A belief inference-based b-s distribution reliability interval estimation method

Abstract

The application provides a B-S distribution reliability interval estimation method based on belief inference, which is realized through the following steps: 1, estimating parameters in an acceleration model; 2, calculating acceleration factors between each acceleration stress and a conventional stress; 3, converting acceleration failure data into failure data under the conventional stress; and 4, calculating a confidence interval of product reliability under the conventional stress. The application provides parameter estimation values of a specific acceleration model, has high estimation accuracy, is simple to calculate, and embodies the superiority in small sample reliability evaluation. The application is suitable for different reliability evaluation situations and has strong expandability.

Description

Technical Field

[0001] This invention relates to a reliability interval estimation method for the BS distribution based on belief inference. It is a method for estimating the reliability interval of the BS distribution based on the least squares principle and belief inference. Taking complete failure data obtained from accelerated life testing of products, it uses the least squares principle to obtain parameter estimates in the accelerated model, and then calculates the acceleration factor between each accelerated stress level and the normal stress level. Using the acceleration factor, the failure data under accelerated stress levels is converted into failure data under normal stress levels, a suitable pivot quantity is constructed, and the confidence interval of reliability is obtained using belief inference. This method is applicable to fields such as reliability assessment of the BS distribution in accelerated life testing.

[0002] The "BS distribution" mentioned above refers to the Birnbaum-Saunders lifetime distribution, if the random variable If it follows a Black-Scholes distribution, then its cumulative failure function is: In the formula, The cumulative failure function is a standard normal distribution. and These are the shape and scale parameters of the BS distribution, respectively.

[0003] The "acceleration factor" refers to the ratio of a product's lifespan under normal operating stress to its lifespan under accelerated stress. Background Technology

[0004] The Birnbaum-Saunders lifetime distribution is derived from the basic characteristics of the fatigue process. Compared with other common lifetime distributions, such as the log-normal distribution and the Weibull distribution, the BS distribution is more suitable for the study of fatigue failure. It has important application value in the reliability research of mechanical products and electronic products.

[0005] For reliability assessment of high-reliability, long-life products, accelerated life testing is a reasonable and feasible method that can obtain sufficient product reliability information while saving time and costs. Compared to point estimation, interval estimation of reliability is more significant in engineering and can provide guidance for the design and verification of practical engineering projects. Currently, relatively mature theoretical systems have been developed for reliability interval estimation methods for common lifetime distribution models such as log-normal and Weibull distributions. However, due to the complexity of the Black-Scholes distribution model, classical statistical methods are difficult to provide interval estimates of its reliability. Belief inference, on the other hand, is a statistical inference method that treats parameters as random variables. The distribution that the parameters follow is called the belief distribution. Furthermore, the belief distribution corresponding to reliability can be obtained from the belief distribution of the parameters, thereby allowing for interval estimation of reliability. By constructing an appropriate pivot quantity, the belief distributions of parameters and reliability can be obtained, providing a new approach for interval estimation of reliability in accelerated life testing based on the Black-Scholes distribution.

[0006] Based on this, the present invention proposes a reliability interval estimation method for the BS distribution based on belief inference, which converts the lifetime failure data under various accelerated stress levels in accelerated life tests into lifetime failure data under normal stress, and gives the confidence interval of reliability and the one-sided confidence lower limit based on the belief inference method.

[0007] The lifetime distribution mentioned above refers to the random distribution that the product lifetime follows; The aforementioned log-normal distribution refers to the distribution where the random variable... If it follows a log-normal distribution, then its cumulative failure function is: In the formula, The cumulative failure function is a standard normal distribution. and These are the logarithmic mean and logarithmic standard deviation of a log-normal distribution, respectively. The Weibull distribution mentioned above refers to the distribution where the random variable... If it follows a Weibull distribution, then its cumulative failure function is: In the formula, and These are the shape and scale parameters of the Weibull distribution, respectively. Summary of the Invention

[0008] The purpose of this invention is to provide a reliability interval estimation method for the BS lifetime distribution model based on belief inference, that is, a reliability interval estimation method for accelerated life testing. It utilizes the least squares principle to convert accelerated failure data into failure data under normal stress, and finally estimates the confidence interval and one-sided confidence lower limit of reliability based on the belief inference method. The principle of least squares refers to the principle of least squares estimation method; The accelerated failure data mentioned above refers to product failure data obtained through accelerated life testing; The failure data mentioned above refers to the lifespan data obtained when the product fails. The aforementioned belief inference method refers to a method of statistical inference that treats the parameters in the product life distribution model as random variables.

[0009] Technical solution: This invention provides a method for estimating the reliability interval of the Black-Scholes distribution based on belief inference, which requires the following basic settings: Setting 1 for different stress levels Accelerated product lifespan All follow a Black-Scholes distribution, with their probability density function and cumulative failure function as follows: ① Probability density function: (1) in, The probability density function is the standard normal distribution. ② Cumulative failure function: (2) in, The cumulative distribution function of the standard normal distribution. and stress levels The scale and shape parameters of the lower BS distribution; The probability density function is a function that describes the likelihood of a random variable taking a value near a certain point. The cumulative failure function mentioned above refers to the integral of the probability density function; The scale parameter mentioned refers to the parameter in the BS distribution that characterizes the product lifetime scale range; The shape parameter mentioned refers to the parameter in the BS distribution that characterizes the product failure mechanism; Setting 2 for different stress levels Assuming that the failure mechanisms of the products remain unchanged, then the shape parameter of the BS distribution... Keep it at a constant value, and denote this constant value as ; The stress level refers to the magnitude of stress applied to the product when it operates in different environments; The failure mechanism refers to the physical, chemical, or other causes and processes that lead to product failure; Setting 3 for stress level and Acceleration factor Defined as: (3) in, and stress levels and The scaling parameters of the lower BS distribution; Setting 4: For fatigue testing, the product's characteristic lifespan. With cyclic stress level It conforms to the inverse power-law model, and its accelerated model is expressed as follows: (4) in, These are the parameters to be estimated in the model, and they have... ; The inverse power law model refers to an acceleration model in which product lifespan is inversely proportional to the power of the acceleration stress. The aforementioned acceleration model refers to a mathematical model that characterizes the relationship between product lifespan and accelerated stress. Based on the above assumptions, this invention provides a method for estimating the reliability interval of the Black-Scholes distribution based on belief inference, which is implemented through the following steps: Step 1: Estimate the parameters in the acceleration model; Accelerated life testing has a total of Group stress level, i.e. , record The full-life failure data obtained from observations at the group stress level are as follows: (5) in, For the first Number of failure samples under the group stress level; The accelerated life test refers to a test technique that places the product in a harsh stress environment, such as high temperature, high pressure and high salt spray, and causes the product to fail rapidly under high stress conditions, so as to obtain a sufficient amount of product reliability information while saving costs and time. The aforementioned full life failure data refers to the life data obtained after all products tested have failed. The number of failed samples refers to the number of failed products; right Make the following transformation, let: (6) in, For the first Scale parameters of the BS distribution under group stress levels; definition If the residual data are equivalent, then they follow a normal distribution with a mean of 0, i.e. ,in, The shape parameter of the BS distribution; The equivalent residual data refers to the data obtained after transformation by equation (6); make The sum of squares of the equivalent residual data, i.e.: (7) in, For the first The scale parameters of the BS distribution under the group stress level, and according to setting 4, have ,in To accelerate the estimation of parameters in the model; According to the principle of least squares, minimize the expression shown in equation (7). Calculated by equation (8) and The estimated value: (8) In the formula: For the first The stress level of the group ; Step 2: Calculate the acceleration factor between each accelerating stress and the normal stress; Let the normal stress level be... According to setting 3, the acceleration factor between each accelerating stress and the normal stress is calculated by equation (9). : (9) in, These are estimated values ​​for the BS distribution scale parameters under normal stress levels. For the first Estimates of the BS distribution scale parameters under the group of accelerated stress levels. ; The accelerated stress refers to the stress level applied to the product during accelerated life testing; The so-called conventional stress refers to the stress level of the product when it is working in a normal environment; Step 3: Convert accelerated failure data to failure data under normal stress; Based on step two, the acceleration factors between each accelerating stress and the normal stress are obtained. Then, the accelerated failure data obtained under various accelerated stress levels will be used. Failure data converted to normal stress levels ,Right now: (10) in, ; Step 4: Calculate the confidence interval for product reliability under normal stress; Based on the failure life data under normal stress levels obtained in step three... postscript The scale parameter for the BS distribution under normal stress levels is set according to setting 2. Let denot be the shape parameter of the BS distribution under normal stress level; According to equation (6), the equivalent residual data under normal stress level are calculated: (11) in, The total number of failure data in accelerated life testing, i.e. ; remember ,but It follows a standard normal distribution, that is ; make , , , ,but and They are independent of each other and follow normal and chi-square distributions respectively, that is: (12) (13) in, Describing the degrees of freedom as The chi-square distribution; The standard normal distribution mentioned above refers to the distribution where the random variable... If it follows a standard normal distribution, then its probability density function is: ; The chi-square distribution mentioned above refers to the distribution where if there is If 3 independent random variables all follow a standard normal distribution, then the distribution of their sum of squares is the chi-square distribution, where , represents the degrees of freedom of the chi-square distribution; ① , Interval estimation of reliability when all are unknown According to equations (12) and (13) and Distribution and unknown parameters , All are irrelevant and can be used as pivotal quantities; a set is obtained through random sampling. ,in, The set number of random samplings is used; the corresponding parameter values ​​are obtained by solving the system of equations shown in equation (14). ; (14) in, , , ; For a given task time Calculate the reliability value of the product under the corresponding normal stress: (15) in, The cumulative distribution function of the standard normal distribution. ; ② unknown, Interval estimation of reliability when known According to equation (12). Distribution and unknown parameters Irrelevant, can be used as a pivot quantity; a set is obtained through random sampling. ,in, The set random sampling number; solve the equation shown in equation (16) to obtain the corresponding parameter values. ; (16) in, ; For a given task time Calculate the reliability value of the product under the corresponding normal stress: (17) in, The cumulative distribution function of the standard normal distribution. ; The reliability value of the product refers to the probability that the product can complete its specified function within a specified time under specified conditions; ③ unknown, Interval estimation of reliability when known According to equation (12). Distribution and unknown parameters Irrelevant, can be used as a pivot quantity; a set is obtained through random sampling. ,in, The set random sampling number; solving the equation shown in equation (18) yields the corresponding parameter values. ; (18) in, , ; For a given task time Calculate the reliability value of the product under the corresponding normal stress: (19) in, The cumulative distribution function of the standard normal distribution. ; The result obtained from equation (15) or equation (17) or equation (19) Sort in ascending order: (20) For a given confidence level The confidence interval for product reliability is calculated according to equation (21): (twenty one) The one-sided confidence lower limit of reliability is calculated according to equation (22): (twenty two) in, Indicates the significance level.

[0010] The aforementioned interval estimation of reliability refers to an estimation of the possible range of product reliability values; The "least squares method" mentioned in step one refers to: set up It is a pair of observations, the first Group independent variables The corresponding observed value is ,in, Let be the dimension of the independent variable. , The number of groups of observation data; and Satisfying the following specific functional relationship:

[0011] in, These are parameters to be determined. The least squares method refers to finding the objective function that makes equation (23) appear as follows: (twenty three) Parameter estimates that reach the minimum value Methods; In step four, the "pivot quantity" refers to: set up It is a sample With unknown parameters A function, if Distribution and If they are irrelevant, then such functions are called pivot quantities; In summary, this invention relates to a reliability interval estimation method for the BS distribution based on belief inference. It is a method for estimating the reliability interval of the BS distribution based on the least squares principle and belief inference. It provides parameter estimates for a specific accelerated model with high estimation accuracy and simple calculation, demonstrating its superiority in small sample reliability assessment. This invention can be applied to different reliability assessment scenarios, and the method has strong scalability.

[0012] (3) Advantages and benefits: The present invention provides a method for estimating the reliability interval of the BS distribution based on belief inference. Its advantages are: ① Based on the principle of least squares, this invention provides parameter estimates for a specific accelerated failure model with high accuracy, and realizes the conversion of accelerated failure data to failure data under normal stress levels.

[0013] ② This invention is based on the belief inference method to estimate the reliability of the BS distribution over an interval. The calculation is simple and the requirement for a small sample size is low while ensuring the accuracy of the assessment, which shows its superiority in small sample reliability assessment.

[0014] ③ The reliability interval estimation method proposed in this invention can be applied to different reliability assessment scenarios by selecting appropriate pivot quantities, and the method has strong scalability. Attached Figure Description

[0015] Figure 1 This is a flowchart of the method described in this invention. Detailed Implementation

[0016] The present invention will now be described in further detail with reference to the embodiments.

[0017] A certain type of mechanical ball bearing has a stress cycle of 0.75 MPa under normal environmental conditions. Forty samples were taken and divided into four groups of ten, and accelerated life tests were conducted on them according to the accelerated stresses shown in Table 1.

[0018] Table 1. Stress levels in accelerated life tests of a certain mechanical ball bearing Stress cycles / MPa 0.87 0.99 1.09 1.18 The collected complete failure data are shown in Table 2, where the failure lifetime data is in units of 10. 3 change: Table 2 Failure data from accelerated life tests of mechanical ball bearings 1 1670 800 12 73 2 2200 1000 180 98 3 2510 1370 200 117 4 3000 2250 240 135 5 3900 2950 260 175 6 4700 3700 320 262 7 7530 6070 320 270 8 14700 6650 420 350 9 27800 7050 440 386 10 37400 7370 880 456 Step 1: Estimate the parameters in the acceleration model; The parameters in the acceleration model are calculated according to equation (8), and the results are as follows: ; Step 2: Calculate the acceleration factor between each accelerating stress and the normal stress; The normal stress level is 0.75 MPa. The acceleration factor between each accelerating stress and the normal stress is calculated according to equation (9), and the result is as follows: ; Step 3: Convert accelerated failure data to failure data under normal stress; After obtaining the acceleration factors between each accelerated stress and the normal stress in step two, the failure data under each accelerated stress level are converted into failure data under the normal stress level according to equation (10). The converted data are shown in Table 3, where the unit of failure life data is 10. 3 change; Table 3. Failure life data under conventional stress levels obtained after conversion. 1 12790 11 36055 21 2024 31 36564 2 16849 12 45069 22 30366 32 49087 3 19223 13 61745 23 33740 33 58603 4 22976 14 101406 24 40488 34 67619 5 29869 15 132954 25 43862 35 87655 6 35996 16 166756 26 53985 36 131231 7 57670 17 273570 27 53985 37 135238 8 112582 18 299710 28 70855 38 175309 9 212911 19 317738 29 74229 39 193341 10 286434 20 332160 30 148458 40 228403 Step 4: Calculate the confidence interval for product reliability under normal stress; Set the task time as (Unit: 10) 3 (Continue), the number of random samplings is Confidence level is The significance level is Therefore, the confidence interval for the reliability of this mechanical ball bearing under normal stress levels is: ① Shape parameters With scale parameters When both are unknown Random sampling is performed according to the distributions shown in equations (12) and (13). Based on equations (14) and (15) as well as equations (20) to (22), we can obtain... and When all factors are unknown, the confidence interval for product reliability is: [0.5193, 0.5479] The lower confidence limit for one-sided confidence is 0.5223; ② Shape parameters Unknown, scale parameter Case when known Random sampling is performed according to the distribution shown in equation (12). Based on equations (16), (17), and (20) to (22), we can obtain... unknown, The confidence interval for the product reliability is known to be: [0.4651, 0.5364] The lower confidence limit for one-sided confidence is 0.4735; ③ Scale parameters Unknown, shape parameters Case when known Random sampling is performed according to the distribution shown in equation (12). Based on equations (18) to (22), we can obtain... unknown, The confidence interval for the product reliability is known to be: [0.5182, 0.5476] The lower confidence limit for one-sided confidence is 0.5218; In summary, this invention relates to a reliability interval estimation method for the BS distribution based on belief inference. It is a method for estimating the reliability interval of the BS distribution based on the least squares principle and belief inference. The specific steps of this method are: 1. Estimating the parameters in the accelerated model; 2. Calculating the acceleration factor between each accelerated stress and the normal stress; 3. Converting accelerated failure data to failure data under normal stress; 4. Calculating the confidence interval of product reliability under normal stress. This invention is applicable to fields such as reliability assessment of accelerated life tests based on the BS distribution, and has high practicality and strong operability.

Claims

1. A method for estimating the reliability interval of the Black-Scholes distribution based on belief inference requires the following basic settings: Setting 1: For different stress levels Accelerated product lifespan All follow a Black-Scholes distribution, with their probability density function and cumulative failure function as follows: ① Probability density function: （1） in, The probability density function is the standard normal distribution. ② Cumulative failure function: （2） in, The cumulative distribution function of the standard normal distribution. and stress levels The scale and shape parameters of the lower BS distribution; The probability density function is a function that describes the likelihood of a random variable taking a value near a certain point. The cumulative failure function mentioned above refers to the integral of the probability density function; The scale parameter mentioned refers to the parameter in the BS distribution that characterizes the product lifetime scale range; The shape parameter mentioned refers to the parameter in the BS distribution that characterizes the product failure mechanism; Setting 2: For different stress levels Assuming that the failure mechanisms of all products remain unchanged, then the shape parameter of the BS distribution... Keep it as a constant value, and denote this constant value as ; The stress level refers to the magnitude of stress applied to the product when it operates in different environments; The failure mechanism refers to the physical and chemical causes and processes that lead to product failure; Setting 3: For stress levels and Acceleration factor Defined as: （3） in, and stress levels and The scaling parameter of the lower BS distribution; Setting 4: For fatigue testing, the product's characteristic lifespan. With cyclic stress level It conforms to the inverse power-law model, and its accelerated model is expressed as follows: （4） in, These are the parameters to be estimated in the model, and they have... ; The inverse power law model refers to an acceleration model in which product lifespan is inversely proportional to the power of the acceleration stress. The aforementioned acceleration model refers to a mathematical model that characterizes the relationship between product lifespan and accelerated stress. Based on the above settings, the specific steps are as follows: Step 1: Estimate the parameters in the acceleration model; Accelerated life testing has a total of Group stress level, i.e. , record The full-life failure data obtained from observations at the group stress level are as follows: （5） in, For the first Number of failure samples under the group stress level; The accelerated life test refers to a test technique that places the product in a harsh stress environment, including high temperature, high pressure and high salt spray, and causes the product to fail rapidly under high stress conditions, so as to obtain a sufficient amount of product reliability information while saving costs and time. The aforementioned full life failure data refers to the life data obtained after all products tested have failed. The number of failed samples refers to the number of failed products; right Make the following transformation, let: （6） in, For the first Scale parameters of the BS distribution under group stress levels; definition If the residual data are equivalent, then they follow a normal distribution with a mean of 0, i.e. ,in, The shape parameter of the BS distribution; The equivalent residual data refers to the data obtained after transformation by equation (6); make The sum of squares of the equivalent residual data, i.e.: （7） in, For the first The scale parameters of the BS distribution under the group stress level, and according to setting 4, have ,in To accelerate the estimation of parameters in the model; According to the principle of least squares, minimize the expression shown in equation (7). Calculated by equation (8) and The estimated value: （8） In the formula: For the first The stress level of the group ; Step 2: Calculate the acceleration factor between each accelerating stress and the normal stress; Let the normal stress level be... According to setting 3, the acceleration factor between each accelerating stress and the normal stress is calculated by equation (9). : （9） in, These are estimated values ​​for the BS distribution scale parameters under normal stress levels. For the first Estimates of the BS distribution scale parameters under the group of accelerated stress levels. ; The accelerated stress refers to the stress level applied to the product during accelerated life testing; The so-called conventional stress refers to the stress level of the product when it is working in a normal environment; Step 3: Convert accelerated failure data to failure data under normal stress; Based on step two, the acceleration factors between each accelerating stress and the normal stress are obtained. Then, the accelerated failure data obtained under various accelerated stress levels will be used. Failure data converted to normal stress levels ,Right now: （10） in, ; Step 4: Calculate the confidence interval for product reliability under normal stress; Based on the failure life data under normal stress levels obtained in step three... postscript The scale parameter for the BS distribution under normal stress levels is set according to setting 2. Let denot be the shape parameter of the BS distribution under normal stress level; According to equation (6), the equivalent residual data under normal stress level are calculated: （11） in, The total number of failure data in accelerated life testing, i.e. ; remember ,but It follows a standard normal distribution, that is ; make , , , ,but and They are independent of each other and follow normal and chi-square distributions respectively, that is: （12） （13） in, Describing the degrees of freedom as The chi-square distribution; If random variable If it follows a standard normal distribution, then its probability density function is: ; The chi-square distribution mentioned above refers to the distribution where if there is If 3 independent random variables all follow a standard normal distribution, then the distribution of their sum of squares is the chi-square distribution, where , represents the degrees of freedom of the chi-square distribution; ① , Interval estimation of reliability when all are unknown According to equations (12) and (13) and Distribution and unknown parameters , All are irrelevant and serve as pivotal quantities; a set is obtained through random sampling. ,in, The set number of random samplings is used; the corresponding parameter values ​​are obtained by solving the system of equations shown in equation (14). ; （14） in, , , ; For a given task time Calculate the reliability value of the product under the corresponding normal stress: （15） in, The cumulative distribution function of the standard normal distribution. ; ② unknown, Interval estimation of reliability when known According to equation (12). Distribution and unknown parameters Irrelevant, as a pivotal quantity; a set is obtained through random sampling. ,in, The set random sampling number; solve the equation shown in equation (16) to obtain the corresponding parameter values. ; （16） in, ; For a given task time Calculate the reliability value of the product under the corresponding normal stress: （17） in, The cumulative distribution function of the standard normal distribution. ; The reliability value of the product refers to the probability that the product can complete its specified function within a specified time under specified conditions; ③ unknown, Interval estimation of reliability when known According to equation (12). Distribution and unknown parameters Irrelevant, as a pivotal quantity; a set is obtained through random sampling. ,in, The set random sampling number; solve the equation shown in equation (18) to obtain the corresponding parameter values. ; （18） in, , ; For a given task time Calculate the reliability value of the product under the corresponding normal stress: （19） in, The cumulative distribution function of the standard normal distribution. ; The result obtained from equation (15) or equation (17) or equation (19) Sort in ascending order: （20） For a given confidence level The confidence interval for product reliability is calculated according to equation (21): （21） The one-sided confidence lower limit of reliability is calculated according to equation (22): （22） in, Indicates the significance level; The aforementioned interval estimation of reliability refers to an estimation of the possible range of product reliability values.

2. The method for estimating the reliability interval of the Black-Scholes distribution based on belief inference according to claim 1, characterized in that: The least squares method mentioned in step one refers to: set up It is a pair of observations, the first Group independent variables The corresponding observed value is ,in, Let be the dimension of the independent variable. , The number of groups of observation data; and Satisfying the following specific functional relationship: ； in, These are parameters to be determined. The least squares method refers to finding the objective function that makes equation (23) appear as follows: （23） Parameter estimates that reach the minimum value The method.

3. The method for estimating the reliability interval of the Black-Scholes distribution based on belief inference according to claim 1, characterized in that: The pivot quantity mentioned in step four refers to: Let It is a sample With unknown parameters A function, if Distribution and If it is irrelevant, then such a function is called a pivot quantity.

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