CN113297790A - High-speed rail response prediction method based on sparse Bayesian width learning - Google Patents

High-speed rail response prediction method based on sparse Bayesian width learning Download PDF

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CN113297790A
CN113297790A CN202110542901.3A CN202110542901A CN113297790A CN 113297790 A CN113297790 A CN 113297790A CN 202110542901 A CN202110542901 A CN 202110542901A CN 113297790 A CN113297790 A CN 113297790A
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王晨岳
黄永
高竞泽
李惠
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Harbin Institute of Technology
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Abstract

The invention provides a high-speed rail response prediction method based on sparse Bayesian width learning, which comprises the steps of carrying out linear and nonlinear feature extraction on an input temperature field variable, carrying out maximum posterior estimation on a weight value of a neuron node output layer of a hidden layer, predicting a structure response output result, preliminarily evaluating a track structure state and the like. The invention adopts a sparse Bayesian width learning method to mine the correlation of the data of the high-speed rail monitoring system, can effectively avoid the over-fitting problem of regression prediction by sparse solving of the weight w reflecting the relationship among data variables, has higher prediction precision, high efficient calculation speed and loose equipment hardware requirements, thereby realizing the mining of the correlation of temperature load and structural strain hidden in a large amount of monitoring data, and can be used as the basis for judging the abnormal service state of the rail structure by discovering the evolution of a monitoring data model in time.

Description

High-speed rail response prediction method based on sparse Bayesian width learning
Technical Field
The invention belongs to the technical field of machine learning and structural health monitoring, and particularly relates to a high-speed rail response prediction method based on sparse Bayesian width learning.
Background
Today, the technology of high-speed rail is widely developed, and the problem of people's concern is more and more becoming to ensure the high-speed rail track to be in service and healthy. At present, a plate-type ballastless track laid by a domestic high-speed railway is used as a continuous track structure, is greatly influenced by temperature and line shape, is easy to cause a disease problem, and effectively evaluating whether the state change occurs in real time is an important subject of high-speed railway operation safety.
In order to solve the problems, scientific research and engineering application aiming at a long-term monitoring technology of a high-speed railway track structure system are developed in recent years at home and abroad, long-term monitoring work points are distributed in key sections of a line, and a temperature field and temperature deformation of a track structure are monitored for a long time. The service state monitoring period of the rails is generally not less than 2 years, and a large amount of monitoring data are obtained in the monitoring process. In the aspect of rail state evaluation, a threshold value evaluation method is mainly adopted at present, namely, the state of a rail is evaluated by monitoring whether data exceed the limit or not. Although the method is simple and direct, the method is too sensitive to local response at a certain moment, misjudgment and missed judgment are easily generated, the change rule of the service state of the track structure cannot be fundamentally disclosed, and the target of track state prediction and evaluation is not really realized.
In addition, the method for evaluating the track state of the high-speed railway by adopting the track structure smoothness is commonly adopted in China, and mainly comprises the steps of adopting methods such as local amplitude overrun deduction, track quality index and the like. The amplitude overrun deduction method is suitable for the condition that the local part has large amplitude irregularity; the track quality index can better reflect the unsmooth quality state of the section track. However, the data objects targeted by the above evaluation method are track geometric form and position parameters (such as track gauge, track direction, level, height, etc.), and are not suitable for track structure service state monitoring data (such as structure temperature, structure strain). Therefore, effective data mining and structural state evaluation methods still need to be researched for long-term monitoring data of the service state of the track structure.
Disclosure of Invention
The invention aims to solve the problem of existing high-speed rail state evaluation, and provides a high-speed rail response prediction method based on sparse Bayesian width learning.
The invention provides a high-speed rail response prediction method based on sparse Bayesian width learning, which is realized by the following technical scheme and specifically comprises the following steps:
establishing a large data set containing temperature and structural response obtained by a high-speed rail monitoring system, and dividing a training set and a test set according to a time sequence, wherein the training set comprises N groups of data, and the test set comprises L groups of data, and is used for high-speed rail response prediction and state evaluation;
step two, establishing a width learning method network framework of input variable temperature and output structure response variable, and randomly generating an input layer weight matrix S of neural network feature nodes based on zero mean Gaussian distributioniAnd bias matrix ΛiInput to training set
Figure BDA0003072405880000021
Use of
Figure BDA0003072405880000022
And
Figure BDA0003072405880000023
mapping to a high dimensional space Zi(x)＝φ(XSi+Λi) Each feature mapping matrix comprises T feature nodes, phi (phi) represents function transformation for abstracting the feature mapping matrix from an input variable x and is called an activation function, phi (x) is taken as x, and M represents the dimension of each group of data of the training set, so that M groups of feature mapping matrices Z (x) is constructed as Z (Z)1,Z2,…,Zm](ii) a Similarly, a weight matrix B and a bias matrix Ψ for the enhanced node are randomly generated based on a zero-mean Gaussian distribution, and
Figure BDA0003072405880000024
constructing an enhanced node matrix H (ZB + Ψ), where Ψ () is also an activation function, representing a functional transformation that abstracts the enhanced node matrix from the feature mapping matrix, and taking
Figure BDA0003072405880000025
Thirdly, operating a sparse Bayes learning method, calculating posterior probability distribution of the output layer weight w of each node, namely, the probability of obtaining various output layer weights under the condition of a known output layer y, and obtaining the maximum value of the posterior distribution probability of the output layer weights, namely the value which is most likely to be obtained by the output layer weights w under the current parameter condition;
step four, substituting the temperature field data x of the test set*Generating a feature mapping matrix and an enhanced node matrix [ Z ] according to the step two*,H*]Calculating the maximum posterior probability distribution of the output layer weight w obtained in the step three to obtain a predicted structural response y, comparing the predicted value of the structural response y with the measured value, and if the structural state is normal, matching the predicted value with the measured value; otherwise, the prediction result of the structural response and the actual measurement result generate obvious deviation, which indicates that the prediction relation between the structural temperature and the response is obviously changed and is used as the basis for judging the structural state abnormity.
Further, the second step is specifically:
step 2.1, the regression problem based on the width learning network architecture can be uniformly written as:
Figure BDA0003072405880000026
wherein, wiRepresenting the output layer weight of the ith characteristic node, n representing the number of generated enhanced nodes, wjRepresenting the output layer weight of the jth enhanced node; si,ΛiAnd Bj,ΨjRespectively representing an input layer weight matrix and a bias matrix randomly generated from zero mean Gaussian distribution by an ith characteristic mapping matrix and a jth enhanced node matrix; psii(.) and psij(.) are respectively the activation functions of the corresponding generated feature node and the enhanced node;
step 2.2, for all N groups of data in the training set, i characteristic nodes and j enhanced nodes are determined and generated, and the following matrixes reflecting the characteristics of input data are obtained by utilizing input layer weights and offsets randomly generated by zero-mean Gaussian distribution:
[Z,H]＝[ψ1(S1,Λ1；X) ... φi(Si,Λi；X) ψ1(B1,Ψ1；Z) ... ψj(jj,Ψj；Z)]；
and 2.3, aiming at the matrix form y of the regression problem in the step 2.1, wherein the matrix form y is [ Z, H ] w, solving the weight w of the output layer can be realized by solving the [ Z, H ] generalized inverse of the characteristic matrix of the input layer, and in the sparse Bayes width learning method, the weight w of the output layer is solved by a sparse method.
Further, in the third step,
the posterior probability distribution of the output layer weight w is calculated by adopting the following formula:
Figure BDA0003072405880000031
is a Gaussian distribution;
wherein:
Figure BDA0003072405880000032
Figure BDA0003072405880000033
Figure BDA0003072405880000034
α＝[α1,α2,...,αiT+j]and σ2Representing a hyperparameter whose maximum a posteriori estimated MAP value is alphaMPAnd
Figure BDA0003072405880000035
estimated by the following equation:
Figure BDA0003072405880000036
Figure BDA0003072405880000037
Figure BDA0003072405880000038
will be above sigmaw、μw、αMPAnd
Figure BDA0003072405880000039
the estimation expression is iterated until convergence, and then the maximum value of the posterior probability distribution of the weight w of the output layer can be obtained, and the posterior probability distribution meets the Gaussian distribution, so that the maximum value of the probability distribution corresponds to the mean value mu of the Gaussian distributionw。
Further, the fourth step is specifically:
4.1, the output layer weight w obtained in the third step reflects the relation between the training set temperature field and the structural response, and is substituted into the test set temperature field data x*From a generated matrix [ Z ] reflecting the characteristics of the input data*,H*]The posterior probability distribution of the structural response prediction is obtained as follows:
Figure BDA00030724058800000310
Figure BDA00030724058800000311
Figure BDA00030724058800000312
wherein the content of the first and second substances,
Figure BDA00030724058800000313
representing the uncertainty of the prediction, is the variance of the gaussian distribution;
the maximum value of the posterior probability distribution of the structural response y obtained by prediction is the mean value of Gaussian distribution
Figure BDA00030724058800000314
Step 4.2, comparing the predicted structural response y with the measured value y*If the two are basically consistent, the relationship between the structure temperature field and the structure response is not changed, and the structure state is normal; otherwise, the structure state is abnormal, and the variable relation deviates; the substantial agreement is evaluated by using the predicted mean and uncertainty, and the two are considered to be substantially in agreement if the measured value is within a range of twice the standard deviation of the predicted value.
The invention has the beneficial effects that:
1. the sparse Bayesian width learning method can be used for mining the mechanical behavior characteristics of environmental temperature load, structural strain and the like hidden in a large amount of monitoring data in the response prediction of the high-speed rail, revealing the relation between the dynamic behavior characteristics and the structural service state, and realizing the identification of structural service state abnormality by discovering the evolution of a monitoring data model in time;
2. compared with the width learning method, the sparse Bayesian width learning method applied by the invention can effectively relieve the problem of overfitting through model node parameter sparsification, and greatly reduces the workload of manual parameter adjustment in application;
3. the sparse Bayesian width learning method has simple network architecture, good prediction precision, high-efficiency calculation speed and loose equipment hardware requirement when applied to engineering with about tens of thousands of groups of data and complex variable relation, and is suitable for the requirement of practical engineering application;
4. the sparse Bayesian width learning method is used as a probability method, the uncertainty of the output result can be quantified, and the uncertainty can be used as a basis for judging the accuracy degree of prediction.
Drawings
FIG. 1 is a network architecture diagram of a sparse Bayesian width learning method from an input layer-a hidden layer-an output layer;
FIG. 2 is a flow chart of sparse Bayesian width learning method for solving from hidden layer-output layer weights w and predicting high-speed rail structure response;
FIG. 3 is a result graph of output layer weights w obtained by prediction using a sparse Bayesian width learning method for high-speed rail strain measurement points according to the present invention;
FIG. 4 is a graph of the data comparison result of the present invention for the 95% confidence interval between the predicted result and the measured result of the strain measuring point of the high-speed rail.
Detailed Description
The technical solutions in the embodiments of the present invention will be described clearly and completely with reference to the accompanying drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The invention aims to solve the problem that the behavior relation among data of the existing high-speed rail monitoring system is difficult to reasonably mine so as to predict the structural response and evaluate the rail state, and provides a high-speed rail response prediction method based on sparse Bayesian width learning. The method is suitable for the regression prediction problem of the correlation among multiple groups of data. The method can be applied to the regression problem of the temperature field, the structural strain and other directions of the high-speed rail.
With reference to fig. 2, the invention provides a high-speed rail response prediction method based on sparse bayes width learning, which specifically comprises the following steps:
establishing a large data set containing temperature and structural response obtained by a high-speed rail monitoring system, and dividing a training set (N groups of data) and a testing set (L groups of data) according to a time sequence for predicting the response and evaluating the state of the high-speed rail;
step two, establishing a width learning method network framework of input variable temperature and output structure response variable, and randomly generating an input layer weight matrix S of neural network feature nodes based on zero mean Gaussian distributioniAnd bias matrix ΛiInput to training set
Figure BDA0003072405880000051
Use of
Figure BDA0003072405880000052
And
Figure BDA0003072405880000053
mapping to a high dimensional space Zi(x)＝φ(XSi+Λi) Here, (-) denotes a functional transformation that abstracts a feature mapping matrix from an input variable x, called an activation function, where phi (x) is generally taken as x; m represents the dimension of each group of data of the training set, so that M groups of feature mapping matrixes (each feature mapping matrix comprises T feature nodes) Z (x) -Z1,Z2,…,Zm](ii) a Similarly, a weight matrix B and a bias matrix Ψ for the enhanced node are randomly generated based on a zero-mean Gaussian distribution, and
Figure BDA0003072405880000054
constructing an enhanced node matrix h (ZB + Ψ), where Ψ () is also the activation function, which represents the functional transformation that abstracts the enhanced node matrix from the feature mapping matrix, and is generally taken to be
Figure BDA0003072405880000055
Thirdly, operating a sparse Bayes learning method, calculating posterior probability distribution of the output layer weight w of each node, namely, the probability of obtaining various output layer weights under the condition of a known output layer y, and obtaining the maximum value of the posterior distribution probability of the output layer weights, namely the value which is most likely to be obtained by the output layer weights w under the current parameter condition;
step four, substituting the temperature field data x of the test set*Generating a feature mapping matrix and an enhanced node matrix [ Z ] according to the step two*,H*]And calculating to obtain the predicted structural response y according to the maximum value of the posterior probability distribution of the output layer weight w obtained in the step three. Comparing the predicted value and the measured value of the structural response y, and if the structural state is normal, the predicted value and the measured value are matched well; otherwise, the prediction result of the structural response and the actual measurement result are obviously deviated, which shows that the prediction relation between the structural temperature and the response is obviously changed and can be used as the basis for judging the structural state abnormity.
Referring to fig. 1, the second step specifically includes:
step 2.1, the regression problem based on the width learning network architecture can be uniformly written as:
Figure BDA0003072405880000056
where n represents the number of enhanced nodes generated, wi(j)Is the output layer weight, Si,ΛiAnd Bj,ΨjRespectively representing input layer weight values and bias parameters randomly generated by the ith characteristic mapping matrix and the jth enhanced node matrix from zero mean Gaussian distribution. Phi is ai(.) and psij(.) are respectively the activation functions of the corresponding generated feature node and the enhanced node, where phii(.) is a linear function, generally taken as phii(x)＝x,ψj(.) is a non-linear function, typically taken
Figure BDA0003072405880000061
Step 2.2, for all N groups of data in the training set, i feature mappings and j enhanced nodes are determined to be generated, and the following matrix reflecting the input data features is obtained by utilizing the input layer weight and the bias randomly generated by zero-mean Gaussian distribution:
[Z,H]＝[φ1(S1,Λ1；X) ... φi(Si,Λi；X) ψ1(B1,Ψ1；Z) ... ψj(Bj,Ψj；Z)]
and 2.3, aiming at the matrix form y of the regression problem in the step 2.1, wherein the matrix form y is [ Z, H ] w, solving the output layer weight w can be realized by solving the [ Z, H ] generalized inverse of the characteristic matrix of the input layer, and in the sparse Bayes width learning method, the output layer weight w is solved by a sparse method.
Solving the output layer weight w by using sparse Bayesian learning specifically comprises the following steps:
the posterior distribution of the output layer weight w is calculated by the following formula:
Figure BDA0003072405880000062
is a Gaussian distribution;
wherein:
Figure BDA0003072405880000063
Figure BDA0003072405880000064
Figure BDA0003072405880000065
α＝[α1,α2,...,αiT+j]and σ2Representing a hyperparameter whose maximum a posteriori estimated MAP value is alphaMPAnd
Figure BDA0003072405880000066
estimated by the following equation:
Figure BDA0003072405880000067
Figure BDA0003072405880000068
Figure BDA0003072405880000069
will be above sigmaw、μw、αMPAnd
Figure BDA00030724058800000610
the estimation expression is iterated until convergence, and then the maximum value of the posterior probability distribution of the weight w of the output layer can be obtained, and the posterior probability distribution meets the Gaussian distribution, so that the maximum value of the probability distribution corresponds to the mean value mu of the Gaussian distributionw。
And step four, finally obtaining a structural response prediction value obtained based on a test set sparse Bayesian width learning method, and carrying out structural state evaluation on the high-speed rail by using the structural response prediction value, specifically:
4.1, the output layer weight w obtained in the third step reflects the relation between the training set temperature field and the structural response, and is substituted into the test set temperature field data x*From a generated matrix [ Z ] reflecting the characteristics of the input data*,H*]The posterior probability distribution of the structural response prediction is obtained as follows:
Figure BDA00030724058800000611
Figure BDA0003072405880000071
Figure BDA0003072405880000072
the maximum value of the posterior probability distribution of the structural response y obtained by prediction is the mean value of the Gaussian distribution
Figure BDA0003072405880000073
Figure BDA0003072405880000074
Representing the uncertainty of the prediction, is the variance of the gaussian distribution.
Step 4.2, comparing the predicted structural response y with the measured value y*If the two are well matched, the relationship between the structure temperature field and the structure response is not changed, and the structure state is normal; otherwise, the structure state is abnormal, and the variable relation is deviated. The predicted mean and uncertainty can be used for evaluation, and if the measured value is within a range of twice the standard deviation of the predicted value, the two values can be considered to be basically consistent.
Examples
The embodiment of the invention is applied to the health monitoring problem of a certain high-speed rail track structure. The track structure monitoring system comprises 6 temperature sensors for respectively measuring the temperature of the atmosphere and the track structure, and 30 structural strain sensors for measuring the strain of the track slab at different positions. And carrying out application analysis by using monitoring data acquired by the monitoring system within three years. And training a temperature-strain regression model by adopting data of the previous two years, and predicting strain by adopting data of the next year and taking the data as a basis for judging the service state of the track structure.
The first step is specifically as follows: the data of 6 measuring points of the temperature field are used as input, the data of 30 strain measuring points are used as output, the data of the first two years are divided into a training set, and the data of the next year are used as a testing set.
The second step is specifically as follows: for each measuring point of the structural response, 6 measuring point data of the temperature field are input, the number of characteristic nodes is set to be 5 multiplied by 5, the number of enhancement nodes is set to be 200, an input layer weight matrix and a bias matrix are randomly generated through standard normal distribution, and a characteristic mapping matrix and an enhancement node matrix are respectively constructed.
The third step and the fourth step are specifically as follows: for the problem, the maximum posterior estimation corresponding value is taken as the gaussian distribution mean value to obtain the output layer weight, as can be seen from fig. 3, the output layer weight which is not 0 for the problem is always not more than 40 and is less than 20% of the total number of nodes, and the sparse bayes width learning method is proved to obtain the effect of sparse weight. The output layer weight obtained from the training set acts on a node matrix representing the characteristics of the input layer randomly generated by the data of the test set to obtain a predicted mean value, an actual measured value and a standard deviation interval twice the predicted mean value, which are shown in fig. 4.
The foregoing describes in detail a high-speed rail response prediction method based on sparse bayes width learning according to the present invention, and the principle and implementation of the present invention are explained herein by applying specific examples, and the description of the foregoing examples is only used to help understanding the method and the core concept of the present invention; meanwhile, for a person skilled in the art, according to the idea of the present invention, there may be variations in the specific embodiments and the application scope, and in summary, the content of the present specification should not be construed as a limitation to the present invention.

Claims (4)

1. A high-speed rail response prediction method based on sparse Bayesian width learning is characterized in that: the method specifically comprises the following steps:
establishing a large data set containing temperature and structural response obtained by a high-speed rail monitoring system, and dividing a training set and a test set according to a time sequence, wherein the training set comprises N groups of data, and the test set comprises L groups of data, and is used for high-speed rail response prediction and state evaluation;
step two, establishing a width learning method network framework of input variable temperature and output structure response variable, and randomly generating an input layer weight matrix S of neural network feature nodes based on zero mean Gaussian distributioniAnd a bias matrix AiInput to training set
Figure FDA0003072405870000011
Use of
Figure FDA0003072405870000012
And
Figure FDA0003072405870000013
mapping to a high dimensional space Zi(x)＝φ(XSi+Ai) Each feature mapping matrix comprises T feature nodes, phi (phi) represents function transformation for abstracting the feature mapping matrix from an input variable x and is called an activation function, phi (x) is taken as x, and M represents the dimension of each group of data of the training set, so that M groups of feature mapping matrices Z (x) is constructed as Z (Z)1，Z2，…，Zm](ii) a Similarly, a weight matrix B and a bias matrix Ψ of the enhanced node are randomly generated based on a zero-mean Gaussian distribution
Figure FDA0003072405870000014
Constructing an enhanced node matrix H (ZB + Ψ), where Ψ () is also an activation function, representing a functional transformation that abstracts the enhanced node matrix from the feature mapping matrix, and taking
Figure FDA0003072405870000015
Thirdly, operating a sparse Bayes learning method, calculating posterior probability distribution of the output layer weight w of each node, namely, the probability of obtaining various output layer weights under the condition of a known output layer y, and obtaining the maximum value of the posterior distribution probability of the output layer weights, namely the value which is most likely to be obtained by the output layer weights w under the current parameter condition;
step four, substituting the temperature field data x of the test set*Generating a feature mapping matrix and an enhanced node matrix [ Z ] according to the step two*，H*]Calculating the maximum posterior probability distribution of the output layer weight w obtained in the step three to obtain a predicted structural response y, comparing the predicted value of the structural response y with the measured value, and if the structural state is normal, matching the predicted value with the measured value; otherwise, the prediction result of the structural response and the actual measurement result generate obvious deviation, which indicates that the prediction relation between the structural temperature and the response is obviously changed and is used as the basis for judging the structural state abnormity.
2. The method of claim 1, wherein: the second step is specifically as follows:
step 2.1, the regression problem based on the width learning network architecture can be uniformly written as:
Figure FDA0003072405870000016
wherein, wiRepresenting the output layer weight of the ith characteristic node, n representing the number of generated enhanced nodes, wjRepresenting the output layer weight of the jth enhanced node; si，ΛiAnd Bj，ΨjRespectively representing an input layer weight matrix and a bias matrix randomly generated from zero mean Gaussian distribution by an ith characteristic mapping matrix and a jth enhanced node matrix; phi i (phi,) and psij(.) are respectively the activation functions of the corresponding generated feature node and the enhanced node;
step 2.2, for all N groups of data in the training set, i characteristic nodes and j enhanced nodes are determined and generated, and the following matrixes reflecting the characteristics of input data are obtained by utilizing input layer weights and offsets randomly generated by zero-mean Gaussian distribution:
[z，H]＝[φ1(S1，Λ1；X)...φi(Si，Λi；X) ψ1(B1，Ψ1；Z)...ψj(Bj，Ψj；Z)]；
and 2.3, aiming at the matrix form y of the regression problem in the step 2.1, wherein the matrix form y is [ Z, H ] w, solving the weight w of the output layer can be realized by solving the [ Z, H ] generalized inverse of the characteristic matrix of the input layer, and in the sparse Bayes width learning method, the weight w of the output layer is solved by a sparse method.
3. The method of claim 2, wherein: in the third step, the first step is carried out,
the posterior probability distribution of the output layer weight w is calculated by adopting the following formula:
Figure FDA0003072405870000021
Figure FDA0003072405870000022
is a Gaussian distribution;
wherein:
Figure FDA0003072405870000023
Figure FDA0003072405870000024
Figure FDA0003072405870000025
α＝[α1，α2，...，αiT+j]and σ2Representing a hyperparameter whose maximum a posteriori estimated MAP value is alphaMPAnd
Figure FDA0003072405870000026
estimated by the following equation:
Figure FDA0003072405870000027
Figure FDA0003072405870000028
Figure FDA0003072405870000029
will be above sigmaw、μw、αMPAnd
Figure FDA00030724058700000210
is iterated through the estimated expressionsThe posterior probability distribution maximum value of the weight w of the output layer can be obtained until convergence, and the posterior probability distribution maximum value is the mean value mu of the Gaussian distribution as the posterior probability meets the Gaussian distributionw。
4. The method of claim 3, wherein: the fourth step is specifically as follows:
4.1, the output layer weight w obtained in the third step reflects the relation between the training set temperature field and the structural response, and is substituted into the test set temperature field data x*From a generated matrix [ Z ] reflecting the characteristics of the input data*，H*]The posterior probability distribution of the structural response prediction is obtained as follows:
Figure FDA00030724058700000211
Figure FDA00030724058700000212
Figure FDA0003072405870000031
wherein the content of the first and second substances,
Figure FDA0003072405870000032
representing the uncertainty of the prediction, is the variance of the gaussian distribution;
the maximum value of the posterior probability distribution of the structural response y obtained by prediction is the mean value of Gaussian distribution
Figure FDA0003072405870000033
Step 4.2, comparing the predicted structural response y with the measured value y*If the two are basically consistent, the relationship between the structure temperature field and the structure response is not changed, and the structure state is normal; otherwise, the structural state is abnormal and changedDeviation of the quantitative relationship; the substantial agreement is evaluated by using the predicted mean and uncertainty, and the two are considered to be substantially in agreement if the measured value is within a range of twice the standard deviation of the predicted value.
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