System reliability estimation and control optimization method based on fusion distribution likelihood ratio

Abstract

The invention provides a system reliability estimation and control optimization method based on a fusion distributed likelihood ratio. The method comprises specific processes of fusion distributed multi-component system reliability modeling, parameter interval estimation and reliability index verification. After a motor train unit fusion distribution model is established, the fusion distribution model is solved by utilizing an expectation maximization algorithm to obtain a point estimation value, and on the basis, a confidence limit estimation value of the fusion distribution model is given by constructing a single-parameter likelihood ratio pivot statistical magnitude and utilizing related properties of the single-parameter likelihood ratio pivot statistical magnitude. And applying an interval to estimate a lower limit value, giving a dynamic change of a reliability index of the fusion distributed system under a conservative condition, and finally quantitatively verifying the feasibility of system maintenance period margin extension. The likelihood ratio test method is good in effect when processing large sample data, key information can be efficiently and accurately extracted, and powerful data support is provided for feasibility verification of prolonging the maintenance cycle interval of an equipment system.

Description

Technical Field

[0001] The present invention designs a reliability interval estimation method suitable for multivariate distributed complex systems. Based on the two-level basic framework of the life fusion distribution model, it is a reliability interval estimation method for fusion distributed systems based on likelihood ratio statistics. For the life data of the fused distributed system, this method comprehensively analyzes the failure data and termination data of the key components of the fused distributed system, and uses the maximum likelihood estimation algorithm based on the maximum expectation algorithm to achieve point estimation of the fused distributed system. The likelihood ratio statistic is constructed using the parameter point estimate, and the reliability interval estimation of the fused distributed system is achieved based on the information transferability of the likelihood ratio pivot quantity. The reliability confidence lower limit is used to explore the change law of the reliable life of the life fusion distributed system. This method is suitable for efficient evaluation of the service reliability and continuous optimization of availability of multi-mode fault equipment in fields such as aviation and rail transportation. Background Art

[0002] For high-reliability systems with complex structures, maintenance support technology plays a vital role in reducing key component failures, ensuring system safety, and improving system operation reliability, availability and economy. As the use of complex equipment systems becomes more and more extensive, and their number continues to increase in fields such as aviation and rail transportation, the importance of maintenance efficiency and cost in the study of operation and maintenance strategies is becoming more and more prominent. Reforming the maintenance interval for complex systems is an effective way to improve the quality of complex systems and reduce costs without affecting the overall quality and safety of complex systems. Optimizing maintenance strategies through reliability modeling and evaluation has become a prominent area of ​​interest. From a statistical point of view, reliability evaluation can make scientific and reasonable estimates of the reliability of complex systems at the data level. In order to ensure economic constraints and safety reliability in an orderly manner during the reform of the maintenance system, the present invention introduces a method for reliability interval evaluation. The key to the reform of the maintenance system is to extend the fixed maintenance cycle interval while maintaining safety standards and reducing maintenance costs. The reliability interval evaluation method can give the changes in reliability indicators before and after the reform of the complex system maintenance process under conservative conditions, and verify the feasibility of maintenance process optimization.

[0003] For systems with highly complex structures, the present invention proposes a systematic reliability assessment model construction strategy. This strategy is rooted in the modeling concept from the overall perspective of the system. The key is that the reliability distribution of the system as a whole is a combination of the reliability distribution of a series of key components within it through complex interactions and fusion effects. Specifically, the present invention adopts a fusion distribution model as the core framework. This model can accurately characterize the overall distribution characteristics of the system formed by the fusion of multiple sub-distributions (i.e., the reliability distribution of each key component), and is a powerful tool for evaluating the reliability of multi-component complex systems.

[0004] The weight value in the fusion distribution model represents the proportion of samples in each sub-distribution in the overall sample, quantifies the relative importance of each component in the overall sample, and accurately reflects the contribution of different sub-distributions to the overall reliability distribution of the system. It shows excellent performance in fitting complex system reliability data. Its adaptability enables the model to flexibly capture the nonlinear relationship and interaction mechanism between the components within the system. Therefore, it can be used to describe the overall distribution constructed by multiple different sub-distributions and dynamically evaluate the change process of system reliability. This advantage provides reliable data support for subsequent system optimization, fault prediction and maintenance strategy formulation.

[0005] The probability density function and reliability function of the fusion distribution are:

[0006]

[0007] f Θ (t) is the probability density function of the fusion distribution, R j (t i θ j ) is the fusion distribution reliability function. t is a random variable, p j is the proportion of each sub-distribution in the overall distribution, which is called the fusion distribution weight. j ,j=1,2,...,k represents the jth distribution probability density function f in the fusion distribution model j Parameters. Θ=(p1,p2,...,p k ,θ1,θ2,...,θ k ) is the parameter space of the fusion distribution model. x is the process variable required for integration and has no specific meaning here.

[0008] The necessary conditions that the fusion distribution model must meet are:

[0009]

[0010] Ω is the value space of t.

[0011] In actual work, after the reliability assessment model is established, it is necessary to estimate the reliability parameters. Reliability estimation includes point estimation and confidence limit estimation methods. Confidence estimation estimates the interval of parameters by maximizing a certain probability distribution, which reduces the estimation error compared to point estimation. However, point estimation needs to be performed before interval estimation. Considering the extension from components to the whole vehicle, the fusion distribution model can describe a population composed of multiple components or sub-populations. The present invention establishes a fusion distribution model for EMUs and solves the point estimation values ​​of parameters through the maximum expectation algorithm. On the basis of point estimation, the confidence limit estimation method of the fusion distribution model is expanded. The fusion distribution model in the present invention is relatively complex. At the same time, facing the problem of a large amount of missing data in the actual engineering data, the general method cannot directly derive the confidence interval analytical solution of the Fisher information matrix based on the asymptotic variance. The present invention uses the log-likelihood ratio statistic to obey the chi-square distribution, and proposes a method for approximate interval estimation, which can well construct the interval estimation pivot of the multi-component fusion distribution model.

[0012] Based on this, the present invention proposes a system reliability estimation and control optimization method based on fusion distribution likelihood ratio, and establishes a multi-component system reliability assessment model based on fusion distribution. On the basis of obtaining the reliability point estimate based on the maximum expectation algorithm, a single unknown parameter likelihood ratio pivot quantity solution model interval estimate is constructed, and the lower limit of the reliability index of the multi-component equipment system during service under conservative conditions and the feasibility of optimizing the maintenance system are given by using the lower limit of the interval estimate. Summary of the invention

[0013] (1) The purpose of the present invention: Aiming at the reliability estimation and control optimization problems of multi-component service equipment systems, and aiming at a method for extending the system maintenance cycle margin for the purpose of improving economy, the present invention provides an interval estimation method for a fused distributed system, and introduces in detail the specific process of reliability modeling, parameter interval estimation, and reliability index verification for a fused distributed multi-component system. After establishing the EMU fusion distribution model, the maximum expectation algorithm is used to solve the fusion distribution model to obtain a point estimate. On this basis, by constructing a single-parameter likelihood ratio pivot statistic, its related properties are used to give the confidence limit estimate of the fusion distribution model. Using the interval estimation lower limit, the dynamic changes of the reliability index of the fusion distributed system under conservative conditions are given, and finally the feasibility of extending the system maintenance cycle margin is quantitatively verified.

[0014] (2) Technical solution:

[0015] Data background of the present invention:

[0016] Consider a batch of multi-component complex systems that actually run, which includes four types of key components. The present invention uses A, B, C, and D to represent these four types of key components, and the failure and maintenance behaviors of the key components are independent. Key assumptions are put forward: the four types of components have fixed preventive maintenance cycles. When these four types of components fail, they are immediately replaced (i.e., repaired as new); when a key component of the system fails, it means that the system is shut down, and it is recorded as a failure of the system; the distributions obeyed by the four types of components A, B, C, and D are all known; the failure data of the four types of components include failure data and termination data. The failure data is the failure mileage when the component fails, and the termination data is randomly truncated data (due to individual differences in each system, the time required for preventive maintenance is not fixed, so it is randomly truncated data).

[0017] The focus of the present invention is to solve the method of estimating the confidence limits of reliability of multi-component systems, and on this basis, the potential for quantitatively extending the preventive maintenance cycle is demonstrated. A comprehensive reliability evaluation model of the system fusion power distribution is established at the whole machine level. The maximum expectation algorithm is used to estimate the parameter points in the fusion distribution model, and on this basis, the likelihood ratio statistics are used to construct an interval estimation method. Finally, the feasibility of extending the preventive maintenance interval is determined by verifying the reliability reduction threshold. Based on the above assumptions and ideas, the system reliability estimation and control optimization method based on the fusion distribution likelihood ratio of the present invention is implemented through the following steps:

[0018] Step 1: Establish a fusion distribution model of multi-component complex equipment system and a likelihood function of truncated data.

[0019] In the framework of the equipment system reliability assessment, this paper constructs a complex system life analysis method based on the finite fusion distribution model. The core of this model is to regard the equipment system as a complex composed of k independent components, and the life of each component is T i All follow a type of distribution function. Therefore, the life distribution of the entire equipment system is defined by the fusion of these k sub-distributions, forming a multi-level fusion distribution model.

[0020] Specifically, it is assumed that the life data of each key component (including complete failure data and termination data generated by observation cutoff) together constitute the n observation points of the system life data set. In this data set, each observation point represents the life information of a component. Suppose the i-th life observation value of a component is t i ′, when the life value t i When it is fully observed i ′=t i (or truncated value c i , when t iThe above process reflects the state of the component at a specific time point. In particular, a random truncation mechanism is adopted according to the data background, that is, the observation cutoff time c is random, and when the actual life X of the component exceeds this cutoff time, its life observation value will be recorded as the truncation time c. i .

[0021] According to the above principles, the probability density function of the fusion distribution model of the life value of the multi-component fusion system and the fusion distribution reliability function expression are defined as follows:

[0022]

[0023] Assuming that the multi-component fusion distribution system contains k key components, it means that the fusion distribution model contains k sub-distributions, j = 1, 2, ..., k. j is the fusion weight (fusion weight value) of the jth component, satisfying And 0≤p j ≤1, f j (t i θ j ) is the density function of the jth component, which depends on the parameter θ j . R j (t i θ j ) is the reliability function of the jth component. R j (t i θ j )The specific expression is as follows:

[0024]

[0025] Since there is termination data in the data background, when the actual life value of the component t i Greater than the censoring time c i When the observed value t i ′ is based on the censoring time c i In this case, the likelihood function including censored data is used for parameter estimation.

[0026] For non-truncated data (actual component life value t i Less than or equal to a certain truncation time c i ), whose likelihood contribution is f Θ (t i ); for censored data (i.e., t i >c i ), whose likelihood contribution is 1-F Θ (c i ), where F Θ (X) is the cumulative distribution function of the fused distribution, and 1-F Θ (ci )=R Θ (c i ).

[0027] The likelihood function for the entire dataset is the product of the likelihood contributions of all observations:

[0028]

[0029] Where I(·) is an indicator function, which takes the value of 1 when the condition is true and 0 otherwise.

[0030] Each observed value of life span can only come from one of the k distributions of the fusion distribution. We introduce the indicator random variable Z, which is an n-dimensional vector, where n is the sample observation capacity. i (i=1,2,...,n) is a k-dimensional random variable indicating whether the observation comes from the jth distribution. Specifically, if the observation comes from the jth distribution, then Z ij =1, otherwise Z ij = 0, and these Z ij Satisfy the constraint: Among all the elements of Z, there is only one that is 1 and the rest are all 0.

[0031] In the case of randomly truncated samples, we only know whether the observation fails before a certain time (i.e., whether it is "truncated"), or we know the specific failure time. For simplicity, assume that there is a sample set {T1, T2, ..., T... Tn} n}, we can get the random truncated sample after introducing the indicator function I(·) and Z i The fusion distribution likelihood function L C (θ|t,z) is:

[0032]

[0033] The parameter t represents the observed value of life span t i 、c i The parameter space is composed of z. z represents the indicator random variable Z i The parameter space formed.

[0034] Step 2: Estimate the fusion distribution model parameters through the maximum expectation algorithm.

[0035] Step 2 mainly uses the maximum expectation algorithm to solve the maximum value of the fusion distribution likelihood function obtained in step 1. The specific steps of the maximum expectation algorithm are:

[0036] I. Initialization: Select the initial proportion of each key component (A, B, C, D) in the total sample (i.e., fusion weight) ( is the fusion weight of key component A; is the fusion weight of key component B; is the fusion weight of the key component C; is the fusion weight of key component D), the sum of these proportions should be kept equal to 1. Select the initial parameters (such as mean, variance, etc.) of the distribution of each key component itself (such as normal distribution, exponential distribution, etc.). The initial parameter space composed of all fusion weights is denoted as p (m-1) , the initial parameter space composed of all parameters is θ 0 .

[0037] II. (E-step): In the E-step, we need to calculate the probability that each observed data point comes from each sub-distribution (or component) under the current parameter estimation (i.e., the indicator variable Z i The expected value of the indicator variable, as a bridge between the observed data and the underlying sub-distribution structure, is calculated based on the posterior probability distribution under the current parameter estimation. For complete failure data (i.e., the exact failure time or state transition point is observed), the expected value of the indicator variable can be directly obtained by weighted summing the conditional probabilities of each sub-distribution under the current observed value, reflecting the exact probability that the observed point originates from each sub-distribution.

[0038] The situation is more complicated when dealing with randomly censored data. Since censored data only provide partial information (such as no failures occurred before a certain time point), the expected value of the indicator variable needs to be derived by considering the combined effects of the censoring mechanism and the sub-distribution characteristics. At this time, the calculation of the expected value needs to take into account the censoring probability to ensure appropriate handling of the incompleteness of the observed data.

[0039] In summary, the indicator variable Z i The expected value of can be expressed as:

[0040]

[0041] In the algorithm, the parameters represents the known fusion weight value estimated for the jth sub-distribution during the m-1th iteration. It represents the known parameter value obtained for the j-th subdistribution at the m-1th iteration. In particular, when As the fusion weight and parameter value of the lth sub-distribution at the m-1th iteration, both appear in the denominator, so its subscript is named l to clearly distinguish it from the parameters in the numerator that need to be solved for the jth sub-distribution (if such a direct solution process exists). j and f l , R j and R l Follow the same naming convention.

[0042] After obtaining the expected value of the indicator variable, the likelihood function can be solved when the parameter is θ (m-1) 、p (m-1) The mathematical expectation Q(θ, p; θ (m-1) , p (m-1) ). In formula (11), Indicates that when the parameter θ is known (m-1) 、p (m-1) Under the condition, we can find the fusion distribution likelihood function L of formula (8) C The mathematical expectation of (θ|t,z).

[0043]

[0044] III. (M-step): Update the model parameters θ and p using the expected value obtained in the E-step. The fusion weights (i.e., the proportion of each fusion component in the total) are updated by calculating the average of the posterior probabilities of all observed data points belonging to each fusion component. At the same time, the parameters of each fusion component are updated by solving the parameter value that maximizes the corresponding expected value function (usually the expected value of the log-likelihood function). Maximize the expected value function so that Q(θ (m) , p (m) θ (m-1) , p (m-1) ) = maxQ(θ, p; θ (m-1) , p (m-1) ). Get the weight of the fusion distribution model in the mth iteration and sub-distribution parameters The estimated value of is:

[0045]

[0046] According to the above process, the E-step and the M-step are performed alternately until the specified convergence condition is met, and the point estimate of the fusion distribution model parameters is obtained.

[0047] Step 3: Construct a single-parameter likelihood ratio statistic to implement interval estimation of the fusion distribution model.

[0048] The approximate interval estimation method for constructing likelihood ratio statistics in the present invention is implemented by hypothesis testing. For the EMU, each component is used as a sub-distribution to form the overall fusion distribution of the EMU. Assuming that the lifespans of k components {T1, T2, ..., T n}Identical and identically distributed in a finite fusion distribution model f(x; θ), θ∈Θ. Let the fusion distribution model parameter space be the parameters of each sub-distribution, Θ=(θ1,θ2,...,θ k ). The null hypothesis is that the estimated parameter is equal to the true parameter value:

[0049] H0:θ=θ0 (14)

[0050] The constructed likelihood ratio statistic λ(T) is:

[0051]

[0052] Based on formula (14), the log-likelihood ratio test statistic Λ is constructed for interval estimation of the fusion distribution model. Λ can be expressed as:

[0053]

[0054] in, represents the parameters that need to be estimated. Indicates that the parameters need to be known L(θ0) represents the likelihood function of the sample under the condition of the true value θ0 of the location parameter.

[0055] For large samples, the following conclusions can be drawn:

[0056]

[0057] Where D represents the convergence of distribution, χ 2 represents the chi-square distribution, d represents the dimension of the chi-square distribution (also known as the degree of freedom of the chi-square distribution), where the dimension of the chi-square distribution is equal to the dimension of the parameter θ, i.e. d = dim(θ). The following will prove that the single-parameter log-likelihood ratio test statistic follows the chi-square distribution.

[0058] Proof (single-parameter procedure):

[0059] According to Taylor's formula, the log-likelihood function L(θ0) is Expand to get:

[0060]

[0061] in, Represents an infinitesimal quantity of order n.

[0062] From the maximum likelihood principle solution process, we can know that the derivative of the likelihood function is 0 at the optimal estimated parameter:

[0063]

[0064] Solving equation (16), equation (18) and equation (19) together yields:

[0065]

[0066] In formula (20), I(θ) is the Fisher information matrix of parameter θ. The maximum likelihood estimate has weak consistency and asymptotic normality. From this property, we can get:

[0067]

[0068] At the same time, according to the law of large numbers, the second-order derivative of the likelihood function of the estimated value of parameter θ can be obtained Except for the Fisher information matrix I(θ) of parameter θ, it converges to 1 with probability, that is:

[0069]

[0070] P means convergence with probability.

[0071] According to the Slutsky theorem, the constructed log-likelihood ratio test statistic Λ converges to the chi-square distribution according to the distribution, which can be expressed as:

[0072]

[0073] D means convergence to the distribution.

[0074] The proof is complete!

[0075] This property can be used to perform interval estimation of the fusion distribution. Formula (7) is the likelihood function of the fusion distribution model. From the theorem of formula (23), it can be seen that the likelihood ratio converges to the chi-square distribution. The confidence interval of its parameter θ when the confidence level is 1-α is:

[0076]

[0077] Represents the chi-square distribution value when the confidence level is 1-α and the dimension is d.

[0078] For the fusion distribution model parameter space Θ=(θ1,θ2,...,θ k ) any parameter θ j , construct its log-likelihood ratio test statistic Λ as:

[0079]

[0080] Set the confidence level to 1-α, then the parameter θ j There are the following inequalities:

[0081]

[0082] According to formula (26), the inequality for interval estimation of the fusion distribution model parameters is:

[0083]

[0084] Substitute the known point estimates of the fused distribution obtained by the maximum expectation algorithm in step 2 to solve the estimated values ​​of any parameter interval of any sub-distribution in the fused distribution model.

[0085] Step 4: Calculate the reliability index of the multi-component complex system and verify the feasibility of extending the maintenance cycle margin.

[0086] The core goal of the present invention is to scientifically and quantitatively verify the feasibility of extending the maintenance cycle margin of the equipment system to a certain percentage above its original cycle. By setting a series of precise reliability calculation steps, the reliability change trajectory of the equipment system during the entire operation cycle is dynamically simulated. In order to clearly define the feasibility boundary of extending the maintenance cycle, the present invention sets a critical reliability threshold W. This threshold represents the minimum reliability level that the equipment system can maintain under a normal maintenance cycle, and is an indispensable safety boundary to ensure the stable operation of the system. By comparing the measured value W' of the percentage reduction in system reliability after the maintenance cycle margin is extended with the set threshold W, the present invention can scientifically determine whether the scheme for extending the maintenance cycle margin is feasible. If the measured reliability reduction percentage W' is not lower than the threshold W, it means that the extension scheme not only has no negative impact on the system reliability and safety, but also achieves the optimal allocation of maintenance resources.

[0087] The formula for calculating the measured value of reliability reduction percentage is as follows:

[0088]

[0089] Among them, R prolong Represents the reliability value when the target maintenance cycle is extended, R pre Indicates the original reliability value when the maintenance period before extension is reached.

[0090] The maximum expectation algorithm in step 2 can be used to solve the fusion weight of any sub-distribution The estimated value of the interval of any sub-distribution parameter in the fusion distribution model can be obtained by solving equation (27) in step 3. Since the positions of the sub-distribution parameters in the fusion distribution formula are different, for example, in the normal distribution, the parameter μ is located in the numerator of the exponential power, and the parameter σ is located in the denominator of the exponential power, the influence on the monotonicity of the overall reliability is different. Therefore, the confidence lower limit of its reliability function can be determined by Monte Carlo sampling, and the reliability value is calculated by randomly sampling in the obtained parameter confidence interval until the boundary value of the reliability is determined. At this time, the obtained parameter combination Θ L The confidence lower limit of the parameter is obtained, and the reliability confidence lower limit function is obtained:

[0091]

[0092] A series of equally spaced time steps are set, and the reliability confidence lower limit of the multi-component system at each preset step node is calculated in detail. The calculated reliability values ​​of each step node are further plotted into a curve graph, and it is calculated whether the measured value W' of the percentage reduction in system reliability after the maintenance cycle margin is extended exceeds the specified threshold W.

[0093] (3) Advantages and effects: The present invention is a system reliability estimation and control optimization method integrating distribution likelihood ratio, which has the following advantages:

[0094] ① The innovative method in the present invention significantly improves the accuracy and flexibility of equipment system reliability assessment. In particular, the present invention constructs a basic fusion distribution reliability assessment model for the interval estimation problem of key components of multi-component complex equipment systems in the context of complete failure data and randomly truncated data, which is difficult to be effectively handled by traditional interval estimation methods. This model not only effectively integrates multi-source heterogeneous data, but also efficiently solves the point estimate of the fusion distribution model through the maximum expectation algorithm, and then uses the likelihood ratio statistic to derive the interval estimate of the model parameters, providing a scientific quantitative basis for the rationality of extending the maintenance interval of the equipment system. This method can significantly reduce the maintenance cost while ensuring the safe and stable operation of the equipment system under conservative conditions, reflecting its innovativeness and wide applicability.

[0095] ② The calculation process of the present invention has low requirements on the initial values ​​of parameters, which means that there is no need to perform complex parameter presets in actual operation, reducing the difficulty of operation and the risk of error. At the same time, the calculation process is concise and clear, the iterative process is fast and convergence is guaranteed, ensuring the timeliness and stability of the evaluation results.

[0096] ③ For more complex models, traditional interval estimation methods cannot directly obtain the analytical solution of the confidence interval based on the Fisher information matrix of the asymptotic variance. This problem is solved by constructing the likelihood ratio pivot statistic. By constructing the pivot quantity of the known distribution, the distribution of the values ​​of the parameter statistics under a certain confidence level can be given, making the interval estimation process interpretable. At the same time, the likelihood ratio test statistic can adapt to a variety of distribution models, including fusion distributions. This means that it can perform effective interval estimation in complex statistical environments, such as when there are multiple sub-populations or heterogeneous data. The likelihood ratio test is not limited to the test of a single parameter, but can also handle the comparison and interval estimation of multiple parameters, which is particularly important in fusion distribution models because fusion distributions usually contain multiple unknown parameters or components. The likelihood ratio test is based on the principle of maximizing the likelihood function, directly comparing the likelihood function values ​​under different assumptions, thereby avoiding the bias that may be caused by the parameter estimation method. When the sample size tends to infinity, the likelihood ratio test is consistent, that is, its results will tend to approach the true parameter value. This property enables the likelihood ratio test to provide more stable and reliable conclusions in the interval estimation of fusion distributions. Its inference process is relatively robust and not easily affected by data outliers or slight deviations from distribution assumptions.

[0097] ④ As the complexity of multi-component equipment systems continues to increase, the amount of data generated during their operation has also increased dramatically. The likelihood ratio test method in the present invention works well when processing large sample data, and can efficiently and accurately extract key information, providing strong data support for the feasibility verification of extending the maintenance cycle interval of the equipment system. BRIEF DESCRIPTION OF THE DRAWINGS

[0098] Figure 1 It is a flow chart of the method of the present invention.

[0099] Figure 2 It is the lower confidence limit curve of EMU reliability.

[0100] Figure 3 It is the percentage curve of the lower confidence limit node reduction of the EMU reliability. DETAILED DESCRIPTION

[0101] In order to explore the core contribution of the present invention, a representative large-scale multi-component equipment system is selected as a specific example.

[0102] In the specific implementation process, the reliability of a certain type of EMU actual operation data was evaluated and verified. The EMU fusion distribution model was established and the point estimate of the model was given by the maximum expectation algorithm. On this basis, the confidence interval of the fusion distribution model parameters was given, and the reliability confidence lower limit was used as the benchmark for quantitatively extending the preventive maintenance cycle margin by 25%, verifying the feasibility of quantitatively extending the maintenance cycle margin, that is, whether it is feasible to extend a certain level of maintenance node of the EMU from the original 1.32 million kilometers to 1.65 million kilometers.

[0103] The data set collected by the present invention is derived from the actual data of the EMU. It contains more than 200 EMUs of the same or similar model (produced under the same technical platform). The data set was collected from June 2017 to June 2022. The EMU-related failure time in the data set is expressed in terms of the EMU running mileage. The selected actual case contains four key components, and components A and C obey the lognormal distribution with known parameters. Components B and D obey the known gamma distribution. The basic data is shown in Table 1:

[0104] Table 1 Data of key components of EMU

[0105] A-Part B Part C-Component D-part Complete vehicle The number of completely failed data 26 25 19 116 186 Number of randomly censored data 280 280 280 280 1120 total 306 305 299 396 1306

[0106] The known initial probability density functions of key components A, B, C, and D are shown below:

[0107]

[0108] See Figure 1 The present invention provides a system reliability estimation and control optimization method based on fusion distribution likelihood ratio, which is implemented by the following steps:

[0109] Step 1: Establish a fusion distribution model of multi-component complex equipment system and a likelihood function of truncated data.

[0110] The probability density function expression of the fusion distribution model is:

[0111]

[0112] p1, p2, p3, p4 are sub-distribution fusion weights, u A ,σ A ,k B ,θ B ,u C ,σ C ,k D ,θ D are the unknown parameters of each sub-distribution. Since the proportion of each component in the overall sample is p j The distribution parameters of each component are known. Set the initial values ​​of the fusion distribution model parameters as follows:

[0113]

[0114] According to formula (8), we can get that after introducing indicator variables I(·) and Z i The likelihood function of the fusion distribution of the EMU after is:

[0115]

[0116] Step 2: Estimate the fusion distribution model parameters through the maximum expectation algorithm.

[0117] The initial value has been given in step 1. Substitute the key fault data of the EMU into the likelihood function (36) and calculate the fusion distribution point estimate using the maximum expectation algorithm in step 2. When the data is a complete failure data, I(t i ≤c i ) takes the value of 1, I(t i >c i ) takes the value of 0. When the data is randomly truncated, I(t i ≤c i ) takes the value of 0, I(t i >c i ) is set to 1. Set the loop termination threshold to 10 -5 , when the parameters converge to the termination threshold, the calculation is stopped. The results are shown in Table 2 and Table 3.

[0118] Table 2 Maximum expectation algorithm weight p j Calculation results

[0119] <![CDATA[p1]]> <![CDATA[p2]]> <![CDATA[p3]]> <![CDATA[p4]]> Estimated value 0.01084 0.82975 0.05755 0.10186

[0120] Table 3 Calculation results of distribution parameters of maximum expectation algorithm

[0121] <![CDATA[u A ]]> <![CDATA[σ A ]]> <![CDATA[k B ]]> <![CDATA[θ B ]]> <![CDATA[u C ]]> <![CDATA[σ C ]]> <![CDATA[k Other ]]> <![CDATA[θ Other ]]> Estimated value -0.9005 0.82941 109.55181 1.86059 2.61431 0.85778 4.69333 18.33670

[0122] Step 3: Construct a single-parameter likelihood ratio statistic to implement interval estimation of the fusion distribution model.

[0123] According to formula (24) in step 3, the likelihood ratio statistics of the interval estimation of each parameter of the fusion distribution model are constructed. According to formula (26), the calculation formula of the interval estimation is obtained. A and σ A The calculation process is taken as an example.

[0124] Parameter u A :

[0125] 1) Construct the likelihood ratio statistic Λ(u A ):

[0126]

[0127] 2) Construct the calculation formula f(u A ):

[0128]

[0129] Parameter σ A :

[0130] 1) Construct the likelihood ratio statistic Λ(σ A ):

[0131]

[0132] 2) Construct the calculation formula f(σ A ):

[0133]

[0134] According to the above process, the remaining parameter k can be given B ,θ B ,u C ,σ C ,k D ,θ D The calculation formula is not repeated here because the process is the same. Take the significance level as α = 0.05, substitute the calculation results of the point estimate of the fusion distribution model parameters based on the maximum expectation algorithm in Table 2 and Table 3 in step 2, and the interval estimate of the fusion distribution model parameters can be calculated.

[0135] Calculate the sub-distribution parameter u of the fusion distribution model A ,σ A ,k B ,θ B ,u C ,σ C ,k D ,θ D The confidence interval values ​​at the significance level α = 0.05 are shown in Tables 4 and 5.

[0136] Table 4 Interval estimation results of sub-distribution parameters of components A and B

[0137]

[0138] Table 5. Interval estimation results of sub-distribution parameters of components C and D

[0139]

[0140] Note: L represents the lower confidence limit and U represents the upper confidence limit

[0141] Step 4: Calculate the reliability index of a complex system with multiple components and verify the feasibility of extending the maintenance cycle margin

[0142] The indicator is selected as the reliability function, and random sampling is performed in the parameter interval calculated in step 3 by the Monte Carlo method. The parameter combination when the lower confidence limit of the reliability (i.e., the lowest reliability) is obtained is shown in Table 6:

[0143] Table 6 Reliability confidence lower limit sub-distribution parameter combinations

[0144] <![CDATA[u AL ]]> <![CDATA[σ AL ]]> <![CDATA[k BL ]]> <![CDATA[θ BL ]]> <![CDATA[u CL ]]> <![CDATA[σ CL ]]> <![CDATA[k DL ]]> <![CDATA[θ DL ]]> Estimated value -0.4110 0.5562 105.7696 1.7961 2.3487 0.6796 4.1540 16.1049

[0145] According to formula (28), the lower confidence limit formula of its reliability is:

[0146]

[0147] In step 4, it is necessary to quantitatively verify whether it is feasible to extend the margin of a maintenance time node of the EMU from 1.32 million kilometers to 1.65 million kilometers. According to the obtained reliability confidence lower limit function, the step length is selected as 10,000 km, and the reliability curve of the EMU from 1.32 million km to 1.65 million km under the confidence lower limit condition is obtained as follows: Figure 2 shown.

[0148] According to formula (27), the reliability reduction percentage is used as the verification threshold, and the threshold is set to 2%:

[0149]

[0150] Where R 165 is the reliability of the EMU when the running mileage is 1.65 million km, R 132 is the reliability of the EMU when the running mileage is 1.32 million km. If the above conditions are met, it is considered that the extension plan of a certain level of maintenance node is feasible. According to formula (27), the percentage reduction of the lower limit value of the EMU reliability at each node is calculated, and the calculation results are as follows: Figure 3 shown.

[0151] Depend on Figure 3 It can be seen that when the reliability confidence lower limit is taken as the verification curve for the sake of conservatism, it is calculated that when a maintenance node cycle is extended to 1.65 million km, the reliability is reduced by 8.6628% compared with 1.32 million km, which is much larger than the original 2%. Therefore, the plan to extend it to 1.65 million km is not feasible. The maximum extension is to 1.53 million km, at which time the reliability is reduced by 1.999%.

[0152] The results show that the method of the present invention can realize the reliability interval estimation of the fusion distribution model of the multi-component equipment system and verify the feasibility of the maintenance system reform, thus achieving the expected purpose.

[0153] In summary, the present invention relates to a system reliability estimation and control optimization method based on fused distribution likelihood ratio. Focusing on the research on the confidence limit estimation and verification method of the fused distribution of multi-component system for the control optimization of extended maintenance system, a confidence limit estimation method of the fused distribution model based on the construction of likelihood ratio statistics is proposed. For a large complex system composed of multiple sub-populations, after establishing a fused distribution model of a multi-component complex system, the fused distribution model is solved by the maximum expectation algorithm to obtain a point estimate. On this basis, by constructing a likelihood ratio statistic, the confidence limit estimate of the fused distribution model can be given by using its pivot transfer property, thereby verifying the feasibility of the maintenance system reform. The specific steps of the method are: establishing a fused distribution model of a multi-component complex equipment system, and calculating its likelihood function under the premise that the data background contains truncated data. Estimate the parameters of the fused distribution model by the maximum expectation algorithm. Construct a single parameter likelihood ratio statistic on the basis of the point estimate, and implement the interval estimation of the fused distribution model. Then calculate the lower confidence limit of the reliability of the multi-component complex system and verify the feasibility of extending the maintenance cycle margin according to the lower limit value. The present invention is applicable to the fields of reliability-centered maintenance strategy optimization, large deletion ratio data reliability assessment, and massive service data processing for multivariate distributed complex systems in the fields of aviation and rail transportation.

Claims

1. A system reliability estimation and control optimization method based on fusion distribution likelihood ratio, characterized by: The steps include: Step 1: Establish a fusion distribution model of multi-component complex equipment system and a likelihood function of truncated data; Step 2: Estimate the fusion distribution model parameters through the maximum expectation algorithm; Step 3: Construct a single parameter likelihood ratio statistic to implement interval estimation of the fusion distribution model; Step 4: Calculate the reliability index of the multi-component complex system and verify the feasibility of extending the maintenance cycle margin.

2. The method for system reliability estimation and control optimization based on fusion distribution likelihood ratio according to claim 1, characterized in that: In step 1, the equipment system is considered to be a complex composed of k independent components, and the life span of each component is T i All follow a type of distribution function; the life distribution of the entire equipment system is defined by the fusion of these k sub-distributions, forming a multi-level fusion distribution model; Assume that the life data of each key component together constitute n observation points of the system life data set; each observation point represents the life information of a component, and assume that the i-th life observation value of a component is t i ′, when the life value t i When it is fully observed i ′=t i , reflects the state of the component at a specific time point; according to the data background, a random truncation mechanism is adopted, that is, the observation cutoff time c is random, and when the actual life X of the component exceeds this cutoff time, its life observation value will be recorded as the truncation time c i .

3. A system reliability estimation and control optimization method based on fusion distribution likelihood ratio according to claim 1 or 2, characterized in that: The probability density function and fusion distribution reliability function of the fusion distribution model of the life value of the multi-component fusion system are defined as follows: Assume that the multi-component fusion distribution system contains k key components, which means that the fusion distribution model contains k sub-distributions, j = 1, 2, ..., k; where p j is the fusion weight of the jth component, satisfying And 0≤p j ≤1, f j (t i θ j ) is the density function of the jth component, which depends on the parameter θ j ; R j (t i θ j ) is the reliability function of the jth component; R j (t i θ j )The specific expression is as follows: Since there is termination data in the data background, when the actual life value of the component t i Greater than the censoring time c i When the observed value t i ′ is based on the censoring time c i In this case, the likelihood function containing censored data is used for parameter estimation; For non-censored data, that is, the actual life value of the component t i Less than or equal to a certain truncation time c i , whose likelihood contribution is f Θ (t i ); for censored data, that is, t i >c i , whose likelihood contribution is 1-F Θ (c i ), where F Θ (X) is the cumulative distribution function of the fused distribution, and 1-F Θ (c i )=R Θ (c i ).

4. The method for system reliability estimation and control optimization based on fusion distribution likelihood ratio according to claim 3 is characterized in that: The likelihood function for the entire dataset is the product of the likelihood contributions of all observations: Where I(·) is an indicator function, which takes the value 1 when the condition is true and 0 otherwise; Each life observation comes from only one of the k distributions in the fusion distribution. The indicator random variable Z is introduced. Z is an n-dimensional vector, where n is the sample observation capacity. Each element Z i is a k-dimensional random variable indicating whether the observation comes from the jth distribution; if the observation comes from the jth distribution, then Z ij =1, otherwise Z ij = 0, and these Z ij Satisfy the constraints: Among all the elements of Z, there is only one that is 1, and the rest are all 0; In the case of randomly truncated samples, we only know whether the observation fails before a certain time, or know the specific failure time. For simplicity, we have a sample set {T1, T2, ..., T n }, we get the random truncated sample after introducing the indicator function I(·) and Z i The fusion distribution likelihood function L C (θ|t,z) is: Among them, the parameter t represents the observed value of life t i 、c i The parameter space formed; z represents the indicator random variable Z i The parameter space formed.

5. The method for system reliability estimation and control optimization based on fusion distribution likelihood ratio according to claim 1, characterized in that: In step 2, it specifically includes: I. Initialization: Select the initial proportion of each key component in the total sample is the fusion weight of key component A; is the fusion weight of key component B; is the fusion weight of the key component C; is the fusion weight of the key component D, and the sum of these proportions should be kept as 1; select the initial parameters of each key component’s own distribution; denote the initial parameter space composed of all fusion weights as p (m-1) , the initial parameter space composed of all parameters is θ 0 ; II. It is necessary to calculate the probability that each observed data point comes from each sub-distribution under the current parameter estimation, that is, the indicator variable Z i The expected value of the indicator variable, as a bridge between the observed data and the potential sub-distribution structure, is calculated based on the posterior probability distribution under the current parameter estimation. For complete failure data, that is, when the exact failure time or state transition point is observed, the expected value of the indicator variable is directly obtained by weighted summing the conditional probabilities of each sub-distribution under the current observed value. III. Update the model parameters θ and p using the expected value obtained. For the fusion weight, that is, the proportion of each fusion component in the total, it is updated by calculating the average of the posterior probabilities of all observed data points belonging to each fusion component. At the same time, for the parameters of each fusion component itself, it is updated by solving the parameter value that maximizes the corresponding expected value function. Maximize the expected value function so that Q(θ (m) , p (m) θ (m-1) , p (m-1) ) = maxQ(θ, p; θ (m-1) , p (m-1) ).

6. The method for system reliability estimation and control optimization based on fusion distribution likelihood ratio according to claim 5, characterized in that: Indicator variable Z i The expected value of is expressed as: parameter Represents the known fusion weight value estimated for the jth sub-distribution during the m-1th iteration; parameter It means the known parameter value obtained for the j-th sub-distribution at the m-1th iteration; when As the fusion weight and parameter value of the lth sub-distribution at the m-1th iteration, both appear in the denominator, so its subscript is named l to clearly distinguish it from the parameters for the jth sub-distribution that need to be solved in the numerator; f j and f l , R j and R l Follow the same naming convention; After obtaining the expected value of the indicator variable, solve the likelihood function with known parameters θ (m-1) 、p (m-1) The mathematical expectation Q(θ, p; θ (m-1) , p (m-1) ); In formula (11), Indicates that when the parameter θ is known (m-1) 、p (m-1) Under the condition, we can find the fusion distribution likelihood function L of formula (8) C The mathematical expectation of (θ|t,z); 7. The method for system reliability estimation and control optimization based on fusion distribution likelihood ratio according to claim 6, characterized in that: The weights of the fusion distribution model in the mth iteration and sub-distribution parameters The estimated value of is:

8. The method for system reliability estimation and control optimization based on fusion distribution likelihood ratio according to claim 6, characterized in that: In step 3, let the lifespans of k components {T1, T2, ..., T n } Independent and identically distributed in a finite fusion distribution model f(x; θ), θ∈Θ; let the fusion distribution model parameter space be the parameters of each sub-distribution, Θ=(θ1,θ2,...,θ k ) ; establish the null hypothesis that the parameter estimate is equal to the true parameter value: H0:θ=θ0 (11) The constructed likelihood ratio statistic λ(T) is: Based on formula (14), the log-likelihood ratio test statistic Λ is constructed for interval estimation of the fusion distribution model, and Λ is expressed as: in, represents the parameters that need to be estimated; Indicates that the parameters need to be known The likelihood function of the sample when θ0 is θ0, and L(θ0) represents the likelihood function of the sample under the condition of the true value of the location parameter θ0; For large samples, the following conclusions can be drawn: Where D represents the convergence of distribution, χ 2 represents the chi-square distribution, d represents the dimension of the chi-square distribution, where the dimension of the chi-square distribution is equal to the dimension of the parameter θ, that is, d=dim(θ).

9. The method for system reliability estimation and control optimization based on fusion distribution likelihood ratio according to claim 1, characterized in that: In step 4, a critical reliability threshold W is set; this threshold represents the minimum reliability level that the equipment system can maintain under normal maintenance cycles, and is an indispensable safety boundary to ensure the stable operation of the system; by comparing the measured value W' of the percentage reduction in system reliability after the maintenance cycle margin is extended with the set threshold W, it is scientifically determined whether the maintenance cycle margin extension plan is feasible; if the measured reliability reduction percentage W' is not lower than the threshold W, it indicates that the extension plan not only has no negative impact on system reliability and safety, but also achieves the optimal allocation of maintenance resources; The formula for calculating the measured value of reliability reduction percentage is as follows: Among them, R prolong Represents the reliability value when the target maintenance cycle is extended, R pre Indicates the original reliability value when the maintenance period before extension is reached.

10. The method for system reliability estimation and control optimization based on fusion distribution likelihood ratio according to claim 9, characterized in that: In step 4, since the positions of the sub-distribution parameters in the fusion distribution are different, the confidence lower limit of the reliability function is determined by Monte Carlo sampling. Random sampling is performed in the obtained parameter confidence interval and the reliability value is calculated until the boundary value of the reliability is determined. At this time, the obtained parameter combination Θ L As the parameter confidence lower limit, the reliability confidence lower limit function is obtained: By setting equally spaced time steps, the reliability confidence lower limit of the multi-component system at each preset step node is calculated in detail. The calculated reliability values ​​of each step node are further plotted into a curve graph, and it is calculated whether the measured value W' of the percentage reduction in system reliability after the maintenance cycle margin is extended exceeds the specified threshold W.

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