Aircraft fuel gear pump confidence degree evaluation method based on Bayesian theory

Abstract

The invention discloses an aviation fuel gear pump confidence degree evaluation method based on the Bayesian theory. The method comprises the following steps: step 1, firstly, determining Weibull distribution as a theoretical basis of reliability modeling according to the life characteristics of the aviation fuel gear pump, and providing probability model support for subsequent analysis; 2, on the basis of Weibull distribution, adopting a Bayesian method to fuse historical data and field test data, and achieving point estimation and interval estimation of shape parameters and scale parameters of the aviation fuel gear pump; and step 3, based on the point estimation and the interval estimation, constructing an MTBF confidence interval and verifying by adopting a Bootstrap method, and finally completing quantitative evaluation of the reliability confidence of the aviation fuel gear pump. According to the method, the specific problems of low evaluation precision, poor confidence interval reliability and insufficient historical data utilization rate of a traditional method under the small sample condition are solved.

Description

Technical Field

[0001] This invention relates to the field of confidence assessment technology for aviation fuel gear pumps, and particularly to a confidence assessment method for aviation fuel gear pumps based on Bayesian theory. Background Technology

[0002] As a key component of aero-engines, the reliability assessment of aviation fuel gear pumps has always faced the problem of insufficient accuracy under small sample data.

[0003] Existing technologies mainly employ statistical methods from the frequentist school, such as classical Weibull distribution analysis. These methods have three significant drawbacks: First, a large amount of failure data (usually more than 30 sets) is needed to obtain stable evaluation results, while in actual engineering, only 5-10 sets of data can often be obtained. Secondly, traditional methods cannot effectively utilize historical experimental information, resulting in a waste of data resources; Finally, the confidence interval construction method is conservative, resulting in intervals that are too wide or insufficient coverage. Summary of the Invention

[0004] To overcome the above-mentioned technical problems, the present invention aims to provide a confidence assessment method for aviation fuel gear pumps based on Bayesian theory. This method solves the specific problems of low assessment accuracy, poor reliability of confidence intervals, and insufficient utilization of historical data under small sample conditions in traditional methods.

[0005] The technical solution adopted in this invention is: A confidence assessment method for aviation fuel gear pumps based on Bayesian theory includes the following steps; Step 1: First, based on the life characteristics of aviation fuel gear pumps, the Weibull distribution is established as the theoretical basis for reliability modeling, providing probabilistic model support for subsequent analysis; Step 2: Based on the Weibull distribution, the Bayesian method is used to fuse historical data and field test data to achieve point estimation and interval estimation of the shape and dimensional parameters of the aviation fuel gear pump; Step 3: Based on the point estimation and interval estimation, the confidence interval of MTBF (Mean Time Between Failures) is constructed and verified using the Bootstrap method, thus completing the quantitative assessment of the reliability confidence of the aviation fuel gear pump.

[0006] Step one specifically involves: The probability density function of the Weibull distribution is given by the following steps: 11) Lifetime distribution If random variable If the variable follows a two-parameter Weibull distribution, then its probability density function is: In the formula: Represents shape parameters ; Representative scale parameter ; Representative position parameters ; Represents a random variable (lifespan of an aviation fuel gear pump); The cumulative failure distribution function is: In the formula: Represents shape parameters ; Representative scale parameter ; Representative position parameters ; This represents a random variable (the lifespan of an aviation fuel gear pump).

[0007] The specific method for estimating the Weibull distribution parameters in step two is as follows: 21) Overview of the Bayes Method Bayesian estimation methods use unknown parameters... Treat it as a known distribution Random variables, thereby mathematically formalizing and utilizing prior information; Bayes' theorem event form is as follows: Let the event be... Incompatible, and Then for any event ,have: In the formula: For mutually exclusive complete event groups; For any event; Indicates an event After it happened The conditional probability; Bayes' theorem density function has the following form: In classical statistics, it depends on the unknown parameters. The density function is denoted as or It represents in parameter space In China, different This corresponds to different distributions. In Bayesian statistics, the density function is denoted as... It means that in a random variable When given a certain value, the overall index The conditional distribution; then, according to Prior information determination prior distribution ; From a Bayesian perspective, the sample The generation requires two steps. First, we assume that the prior distribution... Generate a parameter The second step is given Below, from the overall distribution A sample is generated in the middle. The probability of this sample occurring is proportional to the following joint probability function, i.e. This function is called the likelihood function, denoted as . ; sample and parameters The joint distribution is: After obtaining sample observations, the joint distribution should be used as a basis. right To make a judgment, it is necessary to... The following breakdown is made: In the formula: yes The marginal density function, i.e.: and Irrelevant, or rather Not included Any information; among which, yes The value space. Used to... The inference is based solely on the conditional distribution. Its calculation formula is: In the formula: For unknown parameters; These are sample observations; for The prior distribution; It is the likelihood function; This is the marginal density function.

[0008] In the sample Given, The conditional distribution is called The posterior distribution is a subset of information from the population, the sample, and the prior. All information, while excluding everything related to Results obtained after irrelevant information; when When the variable is a discrete random variable, the prior distribution can be obtained using a probability distribution table. This means that the posterior distribution is also discrete in this case, i.e.: In the formula: For parameters Discrete values; This is the prior distribution; For a given The probability function of the sample at time; If the overall Since it is also discrete, we only need to change the density function. Treat as a probability function That's all; The posterior distribution is a combination of three types of information; the prior distribution... Reflected in the pre-sampling The understanding of the posterior distribution, and the posterior distribution Reflected in the post-sampling Understanding; 22) Bayes point estimation The product lifespan follows a Weibull distribution. When, if shape parameters Given that, according to the properties of the Weibull distribution, the sample data comes from the sample of the exponential estimate, and the parameters... The Bayesian estimation can be transformed into exponential distribution processing. If the parameters Given the unknown, based on the actual situation, the following priors can be considered; If known The range is in the interval Within, it is an interval. A uniform distribution on the surface; if the failure rate is known to be decreasing, then In this case, the Beta distribution can be chosen as the prior; if the failure rate is known to be increasing, then... At this time It follows a Gamma distribution; If shape parameters The prior distribution of is a continuous distribution, that is... In the formula: For shape parameters; Still assuming The prior distribution is uninformative, and under the assumption of parameter independence... The joint prior density is In the formula: For scale parameters; The joint posterior density function is In the formula: The number of samples; Let be the likelihood function of the Weibull distribution; for The sample function; Expiration time; Assuming the loss function is the squared loss function, and the shape parameter... The Bayesian estimate is In the formula: Shape parameters Bayesian estimates; For a given sample At that time, parameters The posterior probability density; The Bayesian estimate is In the formula: yes Bayesian estimates; For a given sample At that time, parameters The posterior probability density; like If prior information is available, conjugate priors can be selected. As its prior distribution, i.e. In the formula: As a hyperparameter, its estimated value can be obtained using prior information; When shape parameters The prior distribution is a continuous distribution, and the scale parameter is... When the prior distribution is a conjugate prior, the joint prior density is The joint posterior density function is In the formula: because By introducing conjugate priors, the exponential term is represented as a combination of prior and sample information. The part corresponding to prior information; 23) Gibbs sampling The Gibbs sampling method is used to obtain samples of the joint posterior density, thereby calculating the Bayesian estimates of the unknown parameters under the squared loss function. In the Gibbs algorithm, the fully conditional probability density of each parameter is sampled. The fully conditional probability density is still very complex and has no explicit expression. The MH algorithm is used to extract the required samples.

[0009] The steps of the MH algorithm are as follows: Step 1: Settings and number of iterations The result of the maximum likelihood estimation is selected as the initial value, i.e. ; Step 2: Use the MH algorithm sampling conditional density function random numbers ; Step 3: Sampling Gamma Density Function random numbers ,but ; Step 4: Repeat steps 2-3 until... .

[0010] According to the Gibbs sampling method, the ascending sequence of random samples is obtained. and Then the Bayesian estimation expression for the unknown parameters is:

[0011] In the formula: Shape parameters Bayesian estimates; scale parameter Bayesian estimation; The number of independent samples; For the first Shape parameters obtained from the Gibbs sampling Sample values; For the first Scale parameters obtained from Gibbs sampling Sample values; 24) Interval estimation The bootstrap method constructs a resampled sample through resampling, and can estimate parameters by intervals. This method relies only on given sample observations and is suitable for problems where it is difficult to find pivotal quantities using conventional methods. The basic idea is: for a sample from the population... A new set of samples is generated through random sampling with replacement. Based on this set of samples, an estimate of the population parameter is obtained. The above steps are repeated B times to obtain B Bayesian point estimates of the population parameter.

[0012] Based on the sample of these parameters, perform statistical analyses such as hypothesis testing or interval estimation. The specific steps are as follows; Step 1: From truncated samples Generate a new set of samples Based on this, the characteristic parameters of the lifetime distribution can be estimated; Step 2: Repeat Step 1B times At a given time t, the reliability estimates of B products are obtained:

[0013] Step 3: Sort the obtained B reliability estimates from smallest to largest to obtain a new reliability ranking:

[0014] Step 4: Reliability The bilateral confidence level is The Bootstrap confidence interval is:

[0015] In the formula: Number of products; The significance level is indicated by .

[0016] Step two can be summarized as follows: First, establish the theoretical foundation of Bayes (prior-likelihood-posterior framework); apply it to the point estimation of Weibull distribution parameters, but due to the complexity of the posterior distribution, there is no analytical solution; introduce Gibbs sampling and the MH algorithm to perform numerical sampling to achieve Bayesian point estimation; finally, independently use the Bootstrap method to construct the confidence interval of reliability through resampling, quantifying the uncertainty of point estimation.

[0017] In step three, the main steps of the confidence assessment of the reliability method are as follows: (1) Based on the product's failure mechanism, an alternative life distribution model is selected. The main failure mode of the fuel gear pump is wear, and the life distribution is usually selected from the Weibull distribution: In the formula: Represents shape parameters ; Representative scale parameter ; Representative position parameters ; Represents a random variable (lifetime); (2) Based on the failure data of the product in the test, perform a goodness-of-fit test on the lifetime distribution to determine the lifetime distribution of the product; (3) Establish the likelihood function based on lifetime distribution and failure data; (4) Based on historical data, obtain the highest prior distribution of unknown parameters in the product life distribution. It is recommended to use the conjugate prior distribution to facilitate calculation and model update. Use the uninformed prior when prior information is insufficient. (5) Calculate the posterior distribution and calculate the Bayesian estimate of the parameters under the squared loss function. When the posterior distribution is very complex or has no explicit expression, Gibbs sampling is used for calculation. The Bootstrap method is used to construct a large number of confidence intervals, and the true values ​​of the corresponding parameters are tested to see if there is a 90% probability that they fall within the confidence intervals.

[0018] The beneficial effects of this invention are: This invention proposes a confidence assessment method for aviation fuel gear pumps based on Bayesian theory. By integrating prior information and field data through Bayesian theory, it significantly improves the accuracy and reliability of aviation fuel gear pump reliability assessment under small sample conditions. Validation using the Bootstrap method shows that its 90% confidence interval coverage can reach 91.4%, providing more efficient and accurate technical support for reliability assessment and maintenance decisions of aviation fuel gear pumps. Attached Figure Description

[0019] Figure 1This is a schematic diagram of the process of this invention.

[0020] Figure 2 It is a confidence interval plot.

[0021] Figure 3 It is a partial confidence interval plot. Detailed Implementation

[0022] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0023] like Figure 1 As shown, a confidence assessment method for aviation fuel gear pumps based on Bayesian theory includes the following steps: Step 1: Theoretical foundation of reliability assessment, including an overview of lifetime distribution and Bayesian methods.

[0024] Commonly used lifetime distributions in reliability assessment include the exponential distribution, log-normal distribution, and Weibull distribution. Here, we study an object whose lifetime follows a two-parameter Weibull distribution, and use Bayesian methods to estimate the point and interval estimates of the unknown parameters.

[0025] 11) Lifetime distribution If random variable If the variable follows a two-parameter Weibull distribution, then its probability density function is:

[0026] In the formula: Represents shape parameters; Represents scale parameters; Represents positional parameters.

[0027] The cumulative failure distribution function is:

[0028] 12) Overview of the Bayes Method Bayesian estimation methods use unknown parameters... Treat it as a known distribution Random variables, thereby mathematically formalizing and utilizing prior information.

[0029] Bayes' theorem event form is as follows: Let the event be... Incompatible, and (A certain event), then for any event ,have:

[0030] Bayes' theorem density function has the following form: In classical statistics, it depends on the unknown parameters. The density function is denoted as or It represents in parameter space In China, different This corresponds to different distributions. In Bayesian statistics, the density function is denoted as... It means that in a random variable When given a certain value, the overall index The conditional distribution; then, according to Prior information determination prior distribution ; From a Bayesian perspective, the sample The generation requires two steps. First, we assume that the prior distribution... Generate a parameter The second step is given Below, from the overall distribution A sample is generated in the middle. The probability of this sample occurring is proportional to the following joint probability function, i.e. This function is called the likelihood function, denoted as . ; sample and parameters The joint distribution is: After obtaining sample observations, the joint distribution should be used as a basis. right To make a judgment, it is necessary to... The following breakdown is made: In the formula: yes The marginal density function, i.e.: and Irrelevant, or rather Not included Any information; among which, yes The value space. Used to... The inference is based solely on the conditional distribution. Its calculation formula is: This formula is the density function form of Bayes' theorem.

[0031] In the sample Given, The conditional distribution is called The posterior distribution is a subset of information from the population, the sample, and the prior. All information, while excluding everything related to The result obtained after irrelevant information.

[0032] when When the variable is a discrete random variable, the prior distribution can be obtained using a probability distribution table. This means that the posterior distribution is also discrete in this case, i.e.: If the overall Since it is also discrete, we only need to change the density function. Treat as a probability function That's all.

[0033] The posterior distribution is a combination of three types of information; the prior distribution... Reflected in the pre-sampling The understanding of the posterior distribution, and the posterior distribution Reflected in the post-sampling The understanding.

[0034] Step 2: Bayesian estimation, i.e., the Bayesian estimation method for parameters.

[0035] 21) Bayes point estimation The product lifespan follows a Weibull distribution. When, if shape parameters Given that, according to the properties of the Weibull distribution, the sample data comes from the sample of the exponential estimate, and the parameters... The Bayesian estimation can be transformed into exponential distribution processing. If the parameters Given the unknown, based on the actual situation, the following priors can be considered.

[0036] If known The range is in the interval Within, it is an interval. A uniform distribution on the surface; if the failure rate is known to be decreasing, then In this case, the Beta distribution can be chosen as the prior; if the failure rate is known to be increasing, then... At this time It follows a Gamma distribution.

[0037] If shape parameters The prior distribution of is a continuous distribution, that is... Still assuming The prior distribution is uninformative, and under the assumption of parameter independence... The joint prior density is The joint posterior density function is Assuming the loss function is the squared loss function, and the shape parameter... The Bayesian estimate is

[0038] The Bayesian estimate is like If prior information is available, conjugate priors can be selected. As its prior distribution, i.e. In the formula: As a hyperparameter, its estimated value can be obtained using prior information.

[0039] When shape parameters The prior distribution is a continuous distribution, and the scale parameter is... When the prior distribution is a conjugate prior, the joint prior density is

[0040] The joint posterior density function is

[0041] exist When the prior distribution is a continuous distribution, the resulting posterior distribution is usually very complex and does not have an explicit expression, requiring special methods for calculation.

[0042] 22) Gibbs sampling Since an analytical expression for the Bayesian estimate is unavailable, the Gibbs sampling method is used to obtain samples of the joint posterior density, thereby calculating the Bayesian estimate of the unknown parameters under the squared loss function. In the Gibbs algorithm, the fully conditional probability density of each parameter needs to be sampled. The fully conditional probability density is still very complex and has no explicit expression, so the MH algorithm is used to extract the required samples.

[0043] The steps are as follows: Step 1: Settings and number of iterations The result of the maximum likelihood estimation is selected as the initial value, i.e. ; Step 2: Use the MH algorithm sampling conditional density function random numbers ; Step 3: Sampling Gamma Density Function random numbers ,but ; Step 4: Repeat steps 2-3 until... .

[0044] According to the Gibbs sampling method, the ascending sequence of random samples is obtained. and Then the Bayesian estimation expression for the unknown parameters is:

[0045] 23) Interval estimation The bootstrap method constructs a resampled sample through resampling, and can estimate parameters by intervals. This method relies only on given sample observations and is suitable for problems where it is difficult to find pivotal quantities using conventional methods. The basic idea is: for a sample from the population... A new set of samples is generated through random sampling with replacement. Based on this set of samples, an estimate of the population parameter is obtained. The above steps are repeated B times to obtain B Bayesian point estimates of the population parameter.

[0046] Based on the sample with these parameters, statistical analyses such as hypothesis testing or interval estimation can be performed.

[0047] The specific steps are as follows; Step 1: From truncated samples Generate a new set of samples Based on this, the characteristic parameters of the lifetime distribution can be estimated.

[0048] Step 2: Repeat Step 1B times At a given time t, the reliability estimates of B products are obtained:

[0049] Step 3: Sort the obtained B reliability estimates from smallest to largest to obtain a new reliability ranking:

[0050] Step 4: Reliability The bilateral confidence level is The Bootstrap confidence interval is:

[0051] Step 3: Reliability Methods and Confidence Assessment. This step verifies whether the true value of the parameter has a certain probability of falling within the confidence interval.

[0052] The main steps of confidence assessment using reliability methods are as follows: Select alternative lifetime distribution models based on the product's failure mechanism.

[0053] The goodness-of-fit test of the lifetime distribution is performed based on the product's failure data in the test to determine the product's lifetime distribution.

[0054] A likelihood function is established based on lifetime distribution and failure data.

[0055] Based on historical data, the highest prior distribution of unknown parameters in the product lifetime distribution is obtained. It is recommended to use the conjugate prior distribution to facilitate calculation and model updates. Uninformative priors can be used when prior information is insufficient.

[0056] Calculate the posterior distribution and then compute Bayesian estimates of the parameters using the squared loss function. When the posterior distribution is very complex or lacks an explicit expression, Gibbs sampling is used for computation.

[0057] The Bootstrap method is used to construct a large number of confidence intervals, and the true values ​​of the corresponding parameters are tested to see if there is a 90% probability that they fall within the confidence intervals.

[0058] Experimental Example: The advantages of this invention can be further illustrated by the following simulation experiments: Table 1 shows the field usage data of a certain model in the reliability assessment specification for aircraft engine fuel systems.

[0059] Table 1. Field usage data for a certain model (normalized data) The table above provides an estimate of the MTBF for a large sample. The calculation method is as follows In the formula: The cumulative working time for the product is measured in hours (h). In order to be in The number of product-related failures that occurred during the period.

[0060] Substituting the data from Table 1 into the above formula yields the result. .

[0061] The product failure data is shown in Table 2.

[0062] Table 2 Product Failure Time (Normalized Data) The meaningless data in Table 2 mainly consists of early failure data, which was caused by human factors. This type of failure data needs to be directly removed. Among the failure data older than 20 hours, there may be data with inconsistent meaning, which also needs to be screened (removed). The principles and methods are as follows: (1) Use the Bootstrap method to repeatedly extract multiple sets of data from Table 2.

[0063] (2) Perform point estimation on the multiple sets of data extracted in (1) and calculate the mean.

[0064] (3) Find the group in (2) whose estimated value is close to the mean, and whose data are consistent with the meaning.

[0065] (4) Add the missing data from the data of more than 20 hours to the set of data obtained in (3) one by one, and perform a goodness-of-fit test. If they do not follow the same distribution, it means that the missing data is inconsistent with the meaning of other data and needs to be removed directly.

[0066] The processed data is shown in Table 3.

[0067] Table 3 Fault Data Table (Normalized) As shown in the table above, the timing end time is 500 hours, the total number of samples is 178, the number of samples after removing invalid data is 122, and the number of invalid samples is 24.

[0068] The van Montfort test was used to test the goodness of fit. The test steps are as follows: As shown in Table 3, there are a total of One failed data point: We establish the following assumptions:

[0069] make , , . , These are the order statistics of the extreme value distribution and the standard extreme value distribution, respectively. , All parameters are unknown. The following statistics are constructed:

[0070] Put each If the groups are evenly divided into two groups, then the statistic is...

[0071] Gradual obedience ,in For a given significance level The rejection domain is:

[0072] If the sample falls into the rejection region, the null hypothesis is rejected; otherwise, the truncated sample can be considered to come from the Weibull distribution.

[0073] Calculate the test quantity The value is 1.75, which is significant at the significance level. , Less than and greater than If the test result is outside the rejection region, then the hypothesis can be considered valid. This is valid. It is assumed that the lifespan of this batch of products follows a Weibull distribution. .

[0074] Bayesian estimation and confidence interval construction based on failure data: (1) Establish the likelihood function based on lifetime distribution and failure data.

[0075] According to the data in Table 3, the failure data... Total number of samples Based on the failure data, the likelihood function of the sample is:

[0076] In the formula: , , .

[0077] (2) The Gibbs sampling method is used to calculate the Bayesian estimate of the parameters under the continuous-conjugate prior distribution. The algorithm flow of the Gibbs sampling method is as follows: Step 1: Settings and number of iterations The result of the maximum likelihood estimation is selected as the initial value, i.e. ; Step 2: Use the MH algorithm sampling conditional density function random numbers ; Step 3: Sampling Gamma Density Function random numbers ,but ; Step 4: Repeat steps 2-3 until... .

[0078] According to the Gibbs sampling method, the ascending sequence of random samples is obtained. and Then the Bayesian estimation expression for the unknown parameters is:

[0079] (3) The Bootstrap method was used to repeatedly sample 1000 times to construct a 90% confidence interval.

[0080] The basic steps of the Bootstrap method are as follows: Step 1: From truncated samples Generate a new set of samples Based on this, the characteristic parameters of the lifetime distribution can be estimated.

[0081] Step 2: Repeat Step 1B times At a given time t, the reliability estimates of B products are obtained:

[0082] Step 3: Sort the obtained B reliability estimates from smallest to largest to obtain a new reliability ranking:

[0083] Step 4: Reliability The bilateral confidence level is The Bootstrap confidence interval is:

[0084] The Bayesian estimate of MTBF is 1960, and its 90% confidence interval is (1160, 1910).

[0085] Table 1 shows the estimated value of MTBF under a large sample. It can be used as The method confidence is evaluated by taking the truth value. Then, the confidence of the reliability assessment method is evaluated. Generally, a reliability assessment method is considered effective when its confidence level is greater than 90%. The specific steps are as follows: Use the Bootstrap method to construct a large number of confidence intervals.

[0086] Confidence level of reliability assessment methods Is it greater than 90%?

[0087] Confidence level of reliability assessment methods The calculation method is as follows

[0088] In the formula: The total number of confidence intervals constructed. For inclusion The number of intervals.

[0089] A large number of confidence intervals are constructed using the Bootstrap method, and the confidence level of the reliability assessment method is calculated. .

[0090] Total number constructed using the Bootstrap method For a confidence interval of 1000 groups, such as Figure 2 As shown in the figure. Point a is the intersection of the MTBF value and the upper limit of the interval, and point b is the intersection of the MTBF value and the lower limit of the interval. That is, the true value of MTBF is contained from the 42nd group to the 955th group.

[0091] Figure 2 Given 1000 confidence intervals for the MTBF, sort the lower bounds of these 1000 intervals in ascending order. Figure 2Zooming in reveals the specific 10 confidence intervals, such as... Figure 3 As shown.

[0092] from Figure 2 It can be seen that the interval from group 42 to group 955 includes Therefore Then the confidence level of the reliability assessment method can be calculated. Therefore, the Bayesian method can be considered reliable.

[0093] Table 4 shows the confidence assessment results of a reliability evaluation method for a certain type of fuel accessory. The experimental data are presented in Tables 1 and 2. The MTBF estimate for the large sample is calculated from the data in Table 1 and is 1785 hours. Table 2 shows the product failure data. Meaningless duplicate data and inconsistent data from the data source were processed to obtain Table 3. Based on the failure data in Table 3, the van Montfort test was used to perform a goodness-of-fit test. If the test result is outside the rejection region, then the product lifespan can be considered to conform to the Weibull distribution.

[0094] Based on the data in Table 3, Bayesian estimation was performed on the product, and confidence intervals were constructed using the Bootstrap method. The 90% confidence interval was (1160, 1910). Using the Bootstrap method, 1000 confidence intervals with a 90% confidence level were constructed, of which approximately 914 contained the true value of the MTBF. This indicates a 91.4% probability of including the MTBF value from a large sample, suggesting that the Bayesian evaluation method is reliable.

[0095] Table 4. Confidence Assessment Results of a Certain Type of Fuel Accessory Reliability Assessment Method

Claims

1. A confidence assessment method for aviation fuel gear pumps based on Bayesian theory, characterized in that, Includes the following steps; Step 1: First, based on the life characteristics of aviation fuel gear pumps, the Weibull distribution is established as the theoretical basis for reliability modeling, providing probabilistic model support for subsequent analysis; Step 2: Based on the Weibull distribution, the Bayesian method is used to fuse historical data and field test data to achieve point estimation and interval estimation of the shape and dimensional parameters of the aviation fuel gear pump; Step 3: Based on the point estimation and interval estimation, the MTBF confidence interval is constructed and verified using the Bootstrap method, thus completing the quantitative assessment of the reliability confidence of the aviation fuel gear pump.

2. The confidence assessment method for aviation fuel gear pumps based on Bayesian theory according to claim 1, characterized in that, Step one specifically involves: The probability density function of the Weibull distribution is given by the following steps: If random variable If the variable follows a two-parameter Weibull distribution, then its probability density function is: In the formula: Represents shape parameters ; Representative scale parameter ; Representative position parameters ; Represents a random variable (lifespan of an aviation fuel gear pump); The cumulative failure distribution function is: In the formula: Represents shape parameters ; Representative scale parameter ; Representative position parameters ; This represents a random variable (the lifespan of an aviation fuel gear pump).

3. The confidence assessment method for aviation fuel gear pumps based on Bayesian theory according to claim 1, characterized in that, The specific method for estimating the Weibull distribution parameters in step two is as follows: 21) Overview of the Bayes Method Bayesian estimation methods take unknown parameters Treat it as a known distribution Random variables, thereby mathematically formalizing and utilizing prior information; Bayes' theorem event form is as follows: Let the event be... Incompatible, and Then for any event ,have: In the formula: For mutually exclusive complete event groups; For any event; Indicates an event After it happened The conditional probability; Bayes' theorem density function has the following form: In classical statistics, it depends on the unknown parameters. The density function is denoted as or It represents in parameter space In China, different This corresponds to different distributions. In Bayesian statistics, the density function is denoted as... It means that in a random variable When given a certain value, the overall index The conditional distribution; then, according to Prior information determination prior distribution ; From a Bayesian perspective, the sample The generation of this requires two steps; first, we assume that it is derived from the prior distribution. Generate a parameter The second step is given Below, from the overall distribution A sample is generated in the middle. The probability of this sample occurring is proportional to the following joint probability function, i.e. This function is called the likelihood function, denoted as . ; sample and parameters The joint distribution is: After obtaining sample observations, the joint distribution should be used as a basis. right To make a judgment, it is necessary to... The following breakdown is made: In the formula: yes The marginal density function, i.e.: and Irrelevant, or rather Not included Any information; among which, yes The value space is used to... The inference is based solely on the conditional distribution. Its calculation formula is: In the formula: For unknown parameters; These are sample observations; for The prior distribution; It is the likelihood function; It is the marginal density function; In the sample Given, The conditional distribution is called The posterior distribution is a subset of information from the population, the sample, and the prior. All information, while excluding everything related to Results obtained after irrelevant information; when When the variable is a discrete random variable, the prior distribution can be obtained using a probability distribution table. This means that the posterior distribution is also discrete in this case, i.e.: In the formula: For parameters Discrete values; This is the prior distribution; For a given The probability function of the sample at time; If the overall Since it is also discrete, we only need to consider the density function. Treat as a probability function That's all; The posterior distribution is a combination of three types of information; the prior distribution... Reflected in the pre-sampling The understanding of the posterior distribution, and the posterior distribution Reflected in the post-sampling Understanding; 22) Bayes point estimation The product lifespan follows a Weibull distribution. When, if shape parameters Given that, according to the properties of the Weibull distribution, the sample data comes from the sample of the exponential estimate, and the parameters... The Bayesian estimation can be transformed into exponential distribution processing. If the parameters Given the unknown, based on the actual situation, the following priors can be considered; If known The range is in the interval Within, it is an interval. A uniform distribution on the surface; if the failure rate is known to be decreasing, then In this case, the Beta distribution can be chosen as the prior; if the failure rate is known to be increasing, then... At this time It follows a Gamma distribution; If shape parameters The prior distribution of is a continuous distribution, that is... In the formula: For shape parameters; Still assuming The prior distribution is uninformative, and under the assumption of parameter independence... The joint prior density is In the formula: For scale parameters; The joint posterior density function is In the formula: The number of samples; Let be the likelihood function of the Weibull distribution; for The sample function; Expiration time; Assuming the loss function is the squared loss function, and the shape parameter... The Bayesian estimate is In the formula: Shape parameters Bayesian estimates; For a given sample At that time, parameters The posterior probability density; The Bayesian estimate is In the formula: yes Bayesian estimates; For a given sample At that time, parameters The posterior probability density; like If prior information is available, conjugate priors can be selected. As its prior distribution, i.e. In the formula: As a hyperparameter, its estimated value can be obtained using prior information; When shape parameters The prior distribution is a continuous distribution, and the scale parameter is... When the prior distribution is a conjugate prior, the joint prior density is The joint posterior density function is In the formula: because By introducing conjugate priors, the exponential term is represented as a combination of prior and sample information. The part corresponding to prior information; 23) Gibbs sampling The Gibbs sampling method is used to obtain samples of the joint posterior density, thereby calculating the Bayesian estimates of the unknown parameters under the squared loss function. In the Gibbs algorithm, the fully conditional probability density of each parameter is sampled. The fully conditional probability density is still very complex and has no explicit expression. The MH algorithm is used to extract the required samples. 24) Interval estimation For a sample from the population A new set of samples is generated through random sampling with replacement. Based on this set of samples, an estimate of the population parameter is obtained. The above steps are repeated B times to obtain B Bayesian point estimates of the population parameter.

4. The confidence assessment method for aviation fuel gear pumps based on Bayesian theory according to claim 3, characterized in that, The steps of the MH algorithm are as follows: Step 1: Settings and number of iterations The result of the maximum likelihood estimation is selected as the initial value, i.e. ; Step 2: Use the MH algorithm sampling conditional density function random numbers ; Step 3: Sampling Gamma Density Function random numbers ,but ; Step 4: Repeat steps 2-3 until... ; According to the Gibbs sampling method, the ascending sequence of random samples is obtained. and Then the Bayesian estimation expression for the unknown parameters is: In the formula: Shape parameters Bayesian estimates; scale parameter Bayesian estimation; The number of independent samples; For the first Shape parameters obtained from the Gibbs sampling Sample values; For the first Scale parameters obtained from Gibbs sampling Sample value.

5. The confidence assessment method for aviation fuel gear pump based on Bayesian theory according to claim 3, in step 24), statistical analysis such as hypothesis testing or interval estimation is performed based on the sample of this set of parameters; The specific steps are as follows; Step 1: From truncated samples Generate a new set of samples Based on this, the characteristic parameters of the lifetime distribution can be estimated; Step 2: Repeat Step 1B times At a given time t, the reliability estimates of B products are obtained: Step 3: Sort the obtained B reliability estimates from smallest to largest to obtain a new reliability ranking: Step 4: Reliability The bilateral confidence level is The Bootstrap confidence interval is: In the formula: Number of products; The significance level is indicated by .

6. The confidence assessment method for aviation fuel gear pumps based on Bayesian theory according to claim 1, wherein the main steps of the confidence assessment method in step three are as follows: (1) Based on the product's failure mechanism, an alternative life distribution model is selected. The main failure mode of the fuel gear pump is wear, and the life distribution is usually selected from the Weibull distribution: In the formula: Represents shape parameters ; Representative scale parameter ; Representative position parameters ; Represents a random variable (lifetime); (2) Based on the failure data of the product in the test, perform a goodness-of-fit test on the lifetime distribution to determine the lifetime distribution of the product; (3) Establish the likelihood function based on lifetime distribution and failure data; (4) Based on historical data, obtain the highest prior distribution of unknown parameters in the product life distribution. It is recommended to use the conjugate prior distribution to facilitate calculation and model update. Use the uninformed prior when prior information is insufficient. (5) Calculate the posterior distribution and calculate the Bayesian estimate of the parameters under the squared loss function. When the posterior distribution is very complex or has no explicit expression, Gibbs sampling is used for calculation. The Bootstrap method is used to construct a large number of confidence intervals, and the true values ​​of the corresponding parameters are tested to see if there is a 90% probability that they fall within the confidence intervals.

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