A product life prediction method based on zero-failure data in sequential testing

Abstract

The present invention discloses a product life prediction method based on zero-failure data in a sequential test in the field of open reliability test evaluation. According to the truncation time from small to large, the product is tested in sequence, and the Bayesian estimation model is used to obtain the Bayesian estimation value of the cumulative failure probability of each sample #imgabs0#. The set #imgabs1# is used as the input of the grey bootstrap model, t i 0 to the longest test duration t k At any time between , a self-service sample set is sampled to generate new grey self-service sample points, forming a set of cumulative failure rate estimation results at multiple times, constructing a square loss function, and obtaining the parameter estimation value that minimizes the square loss function. Using the maximum entropy method, the parameter estimation value with the largest information entropy corresponding to the grey self-service sample point among all parameter estimation values ​​is obtained as the final result, and the prediction result is accurate.

Description

Technical Field

[0001] The present invention belongs to the technical field of reliability test evaluation, and in particular relates to a product life prediction method based on zero-failure data in a sequential test. Background Art

[0002] With the rapid development of science and technology today, the design, production and processing methods of products have also evolved. High-reliability and long-life products have become the key to civil and national defense construction. In order to better evaluate the reliability and predict the life of such high-reliability and long-life products, product reliability testing is very critical. Traditional reliability tests include timed truncation reliability tests. In the test, if the truncated test samples have not failed, they are directly faced with the situation of being abandoned or repaired, refurbished and reused. However, the maintenance cost of the product is very high. For such a type of product that has undergone a long-term reliability test, if it is used, it will not meet the factory standards, and if it is repaired, it will face high costs. Therefore, in order to improve the utilization rate of this type of test samples that have not failed after the truncation time, and at the same time obtain more and richer test data, it is necessary to conduct sequential tests. For sequential tests, how to improve the accuracy of life prediction based on test data is very critical.

[0003] Sequential testing aims to extend the test time of each test sample as much as possible under small sample conditions, incorporate the non-failed samples in the previous group of reliability tests into the next group of tests to continue the test, and obtain zero-failure data after the test to verify the reliability and determine the life of the product. If the zero-failure data under sequential testing is directly expanded according to the small sample sampling rules, since the total number of test samples at each truncation time is different, it is possible that one group of test failure samples is very small, while another group of test samples under the same test conditions has many failure samples. Especially in the case of small samples, the randomness of test sample failure is too strong, and the accuracy of life prediction modeling analysis is not high enough. At the same time, when dealing with zero-failure samples, since the number of failures in each group of test samples in the zero-failure data is 0, when using the general small sample sampling method for modeling, it will cause the prediction matrix to be a singular matrix and thus the modeling failure problem. Summary of the invention

[0004] Technical purpose: In order to solve the problem that it is difficult to establish a relatively accurate life model distribution through zero-failure data in sequential tests under small sample conditions, and the accuracy of life prediction modeling analysis is not high enough, the present invention proposes a product life prediction method based on zero-failure data in sequential tests to improve the prediction accuracy.

[0005] Technical solution: The technical solution adopted by the method for predicting product life based on zero-failure data under sequential testing described in the present invention includes:

[0006] Step 1): Carry out truncation tests on the products in order from the smallest to the largest truncation time, and calculate the longest test time t k and the sum of the time t of all censored trials experienced by each test sample p , k=1,2,…p, p is the number of test samples, and the failure data set {,Y} and the non-failure data set {t p , N}, Y is the sample with failure, N is the sample without failure p Failure samples;

[0007] Step 2): Use the Bayesian estimation model based on the failure data set {t p ,Y} and the zero-failure data set {t p ,N}, calculate the cumulative failure probability p of each sample i And its posterior distribution, the integral calculation obtains the Bayesian estimate of the cumulative failure probability of the i-th sample i=1,2,…p;

[0008] Step 3): Use a small sample sampling method to collect As the input of the grey bootstrap model, t i 0 to the longest test duration t k At any time between , a self-service sample set is obtained by sampling, a new gray self-service sample point is generated, and the expanded sample is obtained to form a cumulative failure rate estimation result set at multiple times

[0009] Step 4): Estimate the result set based on the cumulative failure rate Construct a square loss function Q(β,η), and obtain the life distribution parameter that minimizes the square loss function Q(β,η) as the parameter estimate;

[0010] Step 5): Repeat steps 3)-4) and use the maximum entropy method to obtain the parameter estimate with the largest information entropy corresponding to the gray self-help sample point among all parameter estimates as the final result to complete the life prediction.

[0011] Technical effect: The present invention constructs a life prediction model based on zero-failure data under sequential tests, combines the Bayesian estimation method, the gray prediction model and the maximum entropy method, and proposes a gray self-help and maximum entropy method based on the Bayesian estimation method. The zero-failure data under sequential tests are converted into cumulative failure probability values ​​through the Bayesian estimation model, and then the gray self-help and maximum entropy methods are used to select the sample parameter estimation value with the largest entropy, and the key parameters of the product life distribution are obtained by combining the parameter estimation method. Since the test sample size of each group of timed truncation tests in the sequential truncation experiment under small sample conditions may be different, this causes the failure caused by accidental situations to be unable to be "diluted" by the large number of samples in the same group. Therefore, when performing life prediction based on zero-failure data under sequential tests, it is first necessary to pre-process the non-time-sensitive data. Therefore, in the model, the Bayesian method is first selected to realize the combination of test data and prior information under small sample conditions, and the product-related historical data, expert experience and test data are considered for evaluation, and the influence of sample failure caused by accidental reasons on the evaluation results is reduced, so that the evaluation results are more in line with reality. The selection of the grey prediction model can further explore the data and expand the data, rather than simply using the bootstrap method to repeatedly list the data. In the grey prediction model, if the original matrix elements input are all 0, then the model construction will fail. In order to avoid the situation where all the non-failure data are 0 during modeling, the present invention expands the test sample by using the Bayesian estimation before the grey bootstrap model. The maximum entropy method can screen the results of multiple model evaluations. Since the parameter estimation samples generated by the grey bootstrap method are different each time, the maximum entropy method is used to screen each evaluation result, and the parameter estimation result that best fits the actual situation can be selected. Therefore, the prediction result is accurate. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 A flowchart of a method for predicting product life based on zero-failure data in a sequential test provided by the present invention;

[0013] Figure 2 A schematic diagram of the grey self-service sampling points and the cumulative failure function according to an embodiment of the present invention;

[0014] Figure 3 Schematic diagram of gray self-service sampling points and life distribution according to an embodiment of the present invention. DETAILED DESCRIPTION

[0015] The present invention firstly performs truncation tests on the products in order from the smallest to the largest truncation time, and calculates the longest test time t k and the sum of the time t of all censored trials experienced by each test sample p , k=1,2,…p, p is the number of test samples, and the failure data set {t p,Y} and the zero-failure data set {t p , N}, Y is the sample with failure, N is the sample without failure; then use the Bayesian estimation model, based on the failure data set {t p ,Y} and the zero-failure data set {t p ,N}, calculate the cumulative failure probability p of each sample i And its posterior distribution, the integral calculation obtains the Bayesian estimate of the cumulative failure probability of the i-th sample i=1,2,…p; then use the small sample sampling method to set the set As the input of the grey bootstrap model, t i 0 to the longest test duration t k At any time between , a self-service sample set is obtained by sampling, a new gray self-service sample point is generated, and the expanded sample is obtained to form a cumulative failure rate estimation result set at multiple times Estimation result set based on cumulative failure rate Construct the square loss function Q(β,η), and obtain the life distribution parameter that minimizes the square loss function Q(β,η) as the parameter estimate; finally, repeat the small sample sampling method and construct the square loss function Q(β,η) to obtain all parameter estimates, and use the maximum entropy method to obtain the parameter estimate with the largest information entropy of the gray bootstrap sample point among all parameter estimates as the final result to complete the life prediction. Figure 1 As shown, the specific implementation steps are as follows:

[0016] S1. Conduct sequential life test for products with life distribution. The cut-off time of the designed sequential life is t test1 ,t test2 ……t testn , t test1 ,t test2 ……t testn Divide the total test samples into n groups, and conduct tests in order from small to large censoring times. First, select censoring time t test1 The first set of trials was grouped and censored until completed.

[0017] S2. After the current truncation test is completed in the first test group, the serial number of the failed samples is recorded, and the samples that have not failed in this group are merged into the next test group, and then the next test with a longer truncation time is started. That is, when the truncation time is t test1 After the test, the samples that have not failed in this group are merged into the group with a truncation time of t test2 The experiment is continued in the second group of experiments. And so on, until the censoring time is t testnAfter the nth group of tests in the sequential test are grouped, calculate the total time of all censored tests experienced by each test sample. If the sample with serial number p was originally a test sample in the m1th group of censored tests and finally fails in the m2th group of tests (1 ≤ m1 < m2 ≤ n), the total time of the censored tests it has experienced If the test sample has not failed after experiencing the censored test in the final nth group of test groupings, the total time of the censored test of the test sample

[0018] Collect the non-failure data under sequential tests: When each group of tests is completed, record which samples have failed in this group of tests. If a failure occurs, record it as Y; if no failure occurs, record it as N, and combine it with the total censored test time t experienced by the sample p p and count them together as the failure data set {t p , Y} and the non-failure data set {t p , N}.

[0019] S3. Determine the distribution of the failure probability of the product working until the final moment. This distribution can be obtained from experience and is generally a uniform distribution.

[0020] According to the censoring times t test1 , t test2 ……t testn , t test1 , t test2 ……t testn of each group of test groupings, calculate the total time of all censored tests experienced by the product, that is, the longest test duration is t k = t test1 + t test2 +…t testn , k = 1, 2, … p. Using the uniform distribution as the prior distribution, it is considered that the cumulative failure probability p k of the product at the t k moment follows a uniform distribution on [0, λ k . The prior distribution of the cumulative failure probability p k of the product after the longest test duration t k can be obtained as follows:

[0021]

[0022] In the formula, t k represents the longest test duration of the product: p k is the cumulative failure probability of the product after the test t k ; λ k is the upper bound of the range within which the cumulative failure probability p k may vary.

[0023] S4. Based on the concavity of the product cumulative failure function, a conservative estimate is made for the cumulative failure probability of each sample, which is the prior distribution of the Bayesian estimation model.

[0024] When the cumulative failure probability p is obtained k After the prior distribution of i The cumulative failure probability p i The prior distribution of , according to the logarithm of the cumulative failure function of the Weibull distribution, is:

[0025]

[0026] Among them, t i 0 to t k At any time between i Then is the cumulative failure probability value of the product at that moment, where: i ≤t k , F(t i )=p i , F(t k )=p k , from which we can infer:

[0027]

[0028] In practical engineering, the cumulative failure probability p can be considered i With p k There are the following relationships:

[0029]

[0030] Since the cumulative failure probability p i With p k The relationship between is relatively conservative, and the conclusion obtained from it will also tend to be conservative and more easily accepted by engineers. Using formula (4), the cumulative failure probability p is obtained i The prior distribution of is:

[0031]

[0032] where λ i t i The upper bound of the failure probability of the product at time i=1,2,…p.

[0033] Cumulative failure probability p i The prior distribution of is also the prior distribution of the Bayesian estimation model.

[0034] S5. According to the estimation result of step S4, based on the failure data set {t p ,Y} and the zero-failure data set {tp ,N}, using the Bayesian estimation model, calculate the posterior distribution of the cumulative failure probability of each sample;

[0035] In the failure data set {t p ,Y} and the zero-failure data set {t p , N}, when the sum of the time of all censored trials is t i There are n samples of i When s i =n i -r i If no product fails, the total test time of the product is t i When i The conditional probability that a product has not failed is:

[0036]

[0037] In the formula, s i Represents n i The total number of samples in the test time t i The failure-free data set {t p ,N}, the number of samples that have not failed, r i The failure data set {t p ,Y}, and C represents the number of combinations.

[0038] The number of samples that have not failed can be obtained i The conditional probability distribution of is:

[0039]

[0040] From the cumulative failure probability p i The prior distribution (5) and the Bayesian formula show that the cumulative failure probability p i The posterior distribution of is:

[0041]

[0042] S6. According to the cumulative failure probability p obtained in step S5 i The posterior distribution of i (p i |s i ), and integrate to calculate the Bayesian estimate of the cumulative failure probability for each sample.

[0043] Under squared loss, the cumulative failure probability p can be obtained i The Bayesian estimate of for:

[0044]

[0045] By using the Bayesian estimation model to estimate, the posterior estimate of the cumulative failure probability under each simulation total test time can be calculated, and the probability value obtained at this time is equivalent to the Bayesian estimation result.

[0046] S7, use the small sample sampling method, select the gray bootstrap method, and input the fusion sum test time t i and the posterior estimate of the cumulative failure probability Will gather As the input of the grey bootstrap model, the zero-failure data is expanded to obtain multiple groups of samples.

[0047] First, we can get m bootstrap samples by bootstrapping:

[0048] Y b =(y b (1),y b (2),…,y b (u),…,y b (m)),b=1,2,…,B (10)

[0049] Where Y b is the total test time t for the product i and the cumulative failure probability p i The posterior estimate of The b-th self-help sample set of ; B is the number of gray self-help samples extracted, which is also the upper limit of b which represents the sequence number of the self-help sample set in the formula; each self-help sample y b (u) is a set A repeated sampling of (u=1,2…k); u indicates that the element is a self-service sample set Y b The u-th repeated sampling in , m is the capacity of each bootstrap sample set.

[0050] According to the gray prediction GM(1,1) principle, these data are accumulated once to generate a new sequence vector:

[0051]

[0052] In the formula, x b (k) is the cumulative value of the 1st to uth elements in the bth self-service sample set.

[0053] Then, the accumulated sequence is subjected to neighboring mean generation, that is, the average value of adjacent data is used to construct new data. The calculation method is as follows:

[0054]

[0055] In the formula, z b (u) is X in formula (11)b The mean of adjacent elements in the set, so the range of u is reduced by 1.

[0056] Then construct the data matrix D and data vector Y i :

[0057]

[0058] Y i =[y i (2) y i (3) … y i (m)]i=1,2,…B (14)

[0059] Where I is b A vector with the same capacity and all elements are 1. Here, i means that the i-th group of bootstrap samples is selected for calculation.

[0060] The expression of the gray generation model can be expressed as:

[0061]

[0062] Where k is a continuous variable, c1 and c2 are parameters. The initial condition is x i (1) = y i (1), the least squares solution of the differential equation is:

[0063]

[0064] in:

[0065] (c1,c2) T =(D T D) -1 D T (Y i ) T (17)

[0066] From the cumulative generation formula (11), we can get:

[0067] x i (u+1)-x i (u) = y i (u+1) (18)

[0068] Based on this, a new grey self-service sample point can be generated based on the self-service sample set:

[0069]

[0070] Among the B self-service sample sets, each sample set can predict one gray self-service sample point. By taking different values ​​of i in formula (13), B gray self-service samples, i.e., the expanded samples, can be obtained, forming the following vector, i.e., the cumulative failure rate estimation result set at multiple moments:

[0071]

[0072] In the formula, is a bootstrap sample set Y with m elements b The m+1th point generated based on (w=m+1,b=1,2…B); Since there are B self-service sample sets, the m+1th point generated each time is Almost all are different, that is, gray self-service sample points (w=m+1,b=1,2…B) are almost different, so the superscript (b) is used to indicate the b-th bootstrap sample set Y b The generated points.

[0073] S8, based on the cumulative failure rate estimation result set at multiple moments in step S7 The cumulative failure curve of the product is fitted using the least square method to calculate the key parameter values ​​of the product life distribution.

[0074] For multiple time-posterior probability sample points It is required to get as close as possible to the two-parameter Weibull cumulative failure probability curve of all sample points In Therefore, we need to construct the loss function first. The set of predicted values ​​of the cumulative failure probability points calculated by S7 is Construct the loss function and then use the least squares method to fit the shape parameter β and scale parameter η.

[0075] First construct the square loss function:

[0076]

[0077] Seeking satisfaction The parameter estimates of This is the key parameter of product life distribution.

[0078] S9. Use parameter estimation method to calculate the life distribution parameters of each group of samples: refer to Gibbs sampling method to obtain the key parameter values ​​of product life distribution that minimizes the loss function Q(β,η).

[0079] The function Q(β,η) is used as the analysis object of the Gibbs method for optimization. In the two-dimensional case, the algorithm steps for finding its minimum point are as follows:

[0080] (1) Randomly initialized initial state η = η0;

[0081] (2) Setting the minimum iteration error ε;

[0082] (3) Repeat the following process to collect samples:

[0083] a) Find the point β1 near the minimum value of the loss function Q(β|η0);

[0084] b) Find the point η1 near the minimum value of the loss function Q(η|β1);

[0085] c) Calculate the difference L = |Q(β0,η0)-Q(β1,η1)|;

[0086] d) Record the point (β1,η1) calculated at this time, and let η0=η1.

[0087] (4) Loop until L<ε and output (β1,η1) as the solution.

[0088] S10, repeat S7-S9 operations M times to generate multiple groups of gray self-service sample sets Calculate the information entropy of each group of samples and select the appropriate parameter estimation results. Specifically, use the maximum entropy method to compare and analyze and select the gray bootstrap samples with the largest information entropy, and use their parameter estimation results as the key parameter values ​​of the life prediction model to complete the life prediction.

[0089] Since the generated grey bootstrap sample points vary widely, steps S7-S9 are repeated to generate multiple grey bootstrap sample sets. The optimal grey bootstrap sample set is selected according to the maximum entropy principle, and its parameter estimation result is used as the final life prediction result. First, M groups of grey bootstrap sample points are calculated according to step S7. v=1,2…M, and then use S8-S9 to calculate the parameter estimates of these M groups of gray bootstrap samples respectively (v=1,2…M), and finally, according to the maximum information entropy principle, the parameter estimation value with the maximum information entropy corresponding to the grey bootstrap sample point among all parameter estimation values ​​is obtained as the final result.

[0090] Among all the probability density functions that meet the given conditions, the probability density function with the least subjective bias (i.e., the smallest deviation) should be the one with the largest information entropy, that is,

[0091]

[0092] Where H is the information entropy; Ω∈[Ω min ,Ω max ] is the range of the random variable θ; ξ(θ) is the probability density function of the parameter θ.

[0093] The gray bootstrap sample set generated for the vth time is required The entropy of the gray bootstrap sample set is first calculated to obtain the parameter estimation result Substitute into the original cumulative failure function In, order Constructing functions According to equation (21), let ξ(θ) = p v (t), then the information entropy of the grey bootstrap sample set generated for the vth time is:

[0094]

[0095] In the formula, is the gray bootstrap sample set generated for the current vth time time component.

[0096] Formula (22) can generate M entropy values ​​H = {H1, H2…H M}, select the largest H max The corresponding gray self-help sample And the corresponding cumulative failure function parameter estimation results As output, lifespan prediction is completed.

[0097] An embodiment is provided below, which takes a deep groove ball bearing as an example. The deep groove ball bearing is a component belonging to the product category described in the technical field.

[0098] Example

[0099] S1: After equally dividing the bearing samples according to the truncation time of the designed test, the tests are carried out in order from the smallest truncation time to the largest one. The first group of tests with the smallest truncation time is selected for testing until completion.

[0100] First, the actual life of 30 groups of deep groove ball bearings is obtained by simulation, and then they are divided into 6 groups according to the truncation time. The truncation time and grouping are shown in Table 1.

[0101] Table 1. Truncation test time grouping and simulation life data

[0102]

[0103]

[0104] S2: After each experimental group completes the current experiment, merge the non-failed samples in the group into the next experimental group, and then start the next group experiment with a longer truncation time. Calculate the experimental time experienced by each sample, record whether it fails, and combine and count it as the non-failure data of the experiment.

[0105] The total truncated test time and failure record of each bearing using the sequential test method are shown in Table 2.

[0106] Table 2 Sequential test simulation results

[0107]

[0108] The test data of the bearings are grouped into test samples with the same total actual test time. The failure-free data collected are shown in Table 3.

[0109] Table 3 No failure data

[0110]

[0111] S3: According to the truncation time determined in S1, determine the distribution of the failure probability of the bearing at the final moment of operation.

[0112] This distribution can be derived empirically and is generally uniform.

[0113] The truncation time of each group of truncation tests is set to 150h, 300h, 450h, 750h, 1500h, and 2000h, so that the maximum total time of all the scheduled truncation tests experienced by the bearing is t k =150+300+450+750+1500+2000=5150h. Taking uniform distribution as the prior distribution, it is assumed that the cumulative failure probability of the bearing at the 5150th hour is p k In [0,λ k ] obeys uniform distribution 0, here we take λ k =0.35, we can get p k Prior distribution:

[0114]

[0115] Where, t k Represents the maximum test duration of the bearing, defined in S2: p k The bearing is tested in t k The cumulative failure probability after k For the moment p k The upper bound of the range of possible changes, .

[0116] S4: Based on the concavity of the bearing cumulative failure function, a conservative estimate of the cumulative failure probability of each test bearing is made, which is the prior distribution of the Bayesian estimation model;

[0117] In obtaining t k = Failure probability p at 5150h k After the prior distribution of iThe probability of failure at time p i According to the zero-failure data in Table 3, t i The values ​​of are 1200, 1500, 2000, 3000, 3500, 4250, 4700, 5000 and 5150. Based on the concavity of the logarithm of the cumulative failure function of the Weibull distribution, p is conservatively estimated to be i Prior distribution of :

[0118]

[0119] where λ i t i The upper bound of the failure probability of the bearing at time , i=1,2,…p.

[0120] S5: According to the p obtained in step S4 i The prior distribution of is used to calculate the posterior distribution of the cumulative failure probability of the bearing using the Bayesian estimation model.

[0121] When the sum of the time of all censored trials is t i There are n samples of i When s i =n i -r i If no bearing fails, the total test time of the bearing is t i When i When the bearings are not failed, s is calculated according to formula (7) i Prior distribution of :

[0122]

[0123] In the formula, s i Represents n i samples in the total test time t i The number of samples that have not failed at the time, s i =n i -r i , n i The fourth column of the no-failure data table 3, r i is the number of failed samples, which is the third column in the no-failure data table 3.

[0124] According to formula (8), p i The prior distribution of π(s i |p i ) and the Bayesian formula, p i The posterior distribution of is:

[0125]

[0126] S6: According to the posterior distribution π obtained in step S5 i (p i |s i ), integrate and calculate the expected value of the cumulative failure probability of each sample;

[0127] Under the square loss function, we can get p from formula (9): i The Bayesian estimate of is:

[0128]

[0129] After Bayes estimation, the cumulative failure probability value for each simulation total test time is obtained as follows: Figure 2 As shown by the triangle marked in the figure.

[0130] S7: The total test time t experienced by the fused samples using the grey bootstrap method i and the posterior estimate of the cumulative failure probability Will Figure 2 The set of triangle points As the input of the grey bootstrap model, firstly, m = 45 bootstrap samples can be obtained by bootstrap sampling, and the bootstrap sample points are constructed as shown in formula (10), and a total of B = 80 bootstrap sample sets are constructed:

[0131]

[0132] According to the gray prediction GM(1,1) principle, these data are accumulated once to generate a new sequence vector:

[0133]

[0134] Then, according to formula (12), the neighboring mean of the cumulatively generated sequence is generated:

[0135]

[0136] Then, according to equations (13) and (14), the data matrix D and data vector Y are constructed. b :

[0137]

[0138] According to formula (15), the gray generation model is constructed. According to (16), (17), and (18), 80 gray bootstrap samples can be obtained. The data can form the following vector:

[0139]

[0140] All generated Gray self-service sample points such as Figure 2 As indicated by the circle marked.

[0141] S8: Estimating the cumulative failure rate at multiple moments in step S7 The cumulative failure curve of the product is fitted using the least square method, and the key parameter values ​​of the bearing life distribution are calculated.

[0142] For multiple time-posterior probability sample points (b=1,2…80), it is required to get as close as possible to the bearing two-parameter Weibull cumulative failure probability curve of all sample points In Therefore, we must first construct the loss function and then use some optimization calculation methods for the parameters.

[0143] The set of predicted values ​​of the cumulative failure probability points calculated according to S7 Construct the loss function. Use the least squares method to fit the shape parameter β and the scale parameter η, and first construct the square loss function:

[0144]

[0145] Seeking satisfaction The parameter estimates of This is the key parameter for bearing life distribution.

[0146] S9: Refer to the Gibbs method to obtain the key parameter values ​​of the bearing life distribution that minimizes the loss function Q(β,η).

[0147] Take Q(β,η) as the analysis object and find its minimum point in two dimensions. The steps are as follows:

[0148] (1) Randomly initialized initial state η = η0;

[0149] (2) Setting the minimum iteration error ε;

[0150] (3) Repeat the following process to collect samples:

[0151] e) Find the point β1 near the minimum value of the loss function Q(β|η0);

[0152] f) Find the point η1 near the minimum value of the function in the loss function Q(η|β1);

[0153] g) Calculate the difference L = |Q(β0,η0)-Q(β1,η1)|;

[0154] h) Record the point (β1, η1) calculated at this time, and set η0 = η1.

[0155] (4) Loop until L<ε and output (β1, η1) = (1.0152142, 25269.813) as the solution.

[0156] S10: Repeat S7-S9 operations M = 10 times to generate multiple groups of gray self-service sample sets The maximum entropy method is used for comparative analysis to select the grey bootstrap samples with the largest information entropy, and its parameter estimation results are used as the key parameter values ​​of the life prediction model to complete the bearing life prediction.

[0157] From formula (22), 10 entropy values ​​H = {H1, H2…H 10}, select the largest H max The corresponding gray self-help sample And the corresponding cumulative failure function parameter estimation results As output, the estimated bearing life distribution parameters and information entropy for 10 groups of samples are shown in Table 4.

[0158] Table 4

[0159]

[0160] Therefore, the parameter estimation values ​​of the seventh group of samples are selected as the final parameter estimation results, and the final shape parameter estimation results are The estimated result of scale parameter is The predicted life value of the bearing at a reliability of 0.9 is 2704.8h, which is not much different from the life value of 2635h obtained by LP calculation after correction using the ISO281 standard, indicating the accuracy of the life prediction of the model.

[0161] The solution result Substitute into the cumulative failure function The schematic diagram of the cumulative failure function curve of the bearing is as follows Figure 3 shown.

[0162] Those skilled in the art will appreciate that the model described in the present invention is not limited to the processing of life test results, but can also be applied to the simulation and simulation of reliability data. Those skilled in the art can make various other specific variations and combinations based on the technical inspiration disclosed in the present invention without departing from the essence of the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. A product life prediction method based on zero-failure data in sequential testing, characterized by: include: Step 1): Carry out truncation tests on the products in order from the smallest to the largest truncation time, and calculate the longest test time t k and the sum of the time t of all censored trials experienced by each test sample p , k=1,2,…p, p is the number of test samples, and the failure data set {t p ,Y} and the zero-failure data set {t p ,N}, Y is the sample with failure, N is the sample without failure; Step 2): Use the Bayesian estimation model based on the failure data set {t p ,Y} and the zero-failure data set {t p ,N}, calculate the cumulative failure probability p of each sample i And its posterior distribution, the integral calculation obtains the Bayesian estimate of the cumulative failure probability of the i-th sample i=1,2,…p; First, use uniform distribution as the prior distribution to obtain the longest test time t k The cumulative failure probability p k Based on the prior distribution of the product cumulative failure function, the cumulative failure probability of each sample is conservatively estimated to determine t i The cumulative failure probability p i The prior distribution of the product; then calculate the conditional probability that the product has not failed, and obtain the number of samples that have not failed s i The conditional probability distribution of the final cumulative failure probability p i The posterior distribution of i The posterior distribution of The maximum test duration t k The cumulative failure probability p k The prior distribution of λ k is the cumulative failure probability p k The upper bound of the range; The i The cumulative failure probability p i The prior distribution of 0 <p i <λ i ,λ i t i The upper bound of the failure probability of the product at time The number of samples that have not failed i The conditional probability The number of samples that have not failed i The conditional probability distribution of s i Represents n i The total number of samples in the test time t i The failure-free data set {t p ,N}, the number of samples that have not failed, r i is the set of failure data {t p , the number of failed samples in Y}; The cumulative failure probability p i The posterior distribution of Step 3): Use a small sample sampling method to collect As the input of the grey bootstrap model, t i 0 to the longest test duration t k At any time between , a self-service sample set is obtained by sampling, a new gray self-service sample point is generated, and the expanded sample is obtained to form a cumulative failure rate estimation result set at multiple times Step 4): Estimate the result set based on the cumulative failure rate Construct a square loss function Q(β,η), and obtain the life distribution parameter that minimizes the square loss function Q(β,η) as the parameter estimate; Constructed square loss function: w=m+1, m is the number of self-service sample sets, B is the number of grey self-service sample sets, β is the shape parameter, η is the scale parameter η, and the grey self-service sample points are all different, b = 1, 2, ... B, the superscript (b) represents the point generated by the b-th bootstrap sample set, and we need to find the point that satisfies The parameter estimates of It is the key parameter of product life distribution; Step 5): Repeat steps 3)-4) and use the maximum entropy method to obtain the parameter estimate with the largest information entropy corresponding to the gray self-help sample point among all parameter estimates as the final result to complete the life prediction.

2. The method for predicting product life based on zero-failure data in a sequential test according to claim 1 is characterized by: In step 2), the uniform distribution is used as the prior distribution to obtain the longest test time t k The cumulative failure probability p k Based on the prior distribution of the product cumulative failure function, the cumulative failure probability of each sample is conservatively estimated to determine t i The cumulative failure probability p i The prior distribution of the product; then calculate the conditional probability that the product has not failed, and obtain the number of samples that have not failed s i The conditional probability distribution of the final cumulative failure probability p i The posterior distribution of i The posterior distribution of 3. The method for predicting product life based on zero-failure data in a sequential test according to claim 1 is characterized by: According to the cumulative failure probability p i The cumulative failure probability p is calculated from the posterior distribution of i The Bayesian estimate of 4. The method for predicting product life based on zero-failure data in a sequential test according to claim 1 is characterized by: In step 2), according to the logarithm of the cumulative failure function of the Weibull distribution: t i 0 to t k At any time between i is the cumulative failure probability of the product at that moment, t i ≤t k , F(ti)=pi, F(tk)=pk, we get Determine the cumulative failure probability.

5. The method for predicting product life based on zero-failure data in a sequential test according to claim 1 is characterized by: In step 3), m self-service samples are obtained by self-service sampling, which are accumulated to generate a new sequence vector, and the accumulated sequence vector is subjected to neighbor mean to generate new data, and the data matrix D and data vector Y are constructed. i , get the grey bootstrap model expression and generate new grey bootstrap sample points.

6. The method for predicting product life based on zero-failure data in a sequential test according to claim 1 is characterized by: In step 5), find the minimum of the parameter estimates β and η of the square loss function Q(β,η), randomly initialize the initial state η=η0, find the point β1 near the minimum value of the function in the square loss function Q(β|η0), find the point η1 near the minimum value of the function in the square loss function Q(η|β1), calculate the difference L=|Q(β0,η0)-Q(β1,η1)|, record the calculated point (β1,η1), set η0=η1 and then loop until L<ε, ε is the minimum iterative error, then (β1,η1) is the minimum value of the parameter estimate.

7. The method for predicting product life based on zero-failure data in a sequential test according to claim 5 is characterized by: Generate M sets of cumulative failure rate estimation results By finding the minimum values ​​of the parameter estimates β and η of the square loss function Q(β,η), the minimum value of the M group of parameter estimates is obtained, and the information entropy of the gray bootstrap sample set generated for the vth time is calculated as: is the grey bootstrap sample set generated for the vth time The time component; generate M entropy values ​​H = {H1, H2…H M }, select the largest H max The corresponding grey bootstrap samples and their corresponding parameter estimates as output.

8. The method for predicting product life based on zero-failure data in a sequential test according to claim 6 is characterized by: The parameter estimation results calculated from the grey bootstrap sample set Substitute into the original cumulative failure function In, order Constructing functions

Images