statistical analysis method of dependent competing failure model under middle censoring test

Abstract

The present application relates to the technical field of reliability statistical analysis, in particular to a statistical analysis method of dependent competing failure model under middle censoring test, comprising the following steps: setting product failure related assumptions, designing middle censoring test scheme, performing parameter estimation based on observed samples, and constructing confidence / credible interval of parameters; the present application has the beneficial effects that: the product failure mechanism dependence characteristics and the middle censoring test scene are accurately adapted, the mechanism correlation is described through MOBW distribution, the processing framework of various types of censoring data is unified, and the adaptation pain points of traditional methods are solved; three flexible and optional parameter estimation methods are provided, the accuracy, efficiency and small sample stability are considered, a scientific confidence / credible interval construction method is matched, and the result rigor is ensured; the method has strong cross-field applicability, high calculation efficiency and reliable results, can provide accurate data support for reliability evaluation and decision-making in the fields of medicine and engineering, and has wide practical value and promotion prospect.

Description

Technical Field

[0001] This invention relates to the field of reliability statistical analysis technology, specifically to a statistical analysis method for dependent competing failure models under middle truncation tests. Background Technology

[0002] In modern industrial production, healthcare, and other fields, product reliability assessment (including equipment and biological organisms) is a core step in ensuring operational safety and optimizing resource allocation. However, the failure processes of products in real-world scenarios are often complex, posing numerous technical challenges to reliability analysis, specifically in the following two aspects: On the one hand, the interdependence of product failure mechanisms is widespread, making the traditional independent failure assumption difficult to reflect reality. Most product failures are not caused by a single mechanism, but rather by the combined effect of multiple mechanisms. Furthermore, these mechanisms are not independent but significantly interact—for example, in power-communication network systems, a failure in the information and communication system can trigger a power system runaway, with the two failure mechanisms being interrelated; in the medical field, the risk of blindness in diabetic patients is simultaneously affected by treatment failure and natural disease progression, with these two failure paths having a synergistic effect. Traditional reliability analysis methods often assume that each failure mechanism is independent, neglecting the interdependence between mechanisms, leading to significant deviations between the model and the actual failure process, and insufficient accuracy in the assessment results.

[0003] On the other hand, the inaccurate observation of failure time is a prominent issue in truncated tests, and existing truncated processing solutions have limitations. In product life testing, due to limitations such as test cycle and observation conditions, it is often impossible to obtain the precise failure time of all products: some products fail before the test begins (left truncated), some fail after the test ends (right truncated), and some can only be determined to have failed within a certain time period (interval truncated). Existing truncated test schemes (such as Type I truncated and Type II truncated) are mostly designed for a single truncated type, lacking a unified processing framework, and are difficult to adapt to the precise analysis of "interval data". At the same time, when truncated data is superimposed with the failure mechanism, traditional parameter estimation methods (such as direct maximum likelihood estimation) are prone to large parameter estimation biases or even failure to converge because they cannot effectively handle the implicit information of interval data, further affecting the scientific nature of reliability assessment. Therefore, it is necessary to propose a statistical analysis method for dependent competing failure models under middle truncated tests. Summary of the Invention

[0004] To address the problems in the existing technology, this invention provides a statistical analysis method for dependent competition failure models under middle truncation experiments.

[0005] The technical solution adopted by this invention to solve its technical problem is: a statistical analysis method for dependent competition failure models under middle truncation experiments, comprising the following steps: S1. Set product failure-related assumptions: The product has two dependent failure mechanisms; the occurrence of either mechanism leads to product failure. Let the occurrence times of the two failure mechanisms be denoted as follows: and The product's expiration time is ; Obeying shape parameters Scale parameters are and The MOBW distribution, denoted as This distribution consists of mutually independent , and (and (independent) through \ Constructed from this, its joint survival function is: ; S2. Design a middle truncation test scheme: Select A number of independent, competing products that fail, with a pre-defined number of products whose exact failure time can be observed. ; Each product corresponds to a random time interval that follows an unknown two-dimensional distribution. When the product expires At that time, only the interval failure data is acquired; Define indicator functions ,in Indicates data that is invalid within a range. This indicates that the first failure mechanism leads to failure. This indicates that the failure was caused by a second failure mechanism; Finally, the observation samples were obtained. ,forward One is the exact failure time data, the remaining ones are... Each is an interval failure data; S3. Parameter estimation based on observed samples: Solve for the shape parameters of the MOBW distribution using at least one of the following methods: maximum likelihood estimation (EM algorithm), midpoint approximation estimation, and Bayesian point estimation. Scale parameters and ; S4. Construct confidence / confidence intervals for parameters: Use the asymptotic confidence interval or maximum a posteriori density confidence interval method to determine the confidence / confidence intervals of the parameter estimates obtained in step 3.

[0006] Specifically, the maximum likelihood estimation described in step 3 is implemented using the EM algorithm, and the specific steps are as follows: 1) Construct likelihood and log-likelihood functions based on observed samples to reflect the correlation between samples and parameters; 2) Given initial values ​​for parameters ; 3) No. The next iteration Step: Calculate the expected failure time, logarithmic expected failure time, and failure mechanism priority probability corresponding to the interval failure data; 4) No. The next iteration Step: Based on The step involves maximizing the expectation of the log-likelihood function to obtain parameter estimates. ; 5) Repeat steps 3-4 until the parameter change is less than the preset decimal. The parameter value at this point is the maximum likelihood estimate.

[0007] Specifically, the implementation process of the midpoint approximation described in step 3 is as follows: 1) Using interval data midpoint To establish a complete observation sample as the exact product failure time; 2) Construct a log-likelihood function based on this sample, take the first-order partial derivatives with respect to the parameters and set them equal to zero to obtain the parametric equations; 3) Solve the parametric equations using the fixed-point iteration method to obtain... , , The estimated value, where the scale parameter satisfies: ; .

[0008] in, , This is an approximate value of the midpoint of the failure time corresponding to the two failure mechanisms; Specifically, the implementation process of Bayesian point estimation in step 3 is as follows: 1) Define the prior distribution: Obtaining hyperparameters The gamma distribution, ( , ) obeys the hyperparameters as The Dirichlet distribution; 2) By combining the prior distribution and the likelihood function of the observed samples, the joint posterior density function of the parameters is obtained; 3) Use the Gibbs sampling algorithm to draw N sets of random parameter samples, and take the mean of the samples to obtain the Bayesian point estimate: ; ; .

[0009] Specifically, the method for constructing the asymptotic confidence interval in step 4 is as follows: 1) Construct the Fisher information matrix using the negative second-order partial derivatives of the log-likelihood function, and then solve for the parameter variance-covariance matrix; 2) Based on the asymptotic normality of maximum likelihood estimation, construct the asymptotic confidence intervals for the parameters: , , ,in Upper side of the standard normal distribution quantiles, For confidence level, This represents the variance of the corresponding parameter.

[0010] Specifically, the method for constructing the maximum a posteriori density confidence interval in step 4 is as follows: 1) At that time Select those that meet the confidence level The shortest interval is taken as the reliable interval; 2) When When the parameter samples obtained by Gibbs sampling are sorted in ascending order, the shortest interval that meets the confidence level requirement is selected as the approximate credible interval.

[0011] The beneficial effects of this invention: The statistical analysis method for the dependent competition failure model under the middle truncation test described in this invention: (i) Accurately adapt to complex failure scenarios and solve the dual challenges of dependency and truncated data: For the common dependency relationships between product failure mechanisms (such as the mutual influence between power and communication systems, and the correlation of multiple factors causing failure due to disease), the MOBW distribution is used to intuitively characterize the dependency characteristics of two failure mechanisms. At the same time, common test scenarios such as left truncation, right truncation, and interval truncation are uniformly incorporated into the middle truncation framework, which effectively solves the technical pain point that traditional methods cannot handle "failure mechanism dependency" and "failure time cannot be accurately observed" at the same time, and greatly improves the adaptability of reliability analysis in complex scenarios.

[0012] (II) Multiple parameter estimation methods are available for flexible selection, balancing accuracy and efficiency: Three core parameter solution methods are provided: maximum likelihood estimation (EM algorithm), midpoint approximation estimation, and Bayesian point estimation. The appropriate method can be selected according to actual needs: the EM algorithm completes the implicit information of the interval data through iteration, ensuring the accuracy of parameter estimation; the midpoint approximation simplifies the interval data processing flow and significantly improves the computational efficiency; the Bayesian point estimation combines prior experience and observational data, and can still obtain stable and reliable estimation results in scenarios with small samples or insufficient data. The three methods complement each other to meet the analysis needs under different experimental scales and data quality.

[0013] (III) Scientific construction of confidence / credibility intervals ensures the reliability and practicality of the results: By using two methods, namely asymptotic confidence intervals (based on Fisher's information matrix and asymptotic normality) and maximum posterior density confidence intervals (adapted to Bayesian estimation logic), the parameter estimates are provided with strict statistical confidence guarantees, the range of parameter fluctuations is clearly defined, and the uncertainty risk brought about by a single estimate is avoided; at the same time, the confidence interval construction process takes into account both independent and dependent parameter cases, further improving the rigor of the results and providing a solid data analysis foundation for subsequent reliability assessment.

[0014] (iv) Strong cross-domain applicability and empowers reliability assessment in multiple scenarios: The method is not limited to specific industries and can be directly applied to multiple fields such as medicine (disease failure mechanism analysis, patient prognosis assessment) and engineering (power-communication network, life prediction of complex equipment). It can quantify the impact weight of different failure mechanisms and provide accurate data support for practical decisions such as product design optimization, maintenance plan formulation and treatment plan selection. It has broad practical value and promotion prospects. Attached Figure Description

[0015] The present invention will be further described below with reference to the accompanying drawings and embodiments; Figure 1 This is a flowchart illustrating the statistical analysis method for the dependent competition failure model under the middle truncation test provided by the present invention. Detailed Implementation

[0016] To make the technical means, creative features, objectives and effects of this invention easier to understand, the invention will be further described below in conjunction with specific embodiments.

[0017] like Figure 1 As shown, the present invention provides the following technical solution: Example: Statistical analysis method for dependent competition failure model under middle truncation test, including the following steps: S1. Set product failure-related assumptions: The product has two dependent failure mechanisms; the occurrence of either mechanism leads to product failure. Let the occurrence times of the two failure mechanisms be denoted as follows: and The product's expiration time is ; Obeying shape parameters Scale parameters are and The MOBW distribution, denoted as This distribution consists of mutually independent , and (and (independent) through \ Constructed from this, its joint survival function is: ; S2. Design a middle truncation test scheme: Select A number of independent, competing products that fail, with a pre-defined number of products whose exact failure time can be observed. ; Each product corresponds to a random time interval that follows an unknown two-dimensional distribution. When the product expires At that time, only the interval failure data is acquired; Define indicator functions ,in Indicates data that is invalid within a range. This indicates that the first failure mechanism leads to failure. This indicates that the failure was caused by a second failure mechanism; Finally, the observation samples were obtained. ,forward One is the exact failure time data, the remaining ones are... Each is an interval failure data; S3. Parameter estimation based on observed samples: Solve for the shape parameters of the MOBW distribution using at least one of the following methods: maximum likelihood estimation (EM algorithm), midpoint approximation estimation, and Bayesian point estimation. Scale parameters and ; S4. Construct confidence / confidence intervals for parameters: Use the asymptotic confidence interval or maximum a posteriori density confidence interval method to determine the confidence / confidence intervals of the parameter estimates obtained in step 3. The maximum likelihood estimation in step 3 is implemented using the EM algorithm, and the specific steps are as follows: 1) Construct likelihood and log-likelihood functions based on observed samples to reflect the correlation between samples and parameters; 2) Given initial values ​​for parameters ; 3) No. The next iteration Step: Calculate the expected failure time, logarithmic expected failure time, and failure mechanism priority probability corresponding to the interval failure data; 4) No. The next iteration Step: Based on The step involves maximizing the expectation of the log-likelihood function to obtain parameter estimates. ; 5) Repeat steps 3-4 until the parameter change is less than the preset decimal. The parameter value at this point is the maximum likelihood estimate. The implementation process of the midpoint approximation in step 3 is as follows: 1) Using interval data midpoint To establish a complete observation sample as the exact product failure time; 2) Construct a log-likelihood function based on this sample, take the first-order partial derivatives with respect to the parameters and set them equal to zero to obtain the parametric equations; 3) Solve the parametric equations using the fixed-point iteration method to obtain... , , The estimated value, where the scale parameter satisfies: ; ; in, , This is an approximate value of the midpoint of the failure time corresponding to the two failure mechanisms; The implementation process of Bayesian point estimation in step 3 is as follows: 1) Define the prior distribution: Obtaining hyperparameters The gamma distribution, ( , ) obeys the hyperparameters as The Dirichlet distribution; 2) By combining the prior distribution and the likelihood function of the observed samples, the joint posterior density function of the parameters is obtained; 3) Use the Gibbs sampling algorithm to draw N sets of random parameter samples, and take the mean of the samples to obtain the Bayesian point estimate: ; ; ; The method for constructing the asymptotic confidence interval in step 4 is as follows: 1) Construct the Fisher information matrix using the negative second-order partial derivatives of the log-likelihood function, and then solve for the parameter variance-covariance matrix; 2) Based on the asymptotic normality of maximum likelihood estimation, construct the asymptotic confidence intervals for the parameters: , , ,in Upper side of the standard normal distribution quantiles, For confidence level, Let V be the variance of the corresponding parameter; wherein, the method for constructing the maximum posterior density confidence interval in step 4 is as follows: 1) At that time Select those that meet the confidence level The shortest interval is taken as the reliable interval; 2) When When sampling parameters obtained from Gibbs sampling, sort the samples in ascending order and iterate through them to select the shortest interval that meets the confidence level requirement as the approximate reliable interval. When using this interval, the following steps are included: Step 1: Preliminary Preparation and Assumption Setting 1) Define the object of analysis: Identify the product whose reliability is to be evaluated, and clarify that the product has two mutually influential failure mechanisms (such as mechanical wear failure and circuit failure of equipment, treatment failure and natural disease progression failure of disease), and the occurrence of either failure mechanism will lead to product failure; 2) Define the failure distribution: Define the occurrence times of the two failure mechanisms as follows: and The product's expiration time is ,and Follows the MOBW distribution (denoted as This distribution consists of mutually independent , and (and (independent) through , Construct, its joint survival function is: ,in For shape parameters, and For scale parameters; Step 2: Design and implement the middle truncation experiment 1) Sample selection: Selecting Each product is an independent product to be tested to ensure that the sample is representative (e.g., covering different production batches and usage environments of the product). 2) Test parameter setting: Pre-set the number of products whose exact failure time can be observed. Define the random time interval corresponding to each product. (This interval follows an unknown two-dimensional distribution); 3) Data Collection and Labeling: Continuously monitor the test products and record the failure status of each product: if the product failure time... Accurate observation (front) (Each product), record the exact failure time. ;like (remaining) (Each product), record interval failure data. ; using indicator functions Indicates failure status: Indicates data that is invalid within a range. This indicates that the first failure mechanism leads to failure. This indicates that the second failure mechanism led to the failure; a complete observation sample was then compiled. ; The third step is to perform parameter estimation based on the observed samples (verification of one or a combination of three methods). (a) Maximum Likelihood Estimation (EM Algorithm) 1) Function Construction: Construct the likelihood function and log-likelihood function based on the observed samples, and establish the relationship between samples and parameters. , , The connection; 2) Initialize parameters: Set the initial values ​​of the parameters. (Estimation can be made based on experience with similar products or preliminary data); 3) Iterative calculation: Step (Expectation Calculation): Calculate the expected failure time, logarithmic expectation, and failure mechanism priority probability corresponding to the interval failure data, and complete the implicit information of the interval data; Step (Maximization): Based on The calculation results are then used to maximize the expectation of the log-likelihood function, yielding new parameter estimates. ; 4) Convergence judgment: Repeat Step and The process continues until the parameter changes in two consecutive iterations are both less than a preset decimal point. (like The parameter value at this point is the maximum likelihood estimate. (ii) Approximate estimation of midpoint Data preprocessing: Preprocessing the invalid data within the interval midpoint As the exact time of product failure, (Approximate midpoint of failure time corresponding to the two failure mechanisms), construct a complete observation sample; Equation establishment: Construct a log-likelihood function based on the preprocessed samples, and adjust the parameters... , , By taking the first-order partial derivatives and setting them equal to zero, we obtain the parametric equations. Solving for the parameters: The parametric equations are solved using the fixed-point iteration method, where the scale parameter is calculated according to the following formula: , ; Iterate until the results converge to obtain the final parameter estimates; (III) Bayesian point estimation Setting the prior distribution: The parameters of the prior distribution are set based on domain experience. Obtaining hyperparameters The gamma distribution, ( , ) obeys the hyperparameters as The Dirichlet distribution; Constructing the posterior distribution: Combining the prior distribution with the likelihood function of the observed samples, we obtain the joint posterior density function of the parameters; Sampling calculation: The Gibbs sampling algorithm is used to draw samples from the conditional density function through acceptance-rejection sampling. Group parameters random sample ( (Usually, a value of 10,000 or higher is used to ensure accuracy). Obtaining the estimate: Take the mean of the N samples obtained from the sampling, which gives the Bayesian point estimate. ; ; ; Step 4: Construct the confidence / credibility interval of the parameters (choose one) (a) Asymptotic confidence interval Calculate the Fisher information matrix: Solve the log-likelihood function with respect to the parameters ( , , Using the negative second-order partial derivatives of ), construct the Fisher information matrix, and then solve for the parameter variance-covariance matrix. ; Constructing confidence intervals: Based on the asymptotic normality of maximum likelihood estimation, the confidence level is constructed using the following formula: (like The confidence interval for (=0.95) is: ; ; ; in Upper side of the standard normal distribution quantiles (e.g.) When =0.95), =1.96). (ii) Maximum posterior density confidence interval Sample processing: If (The parameters are independent of each other) and the interval is selected directly based on the known form of the posterior density function; if Arrange the parameter samples obtained by Gibbs sampling in ascending order; Filtering confidence intervals: Selecting intervals that meet the confidence level And the shortest interval is used as the parameter for the maximum posterior density confidence interval (traversing all intervals of length). (Choose the shortest subinterval). Step 5: Product Reliability Assessment and Application Calculate the reliability index: Combining the joint survival function of the MOBW distribution and the parameter estimates obtained in step three, calculate the product's reliability index at any given time. Reliability Failure probability Core indicators; Domain Applications: In the medical field: Analyze the influence weight of different failure mechanisms of diseases and conduct patient prognostic assessments (such as predicting the risk of disease progression under different treatment regimens). In the engineering field: predicting the lifespan of power-communication network systems, complex equipment, etc., to guide the development of equipment maintenance plans and reliability optimization; The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The descriptions in the above embodiments and specification are merely illustrative of the principles of the present invention. Various changes and modifications can be made to the present invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of protection claimed by the present invention. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. A statistical analysis method for dependent competition failure models under middle-truncation experiments, characterized in that, Includes the following steps: S1. Set product failure-related assumptions: The product has two dependent failure mechanisms; the occurrence of either mechanism leads to product failure. Let the occurrence times of the two failure mechanisms be denoted as follows: and The product's expiration time is ; Obeying shape parameters Scale parameters are and The MOBW distribution, denoted as This distribution consists of mutually independent , and (and (independent) through \ Constructed from this, its joint survival function is: ; S2. Design a middle truncation test scheme: Select... A number of independent, competing products that fail, with a pre-defined number of products whose exact failure time can be observed. Each product corresponds to a random time interval that follows an unknown two-dimensional distribution. When the product expires At that time, only interval failure data is acquired; define the indicator function. ,in Indicates data that is invalid within a range. This indicates that the first failure mechanism leads to failure. This indicates that the second failure mechanism led to the failure; ultimately, observation samples were obtained. ,forward One is the exact failure time data, the remaining ones are... Each is an interval failure data; S3. Parameter estimation based on observed samples: Solve for the shape parameters of the MOBW distribution using at least one of the following methods: maximum likelihood estimation (EM algorithm), midpoint approximation estimation, and Bayesian point estimation. Scale parameters and ; S4. Construct confidence / confidence intervals for parameters: Use the asymptotic confidence interval or maximum a posteriori density confidence interval method to determine the confidence / confidence intervals of the parameter estimates obtained in step 3.

2. The statistical analysis method for the dependent competition failure model under the middle truncation test according to claim 1, characterized in that: The maximum likelihood estimation described in step 3 is implemented using the EM algorithm, and the specific steps are as follows: 1) Construct likelihood and log-likelihood functions based on observed samples to reflect the correlation between samples and parameters; 2) Given initial values ​​for parameters ; 3) No. The next iteration Step: Calculate the expected failure time, logarithmic expected failure time, and failure mechanism priority probability corresponding to the interval failure data; 4) No. The next iteration Step: Based on The step results in maximizing the expectation of the log-likelihood function to obtain parameter estimates. ; 5) Repeat steps 3-4 until the parameter change is less than the preset decimal. The parameter value at this point is the maximum likelihood estimate.

3. The statistical analysis method for the dependent competition failure model under the middle truncation test according to claim 1, characterized in that: The implementation process of the midpoint approximation described in step 3 is as follows: 1) Using interval data midpoint To establish a complete observation sample as the exact product failure time; 2) Construct a log-likelihood function based on this sample, take the first-order partial derivatives with respect to the parameters and set them equal to zero to obtain the parametric equations; 3) Solve the parametric equations using the fixed-point iteration method to obtain... , , The estimated value, where the scale parameter satisfies: ； ； in, , This is an approximate value of the midpoint of the failure time corresponding to the two failure mechanisms.

4. The statistical analysis method for dependent competition failure models under middle-truncation experiments according to claim 1, characterized in that: The implementation process of Bayesian point estimation in step 3 is as follows: 1) Define the prior distribution: Obtaining hyperparameters The gamma distribution, ( , ) obeys the hyperparameters as The Dirichlet distribution; 2) By combining the prior distribution and the likelihood function of the observed samples, the joint posterior density function of the parameters is obtained; 3) Use the Gibbs sampling algorithm to draw N sets of random parameter samples, and take the mean of the samples to obtain the Bayesian point estimate: ； ； 。 5. The statistical analysis method for the dependent competition failure model under the middle truncation test according to claim 1, characterized in that: The method for constructing the asymptotic confidence interval in step 4 is as follows: 1) Construct the Fisher information matrix using the negative second-order partial derivatives of the log-likelihood function, and then solve for the parameter variance-covariance matrix; 2) Based on the asymptotic normality of maximum likelihood estimation, construct the asymptotic confidence intervals for the parameters: , , ,in Upper side of the standard normal distribution quantiles, For confidence level, This represents the variance of the corresponding parameter.

6. The statistical analysis method for dependent competition failure models under middle-truncation experiments according to claim 1, characterized in that: The method for constructing the maximum a posteriori density confidence interval in step 4 is as follows: 1) At that time Select those that meet the confidence level The shortest interval is taken as the reliable interval; 2) When When the parameter samples obtained by Gibbs sampling are sorted in ascending order, the shortest interval that meets the confidence level requirement is selected as the approximate credible interval.

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