Logarithmic normal distribution reliability sequential probability ratio dynamic verification test method and system

Abstract

The invention relates to a logarithmic normal distribution reliability sequential probability ratio dynamic verification test method and a logarithmic normal distribution reliability sequential probability ratio dynamic verification test system in the technical field of mechanical engineering, and aims to solve the problem that the change of logarithmic standard deviation can influence the intercept of a sequential judgment equation in an existing sequential verification test model. The method comprises the following steps: acquiring test data of a work effectiveness test on an (n + 1) th target product, and when the product is determined to be invalid, acquiring the failure time of the (n + 1) th target product, the number of invalid products after the work effectiveness test of the first n target products, the failure time, and prior distribution of a logarithmic mean and a logarithmic variance, and determining the joint probability density and the reliability sequential probability ratio, verifying the test model according to the updated reliability sequential probability ratio, and making a sequential decision. According to the method, historical data and new failure data information generated in the test process are fully utilized, and a theoretical basis is provided for formulating a reliability verification test model.

Description

TECHNICAL FIELD

[0001] The present application relates to a lognormal distribution reliability sequential probability ratio dynamic verification test method and system, and belongs to the technical field of mechanical engineering. BACKGROUND

[0002] In order to evaluate whether the mean time between failures (MTBF) of a product meets the design life requirement, a reliability verification test model is usually used. Reliability verification test includes two parts of qualification and acceptance. The reliability verification test model of the life subject to exponential distribution is given in GJB-899A 2009 "Reliability Qualification and Acceptance Test" and MIL-STD-781D "Military Standard Reliability Testing for Engineering Development, Qualification, and Production" standards. However, the failure rate of fatigue life, early failure and integrated circuit failure of some mechanical products are more suitable for describing the life distribution of lognormal distribution. Therefore, the method in GJB-899A and MIL-STD-781D has limitations.

[0003] In the sequential verification test model, the change of the log standard deviation will affect the intercept of the sequential decision equation. If the log standard deviation is too large or too small, the qualified product may be judged as rejected or continued test, or the unqualified product may be judged as accepted or continued test, or the decision of continued test in the actual situation may be determined as accepted or rejected in advance. Generally, the log mean and log standard deviation need to be obtained from historical data. For new research products, the failure history data sample is small, and the parameter estimation result obtained from the historical data is often not accurate enough. SUMMARY

[0004] The purpose of the present application is to use the failure information as a supplement to the failure sample in the reliability sequential verification test to realize the dynamic update of the decision equation.

[0005] To achieve the above purpose, the present application is implemented by using the following technical solutions:

[0006] In the first aspect, the present application provides a lognormal distribution reliability sequential probability ratio dynamic verification test method, comprising:

[0007] Obtaining test data of a work effectiveness test on an n+1th target product, and determining the failure of the product according to the test data,

[0008] obtain the failure time of the (n+1)th target product, the number of failed products, the failure time, the prior distribution of the logarithmic mean and the logarithmic variance after the working effectiveness test of the first n target products, determine the joint probability density and the reliability sequential probability ratio based on the failure time of the (n+1)th target product, the number of failed products, the failure time, the prior distribution of the logarithmic mean and the logarithmic variance after the working effectiveness test of the first n target products, input the joint probability density into the pre-constructed reliability sequential probability ratio verification test model to obtain an updated reliability sequential probability ratio verification test model, and determine the sequential decision by inputting the reliability sequential probability ratio into the updated reliability sequential probability ratio verification test model.

[0009] Further, the inputting of the joint probability density into the pre-constructed reliability sequential probability ratio verification test model to obtain the updated reliability sequential probability ratio verification test model comprises:

[0010] combining the new failure data and the pre-determined prior distribution of the logarithmic mean and the logarithmic variance, and calculating the joint posterior distribution of the logarithmic mean and the logarithmic variance by using the Bayes theorem;

[0011] obtaining the marginal posterior distribution of the logarithmic mean and the logarithmic variance from the joint posterior distribution, and taking the posterior expectation of the marginal posterior distribution of the logarithmic mean and the logarithmic variance as the new logarithmic mean and the logarithmic variance;

[0012] substituting the new logarithmic mean and the logarithmic variance into the reliability sequential probability ratio verification test model to obtain the updated reliability sequential probability ratio verification test model.

[0013] Further, the reliability sequential probability ratio verification test model is realized by the following steps:

[0014] obtaining the life index and the risk parameter of the target product, wherein the life index comprises the test upper limit based on the average life of the product , the test lower limit , and the risk parameter comprises the producer's risk α and the user's risk β;

[0015] obtaining the historical failure data, wherein the failure time t in the historical failure data is subject to a logarithmic normal distribution, and the prior distribution of the logarithmic mean and the logarithmic variance of the logarithmic normal distribution is determined based on the historical failure data;

[0016] establishing the sequential decision basis based on the life index and the risk parameter;

[0017] establishing the corresponding decision equation according to the sequential decision basis to obtain the reliability sequential probability ratio verification test model.

[0018] Furthermore, the prior distributions of the logarithmic mean and logarithmic variance include:

[0019] At the start of the process, the prior distribution is a prior distribution based on the initial logarithmic mean and logarithmic variance;

[0020] After at least one round of updates, the prior distribution is a joint distribution consisting of the logarithmic mean and logarithmic variance inherited from the previous posterior distribution.

[0021] Log-variance in log-normal distribution The prior distribution follows a degree of freedom of The variance is The reverse The distribution, and its distribution form is as follows:

[0022] (1)

[0023] In the formula, x is a random variable with logarithmic variance. For the Gamma function, .

[0024] The prior distribution of the logarithmic mean follows a mean of logarithmic variance given that the prior value is logarithmic variance. The variance is It follows a normal distribution, and its distribution form is:

[0025] (2)

[0026] In the formula, A random variable with logarithmic mean. The variance coefficient, The standard deviation is the logarithm.

[0027] The logarithmic mean and logarithmic variance of the newly collected m failed data points are as follows:

[0028] (3)

[0029] (4)

[0030] In the formula, Let i be the failure time of the newly collected i-th failure data, i=1,2,…,m, where m is the number of newly collected failure data.

[0031] Furthermore, the probability density function of the log-normal distribution is f(t), the failure distribution function is F(t), and the joint probability density function of the k failed products is... ;

[0032] The probability density function f(t) is:

[0033] (5)

[0034] The failure distribution function F(t) is:

[0035] (6)

[0036] Joint probability density function :

[0037] (7)

[0038] In the formula, The logarithmic mean is... For logarithmic variance, The standard deviation is denoted by n; n is the number of product trials conducted, and k is the number of failures among the product trials conducted. The failure time of the defective product. and Let be the failure times of the i-th and r-th failed products.

[0039] Furthermore, the sequential decision is based on:

[0040] If the upper limit of the test and lower limit of testing The joint probability density function satisfies:

[0041] (8)

[0042] Then accept hypothesis H1 and reject it with a high probability;

[0043] If the probability ratio satisfies:

[0044] (9)

[0045] Then accept hypothesis H0 with high probability;

[0046] If the probability ratio is between A and B, that is:

[0047] (10)

[0048] Therefore, no judgment can be made, and further experimentation is required;

[0049] In the formula, and To test the upper limit and lower limit of testing The joint probability density function at; and In the theory of sequential probability ratio tests and Two assumptions: A and B are constants;

[0050] when When, the joint probability density function Represented as:

[0051] (11)

[0052] when When, the joint probability density function Represented as:

[0053] (12)

[0054] Constants A and B are respectively:

[0055] (13)

[0056] (14)

[0057] Furthermore, the reliability sequential probability ratio verification test model is as follows:

[0058] (15)

[0059] In the formula, For reliability sequential probability ratio, .

[0060] Furthermore, the joint probability function of the posterior distributions of the logarithmic mean and logarithmic standard deviation of the updated model, as expressed by Bayes' theorem, is:

[0061] (16)

[0062] The joint probability density function is expressed as:

[0063] (17)

[0064] In the formula, , , ,

[0065] .

[0066] Furthermore, the posterior marginal distribution of the log-mean in the log-normal distribution follows a t-distribution, which is expressed as:

[0067] (18)

[0068] The posterior marginal distribution of the log-variance is:

[0069] (19)

[0070] Secondly, this invention proposes a dynamic verification test system for the sequential probability ratio of log-normal distribution reliability, comprising:

[0071] The initialization module is used to initialize lifespan indicators and risk parameters;

[0072] The historical data acquisition module is used to acquire historical failure data of the product;

[0073] The sequential decision module is used to establish the basis for sequential decisions and the corresponding decision equations, and to generate a reliability sequential probability ratio verification test model.

[0074] The dynamic update module is used to dynamically update parameters using Bayes' theorem and to make sequential decisions using the reliability sequential probability ratio verification test model.

[0075] Compared with the prior art, the beneficial effects achieved by the present invention are as follows:

[0076] For products whose lifetime distribution follows a log-normal distribution, existing standards lack corresponding reliability verification test models. This invention, based on the concept of a sequential probability ratio verification test model, derives decision equations for "acceptance," "rejection," or "continue testing" during the testing process. Considering that changes in the logarithmic standard deviation will affect the sequential decision model, the prior distributions of the logarithmic mean and logarithmic variance are obtained from historical failure data. Based on newly collected failure data, the posterior distributions of the logarithmic mean and logarithmic variance are updated using Bayesian methods, thereby dynamically updating the sequential verification test model. The sequential probability ratio reliability verification test method for log-normally distributed lifetime products provided by this invention can fully utilize historical data and new failure data generated during the testing process, providing a theoretical basis for the formulation of reliability verification test models. Attached Figure Description

[0077] To more clearly illustrate the technical solutions in this invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0078] Figure 1 This is a flowchart of the dynamic verification test method for the sequential probability ratio of the reliability of the log-normal distribution. Detailed Implementation

[0079] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The following description of at least one exemplary embodiment is merely illustrative and is in no way intended to limit the present invention or its application or use.

[0080] Example 1:

[0081] This embodiment provides a dynamic verification test method for the sequential probability ratio of the reliability of a log-normal distribution, such as... Figure 1 As shown, the specific steps include:

[0082] To obtain test data for the (n+1)th mechanical or electronic product (such as a fluorescent lamp) during a functional effectiveness test, and to determine the failure of the fluorescent lamp based on the test data, assuming the failure time t follows a log-normal distribution, obtain the prior distribution of the failure time of the (n+1)th fluorescent lamp, the number of failed products after the functional effectiveness tests of the previous n fluorescent lamps, the failure time, and the logarithmic mean and logarithmic variance.

[0083] Establish a sequential decision-making basis based on the sequential probability ratio test theory;

[0084] Based on the sequential decision criteria, a corresponding decision equation is established to obtain a reliability sequential probability ratio verification test model;

[0085] Obtain experimental data from the sequential probability ratio verification experiment of the target product, and make a decision based on the experimental data:

[0086] If new failure data is generated during the experiment, the sequential verification test model is dynamically updated using Bayes' theorem, and the sequential decision is made using the reliability sequential probability ratio verification test model.

[0087] If no new invalid data emerges, maintain the current logarithmic mean and logarithmic variance, and revise the corresponding decision equations.

[0088] In this embodiment, the prior distributions of the logarithmic mean and logarithmic variance include:

[0089] At the start of the process, the prior distribution is based on the initial log mean and log variance;

[0090] After at least one round of updates, the prior distribution is a joint distribution consisting of the logarithmic mean and logarithmic variance inherited from the previous posterior distribution.

[0091] In this embodiment, the failure time t is assumed to follow a log-normal distribution, and its probability density function f(t) is:

[0092] (1)

[0093] In the formula, μ is the logarithmic mean of the variable, and σ is the logarithmic standard deviation. The variance is logarithmic.

[0094] The failure distribution function F(t) is:

[0095] (2)

[0096] n+1 fluorescent lamps are tested, of which k are defective. Their failure times are denoted as t1, t2, ..., t3. k Then the joint probability density function of the k failed products is:

[0097] (3)

[0098] Substituting equations (1) and (2) into equation (3), we get:

[0099] (4)

[0100] In this embodiment, it is assumed that the upper limit for the inspection of the mean time between failures (MTBF) or mean life of the fluorescent lamp is... The lower limit of the test is In the sequential reliability verification test, two assumptions exist: , , Due to the randomness of sampling, if a qualified product is judged as unqualified, it will generate producer risk α; if an unqualified product is judged as qualified, it will generate user risk β.

[0101] if The joint probability density is:

[0102] (5)

[0103] Received with a high probability.

[0104] if The joint probability density is:

[0105] (6)

[0106] They will be rejected with a high probability.

[0107] Based on the sequential experiment concept, let constants A and B be:

[0108] (7)

[0109] (8)

[0110] If the probability ratio satisfies:

[0111] (9)

[0112] Then accept hypothesis H1 and reject it with a high probability.

[0113] If the probability ratio satisfies:

[0114] (10)

[0115] Then accept hypothesis H0 with a high probability.

[0116] If the probability ratio is between A and B, that is:

[0117] (11)

[0118] Therefore, a judgment cannot be made, and further experimentation is required.

[0119] In this embodiment, equation (4) is substituted into the probability ratio. , can be obtained

[0120] (12)

[0121] Furthermore, substituting the above equation into equation (11), we can obtain:

[0122] (13)

[0123] The above expression can be written in the following form:

[0124] (14)

[0125] Therefore, for each failure data point generated during the experiment, the log-normal probability ratio can be obtained from the number of failures and the failure time, as shown in the following reliability sequential verification test model:

[0126] (15)

[0127] In the formula, For reliability sequential probability ratio, .

[0128] In this embodiment, dynamically updating the sequential verification test model includes:

[0129] By combining the new failure data with the predetermined prior distributions of the logarithmic mean and logarithmic variance, the joint posterior distribution of the logarithmic mean and logarithmic variance is calculated using Bayes' theorem.

[0130] Marginalize the log-mean and log-variance marginal posterior distributions from the joint posterior distribution, and use their respective posterior expectations as the new log-mean and log-variance;

[0131] Substitute the new logarithmic mean and logarithmic variance into the reliability sequential probability ratio verification test model to make sequential decisions.

[0132] Among them, the log-variance in the log-normal distribution The prior distribution follows a degree of freedom of The variance is The reverse The distribution, and its distribution form is as follows:

[0133] (16)

[0134] In the formula, x is a random variable with logarithmic variance. For the Gamma function, .

[0135] The prior distribution of the logarithmic mean follows a mean of logarithmic variance given that the prior value is logarithmic variance. The variance is It follows a normal distribution, and its distribution form is:

[0136] (17)

[0137] In the formula, A random variable with logarithmic mean. The variance coefficient, The standard deviation is the logarithm.

[0138] The logarithmic mean and logarithmic variance of the newly collected m failed data points are as follows:

[0139] (18)

[0140] (19)

[0141] In the formula, Let m be the failure times of the newly collected m failure data.

[0142] In this embodiment, the joint probability function of the posterior distributions of the logarithmic mean and logarithmic standard deviation of the model after Bayes' theorem update is expressed as:

[0143] (20)

[0144] The joint probability density function is expressed as:

[0145] (twenty one)

[0146] In the formula, , , ,

[0147] .

[0148] In this embodiment, the posterior marginal distribution of the log-mean in the log-normal distribution follows a t-distribution, and its expression is:

[0149] (twenty two)

[0150] The posterior marginal distribution of the log-variance is:

[0151] (twenty three)

[0152] Example 2

[0153] Based on Example 1, this embodiment proposes a dynamic verification test system for the sequential probability ratio of log-normal distribution reliability, including:

[0154] The initialization module is used to initialize lifespan indicators and risk parameters;

[0155] The historical data acquisition module is used to acquire historical failure data of the product;

[0156] The sequential decision module is used to establish the basis for sequential decisions and the corresponding decision equations, and to generate a reliability sequential probability ratio verification test model.

[0157] The dynamic update module is used to dynamically update parameters using Bayes' theorem and to make sequential decisions by verifying the reliability sequential probability ratio of the experimental model.

[0158] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0159] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0160] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0161] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0162] The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims. All of these forms are within the protection scope of the present invention.

Claims

1. A dynamic verification test method for the sequential probability ratio of the reliability of a log-normal distribution, characterized in that, include: Obtain test data for the (n+1)th target product during a functional effectiveness test. When the product is determined to be faulty based on the test data, Obtain the prior distributions of the failure time of the (n+1)th target product, the number of failed products after the operational effectiveness test of the previous n target products, the failure time, and the logarithmic mean and logarithmic variance. Based on these prior distributions, determine the joint probability density and the reliability sequential probability ratio. Input the joint probability density into a pre-constructed reliability sequential probability ratio verification test model to obtain an updated reliability sequential probability ratio verification test model. Input the reliability sequential probability ratio into the updated reliability sequential probability ratio verification test model to determine the sequential decision.

2. The dynamic verification test method for the sequential probability ratio of log-normal distribution reliability according to claim 1, characterized in that, The step of inputting the joint probability density into a pre-built reliability sequential probability ratio verification test model to obtain an updated reliability sequential probability ratio verification test model includes: The new failure data is combined with the predetermined prior distributions of logarithmic mean and logarithmic variance, and the joint posterior distribution of logarithmic mean and logarithmic variance is calculated using Bayes' theorem. Marginalize the marginal posterior distributions of the log-mean and log-variance from the joint posterior distribution, and use the posterior expectations of the marginal posterior distributions of the log-mean and log-variance as the new log-mean and log-variance; Substituting the new logarithmic mean and logarithmic variance into the reliability sequential probability ratio verification test model yields the updated reliability sequential probability ratio verification test model.

3. The dynamic verification test method for the sequential probability ratio of log-normal distribution reliability according to claim 1, characterized in that, The reliability sequential probability ratio verification test model is implemented through the following steps: Obtain the lifespan indicators and risk parameters of the target product, wherein the lifespan indicators include the upper limit of inspection based on the average product lifespan. Lower limit of test The risk parameters include producer risk α and user risk β; Acquire historical failure data, wherein the failure time t in the historical failure data follows a log-normal distribution, and determine the prior distribution of the log-normal mean and log-normal variance based on the historical failure data; Based on the aforementioned lifespan indicators and risk parameters, a sequential decision-making basis is established; Based on the sequential decision criteria, a corresponding decision equation is established to obtain a reliability sequential probability ratio verification test model.

4. The dynamic verification test method for the sequential probability ratio of log-normal distribution reliability according to claim 1, characterized in that, The prior distributions of the logarithmic mean and logarithmic variance include: At the start of the process, the prior distribution is a prior distribution based on the initial logarithmic mean and logarithmic variance; After at least one round of updates, the prior distribution is a joint distribution consisting of the logarithmic mean and logarithmic variance inherited from the previous posterior distribution. Log-variance in log-normal distribution The prior distribution follows a degree of freedom of The variance is The reverse The distribution, and its distribution form is as follows: (1) In the formula, x is a random variable with logarithmic variance. For the Gamma function, ; The prior distribution of the logarithmic mean follows a mean of logarithmic variance given that the prior value is logarithmic variance. The variance is It follows a normal distribution, and its distribution form is: (2) In the formula, A random variable with logarithmic mean. The variance coefficient, The standard deviation is the logarithm. The logarithmic mean and logarithmic variance of the newly collected m failed data points are as follows: (3) (4) In the formula, Let i be the failure time of the newly collected i-th failure data, i=1,2,…,m, where m is the number of newly collected failure data.

5. The dynamic verification test method for the sequential probability ratio of log-normal distribution reliability according to claim 1, characterized in that, The probability density function of the log-normal distribution is f(t), the failure distribution function is F(t), and the joint probability density function of the k failed products is... ; The probability density function f(t) is: (5) The failure distribution function F(t) is: (6) Joint probability density function : (7) In the formula, The logarithmic mean is... For logarithmic variance, The standard deviation is denoted by n; n is the number of product trials conducted, and k is the number of failures among the product trials conducted. The failure time of the defective product. and Let be the failure time of the i-th and r-th failed products.

6. The dynamic verification test method for the sequential probability ratio of log-normal distribution reliability according to claim 3, characterized in that, The basis for the sequential decision is: If the upper limit of the test and lower limit of testing The joint probability density function satisfies: (8) Then accept hypothesis H1 and reject it with a high probability; If the probability ratio satisfies: (9) Then accept hypothesis H0 with high probability; If the probability ratio is between A and B, that is: (10) Therefore, no judgment can be made, and further experimentation is required; In the formula, and To test the upper limit and lower limit of testing The joint probability density function at; and In the theory of sequential probability ratio tests and Two assumptions: A and B are constants; when When, the joint probability density function Represented as: (11) when When, the joint probability density function Represented as: (12) Constants A and B are respectively: (13) (14)。 7. The dynamic verification test method for the sequential probability ratio of log-normal distribution reliability according to claim 6, characterized in that, The reliability sequential probability ratio verification test model is as follows: (15) In the formula, For reliability sequential probability ratio, .

8. The dynamic verification test method for the sequential probability ratio of the log-normal distribution reliability according to claim 2 or 4, characterized in that, The joint probability function of the posterior distributions of the log-mean and log-standard deviation of the model after Bayes' theorem update is expressed as: (16) The joint probability density function is expressed as: (17) In the formula, , , , 。 9. The dynamic verification test method for the sequential probability ratio of log-normal distribution reliability according to claim 8, characterized in that, The posterior marginal distribution of the log-mean in a log-normal distribution follows a t-distribution, which is expressed as: (18) The posterior marginal distribution of the log-variance is: (19)。 10. A dynamic verification test system for the sequential probability ratio of log-normal distribution reliability, characterized in that, include: The initialization module is used to initialize lifespan indicators and risk parameters; The historical data acquisition module is used to acquire historical failure data of the product; The sequential decision module is used to establish the basis for sequential decisions and the corresponding decision equations, and to generate a reliability sequential probability ratio verification test model. The dynamic update module is used to dynamically update parameters using Bayes' theorem and to make sequential decisions using the reliability sequential probability ratio verification test model.

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