A method for combining batch product composite sampling compatibility testing and reliability assessment

Abstract

This invention provides a method for the compatibility testing and reliability evaluation of batch-product composite sampling data, specifically a method for the compatibility testing and reliability evaluation of batch-product composite sampling data based on Bayesian theory. The steps are as follows: 1. Failure probability distribution point estimation and interval estimation; 2. Compatibility testing of specific test data and actual use data; 3. Determination of the prior distribution of failure probability and update of the posterior distribution; 4. Batch product reliability calculation. Through these steps, the reliability of batch-products is evaluated by fusing information from specific test data and actual test data, solving the problems of small sample evaluation and fuzzy sample processing for batch-products. The method is simple and easy to operate.

Description

[Technical Field]

[0001] This invention provides a method for compatibility testing and reliability fusion evaluation of batch-product composite sampling data. Specifically, it is a method for compatibility testing and reliability fusion evaluation of batch-product composite sampling data based on Bayesian theory. It conducts batch data compatibility testing and fuzzy failure sample randomization processing on both specialized sampling test data and actual usage data of batch-products. The specialized sampling test data is used as prior information, and the actual usage data is used as observation information. The failure probability of batch-products is iteratively updated using Bayesian principles, thereby improving the accuracy of batch-product reliability evaluation. This method is applicable to fields such as composite sampling batch-product reliability evaluation and multi-source information fusion evaluation. [Background Technology]

[0002] After mass production, products need to undergo quality assessment. The first method is to conduct random sampling tests on each batch of products and evaluate the reliability of the batch based on the results. However, this method has a limited sample size and the confidence level of the evaluation results is low. The second method is to conduct statistical evaluation of the reliability of the batch products based on actual usage data. However, the testing conditions for actual usage data are complex and it is not easy to obtain accurately. Therefore, it is necessary to combine the special test data and the actual usage data to conduct a comprehensive batch product reliability assessment.

[0003] In special sampling tests, the testing conditions and failure criteria are the same for different samples, so the number of successful samples and quality problem samples can be accurately obtained. Based on this, a binomial distribution or hypergeometric distribution can be selected to estimate the failure probability of batch products at a point or with confidence limits, depending on the total sample size. However, in actual use data, the usage and testing conditions of different products vary. In addition to samples that can be accurately judged as successful, quality problem, or non-quality problem, there are also ambiguous problem samples whose failure causes cannot be determined. Therefore, the success or failure data evaluation method cannot be directly used to process the actual use data. A feasible approach is to randomize the number of ambiguous problem samples, and based on historical experience information, use a uniform distribution or a truncated normal distribution to describe the number of failure samples. Then, the point estimate and confidence limit estimate of the failure probability can be obtained using the law of total probability.

[0004] Before integrating the data from special tests and actual use, compatibility tests need to be conducted on samples from different batches to determine whether the overall distribution of samples from different batches is consistent. Based on this, using Bayesian theory, the data from special sampling tests is used as prior information to determine the prior distribution of failure probability. Then, the actual use data is used as observation information to update the failure probability and obtain the posterior distribution. Finally, the reliability of the batch of products is calculated based on the posterior distribution.

[0005] Based on the above ideas, this invention provides a batch product composite sampling compatibility test and reliability fusion evaluation method, namely a batch product composite sampling compatibility test and reliability fusion evaluation method based on Bayesian theory; it comprehensively utilizes composite sampling data of special sampling test samples and actual use samples, which can effectively solve the problems of small sample size in special tests and vague and unusable information of actual use samples. [Summary of the Invention]

[0006] (1) Purpose of the present invention: In view of the problems of small sample size of special tests and fuzzy information of actual use samples in the quality assessment of composite sampling products, the present invention provides a batch product composite sampling compatibility test and reliability fusion assessment method, that is, a batch product composite sampling compatibility test and reliability fusion assessment method based on Bayesian theory, which integrates batch product special data and actual data through compatibility test and Bayesian update, and finally assesses the reliability of batch products.

[0007] (2) Technical solution:

[0008] The following basic settings are required for this invention:

[0009] Set up 1. Both the special test data and the actual use data are sampling data; set the total number of samples to be sampled as N, including the number of failures as D; set the total number of special sampling samples as n, the number of failure samples as r0, and the number of success samples as n-r0.

[0010] When N is large, the number of failure samples X is described by a binomial distribution, i.e.

[0011]

[0012] When N is small, the number of failure samples X can be described by a hypergeometric distribution, i.e.

[0013]

[0014] Setting 2: Let the total number of samples in the actual use process be M, the number of quality problem samples be r1, the number of non-quality problem samples be r2, the number of ambiguous problem samples be r3, and the number of successful samples be M-r1-r2-r3; Treat the number of failed samples r as a random variable, and its distribution is described by uniform distribution or truncated normal distribution, then r1≤r≤r1+r3.

[0015] If the probability density function of the number of failed samples is f(r), for a uniform distribution...

[0016]

[0017] If the discrete characteristics of the sample are considered, a discrete uniform distribution is adopted.

[0018]

[0019] For the truncated normal distribution

[0020]

[0021] In the formula: φ(·) represents the standard normal probability density function, and Φ(·) represents the standard normal cumulative distribution function;

[0022] The method proposed in this invention mainly includes point estimation and interval estimation of failure probability distribution, compatibility test between special test data and actual use data, determination of prior distribution of failure probability and update of posterior distribution, and calculation of batch product reliability.

[0023] This invention provides a method for the integrated evaluation of batch product composite sampling compatibility and reliability, namely, a method for the integrated evaluation of batch product composite sampling compatibility and reliability based on Bayesian theory, the steps of which are as follows:

[0024] Step 1: Point and Interval Estimation of Failure Probability Distribution

[0025] For a specific sampling test, if the number of failed samples follows a binomial distribution, then the point estimate of the failure probability p is:

[0026]

[0027] Given a confidence level of 1-α, the lower confidence limit p of a one-sided confidence test L Determined by the following formula:

[0028]

[0029] One-sided confidence limit p of the parameter U Determined by the following formula:

[0030]

[0031] Two-sided confidence interval of the parameter (p L ,p U It is determined by the following formula:

[0032]

[0033]

[0034] Where: n is the total number of samples in the special inspection, and r is the number of invalid samples in the special inspection;

[0035] If the number of failed samples in a special sampling inspection follows a hypergeometric distribution, then the point estimate of the failure probability p is:

[0036]

[0037] Given a confidence level of 1-α, the one-sided lower confidence limit p of the parameter L Determined by the following formula:

[0038]

[0039] In the formula: the symbol ! represents the factorial function;

[0040] Based on the values ​​of N, n, r, and α, p can be calculated from the above formula. L The value can also be directly queried through GJB 376-87;

[0041] Based on actual usage data, the data is divided into K groups by batch, with N samples in each group. i Let i = 1, 2, ..., K; count the number of failure samples in each group caused by quality problems, non-quality problems, and fuzzy problems, and obtain the failure probability point estimate for each group based on the randomization assumption and the theory of total probability. The point estimate of the failure probability p can be calculated using the following formula:

[0042]

[0043] In the formula: f(r) is the probability density function of the number of failed samples;

[0044] If the number of failures follows a continuous uniform distribution, then

[0045]

[0046] If the number of failures follows a discrete uniform distribution, then

[0047]

[0048] If the number of failures follows a truncated normal distribution, then

[0049]

[0050] In the formula: M is the total number of actual samples, r1 is the number of samples with quality problems, r3 is the number of samples with fuzzy problems, μ is the mean of the truncated normal distribution, and σ is the standard deviation of the truncated normal distribution.

[0051] Step Two: Compatibility Verification of Specific Test Data and Actual Use Data

[0052] The compatibility issue between specialized test data and actual usage data boils down to whether the two types of data originate from the same population. Therefore, compatibility testing can be transformed into a test to determine whether the population distributions of the two types of data are statistically consistent. If the quality status of each batch of products is consistent upon leaving the factory, then the failure probability value obtained from the statistical analysis of K groups of actual usage data should follow a normal distribution, and its mean and variance can be determined based on the point and interval estimates of the p-value obtained from the specialized sampling test. Based on this, the compatibility issue between specialized sampling data and actual usage data can be transformed into... Whether a data follows a normal distribution with a given mean and variance can be tested for data compatibility using statistical tests.

[0053] First, a T-test is used to determine whether there is a difference between the distribution mean of the special test data and the actual data used. The null hypothesis H0 is established: the two sample means are equal. The T-statistic is then calculated.

[0054]

[0055] Degrees of freedom can be expressed as

[0056]

[0057] In the formula: s1 and s2 are the means of the two test samples, s1 and s2 are the standard deviations of the two samples, and n1 and n2 are the sizes of the two samples.

[0058] Given a confidence level of 1-α, if T>t α / 2,f If the null hypothesis is true, then we reject the null hypothesis and conclude that the two sample means are not equal; otherwise, we do not reject the null hypothesis. Then, we use the F-test to determine whether the population variances of the two normal random variables are equal, establishing the null hypothesis H0: the population variances of the two samples are equal, and calculating the statistic.

[0059]

[0060] In the formula: s1 and s2 are the standard deviations of the two groups of samples; the degrees of freedom are (n1-1, n2-1);

[0061] Given a confidence level of 1-α, if the following conditions are met... If the null hypothesis is rejected, the conclusion is that the population variances of the two samples are not equal; otherwise, the null hypothesis is not rejected. The few samples that do not meet the compatibility test are removed, and the remaining batch of samples can be used for information fusion in step three.

[0062] Step 3: Determining the Prior Distribution and Updating the Posterior Distribution of Failure Probability

[0063] The specific steps are as follows:

[0064] I. Determination of the prior distribution of failure probability

[0065] Using specific sampling test data as prior information, the prior distribution of the failure probability p is given (taking the truncated normal distribution as an example):

[0066]

[0067] In the formula: This represents the mean estimate of the probability of invalidity of data from special sampling inspections. Variance estimate representing the probability of failure;

[0068] II. Update of the posterior distribution of failure probability

[0069] Using actual usage data as observation information, the posterior distribution of the failure probability p is updated based on Bayesian theory:

[0070]

[0071] In the formula: Θ represents the actual dataset used;

[0072] Step 4: Batch Product Reliability Calculation

[0073] Calculate the reliability of the batch of products based on the posterior distribution of the failure probability p:

[0074]

[0075] In the formula: E(·) represents the expected function;

[0076] By integrating the data from the special test and the actual test through the above steps, the reliability of the batch of products was evaluated, and the problems of small sample evaluation and fuzzy sample processing of batch products were solved. The method is simple and easy to operate.

[0077] (3) Advantages and benefits:

[0078] ①This method integrates batch product-specific test information and actual use information, solving the problem of small sample size in specific tests and ambiguous classification of actual use data samples;

[0079] ②This method is scientific, effective, and easy to operate, and has broad application value. [Attached Image Description]

[0080] Figure 1 This is a flowchart of the method described in this invention.

[0081] Figure 2 Let be the probability density function of the failure probability after each update.

[0082] The serial numbers, symbols, and codes in the diagram are explained as follows:

[0083] p represents the failure probability; The distribution representing the probability of failure;

Detailed Implementation Methods

[0084] The invention will now be described in further detail with reference to examples.

[0085] Assume that the data types for both the product-specific test and the actual use data are sampling data, corresponding to Table 1 and Table 2 respectively; the total number of samples to be sampled in the product-specific test is 200, including 10 non-conforming products;

[0086] Table 1. Specific Test Data

[0087]

[0088] Table 2 Actual Usage Data

[0089]

[0090] This invention provides a method for the integrated evaluation of batch product composite sampling compatibility and reliability, specifically a method based on Bayesian theory for the integrated evaluation of batch product composite sampling compatibility and reliability. Figure 1 As shown, this can be achieved through the following steps:

[0091] Step 1: Point and Interval Estimation of Failure Probability Distribution

[0092] The sample size of the special test data is relatively large. Therefore, based on equation (6), the binomial distribution is used to calculate the point estimate of the failure probability of each group, and the mean and sample variance of the failure probability are obtained as follows:

[0093]

[0094] Based on actual usage data, a uniform distribution is used to characterize the randomness of failure samples. Based on equations (3) and (13), the point estimate of the failure probability is calculated as follows:

[0095]

[0096] In the formula: M is the total number of actual samples, r1 is the number of samples with quality problems, and r3 is the number of samples with ambiguous problems;

[0097] The calculation results for each group are shown in Table 3.

[0098] Table 3. Failure probability point estimation for each group

[0099]

[0100] Table 3 shows that the mean and sample variance of the failure probability are respectively...

[0101]

[0102] Step Two: Compatibility Verification of Specific Test Data and Actual Use Data

[0103] Given a confidence level of α = 0.05, and since the sample size is less than 30, the difference between the two sample means can be determined using the T-test method. The T-statistic and degrees of freedom can be calculated using equations (17) and (18).

[0104]

[0105] In the formula: and These are the sample means of data from specific experiments and data from actual use, respectively. and n1 and n2 are the sample variances of the special test data and the actual use data, respectively, and the sample sizes of the actual use data and the special test data are the sample sizes of the special test data and the actual use data, respectively.

[0106] Because of t α / 2,5 =2.5706>T, therefore the means of the two groups of samples can be considered equal;

[0107] Then, based on the F-test to determine the variance difference between the two groups of samples, the F-statistic can be calculated using equation (19).

[0108]

[0109] In the formula: and These are the sample variances of the data from the special test and the data from actual use, respectively.

[0110] Due to F α,(3,5) =3.62>F, therefore the variances of the two groups of samples can be considered equal;

[0111] In summary, the data from the special test and the data from actual use meet the compatibility requirements;

[0112] Step 3: Determining the Prior Distribution and Updating the Posterior Distribution of Failure Probability

[0113] I. Determination of the prior distribution of failure probability

[0114] Selecting specific experimental data as prior information and using a truncated normal distribution for description, we can obtain the following from equations (20) and (23).

[0115]

[0116] In the formula: φ(a,b;c) and Φ(a,b;c) are the probability density function and cumulative distribution function values ​​respectively when the mean parameter is a, the variance parameter is b, and the value is c;

[0117] II. Update of the posterior distribution of failure probability

[0118] Using actual usage data as observation information, the actual failure probability is described by a discrete uniform distribution. The posterior distribution of the failure probability is updated sequentially by equations (4), (21), and (28):

[0119]

[0120] In the formula: Let be the prior distribution of the failure probability. To integrate the failure probability Θ of the actual usage dataset; r1 is the number of quality problem samples in the actual usage data, and r3 is the number of fuzzy problem samples in the actual usage data;

[0121] The results of each update are as follows Figure 2 As shown;

[0122] Step 4: Batch Product Reliability Calculation

[0123] From equation (22), the reliability after successive updates can be obtained, as shown in Table 4.

[0124] Table 4 Reliability after each update

[0125]

[0126] Depend on Figure 2 As shown in Table 4, with the increase of actual usage data, the reliability assessment results are closer to the true value and the estimation variance is smaller. Therefore, the method of this invention can gradually reduce the dispersion of batch product reliability assessment results.

[0127] In summary, this invention provides a method for the integrated evaluation of batch product composite sampling compatibility and reliability, namely, a method based on Bayesian theory for the integrated evaluation of batch product composite sampling compatibility and reliability. It fully utilizes fuzzy sample information from actual usage data, comprehensively employing fuzzy failure sample randomization methods and Bayesian theory to iteratively update the reliability of batch products, thereby improving the accuracy of reliability assessment. The specific steps of this method are: 1. Failure probability distribution point estimation and interval estimation; 2. Compatibility testing of specific test data and actual usage data; 3. Determination of the prior distribution of failure probability and updating of the posterior distribution; 4. Batch product reliability calculation. This method is applicable to the reliability assessment of composite sampling products containing fuzzy sample information and features high robustness and low dispersion.

Claims

1. A method for integrating batch product composite sampling compatibility testing and reliability evaluation requires the following settings: Setting 1: Both the special test data and the actual usage data are sampling data; the total number of samples to be sampled is set to be... The number of failures is The total number of samples in the special sampling inspection was The number of failed samples is The number of successful samples is ; Setting 2: Let the total number of samples in actual use be... The number of quality problem samples is The number of samples without quality issues is The number of samples for the fuzzy problem is The number of successful samples is ; Number of failed samples If we consider it as a random variable, and its distribution is described by a uniform distribution or a truncated normal distribution, then we have: ; If the probability density function of the number of failed samples is For uniform distribution (3) If the discrete characteristics of the sample are considered, a discrete uniform distribution is adopted. (4) For the truncated normal distribution (5) In the formula: This represents the standard normal probability density function. Represents the standard normal cumulative distribution function; Its features are as follows: Step 1: Point and Interval Estimation of Failure Probability Distribution For specific sampling tests, if the number of failed samples follows a binomial distribution, then the failure probability is... The point estimate is (6) Given confidence level One-sided confidence lower limit Determined by the following formula: (7) Upper confidence limit of the parameter Determined by the following formula: (8) Two-sided confidence interval of the parameter Determined by the following formula: (9) (10) In the formula: This represents the total number of samples collected in the special inspection. This refers to the number of invalid samples in the special random inspection. according to , , and The value is obtained from the above formula. The value can be found directly through GJB 376-87; Based on actual usage data, it is divided into batches. Groups, with the number of samples in each group being , The number of failure samples in each group due to quality issues, non-quality issues, and fuzzy issues was counted separately. Based on the randomization assumption and the theory of total probability, the failure probability point estimate for each group of samples was obtained. Failure probability The point estimate is calculated using the following formula: (13) In the formula: Let be the probability density function of the number of failed samples; If the number of failures follows a continuous uniform distribution, then (14) If the number of failures follows a discrete uniform distribution, then (15) If the number of failures follows a truncated normal distribution, then (16) In the formula: This represents the total number of samples actually collected. For the number of samples with quality issues, For the number of samples in the fuzzy problem, To truncate the mean of the parent normal distribution, To truncate the standard deviation of the parent distribution of the normal distribution; Step Two: Compatibility Verification of Specific Test Data and Actual Use Data The compatibility issue between specialized test data and actual usage data boils down to whether the two types of data originate from the same population. Therefore, the compatibility test is transformed into a test to determine whether the population distributions of the two types of data are statistically consistent. If the quality status of each batch of products is consistent upon leaving the factory, then according to... The failure probability values ​​obtained from the actual usage data of the group should follow a normal distribution, and their mean and variance should be obtained from the special sampling test. The point and interval estimates of the values ​​are determined; the compatibility problem between special sampling data and actual usage data is transformed into... To determine whether a data follows a normal distribution with a given mean and variance, statistical tests can be used to verify data compatibility. First, a T-test is used to determine whether there is a difference between the distribution mean of the special test data and the actual usage data, and the null hypothesis is established. If the means of two samples are equal, calculate the T-statistic. (17) Degrees of freedom are represented as (18) In the formula: The mean of the two test samples. The standard deviations of the two groups of samples are: The size of the two groups of samples; Given confidence level If satisfied If the sample means are equal, then the null hypothesis is rejected, leading to the conclusion that the two sample means are not equal; otherwise, the null hypothesis is not rejected. Then, the F-test is used to determine whether the population variances of the two normal random variables are equal, thus establishing the null hypothesis. If the population variances of the two samples are equal, calculate the statistic. (19) In the formula: The standard deviations of the two samples are given; the degrees of freedom are given. ; Given confidence level If satisfied If the population variances of the two samples are not equal, then the null hypothesis is rejected and the conclusion is that the population variances of the two samples are not equal; otherwise, the null hypothesis is not rejected. A few samples that do not meet the compatibility test are removed, and the remaining batch of samples are used for information fusion in step three. Step 3: Determining the Prior Distribution and Updating the Posterior Distribution of Failure Probability The specific steps are as follows: 3.1 Determination of the prior distribution of failure probability Selecting specific sampling test data as prior information, the failure probability is given. Prior distribution: (20) In the formula: This represents the mean estimate of the probability of invalidity of data from special sampling inspections. Variance estimate representing the probability of failure; 3.2 Update of Posterior Distribution of Failure Probability Actual usage data is selected as the observation information, and the failure probability is updated based on Bayesian theory. The posterior distribution: (21) In the formula: This indicates the actual dataset used; Step 4: Batch Product Reliability Calculation Based on failure probability p Calculate the reliability of the batch of products using the posterior distribution: (22) In the formula: This represents the expected function.

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