayesian, the credibility of the prior data is not considered, and the contribution of the prior data to the posterior model is consistent regardless of the credibility of the prior data. The invention improves this deficiency. Assuming that the prior credibility is B, the posterior information update of the prior data credibility is:

[0102] ;

[0103] ;

[0104] 4. Update of the distribution. The previous assumption is that the expectation of is:

[0105]

[0106] where is the expectation of the distribution computed at each iteration of the EM algorithm.

[0107] Substituting the expectation of into the probability density function we obtain:

[0108]

[0109] where is the probability density function with the introduction of the prior credibility.

[0110] The reliability function is updated as:

[0111]

[0112] where D is the failure threshold of the product, is the reliability function with the introduction of the prior credibility.

[0113] The distribution function of the product lifetime is updated as:

[0114]

[0115] where is the cumulative distribution function of the failure with the introduction of the prior credibility.

[0116]

[0117] where is the average lifetime of the product.

[0118] The expression of the product lifetime is updated as:

[0119]

[0120] where is the expression of the product lifetime of the BS distribution with the introduction of the prior credibility, is a function of time.

[0121] Case analysis:

[0122] ​​​​​The application will be analyzed by taking a case. The case is only used to help understand the application and should not limit the application in any applicable scope. It should be pointed out that any modification and improvement of the application without departing from the concept of the application is within the protection scope of the application.

[0123] The parameters in the gamma process are set as random parameters to describe the differences between different individuals, and when the model is a conjugate prior distribution, only the parameter value changes between the posterior and the prior.

[0124] Based on the above, the degradation model of is established, and .

[0125] wherein, is a shape parameter, is a scale parameter, is a gamma function.

[0126] The EM algorithm is used to estimate the hyperparameter value in the prior distribution, and the full likelihood function is established:

[0127] ;

[0128] wherein, N is the set of the N th accelerated stress level, n is the set of the th sample under each stress level, L is the set of the target likelihood function, is the prior hyperparameter value to be estimated, is the hyperparameter value in each iteration process, is the measurement interval between the th sample and the previous sample, is the increment of the th product performance measurement value under the th stress level.

[0129] ;

[0130] wherein, is a gamma function.

[0131] The estimator is obtained:

[0132] ;

[0133] ;

[0134] ;

[0135] wherein, The reciprocal of the gamma function times the derivative of the gamma function.

[0136] The updating of the posterior parameters is obtained by Bayesian inference:

[0137] ;

[0138] ;

[0139] where, and are the hyperparameters in the posterior distribution.

[0140] Assume that the initial value of is , and the value after k iterations is denoted as

[0141] The E-step algorithm (solving the expectation of and ):

[0142] ;

[0143] ;

[0144] where, is the expected value of the hypothetical distribution, is the expected value of the logarithm of the hypothetical distribution.

[0145] In the M-step, replace in the three estimators with Assume that after k+1 iterations, the hyperparameter values reach the convergence accuracy, and at this time, , , are the estimated values of the hyperparameters as the prior distribution:

[0146] ;

[0147] ;

[0148] ;

[0149] The case will use the test data of a certain type of carbon film resistor in CSADT provided by Meeker et al., Figure 2 , Figure 3 , Figure 4 The test data under the accelerated stresses of 83℃, 133℃, and 173℃ are shown.

[0150] The EM algorithm is used to analyze the accelerated data at 83℃, and The values of 2.605, 0.118, 0.534 and 0.466, respectively, i.e. .

[0151] The degradation data of a single resistor obtained by simulation is shown in Table 1:

[0152] Table 1 Simulation degradation data of a single resistor

[0153]

[0154] As Figure 5 shown is a residual life schematic diagram without considering the prior credibility, in the case of not considering the prior credibility, assuming that the failure threshold of the product is 5, the posterior estimation of the parameters is performed, and the updating process of the parameters is shown in Table 2:

[0155] Table 2 Posterior parameter estimation process without considering prior credibility

[0156]

[0157] As Figure 6 shown is a residual life schematic diagram considering the prior credibility, in the case of considering the prior credibility, assuming that the failure threshold of the product is 5 and the prior data credibility is 0.87, the posterior estimation of the parameters is performed, and the updating process of the parameters is shown in Table 3:

[0158] Table 3 Posterior parameter estimation process considering prior credibility

[0159]

[0160] Through simulation verification, the time used for the resistance value increment to reach 5% is 118225 hours, and the prediction life results before and after considering the prior credibility are listed in Table 4.

[0161] Table 4 Prediction life before and after considering prior credibility

[0162]

[0163] The accuracy of the prediction life without considering the prior credibility is , and the accuracy of the prediction life considering the prior credibility is .

[0164] Table 5 Accuracy and error before and after considering prior credibility

[0165]

[0166] According to the experimental results, in the accelerated degradation life prediction of carbon film resistance, the introduction of prior credibility significantly improves the prediction accuracy. Without considering the prior credibility, the parameter estimation of the prediction life is seriously overestimated (such as the prediction value of 25000 hours observation point is 2.013×10 6 hours), which leads to the accuracy of only 5.91% and the error of 94.09%; and after adding the prior information with the credibility of 0.87, the prediction value (such as 0.946×10 5 hours) is closer to the actual failure time 118225 hours, the accuracy is improved to 80.03%, and the error is reduced to 19.97%. The experiment proves that the reasonable use of prior credibility can effectively correct the deviation of posterior parameter estimation, greatly improve the reliability of life prediction, and provide an important reference for the modeling optimization of accelerated degradation test in engineering.

[0167] The present application encompasses any substitutions, modifications, equivalent methods and schemes made on the essence and scope of the present application. In order to make the public have a thorough understanding of the present application, specific details are described in the following preferred embodiments of the present application, and the present application can also be fully understood without the description of these details for those skilled in the art. In addition, in order to avoid unnecessary confusion to the essence of the present application, well-known methods, processes, procedures, elements and circuits are not described in detail.

[0168] The above is only the preferred embodiment of the present application, and it should be pointed out that for ordinary skilled in the art, without departing from the principle of the present application, a number of improvements and refinements can also be made, which should be considered as the protection scope of the present application.

Claims

1. A method of predicting the remaining life taking into account the a priori credibility, characterized in that, Includes the following steps: S1, Constructing a conjugate prior distribution: To address the degradation differences between individual products, a conjugate prior distribution is used to model the degradation model parameters, and the hyperparameter values ​​of the prior distribution are determined. S2, Posterior weighted representation: The posterior parameters of the degradation model are represented as a weighted combination of field observation information and prior information; S3, Credibility Calibration Weight: Introducing a prior credibility coefficient to calibrate the weight of prior information in the posterior distribution; S4, Posterior Distribution Update: Based on the calibrated weights, update the posterior distribution of parameters in the degradation model to reflect the actual characteristics of individual degradation behavior, and use the updated posterior distribution to predict remaining lifespan. The posterior weighted representation in S2 includes: S21, Define field observation data and measurement interval: Set the field-measured data and measurement interval time, where the field-measured data is... The measurement interval is Define the measurement interval and increment as follows: ； ； in, and For on-site data; S22, Bayesian derivation and parameter update of the posterior distribution: By combining field observation data with the gamma distribution using the Bayesian derivation method, the posterior distribution is derived, expressed as: in, Let be the posterior distribution function. and Let be the hyperparameter in the posterior distribution. As prior information, To estimate the prior hyperparameter values, β For the scale parameters in the gamma process; S23, Update the posterior distribution parameters: Since the prior and posterior are conjugate, the updated posterior distribution parameters are expressed as follows: ； ； The confidence calibration weights in S3 include: S31, Clarify the weighted composition of the posterior parameters: In the update of the posterior distribution parameters, the parameters... and All are obtained by weighted superposition of prior information and field-measured data, among which, In the calculation, For data measured on site, It is a priori hyperparameter estimation. In the calculation, For data measured on site, It is a prior hyperparameter estimation; S32, Introducing a credibility factor and reconstructing the weighting: Introducing a credibility factor B, and by adjusting the prior weights, suppressing unreliable prior information, resulting in a calibrated update of the posterior parameters, expressed as: ； 。 2. The remaining lifetime prediction method considering prior confidence level according to claim 1, characterized in that, The construction conjugate prior distribution in S1 includes: S11, Determine the prior distribution form: Due to manufacturing and storage factors, products in the same batch vary. The differences in degradation parameters between individuals are described by the gamma distribution. The prior distribution is set as follows: ； in, For shape parameters, For drift parameters, β For the scale parameters in the gamma process. It is a gamma function; S12, Define the prior probability density function: Based on the gamma distribution, the prior probability density function is expressed as: ； in, This is the prior distribution function; S13, Construct the full likelihood function: The full likelihood function is used to estimate the hyperparameters of the prior distribution, expressed as: ； in, L Let the target likelihood function be defined. To estimate the prior hyperparameter values, These are the hyperparameter values ​​during each iteration. For the first The measurement interval between each sample and the previous sample For the first Under the stress level, the first The increment of a product performance measurement value N For the set number N One level of accelerating stress, n The first set at each stress level One sample, It is a gamma function; S14, take the logarithm of the likelihood function: take the logarithm of the likelihood function to obtain the log-likelihood function; S15, Differentiation of the log-likelihood function: Perform differentiation; S16, gives the maximum likelihood estimate of the hyperparameters: the maximum likelihood estimate of the hyperparameters is calculated by taking the derivative; S17, Establish iterative update conditions: for Set a set of initial values ​​and iteratively update them, setting the current... The iteration stops when the difference between the new iteration value and the previous iteration value reaches the predetermined precision, and the final iteration value is set to the hyperparameter value of the prior distribution.

3. The remaining lifetime prediction method considering prior confidence level according to claim 2, characterized in that, The iterative update condition is expressed as follows: 。 4. The remaining lifetime prediction method considering prior confidence level according to claim 3, characterized in that, The posterior distribution update in S4 includes: S41, Calculate the posterior expectation as the distribution parameter: Let the degenerate parameter be... Combined with posterior parameters after confidence calibration and The posterior expectation is calculated and expressed as: ； in, It is the expected value of the assumed distribution; S42, Construct the probability density function of the degradation process: Substituting the expected value into the probability density function Construct the probability density function of the product under degradation state; S43, Update the reliability function and lifetime expression: Based on the calibrated posterior parameters, derive and update the product's reliability function, lifetime distribution function, and expected lifetime function to achieve quantitative prediction of remaining lifetime.

5. The remaining lifetime prediction method considering prior confidence level according to claim 4, characterized in that, The probability density function of the product in its degraded state is expressed as: ； in, It is the probability density function after introducing prior confidence.

6. The remaining lifetime prediction method considering prior confidence level according to claim 5, characterized in that, The updated reliability function and lifetime expression in S43 include: Reliability function update: ; Where D is the product's failure threshold. It is the reliability function after introducing prior confidence; Lifetime distribution function update: ; in, It is the cumulative fault distribution function after introducing prior confidence; Expected lifetime function update: ; in, It is the product lifetime expression of the BS distribution after introducing prior confidence.

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