A reliability evaluation method for nonlinear random effect parameter dependent degradation process

Abstract

The application discloses a reliability evaluation method for a nonlinear random effect parameter related degradation process, obtains sample data; according to the sample data, calculates parameter estimation values of sample distribution; establishes distribution models of various parameters; utilizes the parameter estimation values of the sample distribution and maximum likelihood estimation to calculate various parameters in the distribution models; utilizes a total probability formula to calculate a degradation amount distribution density; after linear change of the degradation amount, T distribution is obtained. According to the T distribution, a confidence interval of the degradation amount is calculated; a failure threshold of degradation is set, and reliability of a product is calculated. The application considers the correlation between parameters in distribution, obtains simple form degradation amount distribution, and reduces reliability evaluation calculation difficulty.

Description

Technical Field

[0001] This invention relates to a reliability assessment method for nonlinear random effect parameter-related degradation processes, belonging to the field of reliability assessment technology. Background Technology

[0002] Current reliability assessments often employ fixed-effects degradation models, which frequently fail to capture the randomness of the degradation process. Furthermore, the statistically or physically meaningful correlation between scale and shape parameters, and between diffusion and drift parameters during degradation is a rarely considered issue in current research. Moreover, current engineering methods for measuring product degradation often require destructive testing, making it impossible to model individual degradation levels using data from the same batch measured at different times.

[0003] The following methods exist in the prior art:

[0004] (1) Reliability assessment method for degraded data based on non-destructive measurement

[0005] This method directly models the degradation data and fits the trajectory in a stochastic process or distribution manner. It directly infers the overall product reliability through the properties of the performance degradation. However, this method can only be applied to degradation data of the same individual and is non-destructive degradation data. It cannot be applied to degradation data that is commonly used in engineering for destructive measurements.

[0006] (2) Reliability assessment method based on regression fitting of product degradation trajectory

[0007] These methods fit the product degradation trajectory using different regression methods. The models are highly susceptible to data influence, prone to overfitting and underfitting, and have insufficient generalization ability. Alternatively, they may use regression models that independently fit the product degradation distribution parameters without considering further research on parameter correlation.

[0008] (3) Degradation-related product reliability assessment based on Copula method

[0009] This type of method selects an appropriate Copula function to describe the correlation between individuals based on different data characteristics, and has a wide range of applications. However, it introduces new parameters, which increases the difficulty of parameter estimation and often requires the use of complex numerical calculation methods, making it unsuitable for practical engineering applications. Summary of the Invention

[0010] The technical problem this invention aims to solve is to overcome the shortcomings of existing technologies and provide a nonlinear random-effects parameter-related reliability assessment method for degradation processes. Considering the time-varying nature of the overall characteristics, a power function is used to construct a nonlinear time-varying model to improve the model's generalization ability. The mean degradation amount is assumed to follow a normal distribution with a nonlinear mean and a variance correlated with the degradation amount variance. The degradation amount variance parameter follows a Gamma distribution at any given time. The maximum likelihood estimate of the parameter is calculated step by step. The density function of degradation amount over time is derived using the law of total probability. It is found that after linear change, the degradation amount follows a T-distribution at a given time, and the confidence interval for degradation amount and the product reliability function are given based on this result. To achieve the above objective, this invention provides a nonlinear random-effects parameter-related reliability assessment method for degradation processes, including:

[0011] Obtain sample data;

[0012] Based on the sample data, calculate the parameter estimates of the sample distribution;

[0013] Establish distribution models for each parameter;

[0014] Calculate the parameters in the distribution model using the parameter estimates and maximum likelihood estimates of the sample distribution;

[0015] The probability density of degradation is calculated using the law of total probability.

[0016] After linear transformation, the degradation amount yields a T-distribution.

[0017] Firstly, calculate the confidence interval for the degradation amount based on the T-distribution;

[0018] Set a failure threshold for degradation and calculate the product's reliability.

[0019] First, obtain sample data, which is achieved through the following steps:

[0020] Obtain n products whose degradation level is to be measured;

[0021] The degradation was measured at equal time intervals and used as sample data x. ij i = 1, 2, ..., n; j = 1, 2, ..., m, where m represents the total number of measurement batches; the sample data distribution follows a normal distribution.

[0022]

[0023] In the formula, f(x) ij |μ j , σ j ) represents the density of the sample data distribution. Let μ be the variance of the sample data in the j-th batch. jLet be the mean of the sample data for the j-th batch.

[0024] First, based on the sample data, the parameter estimates of the sample distribution are calculated through the following steps:

[0025] Based on the sample data, the parameter estimates of the sample distribution are obtained by maximum likelihood estimation:

[0026]

[0027] In the formula, Let the mean of the sample data of n products be _____. Let n be the variance estimates of the sample data for n products. This is an estimate of the mean of a sample of n products.

[0028] First, establish the distribution model for each parameter, which is achieved through the following steps:

[0029] Establish a distribution model: Let μ be the mean of the sample data at any given time t. t Given a sample of data that follows a normal distribution at any time t, the mean μ is... t The sample data exhibits non-linear properties over time, with the mean μ t Distribution variance and degradation amount x t variance Same, and The time-varying characteristics are:

[0030] The reciprocal follows a Gamma distribution at any given time t: v t ~Gamma(αt) -1 ,β), where α, β, b, and μ are all distribution parameters of the variance of the distribution at any given time t, which are unknown.

[0031] First, using the parameter estimates and maximum likelihood estimates of the sample distribution, the parameters in the distribution model are calculated through the following steps:

[0032] Will and Using the Newton-Raphson iterative method, obtain the maximum likelihood estimates of the distribution parameters α and β of the variance of the distribution at any given time t. but From the law of total probability and v t The distribution of μ is obtained. t Distribution:

[0033]

[0034] In the formula, f(.) is the probability density function, and f(μ) is the probability density function.t ) is μ t The distribution density, f(μ) t |v t ) is μ t The conditional distribution density, f(v) t ) for v t The distribution density, where Γ is the Gamma function notation;

[0035] The maximum likelihood estimates of the distribution parameters μ and b of the variance of the distribution at any given time t can be obtained using Newton's iterative method. based on Get μ t σ t 2 The distribution over time is as follows:

[0036]

[0037] In the formula, Labeled as normal distribution Label the Gamma distribution.

[0038] Firstly, the degradation distribution density is calculated using the law of total probability, through the following steps:

[0039]

[0040] In the formula, f(x) t Let f(x) be the data distribution density function at any given time t. t ,μ t ,v t ) is x t μ t and v t The joint probability density function, f(x) t |μ t ,v t Let be the conditional probability density function of the data at any given time t;

[0041] After the degradation amount undergoes a linear transformation, a T-distribution is obtained, which is achieved through the following steps:

[0042] For x t Perform variable substitution:

[0043]

[0044] In the formula, γ and y t All are substitution variables;

[0045] y t The expression for the distribution density function is:

[0046]

[0047] In the formula, f(y) t ) is y t The probability density function of the distribution;

[0048] By y t The probability density function of y is obtained. t ~T(γ) is a T-distribution.

[0049] Firstly, based on the T-distribution, the confidence interval for the degradation is calculated through the following steps:

[0050] Calculate x t The 1-δ confidence interval is

[0051]

[0052] In the formula, δ is the confidence requirement, and T δ / 2 (γ) is the δ / 2 quantile of the T distribution, T 1-δ / 2 (γ) is the 1-δ / 2 quantile of the T distribution;

[0053] Setting a degradation failure threshold and calculating product reliability are achieved through the following steps:

[0054] If the degradation failure threshold is set to D, then the reliability of the product at any given time t is:

[0055]

[0056] In the formula, R(t) is the reliability function, and P(x) is the reliability function. t >D) represents the probability that the product degradation exceeds the failure threshold.

[0057] For P(x) t >D) Using y t Replace x t The expression obtained afterwards.

[0058] An electronic device includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the steps of any of the methods described above.

[0059] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of any of the methods described above.

[0060] The beneficial effects achieved by this invention are as follows:

[0061] This invention considers the time-varying nature of overall characteristics. To enhance the model's generalization ability, a power function is used to construct a nonlinear time-varying model. The mean degradation amount is assumed to follow a normal distribution with a nonlinear mean and a variance correlated with the degradation amount variance. The degradation amount variance parameter follows a Gamma distribution at any given time. The maximum likelihood estimates of the parameters are calculated step-by-step. The density function of degradation amount over time is derived using the law of total probability. It is found that after a linear change at a given time, the degradation amount follows a T-distribution. Based on this result, confidence intervals for degradation amount and a product reliability function are provided.

[0062] The method of this invention is applicable to degradation data from both destructive and non-destructive measurements. It takes into account the correlation between parameters in the distribution, obtains a degradation distribution with a simple form, and reduces the difficulty of reliability assessment calculation. Attached Figure Description

[0063] Figure 1 This is a flowchart of the present invention. Detailed Implementation

[0064] The following embodiments are only used to illustrate the technical solutions of the present invention more clearly, and should not be used to limit the scope of protection of the present invention.

[0065] A reliability assessment method for nonlinear random effects parameter-related degradation processes, comprising:

[0066] Obtain sample data;

[0067] Based on the sample data, calculate the parameter estimates of the sample distribution;

[0068] Establish distribution models for each parameter;

[0069] Calculate the parameters in the distribution model using the parameter estimates and maximum likelihood estimates of the sample distribution;

[0070] The probability density of degradation is calculated using the law of total probability.

[0071] After linear transformation, the degradation amount is distributed as a T-distribution.

[0072] Calculate the confidence interval for the degradation amount based on the T-distribution;

[0073] Set a failure threshold for degradation and calculate the product's reliability.

[0074] To achieve the above objectives, the present invention employs the following technical solution, as follows: Figure 1 As shown:

[0075] Step 1: Measure the degradation of each of the n products to obtain sample data;

[0076] The obtained sample data is recorded as xij ,i=1,2,…,n;j=1,2,…,m, where m represents the total number of measurement batches, and the sample data distribution follows a normal distribution, i.e.:

[0077] The density function of the sample data distribution is

[0078]

[0079] In the formula, f(x) ij |μ j , σ j ) represents the density (function) of the sample data distribution. Let μ be the variance of the sample data in the j-th batch. j Let be the mean of the sample data for the j-th batch.

[0080] Step 2: Based on the sample data, obtain the parameter estimates of the sample distribution using maximum likelihood estimation:

[0081]

[0082] In the formula, Let the mean of the sample data of n products be _____. Let n be the variance estimates of the sample data for n products. This is an estimate of the mean of a sample of n products;

[0083] Step 3: Establish the distribution model for each parameter: Let μ be the mean of the sample data at any given time t. t Given a sample of data that follows a normal distribution at any time t, the mean μ is... t The sample data exhibits non-linear properties over time, with the mean μ t Distribution variance and degradation amount x t variance Same, and The time-varying characteristics are:

[0084] The reciprocal at any given time t follows a Gamma distribution, i.e. v t ~Gamma(αt) -1 ,β), where α, β, b, and μ are all distribution parameters of the variance of the distribution at a given arbitrary time t, which are unknown.

[0085] Step 4: and Using the Newton-Raphson iterative method, obtain the maximum likelihood estimates of the distribution parameters α and β of the variance of the distribution at any given time t. but

[0086] Step 5: Using the law of total probability and v t The distribution of μ is obtained. t Distribution:

[0087]

[0088] In the formula, f(μ) t ) is μ t The distribution density, f(μ) t |v t ) is μ t The conditional distribution density (function), f(v) t ) for v t The distribution density (function), where Γ is the Gamma function notation;

[0089] Step 6: Use Newton's iterative method to obtain the maximum likelihood estimates of the distribution parameters μ and b of the variance of the distribution at any given time t.

[0090] based on Get μ t σ t 2 The distribution that changes over time is

[0091]

[0092] In the formula, Labeled as normal distribution Label the Gamma distribution;

[0093] Step 7: Calculate the degradation amount x at any given time t using the law of total probability. t Distribution density:

[0094]

[0095] In the formula, f(x) t Let f(x) be the data distribution density function at any given time t. t ,μ t ,v t ) is x t μ t and v t The joint probability density function, f(x) t |μ t ,v t Let be the conditional probability density function of the data at any given time t;

[0096] Step 8: For x t Perform variable substitution

[0097]

[0098] In the formula, γ and y t All are substitution variables;

[0099] Step 9: y t The expression for the distribution density function is:

[0100]

[0101] In the formula, f(y) t ) is y t The probability density function of the distribution;

[0102] By y t The probability density function of y is obtained. t ~T(γ) is a T-distribution;

[0103] Step 10: Calculate x t The 1-δ confidence interval is

[0104]

[0105] In the formula, δ is the confidence requirement, and T δ / 2 (γ) is the δ / 2 quantile of the T distribution, T 1-δ / 2 (γ) is the 1-δ / 2 quantile of the T distribution;

[0106] Step 11: Set the degradation failure threshold to D, then the product's reliability at any given time t is...

[0107]

[0108] In the formula, R(t) is the reliability function, and P(x) is the reliability function. t >D) represents the probability that the product degradation exceeds the failure threshold. For P(x) t >D) Using y t Replace x t The expression obtained afterwards.

[0109] This invention considers the time-varying nature of overall characteristics. To enhance the model's generalization ability, a power function is used to construct a nonlinear time-varying model. The mean degradation amount is assumed to follow a normal distribution with a nonlinear mean and a variance correlated with the degradation amount variance. The degradation amount variance parameter follows a Gamma distribution at any given time. The maximum likelihood estimates of the parameters are calculated step-by-step. The density function of degradation amount over time is derived using the law of total probability. It is found that after a linear change, the degradation amount at a given time follows a T-distribution. Based on this result, confidence intervals for degradation amount and a product reliability function are provided.

[0110] This invention performs maximum likelihood estimation on product degradation data to obtain degradation distribution parameters at each measurement time, models the time-varying nature of the parameters, considers the correlation between parameters during the modeling process, simplifies the subsequent full probability calculation results, and performs reliability analysis on the degradation data based on the simplified results.

[0111] Suppose that at any given time, the mean parameter of degradation is also normally distributed, and the mean of the distribution changes with time as a power function. At the same time, the variance parameter of degradation follows a Gamma distribution at any given time.

[0112] In the time-varying law modeling of degradation distribution parameters, it is assumed that the mean parameter of degradation has a related variance with the degradation. Based on this assumption, it is calculated by the law of total probability. Given any time, the degradation distribution is a T-distribution, which simplifies the subsequent reliability inference.

[0113] This invention conveniently provides the analytical expression of the confidence interval of degradation at a given confidence level at each time point based on the quantile table of the T-distribution, without the need for complex integral calculations. Similarly, the analytical expression of product reliability at a given failure threshold can be obtained simply by using the distribution function of the T-distribution.

[0114] Other alternative solutions can also achieve the purpose of the invention:

[0115] (1) The parameter estimation method can be replaced by point estimation, etc.

[0116] (2) In addition to numerical methods, the MCMC algorithm can be used to solve the maximum likelihood estimation (MLE).

[0117] (3) Nonlinear models can use exponential models instead of power functions.

[0118] Compared with the prior art, the present invention has the following advantages:

[0119] (1) It can be applied to degraded data obtained by destructive measurements.

[0120] (2) Applicable to nonlinear degradation processes.

[0121] (3) A simple degradation distribution is obtained, which reduces the difficulty of reliability assessment calculation.

[0122] (4) The correlation between parameters in the distribution is taken into account.

[0123] Definitions of abbreviations and key terms:

[0124] MLE – Maximum Likelihood Estimation;

[0125] Random effects—distributions in which parameters of degradation exhibit time-varying characteristics;

[0126] Fixed effect—the parameters in the degradation distribution remain constant;

[0127] Nonlinear time-varying—a process in which the change over time is nonlinear;

[0128] Destructive measurement – ​​the measurement of degradation values ​​obtained from changes in the strength limit.

[0129] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. 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... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0130] 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.

[0131] 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.

[0132] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the technical principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A reliability assessment method for nonlinear random effects parameter-related degradation processes, characterized in that, include: Obtain sample data of construction machinery products; Based on the sample data, calculate the parameter estimates of the sample distribution; Establish distribution models for each parameter; Calculate the parameters in the distribution model using the parameter estimates and maximum likelihood estimates of the sample distribution; The probability density of degradation is calculated using the law of total probability. After linear transformation, the degradation amount is distributed as a T-distribution. Calculate the confidence interval for the degradation amount based on the T-distribution; Set a degradation failure threshold and calculate the product's reliability; The distribution model for each parameter is established through the following steps: Establish a distribution model: Suppose that at any given time... The mean of the sample data μ t Following a normal distribution, given any time... The mean of the sample data μ t The sample data exhibits non-linear properties over time, with the mean μ t Distribution variance and degradation amount x t variance Same, and The time-varying characteristics are: The reciprocal follows a Gamma distribution at any given time t: ,in , α、 b and All are distribution parameters of the variance of the distribution at any given time t, which are unknown. The parameters in the distribution model are calculated using the parameter estimates and maximum likelihood estimates of the sample distribution, through the following steps: Will and Using the Newton-Raphson iteration method, obtain the distribution parameter α of the variance of the distribution at any given time t. Maximum likelihood estimate , ,but ; From the law of total probability and The distribution of the data is obtained. Distribution: ； In the formula, f(.) is the probability density function, and f(μ) is the probability density function. t ) is μ t Distribution density, For μ t Conditional distribution density, for The distribution density, where Γ is the Gamma function notation; The distribution parameter μ of the variance of the distribution at any given time t can be obtained using Newton's iterative method. Maximum likelihood estimate , ; based on , ,get , The distribution over time is as follows: ， In the formula, Labeled as normal distribution Label the Gamma distribution; The degradation distribution density is calculated using the law of total probability through the following steps: ； In the formula, Given the data distribution density function at any time t, for , and The joint distribution density function, Let be the conditional probability density function of the data at any given time t; After the degradation amount undergoes a linear transformation, a T-distribution is obtained, which is achieved through the following steps: right Perform variable substitution: ； In the formula, γ and All are substitution variables; The expression for the distribution density function is: ； In the formula, For y t The probability density function of the distribution; Depend on The distribution density function, to know It follows a T-distribution; To obtain sample data, follow these steps: Obtain n products whose degradation level is to be measured; The amount of degradation was measured at equal time intervals and used as sample data. m represents the total number of measurement batches; The sample data distribution follows a normal distribution: ； In the formula, The density of the sample data distribution. Let be the variance of the sample data in the j-th batch. Let be the mean of the sample data for the j-th batch.

2. The reliability assessment method for nonlinear random effect parameter-related degradation processes according to claim 1, characterized in that, Based on the sample data, the parameter estimates of the sample distribution are calculated through the following steps: Based on the sample data, the parameter estimates of the sample distribution are obtained by maximum likelihood estimation: ， ； In the formula, Let the mean of the sample data of n products be _____. Let n be the variance estimates of the sample data for n products. This is an estimate of the mean of a sample of n products.

3. The reliability assessment method for nonlinear random effect parameter-related degradation processes according to claim 2, characterized in that, Based on the T-distribution, the confidence interval for the degradation amount is calculated through the following steps: calculate of Confidence interval is ； In the formula, For confidence requirements, T-distribution Quantiles T-distribution quantiles; Setting a degradation failure threshold and calculating product reliability are achieved through the following steps: Set the degradation failure threshold to Then the product at any given time The reliability is: ； In the formula, For reliability function, This represents the probability that the product degradation exceeds the failure threshold. for use replace The expression obtained afterwards.

4. An electronic device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the steps of the method according to any one of claims 1 to 3.

5. A computer-readable storage medium having a computer program stored thereon, characterized in that, When executed by a processor, the computer program implements the steps of the method according to any one of claims 1 to 3.

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