Small sample degradation quantity prediction method based on data and model fusion

Abstract

This invention discloses a small-sample degradation prediction method that integrates data and models. First, degradation samples D of key equipment during operation are acquired using sensors or equipment recorders. These degradation samples are then processed using a normalization method, dividing them into degradation training samples and degradation test samples. Finally, a neural network is used to learn the degradation amount of the training samples corresponding to each moment of the key equipment, resulting in the encoded degradation amount and the relationship function h between different moments. i Using an aggregator to analyze the relation function h at different times i The method involves integrating the data and calculating the mean and variance of the integrated function. The mean and variance are then sampled to obtain the latent variable z. Finally, a neural network is used to learn the function g between the latent variable z and the degradation test samples to obtain the predicted value of the degradation amount of the key equipment. This invention solves the problem that existing technologies using single traditional stochastic processes and neural network methods have poor performance in modeling and predicting degradation amounts.

Description

Technical Field

[0001] This invention belongs to the technical field of degradation analysis of key equipment, specifically involving a small-sample degradation prediction method that integrates data and models. Background Technology

[0002] With the rapid development of industrial systems, highly reliable and long-life critical equipment has become a crucial direction for stable system operation and cost reduction. However, since the analysis of equipment degradation processes is the foundation of reliability analysis and remaining life prediction, it is generally necessary to establish degradation process models to predict future degradation levels and further obtain a correlation model between degradation levels and equipment failure. Therefore, establishing accurate degradation process models is of paramount importance for improving the safe operation of systems.

[0003] In predicting the degradation of critical equipment, since the degradation amount is a time-series sample, it is first necessary to establish a correlation model between the degradation amount of each piece of equipment and time. Then, based on the correlation model between degradation amount and time, the degradation amount at future times is predicted. Generally, the degradation process of a piece of equipment is mainly composed of different sample degradation amounts, which easily introduces uncertainties, namely heterogeneity between samples, due to deviations that exist when the equipment leaves the factory. When establishing a degradation process model, traditional stochastic process models require prior assumptions about the distribution of certain parameters, which deviate from the actual situation to some extent, further affecting the reliability of subsequent analysis results. Machine learning methods mainly analyze data to obtain a model of the relationship between degradation amount and monitoring time, without requiring prior assumptions about model parameters; however, the uncertainty of degradation samples may not be learnable. If the advantages of neural networks and stochastic process methods can be combined, the accuracy of the degradation process model can be improved, further enhancing the reliability analysis results of critical equipment. Summary of the Invention

[0004] The purpose of this invention is to provide a small-sample degradation prediction method that integrates data and models, which solves the problem that existing technologies using single traditional stochastic processes and neural network methods have poor performance in modeling and predicting degradation.

[0005] The technical solution adopted in this invention is a small-sample degradation prediction method that integrates data and models, specifically implemented according to the following steps:

[0006] Step 1: Obtain degradation samples D of key equipment during operation using sensors or equipment recorders;

[0007] Step 2: Process the degraded samples using a normalization method and divide the samples into degraded training samples and degraded test samples;

[0008] Step 3: Learn the degradation amount of the training samples corresponding to each moment of the key equipment through a neural network, and obtain the relationship function h between the encoded degradation amount and different moments. i ;

[0009] Step 4: Use an aggregator to analyze the relation function h at different times. i The functions are integrated, and the mean and variance of the integrated function are calculated. The mean and variance are sampled to obtain the latent variable z.

[0010] Step 5: Finally, combining the degradation test samples obtained in Step 2 with the latent variable z obtained in Step 4, the function g of the latent variable z and the degradation test samples is learned through a neural network to obtain the predicted value of the degradation amount of the key equipment.

[0011] The invention is further characterized in that,

[0012] Step 2 is as follows:

[0013] Step 2.1: Let the original degraded sample D be... 1:m ={T 1:m ,X 1:m},

[0014]

[0015] Among them, X 1:m The sample represents the overall degradation of critical equipment, where j = 1, ..., l represents the j-th degradation sample of critical equipment. l Let X represent the total number of degraded samples, i = 1, ..., m represent the i-th monitoring time, and m be the total monitoring duration. j,i T represents the degradation of the j-th critical device at time i; 1:m T represents the total observation time of the entire sample. i Indicates the i-th observation time;

[0016] Step 2.2: Using the max-min normalization method, the degradation samples X of key equipment are... 1:m Normalization to the same dimension is performed as shown in formula (1):

[0017]

[0018] Among them, X j,i This indicates that the j-th critical device is in T. i The degradation amount corresponding to time, max(X) j,： ) represents the maximum degradation of the j-th critical device during the entire monitoring period, max(X) j,： X' is the minimum degradation of the j-th critical device during the entire monitoring period. j,i It is the jth key device in T iThe normalized degradation amount corresponding to the given time;

[0019] Therefore, the normalized degenerate sample D1' is obtained. :m ={T 1:m ,X' 1:m},

[0020]

[0021] Where X' 1:m This represents the normalized overall degradation sample of key equipment.

[0022] Step 2.3: Assuming the duration of the degradation training samples is n, then the duration of the test samples is mn-1. Therefore, the normalized degradation training samples are obtained as D'. 1:n＝ {T 1:n ,X' 1:n The normalized degradation test sample is D'. n+1:m ={T n+1:m ,X' n+1:m}

[0023] Step 3 is as follows:

[0024] The normalized training sample D' is obtained through step 2. 1:n＝ {T 1:n ,X' 1:n}, the degradation amount corresponding to each time step is (T1,X'1), ...,(T i ,X' i ), …, (T n ,X' n ), where X' i =[X' 1,i ,...,X' j,i ,...,X' l,i That is, at each time T i Corresponding to the degradation amount of training samples for l key devices, then, the neural network learns the degradation amount of each time step T. i Degradation amount of training samples X' i The relationship function h i =NN(T) i ,X' i ), i = 1,...,n, which means we obtain n relational functions h1, h2,...h n , where X' i It is T i The device training sample degradation amount is n, where n is the degradation amount training sample duration, and NN represents a neural network, specifically a backpropagation neural network (BPNN) or a recurrent time network (RNN).

[0025] Step 4 is as follows:

[0026] Step 4.1: Obtain the time step T at each moment after training the neural network, as described in Step 3. i Degradation amount of training samples X' i The relationship function h i =NN(T) i ,X' i ), i = 1,...,n, using an average aggregator for n relation functions h1, h2,...h n Aggregation is performed as shown in formula (2):

[0027]

[0028] Where r is the aggregation result and n is the duration of the degraded training samples;

[0029] Step 4.2: Calculate the mean μ(r) and variance σ of the aggregation results. 2 (r), and then sample the mean and variance to obtain the latent variable p(z|T) that follows a normal distribution. 1:n ,X' 1:n )~N(μ(r),σ 2 (r)).

[0030] Step 5 is as follows:

[0031] Step 5.1: Assuming the normalized degenerate sample D' follows a Gaussian process f(T)~GP, the degenerate sample X' based on the Gaussian process is obtained according to Bayes' theorem. 1:m The distribution is shown in formula (3):

[0032] p(X' 1:m |T 1:m )=∫p(f(T 1:m ))p(X' 1:m |f(T 1:m ),T 1:m )df(T 1:m (3)

[0033] Where X' 1:m It is the normalized overall degenerate sample, f(T) 1:m ) represents the normalized degenerate sample D1' :m The probability density function, T 1:m It is the overall degradation sample monitoring time, p(f(T) 1:m The distribution of monitoring time for degraded samples is given by f(T) ~ GP, which follows a Gaussian process. p(f(T)) 1:m The distribution follows a Gaussian distribution, therefore, p(X') 1:m |f(T 1:m ),T1:m As shown in formula (4):

[0034]

[0035] Substituting formula (4) into formula (3), we obtain the degenerate sample distribution, as shown in formula (5):

[0036]

[0037] Therefore, the degenerate sample X' based on the Gaussian process 1:m Distribution p(X') 1:m |T 1:m It also follows a Gaussian distribution;

[0038] Step 5.2: Learn each monitoring time T in the test samples using a neural network NN. n+1:m The relationship function g with the latent variable z m-n-1 =NN(T) n+1:m ,z), the stochastic process f(T) is transformed using g m-n-1 =NN(T) n+1:m The description of z) is that the relation function g m-n-1 =NN(T) n+1:m Replace f(T) with z) and combine it with the test degenerate sample D n+1:m ={T n+1:m ,X' n+1:m Formula (5) becomes:

[0039]

[0040] Where p(z,X') n+1:m |T 1:m ) is the joint distribution of the latent variable and the degradation of the test sample, p(z) is the prior distribution of the latent variable, and g is the joint distribution of the latent variable and the degradation of the test sample. v (T v (z) represents the v-th monitoring time T in the test sample. n+1:m The relationship function between the latent variable z;

[0041] Step 5.3: Based on the integral and relation p(X') n+1:m |T 1:m )=∑ z p(z,X' n+1:m |T 1:m Based on the overall degradation sample monitoring time T 1:m Degradation amount of training samples X' 1:n Test sample degradation X' n+1:m The distribution is shown in formula (7):

[0042] p(X' n+1:m |T1:m ,X' 1:n )=∑ z p(z,X' n+1:m |T 1:m ,X' 1:n (7)

[0043] Where, ∑ z p(z,X' n+1:m |T 1:m ,X ' 1:n The test sample degradation X' is based on the latent variable z. n+1:m Distribution integral value, p(X') n+1:m |T 1:m ,X' 1:n ) is the overall degradation sample monitoring time T 1:m Degradation amount of training samples X' 1:n Under these conditions, the degradation amount X' of the test sample n+1:m The distribution is then used to derive the degradation amount X' of the test sample. n+1:m Related to the latent variable z;

[0044] Step 4 yields the latent variable p(z|T) that follows a normal distribution. 1:n ,X' 1:n )~N(μ(r),σ 2 (r)), combining the logarithm and KL divergence rules, to obtain the degradation amount X' of the test sample. n+1:m The lower bound function of the evidence distribution is calculated as shown in formula (8):

[0045]

[0046] in, It is the distribution of the latent variable z that follows a normal distribution, q(z|T) 1:m ,X' 1:m () is an approximation of the distribution of the latent variable z, which follows a normal distribution, under the condition of overall degraded samples. It is an approximation of the distribution of the latent variable z, which follows a normal distribution, q(z|T). 1:m ,X' 1:m ) expectations;

[0047] By using the lower bound function of evidence Find the extreme value to obtain the degradation amount X' of the test sample. n+1:m The distribution results are then used to determine the degradation amount X' of the test sample. n+1:m The distribution is used for sampling to obtain the degradation amount X' of the test sample. n+1:m Predicted value.

[0048] The beneficial effects of this invention are that the small-sample degradation prediction method that integrates data and models can effectively mitigate the problem of prior distribution of model parameters required for stochastic processes. It can also improve the accuracy of degradation process models by combining the degradation trends of real degradation samples and learning the heterogeneity between samples. This provides theoretical support and analysis for the reliability analysis and remaining life prediction of key equipment, and reduces system operation and maintenance costs. Attached Figure Description

[0049] Figure 1 This is an overall flowchart of a small-sample degradation prediction method based on data and model fusion according to the present invention;

[0050] Figure 2 This invention relates to the encoder network process in a small-sample degradation prediction method that integrates data and models.

[0051] Figure 3 This invention relates to a method for predicting small-sample degradation by fusing data and models, which involves aggregating encoded results and sampling latent variables. Detailed Implementation

[0052] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0053] The flowchart of the small-sample degradation prediction method based on data and model fusion of this invention is as follows: Figure 1 As shown, please follow these steps:

[0054] Step 1: Obtain degradation samples D of key equipment during operation using sensors or equipment recorders;

[0055] Step 2: Process the degraded samples using a normalization method and divide the samples into degraded training samples and degraded test samples;

[0056] Step 2 is as follows:

[0057] Step 2.1: Let the original degraded sample D be... 1:m ={T 1:m ,X 1:m},

[0058]

[0059] Among them, X 1:m The sample represents the overall degradation of critical equipment, j = 1,...,l represents the j-th degradation sample of critical equipment, l represents the total number of degradation samples, i = 1,...,m represents the i-th monitoring time, m is the total monitoring duration, and X j,i T represents the degradation of the j-th critical device at time i; 1:m T represents the total observation time of the entire sample.i Indicates the i-th observation time;

[0060] Step 2.2: Using the max-min normalization method, the degradation samples X of key equipment are... 1:m Normalization to the same dimension is performed as shown in formula (1):

[0061]

[0062] Among them, X j,i This indicates that the j-th critical device is in T. i The degradation amount corresponding to time, max(X) j,： ) represents the maximum degradation of the j-th critical device during the entire monitoring period, max(X) j,： X' is the minimum degradation of the j-th critical device during the entire monitoring period. j,i It is the jth key device in T i The normalized degradation amount corresponding to the given time;

[0063] Therefore, the normalized degenerate sample D1' is obtained. :m ={T 1:m ,X' 1:m},

[0064]

[0065] Where X' 1:m This represents the normalized overall degradation sample of key equipment.

[0066] Step 2.3: Assuming the duration of the degradation training samples is n, then the duration of the test samples is mn-1. Therefore, the normalized degradation training samples are obtained as D'. 1:n＝ {T 1:n ,X' 1:n The normalized degradation test sample is D'. n+1:m ={T n+1:m ,X' n+1:m}

[0067] Step 3: Learn the degradation amount of the training samples corresponding to each moment of the key equipment through a neural network, and obtain the relationship function h between the encoded degradation amount and different moments. i ;

[0068] Step 3 is as follows:

[0069] The normalized training sample D' is obtained through step 2. 1:n＝ {T 1:n ,X' 1:n},like Figure 2 As shown, the degradation amount corresponding to each time step is (T1,X'1), ...,(Ti ,X' i ), …, (T n ,X' n ), where X' i =[X' 1,i ,...,X' j,i ,...,X' l,i That is, at each time T i Corresponding to the degradation amount of training samples for l key devices, then, the neural network learns the degradation amount of each time step T. i Degradation amount of training samples X' i The relationship function h i =NN(T) i ,X' i ), i = 1,...,n, which means we obtain n relational functions h1, h2,...h n , where X' i It is T i The device training sample degradation amount is n, where n is the degradation amount training sample duration, and NN represents a neural network, specifically a backpropagation neural network (BPNN) or a recurrent time network (RNN).

[0070] Step 4: Use an aggregator to analyze the relation function h at different times. i The functions are integrated, and the mean and variance of the integrated function are calculated. The mean and variance are sampled to obtain the latent variable z.

[0071] Step 4 is as follows:

[0072] Step 4.1: Obtain the time step T at each moment after training the neural network, as described in Step 3. i Degradation amount of training samples X' i The relationship function h i =NN(T) i ,X' i ), i = 1,...,n, using an average aggregator for n relation functions h1, h2,...h n Aggregation is performed as shown in formula (2):

[0073]

[0074] Where r is the aggregation result and n is the duration of the degraded training samples;

[0075] Step 4.2: Calculate the mean μ(r) and variance σ of the aggregation results. 2 (r), and then sample the mean and variance to obtain the latent variable p(z|T) that follows a normal distribution. 1:n ,X' 1:n )~N(μ(r),σ2 (r)).

[0076] The overall process of step 4 is as follows: Figure 3 As shown.

[0077] Step 5: Finally, combining the degradation test samples obtained in Step 2 with the latent variable z obtained in Step 4, the function g of the latent variable z and the degradation test samples is learned through a neural network to obtain the predicted value of the degradation amount of the key equipment.

[0078] Step 5 is as follows:

[0079] Step 5.1: Assuming the normalized degenerate sample D' follows a Gaussian process f(T)~GP, the degenerate sample X' based on the Gaussian process is obtained according to Bayes' theorem. 1:m The distribution is shown in formula (3):

[0080] p(X' 1:m |T 1:m )=∫p(f(T 1:m ))p(X' 1:m |f(T 1:m ),T 1:m )df(T 1:m (3)

[0081] Where X' 1:m It is the normalized overall degenerate sample, f(T) 1:m ) represents the normalized degenerate sample D1' :m The probability density function, T 1:m It is the overall degradation sample monitoring time, p(f(T) 1:m The distribution of monitoring time for degraded samples is given by f(T) ~ GP, which follows a Gaussian process. p(f(T)) 1:m The distribution follows a Gaussian distribution, therefore, p(X') 1:m |f(T 1:m ),T 1:m As shown in formula (4):

[0082]

[0083] Substituting formula (4) into formula (3), we obtain the degenerate sample distribution, as shown in formula (5):

[0084]

[0085] Therefore, the degenerate sample X' based on the Gaussian process 1:m Distribution p(X') 1:m |T 1:m It also follows a Gaussian distribution;

[0086] Step 5.2: Learn each monitoring time T in the test samples using a neural network NN. n+1:m The relationship function g with the latent variable z m-n-1 =NN(T) n+1:m ,z), the stochastic process f(T) is transformed using g m-n-1 =NN(T) n+1:m The description of z) is that the relation function g m-n-1 =NN(T) n+1:m Replace f(T) with z) and combine it with the test degenerate sample D n+1:m ={T n+1:m ,X' n+1:m Formula (5) becomes:

[0087]

[0088] Where p(z,X') n+1:m |T 1:m ) is the joint distribution of the latent variable and the degradation of the test sample, p(z) is the prior distribution of the latent variable, and g is the joint distribution of the latent variable and the degradation of the test sample. v (T v (z) represents the v-th monitoring time T in the test sample. n+1:m The relationship function between the latent variable z;

[0089] Step 5.3: Based on the integral and relation p(X') n+1:m |T 1:m )=∑ z p(z,X' n+1:m |T 1:m Based on the overall degradation sample monitoring time T 1:m Degradation amount of training samples X' 1:n Test sample degradation X' n+1:m The distribution is shown in formula (7):

[0090] p(X' n+1:m |T 1:m ,X' 1:n )=∑ z p(z,X' n+1:m |T 1:m ,X' 1:n (7)

[0091] Where, ∑ z p(z,X' n+1:m |T 1:m ,X' 1:n The test sample degradation X' is based on the latent variable z. n+1:m Distribution integral value, p(X') n+1:m |T 1:m ,X' 1:n ) is the overall degradation sample monitoring time T1:m Degradation amount of training samples X' 1:n Under these conditions, the degradation amount X' of the test sample n+1:m The distribution is then used to derive the degradation amount X' of the test sample. n+1:m Related to the latent variable z;

[0092] Step 4 yields the latent variable p(z|T) that follows a normal distribution. 1:n ,X' 1:n )~N(μ(r),σ 2 (r)), combining the logarithm and KL divergence rules, to obtain the degradation amount X' of the test sample. n+1:m The lower bound function of the evidence distribution is calculated as shown in formula (8):

[0093]

[0094] in, It is the distribution of the latent variable z that follows a normal distribution, q(z|T) 1:m ,X' 1:m () is an approximation of the distribution of the latent variable z, which follows a normal distribution, under the condition of overall degraded samples. It is an approximation of the distribution of the latent variable z, which follows a normal distribution, q(z|T). 1:m ,X' 1:m ) expectations;

[0095] By using the lower bound function of evidence Find the extreme value to obtain the degradation amount X' of the test sample. n+1:m The distribution results are then used to determine the degradation amount X' of the test sample. n+1:m The distribution is used for sampling to obtain the degradation amount X' of the test sample. n+1:m Predicted value.

[0096] This invention addresses the problem of inaccurate modeling of the relationship between monitoring time and degradation by preprocessing recorded degradation samples. It combines neural networks to learn the relationship model, improving the accuracy of the model. Then, by introducing latent variable methods and combining variational inference strategies with stochastic processes—specifically, distributed sampling—the future degradation prediction value is obtained. This method overcomes the shortcomings of traditional stochastic process methods and neural network methods, improving the accuracy of degradation prediction and providing technical support for subsequent equipment reliability analysis and lifespan prediction.

[0097] The basic idea for predicting the degradation of critical equipment is as follows: First, degradation samples D of the critical equipment during operation are acquired through sensors or equipment recorders. These samples are then processed using a normalization method and divided into training and testing samples. Next, a neural network learns the degradation amount of the training samples corresponding to each moment of the critical equipment, obtaining the encoded relationship function h between the degradation amount and time. Then, an aggregator integrates the relationship functions at different times and calculates the mean and variance of the integrated function, sampling it to obtain the latent variable z. Finally, combining the degradation test samples and the latent variable z, a neural network learns the function g between the latent variable and the test samples to obtain the predicted value of the critical equipment degradation. Currently, most methods for predicting degradation primarily rely on establishing degradation process models, such as Wiener processes and gamma processes. These methods require pre-assertions of parameter distributions; if the assumptions deviate from the actual state, the prediction results will have greater errors. Other data analysis-based methods also require a large number of degradation samples. The neural process method, however, can combine a neural network to learn the process of fewer degradation samples and obtain future degradation values ​​through distributed sampling.

Claims

1. A method for predicting degradation in small samples by fusing data and models, characterized in that, The specific steps are as follows: Step 1: Obtain degradation samples of key equipment during operation using sensors or equipment recorders. D ; Step 2: Process the degraded samples using a normalization method and divide the samples into degraded training samples and degraded test samples; Step 2 is described in detail below: Step 2.1: Assume the original degraded sample D 1:m ={T 1:m ,X 1:m }, ， ， in, This represents a sample of overall degradation of critical equipment. Indicates the first j A sample of key equipment degradation This represents the total number of degraded samples. Indicates the first i Each monitoring moment, For the entire monitoring duration recorded, Indicates the first j The key equipment is in i The amount of degradation at each time point; Represents the total observation time of the entire sample. Indicates the first i Each observation time; Step 2.2: Using the max-min normalization method, the degradation samples X of key equipment are... 1:m Normalization to the same dimension is performed as shown in formula (1): （1） in, Indicates the first j Key equipment T i The amount of degradation at any given moment. It is the first j The maximum degradation of key equipment during the entire monitoring period It is the first j Minimum degradation of key equipment over the entire monitoring period It is the first j Key equipment T i The normalized degradation amount corresponding to the given time; Therefore, the normalized degenerate samples are obtained. ={ }, ， ， in, This represents the normalized overall degradation sample of key equipment. Step 2.3, assuming the duration of the degradation training samples is... n The test sample duration is m - n-1 Therefore, the normalized degenerate training samples are obtained as follows: = { The normalized degradation test samples are: ={ }; Step 3: Learn the degradation amount of the training samples corresponding to each moment of the key equipment through a neural network, and obtain the relationship function between the encoded degradation amount and different moments. h i ; Step 3 is described in detail below: The normalized training samples are obtained through step 2. = { The degradation amount corresponding to each moment is () ), …,( ), …,( ),in, That is, every moment T i Corresponding to l The degradation of training samples for key equipment is measured, and then a neural network is used to learn the degradation at each moment. T i Degradation of training samples Relationship functions In other words, it was obtained n Relational functions ,in, yes T i The amount of equipment training sample degradation at that time. n It is the duration of training samples with degradation. NN Represents a neural network. NN This can be a backpropagation neural network (BPNN) or a recurrent time network (RNN). Step 4: Use an aggregator to analyze the relational functions at different times. h i The functions are integrated, and the mean and variance of the integrated function are calculated. The mean and variance are then sampled to obtain the latent variables. z ; Step 4 is as follows: Step 4.1: Obtain each moment after training the neural network using Step 3. T i Degradation of training samples Relationship functions Using an average aggregator n Relational functions Aggregation is performed as shown in formula (2): （2） in, r It is the result of aggregation. n It refers to the duration of training samples with degradation. Step 4.2: Calculate the mean of the aggregation results. With variance Then, by sampling the mean and variance, we obtain the latent variables that follow a normal distribution. ; Step 5: Finally, combining the degraded test samples obtained in Step 2 with the latent variable z obtained in Step 4, the latent variable is learned through a neural network. z Functions with degradation test samples g The predicted values ​​of degradation of key equipment were obtained. Step 5 is described in detail below: Step 5.1: Assume the degenerate samples after normalization. Obeying a Gaussian process At that time, according to Bayes' theorem, degenerate samples based on Gaussian processes are obtained. The distribution is shown in formula (3): （3） in, It is a normalized overall degraded sample. Represents the degenerate samples after normalization The probability density function, It refers to the overall monitoring time of degraded samples. It is the distribution of monitoring time for degraded samples, due to Follows a Gaussian process. It follows a Gaussian distribution, therefore, As shown in formula (4): （4） Substituting formula (4) into formula (3), we obtain the degenerate sample distribution, as shown in formula (5): （5） Therefore, degenerate samples based on Gaussian processes distributed It also follows a Gaussian distribution; Step 5.2: Using a neural network NN Each monitoring time in the learning test sample With latent variables z Relationship functions random process f ( T )use Description, that is, the relational function replace f ( T ), combined with test degradation samples D n+1:m ={ Formula (5) becomes: （6） in, It is the joint distribution of latent variables and the degradation of test samples. It is a prior distribution of latent variables. Indicates the first in the test sample v Each monitoring time With latent variables z Relationship functions; Step 5.3: Based on integrals and relationships Based on the overall degradation sample monitoring time Degradation of training samples Test sample degradation The distribution is shown in formula (7): （7） in, Based on latent variables z Degradation of test samples Distribution integral value, It is the overall degradation sample monitoring time Degradation of training samples Under these conditions, the amount of degradation of the test sample The distribution was used to determine the degradation level of the test samples. With latent variables z Related; Step 4 yields the latent variables that follow a normal distribution. Combining logarithms and KL Divergence algorithm to obtain the degradation of test samples The lower bound function of the evidence distribution is calculated as shown in formula (8): （8） in, It is the distribution of the latent variable z that follows a normal distribution. It is an approximation of the distribution of the latent variable z, which follows a normal distribution under the condition of overall degraded samples. It is based on latent variables that follow a normal distribution. z Approximate distribution of Expectations; By using the lower bound function of evidence Find the extreme value to obtain the degradation amount of the test sample. The distribution results were then analyzed, and the degradation of the test samples was assessed. The distribution was used to sample and obtain the degradation amount of the test sample. Predicted value.

Images