A method for transferring environmental adaptability indicators based on Box-Cox transform

Abstract

This invention proposes a method for transferring environmental adaptability indicators based on Box-Cox transform. First, the environmental maladaptation function of the system is obtained from component data based on the system structure. Then, simulated system-level product environmental failure data is obtained through Monte Carlo sampling. The prior distribution of the system is approximated based on Box-Cox transform, and the posterior distribution of the system is updated according to Bayesian theory, thereby realizing the transfer of environmental adaptability indicators. This method is applicable to fields such as multi-level product environmental adaptability indicator transfer and evaluation. This invention uses Box-Cox transform to approximate the system's prior distribution using a normal distribution, resulting in low computational complexity and ease of operation. Based on Bayesian theory, it completes the fusion and updating of multi-level product environmental failure data, achieving high accuracy in system environmental adaptability indicator evaluation. The environmental adaptability indicator transfer method established in this invention is applicable to various different system structure types and has strong operability.

Description

Technical Field

[0001] This invention relates to a method for transferring environmental adaptability indicators based on Box-Cox transform. It is a method for transferring environmental adaptability indicators from components to system-level environmental adaptability indicators. First, based on the system structure, the environmental maladaptation function of the system is obtained from component data. Then, simulated system-level product environmental failure data is obtained through Monte Carlo sampling. Based on this, the prior distribution of the system is approximated using Box-Cox transform. Next, the posterior distribution of the system is updated according to Bayesian theory, thereby realizing the transfer of environmental adaptability indicators. This method is applicable to fields such as multi-level product environmental adaptability indicator transfer and evaluation. Background Technology

[0002] For complex equipment systems, limitations such as time and economic constraints make it difficult to conduct large-scale environmental tests, hindering the provision of sufficient statistical data for environmental adaptability analysis. This results in low accuracy and limited application scope for system environmental adaptability indicators. In contrast, component-level environmental test data is abundant, leading to more accurate environmental adaptability assessments. Therefore, environmental adaptability indicators for system-level products can be obtained from component-level data. However, traditional multi-level indicator transfer methods, such as the Analytic Hierarchy Process (AHP), are simplistic and crude in their weight allocation, lacking scientific basis and failing to effectively integrate multi-level, multi-source data, thus affecting the accuracy and rationality of indicator transfer. The Box-Cox transformation can handle complex data types and unify different distributions into a normal distribution, while Bayesian theory can effectively achieve the fusion and updating of data at different levels. This provides a new approach for the transfer and evaluation of multi-level product environmental adaptability indicators.

[0003] Based on this, the present invention proposes an environmental adaptability index transfer method based on Box-Cox transformation. First, the environmental maladaptation function of the system is obtained from the component data according to the system structure. Based on the simulation sampling, the prior distribution of the system is approximated by Box-Cox transformation. Then, Bayesian theory is used to update the prior distribution, thereby realizing the transfer and evaluation of multi-level product environmental adaptability indexes. Summary of the Invention

[0004] The purpose of this invention is to address the problem of low accuracy in evaluating multi-level product environmental adaptability indicators by providing a method for transferring and evaluating environmental adaptability indicators based on Box-Cox transformation and Bayesian theory. By fully utilizing component environmental test data, the scientific validity and rationality of transferring system-level product environmental adaptability indicators are improved, thus perfecting the methodology for evaluating multi-level product environmental adaptability indicators.

[0005] Therefore, the present invention needs to establish the following basic settings: Setting 1: Product under specific environmental conditionss With a specific time t Environmental fitness is expressed as follows: (1) in, This represents the product's environmental tolerance time, and is a random variable. S Indicates environmental stress. Indicates model parameters, Indicates that given model parameters Environmental stress conditions With time conditions The environmental adaptability of the product. Based on this, the environmental maladaptability function of the product is: (2) in, Indicates that given model parameters Environmental stress conditions With time conditions Environmental incompatibility of the product.

[0006] The corresponding probability density function is: (3) in, Indicates that given model parameters Environmental stress conditions With time conditions The probability density function of the product at that time. This represents the derivative of the maladaptation function with respect to time.

[0007] Setting 2: The product's time-related environmental adaptability index is the average time before environmental maladaptation, or simply the average time before maladaptation, which can be expressed as: (4) in, This indicates the average time before the product was adapted to the environment.

[0008] Based on the above assumptions, the present invention provides an environmental adaptability index transfer method based on Box-Cox transformation, which is implemented through the following steps: Step 1: Calculate the component's environmental maladaptation function; For specific environmental conditions s Environmental failure data of lower components Based on the form of the environmental maladaptation function of each component, the model parameters in the environmental maladaptation function of each component are obtained through the maximum likelihood estimation method, i.e.: (5) in, Indicates the firsti Estimated values ​​of parameters for class component models. Indicates the first i The probability density function of class components, The types of display components, Indicates the first i The number of failure data in the class component environment.

[0009] Step 2: Approximate the prior distribution of the system; Based on the system structure (e.g., series, parallel, etc.), the original prior distribution of the system's environmental maladaptation function is obtained from the environmental maladaptation function of the components, i.e.: (6) in, This represents the original prior distribution of the system's fitness function. Indicates the first i The environment incompatibility function of class components Indicates the first i Estimated values ​​of parameters for class component models. The types of display components, This represents the system structure function, which is determined by the system structure (such as series, parallel, etc.).

[0010] Based on the original prior distribution obtained from equation (6), random sampling is performed using its inverse function. One sample: The sample data obtained by sampling is subjected to Box-Cox transformation, that is: (7) in, Represents the Box-Cox transform coefficients. Represents the transformed th a Data.

[0011] The transformed data approximately follows a normal distribution, and the corresponding mean estimate is denoted as . The standard deviation estimate is That is, the probability density function of the transformed data can be approximately expressed as: (8) in, This represents the data after the Box-Cox transformation. This represents the estimated mean of the transformed data. This represents the estimated standard deviation of the transformed data. This represents the approximate probability density function of the transformed data.

[0012] The approximate cumulative distribution function of the transformed data is: (9) in, The cumulative distribution function represents the standard normal distribution. Indicates a specific time Box-Cox transform, This represents the approximate cumulative distribution function of the transformed data.

[0013] Next, the mean of the transformed data will be... Randomization yields a prior distribution that is also normally distributed, with a mean of [value missing]. The standard deviation of the prior distribution is The corresponding probability density function is: (10) in, This represents the estimated mean of the transformed data. This represents the estimated standard deviation of the transformed data. This represents the probability density function of the prior distribution of the mean.

[0014] Step 3: Update the system's posterior distribution; For specific environmental conditions s Next-level system products Each environmental failure data point is denoted as: ,by Perform a Box-Cox transformation on the coefficients, and denote the transformed data as follows: The transformed data follows a normal distribution.

[0015] Due to the conjugate property of the prior distribution of the normal distribution mean, the posterior distribution of the mean of the transformed data is still a normal distribution, and its posterior mean and posterior variance are respectively: (11) (12) in, This represents the mean estimate of system-level product environmental failure data after Box-Cox transformation. This represents the estimated standard deviation of system-level product environmental failure data after Box-Cox transformation.

[0016] According to the law of total probability, the posterior probability density function of the marginal distribution of the transformed data is... for: (13) in, and Let represent the posterior mean and posterior variance calculated by equations (11) and (12), respectively. This represents the data after the Box-Cox transformation.

[0017] Therefore, the data after Bayesian update and fusion still follows a normal distribution, with a mean of 1 / 2. The variance is .

[0018] For the system environment failure time, its probability density function for: (14) in, Indicates a specific time Box-Cox transform, Represents the Box-Cox transform coefficients. Indicates a specific time of Power of 1.

[0019] Step 4: Calculate the system's environmental adaptability indicators; According to equation (14) and setting 2, the average environment of the system is not adapted to the previous time. MTTEI S It can be represented as: (15) in, The probability density function represents the system's environmental failure time.

[0020] The advantages and beneficial effects of this invention are as follows: ① This invention approximates the prior distribution of the system using a normal distribution through Box-Cox transformation, which requires little computation and is easy to operate; ② Based on Bayesian theory, this invention has completed the fusion and updating of multi-level product environmental failure data, and the system's environmental adaptability index assessment has high accuracy; ③ The environmental adaptability index transfer method established in this invention is applicable to a variety of different system structure types and is highly operable. Attached Figure Description

[0021] Figure 1 This is a flowchart of the method described in this invention.

[0022] Figure 2 This is a comparison chart of the original prior distribution and the approximate prior distribution. Detailed Implementation

[0023] The present invention will be further described in detail below with reference to embodiments; A certain type of seawater cooling system consists of three sea-entry valves, two pipelines, and one seawater pump connected in series. Under environmental conditions of seawater temperature of 15℃, seawater flow velocity of 2m / s, seawater salinity of 3.5%, and seawater pressure of 0.2MPa, environmental failure data of various components such as sea-entry valves, pipelines, and seawater pump were recorded, as shown in Table 1.

[0024] Table 1 Environmental Failure Data of Components in a Certain Type of Seawater Cooling System

[0025] Among them, the failure times of the three sea-access valves follow a Weibull distribution, and the failure times of the two pipelines and one seawater pump follow a log-normal distribution. The environmental fitness function expressed by the Weibull distribution can be represented as:

[0026] in, The scale parameter of the Weibull distribution is denoted by . This represents the shape parameter of the Weibull distribution.

[0027] The fitness function represented by the log-normal distribution can be expressed as:

[0028] in, This represents the logarithmic mean parameter of the log-normal distribution. The log-standard deviation parameter represents the logarithmic normal distribution. The cumulative distribution function represents the standard normal distribution.

[0029] Under the same environmental conditions, environmental failure data (in months) of the seawater cooling system were collected, which were: 0.51, 2.79, 2.94, 3.99, 4.87, 5.92, 6.65, and 9.18, respectively.

[0030] Step 1: Calculate the component's environmental maladaptation function; The model parameters in the environmental maladaptation function of each component are estimated according to equation (5), and the results are shown in Table 2.

[0031] Table 2. Estimated values ​​of parameters for the component environmental maladaptation function model.

[0032] Step 2: Approximate the prior distribution of the system; Since the system structure is serial, according to equation (6), the original prior distribution of the system's environmental fitness function can be obtained as follows:

[0033] Based on this original prior distribution, random sampling is performed using its inverse function. A sample is obtained, and then the sample data is subjected to Box-Cox transformation using equation (7). The transformed data approximately follows a normal distribution, and the transformation coefficients can be obtained as follows: The estimated mean of the transformed data is The estimated standard deviation of the transformed data is .

[0034] According to equation (9), the cumulative distribution function of the approximate prior distribution can be obtained as follows:

[0035] in, The cumulative distribution function represents the standard normal distribution. This indicates a specific time when the transformation coefficient is 0.93. The Box-Cox transform.

[0036] The comparison between the original prior distribution and the approximate prior distribution is as follows: Figure 2 As shown, the approximate prior distribution is quite close to the original prior distribution.

[0037] Step 3: Update the system's posterior distribution; Under given environmental conditions, with transformation coefficients A Box-Cox transformation was performed on the environmental failure data of the seawater cooling system. The transformed data approximately follows a normal distribution. The estimated mean of the transformed data for the seawater cooling system is: The estimated standard deviation of the data after the transformation of the seawater cooling system is: According to equations (11) and (12), the posterior mean and posterior variance are respectively: .

[0038] According to equation (14), the probability density function of the environmental failure time of the seawater cooling system after Bayesian fusion update is:

[0039] in, .

[0040] Step 4: Calculate time-related environmental adaptability indicators; According to equation (15), the average time before environmental maladaptation of the seawater cooling system is:

[0041] The results show that the method proposed in this invention can realize the transmission and evaluation of multi-level product environmental adaptability indicators, thus achieving the expected purpose.

[0042] In summary, this invention relates to a method for transferring environmental adaptability indicators based on Box-Cox transform. It is a multi-level product environmental adaptability indicator transfer and evaluation method based on Box-Cox transform and Bayesian theory. The specific steps of this method are: 1. Calculation of component environmental maladaptation function; 2. Approximation of system prior distribution; 3. Update of system posterior distribution; 4. Calculation of time-based environmental adaptability indicators. This invention is applicable to fields such as multi-level product environmental adaptability indicator transfer and evaluation, and has high practicality and operability.

Claims

1. A method for transferring environmental adaptability indicators based on Box-Cox transformation requires the following settings: Setting 1 : Product in specific environmental conditions s With specific time t under the environmental fitness is expressed as: （1） in, environmental resistance time of a product, is a random variable, S environmental stress, model parameters, denotes the environmental fitness of a product given the model parameters , environmental stress conditions and time conditions ; the environmental unfitness function of a product is （2） wherein, represents the environmental inadaptability of the product at a given model parameter , environmental stress condition and time condition ; The corresponding probability density function is: （3） wherein, denotes the probability density function of the product at given model parameters , environmental stress conditions and time conditions , denotes the derivative of the environmental fitness function with respect to time; Setting 2: The product's time-related environmental adaptability indicator is the average time before environmental maladaptation, or simply the average time before maladaptation, expressed as: （4） wherein, represents the average pre-environmental maladaptation time of the product; Based on the above settings, the feature is that it includes the following steps: Step 1: Calculate the component's environmental maladaptation function; For certain environmental conditions s Environmental failure data of lower component The model parameters in the environmental inadaptability function of each component are obtained by maximum likelihood estimation method according to the form of the environmental inadaptability function of each component. Step 2: Approximate the prior distribution of the system; Based on the system structure, the original prior distribution of the system's environmental maladaptation function is obtained from the environmental maladaptation function of the components; Step 3: Update the system's posterior distribution; For specific environmental conditions s The system level product of The environmental failure data is recorded as: The Box-Cox transformation is performed with the coefficient of The transformed data is recorded as: The transformed data is subject to normal distribution; Step 4: Calculate the system's environmental adaptability index.

2. The Box-Cox transformation based environmental suitability index transfer method according to claim 1, characterized in that: In step one, the formula is: （5） wherein, represents the number of the i estimated value of the class component model parameter, represents the number of the i probability density function of the class component, represents the class of the component, represents the number of the i number of the class component environmental failure data.

3. The Box-Cox transformation based environmental suitability index transfer method according to claim 1, characterized in that: In step two, the formula is: （6） wherein, represents a raw prior distribution of the system environment misfit function, represents an estimate of the model parameters of the i class subcomponent, represents an estimate of the model parameters of the i class subcomponent, represents the class of the subcomponent, represents a system structure function, determined by the system structure.

4. The method for transferring environmental adaptability indicators based on Box-Cox transformation according to claim 3, characterized in that: In step two, based on the original prior distribution obtained from equation (6), random sampling is performed using the inverse function. One sample: The sample data obtained by sampling is subjected to Box-Cox transformation, that is: （7） in, Represents the Box-Cox transform coefficients. Represents the transformed th a Data.

5. A method for transferring environmental adaptability indicators based on Box-Cox transformation according to claim 3 or 4, characterized in that: In step two, the transformed data approximately follows a normal distribution, and the corresponding mean estimate is denoted as... The standard deviation estimate is That is, the probability density function of the transformed data can be approximately expressed as: （8） in, This represents the data after the Box-Cox transformation. This represents the estimated mean of the transformed data. This represents the estimated standard deviation of the transformed data. This represents the approximate probability density function of the transformed data. The approximate cumulative distribution function of the transformed data is: （9） in, The cumulative distribution function represents the standard normal distribution. Indicates a specific time Box-Cox transform, This represents the approximate cumulative distribution function of the transformed data.

6. The method for transferring environmental adaptability indicators based on Box-Cox transformation according to claim 5, characterized in that: In step two, the mean of the transformed data is... Randomization yields a prior distribution that is also normally distributed, with the mean of the prior distribution being... The standard deviation of the prior distribution is The corresponding probability density function is: （10） in, This represents the estimated mean of the transformed data. This represents the estimated standard deviation of the transformed data. This represents the probability density function of the prior distribution of the mean.

7. The method for transferring environmental adaptability indicators based on Box-Cox transformation according to claim 1, characterized in that: In step three, due to the conjugate property of the prior distribution of the normal distribution mean, the posterior distribution of the transformed data mean is still a normal distribution, and the posterior mean and posterior variance are respectively: （11） （12） in, This represents the mean estimate of system-level product environmental failure data after Box-Cox transformation. This represents the estimated standard deviation of system-level product environmental failure data after Box-Cox transformation.

8. The method for transferring environmental adaptability indicators based on Box-Cox transformation according to claim 7, characterized in that: In step three, according to the law of total probability, the posterior probability density function of the marginal distribution of the transformed data is... for: （13） in, and Let represent the posterior mean and posterior variance calculated by equations (11) and (12), respectively. This represents the data after the Box-Cox transformation; Therefore, the data after Bayesian update and fusion still follows a normal distribution with a mean of . The variance is .

9. A method for transferring environmental adaptability indicators based on Box-Cox transformation according to claim 7 or 8, characterized in that: In step three, for the system environment failure time, the probability density function... for: （14） in, Indicates a specific time Box-Cox transform, Represents the Box-Cox transform coefficients. Indicates a specific time of Power of 1.

10. The method for transferring environmental adaptability indicators based on Box-Cox transformation according to claim 1, characterized in that: In step four, the system's average environmental condition does not adapt to the previous time. MTTEI S Represented as: （15） in, The probability density function represents the system's environmental failure time.

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