Bridge support displacement online probability prediction method

Abstract

The invention discloses a bridge support displacement online probability prediction method, which comprises the following steps: acquiring historical monitoring data of main beam temperature and support displacement, and introducing a group of false points to construct a probability prediction model based on a flow sparse Gaussian process, according to the approximate posterior distribution and the real posterior distribution, constructing a variational free energy boundary as a lower bound of a logarithmic marginal likelihood function, deriving an optimal variational free energy boundary and optimal approximate posterior distribution, and obtaining a specific expression of a probability prediction model; when new monitoring data is obtained, new monitoring data and old historical monitoring data are summarized in combination with new false points and old posterior distribution, new approximate posterior distribution is derived, new real posterior distribution is derived by using the Bayesian theorem, then model parameters and posterior distribution are updated online, and a new specific expression of the probability prediction model is obtained. According to the method, massive health monitoring flow data of the bridge is processed based on the flow sparse Gaussian process model, online probability prediction of bridge support displacement is achieved, and the method has high engineering application value.

Description

Technical Field

[0001] The invention relates to the technical field of bridge service status assessment, and in particular to an online probability prediction method for bridge support displacement. Background Art

[0002] The bearing is an important component that connects the upper and lower structures of the bridge and transfers loads. Its health status affects the operational safety of the bridge. Under the influence of the environment and vehicle loads, the daily cumulative displacement of the bearing continues to increase, resulting in the degradation of the bearing performance and the occurrence of defects such as excessive wear. The deterioration of the bearing performance can easily lead to the restriction of the relative movement between the main beam and the pier, and the bridge structure is subjected to large additional internal forces, which can cause defects such as cracking of the main beam. The bearing displacement is one of the health monitoring indicators of long-span bridges. The operating status of the bearing can be evaluated by comparing the difference between the measured bearing displacement under the action of ambient temperature and the theoretically predicted displacement. Therefore, it is crucial to establish an accurate ambient temperature-bearing displacement model and evaluate its service performance to ensure the safe operation of the bridge structure and extend its service life.

[0003] Existing studies often use deterministic models to predict the displacement response of bearings under temperature, which makes it difficult to consider the uncertainty of monitoring data. The probabilistic prediction model estimates the target value in the form of probability distribution and provides more information in the form of probability density, quantiles and prediction intervals. At present, the probability prediction methods of bearing displacement are mostly based on offline learning algorithms that assume that the model coefficients are unchanged, that is, the optimal values ​​of the model parameters are estimated once on the training data set. However, the aging of bearing materials and changes in the service environment can easily cause the parameters of the bearing displacement prediction model to vary with time. The offline learning algorithm can only guarantee the prediction accuracy within a certain time range and cannot achieve accurate prediction of bridge bearing displacement during service. The Gaussian process model is a non-parametric supervised learning model with a specific probabilistic interpretation under the Bayesian framework. It can be extended to an online learning framework by combining Bayesian inference. Therefore, how to combine the online Gaussian process model and the bridge health monitoring stream data to gradually update the model parameters and propose a real-time probabilistic prediction method for bridge bearing displacement is a difficult problem that needs to be solved urgently. Summary of the invention

[0004] In order to overcome the deficiencies in the prior art, the present invention provides an online probability prediction method for bridge bearing displacement to solve the problem that the displacement changes of bridge bearings during long-term operation are difficult to accurately predict due to material aging and changes in the service environment.

[0005] To achieve the above object, the present invention adopts the following technical solutions:

[0006] A bridge bearing displacement online probability prediction method comprises the following steps:

[0007] Step 1: Use the standard Gaussian process model as the baseline probability prediction model: y n=f(x n )+ε n , where ε n is the noise term, is the variance; f is the training function, and the training function f obeys the Gaussian process prior distribution with mean 0 and covariance k: Where θ is the hyperparameter on which the covariance k depends and The set of hyperparameters composed of

[0008] Definition: Given a training function f and variance After that, when the main beam temperature input vector is X, the support displacement output vector of the probability prediction model is y;

[0009] Step 2: Obtain the main beam temperature historical monitoring data through the bridge health monitoring system And support displacement historical monitoring data according to Introducing a set of pseudo points in u i =f(z i ), M<<N, M is the number of pseudo points, N is the number of samples; Definition: The set of training function values ​​of pseudo point P The training function value set of historical monitoring data The remaining training function value set f ≠u =fu;

[0010] According to u and f, f ≠u The relationship between , using the product rule to obtain the true prior distribution of the training function f is The expansion of u, and according to Get the expansion of the approximate prior distribution p(f|θ) of the training function f with respect to u, and get the expansion of the approximate joint distribution p(f,y|θ) of the training function f and the output vector y with respect to u based on the original likelihood function of p(f|θ) and the output vector y;

[0011] Define the true prior distribution of u under the output vector y as p(u|y,θ). According to p(u|y,θ) and the relationship between u and f, f ≠u The product rule is used to obtain the expansion of the true posterior distribution p(f|y,θ) of the training function f under the output vector y with respect to u, and the expansion of the approximate posterior distribution q(f|θ) of the training function f with respect to u is obtained based on p(f|y,θ);

[0012] It should be noted that in step 2, u is f={f ≠u ,u}, satisfying u=ff ≠u , according to u and f, f ≠uThe relationship between the true prior distribution of the training function f The expansion of u is obtained by using the product rule according to formula (2.1):

[0013]

[0014] Where: p(u|θ) is the prior distribution of u; p(f|u,θ) is the prior distribution of f under u; p(f ≠u |f,u,θ) is f ≠u Prior distribution under the conditions of f and u;

[0015] Among them, p(u|θ) is determined according to formula (2.2):

[0016]

[0017] Where: K uu is the covariance matrix between u and u;

[0018] Among them, p(f|u,θ) is determined according to formula (2.3):

[0019]

[0020] Where: K fu is the covariance matrix between f and u; K ff is the covariance matrix between f and f; K uf is the covariance matrix between u and f; K uu The inverse matrix of .

[0021] It should be noted that in step 2, in order to reduce K fu and K ff To reduce the complexity of the derivation process, we define the approximate posterior distribution q(f|u,θ) of the training function value set f under the training function value set u of the pseudo point P as the approximation of p(f|u,θ), that is, q(f|u,θ)≈p(f|u,θ); since u and f ≠u are independent of each other, and u is related to f and f ≠u u=ff ≠u The relationship, that is, p(f ≠u |u,θ)≈p(f ≠u |f,u,θ); q(f|u,θ), p(f ≠u |u,θ) replace p(f|u,θ) and p(f ≠u |f,u,θ), the expansion of the approximate prior distribution p(f|θ) of the training function f with respect to u is obtained according to formula (2.4):

[0022] p(f|θ)=p(u|θ)q(f|u,θ)p(f ≠u |u,θ) (2.4)

[0023] Where: p(f|θ) is the approximate prior distribution of the training function f; q(f|u,θ) is the approximate posterior distribution of the training function value f under condition u, which is an approximation of p(f|u,θ); p(f ≠u |u,θ) is the prior distribution p(f ≠u |f,u,θ) is simplified.

[0024] It should be noted that in step 2, the approximate joint distribution p(y,f|θ) of the training function f and the output vector y is obtained according to formula (2.5):

[0025]

[0026] Where: p(y,f|θ) is the approximate joint distribution of the training function f and the output vector y; is the original likelihood function.

[0027] Substituting equation (2.4) into equation (2.5), the expansion of the approximate joint distribution p(y,f|θ) with respect to u is obtained according to equation (2.6):

[0028] p(y,f|θ)=p(y|f,θ)p(f ≠u |u,θ)p(u|θ) (2.6)

[0029] Where p(y|f,θ) is the prior distribution of the output vector y under the training function f.

[0030] It should be noted that in step 2, the expansion of the true posterior distribution p(f|y,θ) of the training function f under the output vector y with respect to u is obtained according to formula (2.7):

[0031] p(f|y,θ)=p(u|y,θ)p(f ≠u |y,u,θ) (2.7)

[0032] Where p(f|y,θ) is the true posterior distribution of f under the condition of y; p(u|y,θ) is the prior distribution of u under the condition of y; p(f ≠u |y,u,θ) is f ≠u Prior distribution under u and y conditions;

[0033] According to p(u|y,θ) and u and f, f ≠u The relationship between the training function f and the approximate posterior distribution q(f|θ) of the training function f is obtained by the expansion of formula (2.8) with respect to u:

[0034] q(f|θ)=p(f≠u |u,θ)q(u|θ) (2.8)

[0035] Where: q(u|θ) is the posterior distribution of u; p(f ≠u |u,θ) is the prior distribution p(f ≠u |f,u,θ) is simplified.

[0036] Step 3: Define the marginal likelihood function of the output vector y as p(y|θ), and construct the lower bound of the logarithmic marginal likelihood function logp(y|θ) based on the approximate posterior distribution q(f|θ) and the approximate joint distribution p(y,f|θ), combined with the principle that p(y|θ) has an analytical solution; define the KL divergence between the approximate posterior distribution q(f|θ) and the true posterior distribution p(f|y,θ), and construct the variational free energy bound based on the difference between the logarithmic marginal likelihood function logp(y|θ) and the KL divergence according to the variational inference principle. As a lower bound for logp(y|θ), we derive Expression about u; according to Select different pseudo points P and use numerical iteration method to iterate u, so that the approximate posterior distribution q(f|θ) approaches the true posterior distribution p(f|y,θ) so that Maximize and obtain the optimal variational free energy bound and the best approximate posterior distribution q vfe (f|θ);

[0037] It should be noted that in step 3, the lower bound of the logarithmic marginal likelihood function logp(y|θ) is constructed according to formula (3.1):

[0038]

[0039] Where: logp(y|θ) is the logarithmic marginal likelihood function of the output vector y; q(f|θ) is the approximate posterior distribution of the training function f; is the variational free energy bound that depends on the approximate posterior distribution q(f|θ); p(y,f|θ) is the approximate joint distribution of the training function f and the output vector y;

[0040] Among them, the variational free energy bound Determine according to formula (3.2):

[0041]

[0042] Where: KL[q(f|θ)||p(f|y,θ)] is the KL divergence between the approximate posterior distribution q(f|θ) and the true posterior distribution p(f|y,θ); p(f|y,θ) is the true posterior distribution of the training function f under the condition of the output vector y;

[0043] Substituting equations (2.6) and (2.8) into equation (3.1), The expression of u is obtained according to formula (3.3):

[0044]

[0045] Where: f n =f(x n ) is x n The implicit function value at x n That is the historical monitoring data of the main beam temperature The nth sample data in ; To eliminate items.

[0046] It should be noted that in step 3, the optimal variational free energy bound and the best approximate posterior distribution q vfe (f|θ) is determined according to formula (3.4) and (3.5) respectively:

[0047]

[0048] Where: I is the identity matrix; trace(·) is the regularized tracking term to prevent the model from overfitting.

[0049] Step 4: According to the optimal variational free energy bound Get the value of the hyperparameter set θ, according to the optimal approximate posterior distribution q vfe (f|θ) and the hyperparameter set θ get the specific expression and variance of the training function f Specific values ​​of , substitute the two into step 1 to obtain the specific expression of the benchmark probability prediction model;

[0050] Step 5: Obtain new monitoring data of girder temperature and support displacement through the bridge health monitoring system, and compare the new monitoring data with the old historical monitoring data in step 2. Composing new historical monitoring data according to Introduce a new pseudo point P according to the method in step 2 new and P new The training function value set u new ; Based on the old true posterior distribution p(f|y,θ), use Bayes’ theorem to derive the new true posterior distribution p(f|y,y new ,θ new );Define the new approximate posterior distribution of the training function f as q new (f|θ new ), define a new approximate posterior distribution q new (f|θ new ) and the new true posterior distribution p(f|y,y new ,θnew ) and construct a new variational free energy bound based on the new KL divergence according to the variational inference principle According to u and u new The relationship between About u, u new The expression of Select multiple different pseudo points P new , using numerical iteration method to calculate u new Perform iterative calculations to make the new approximate posterior distribution q new (f|θ new ) approximates the new true posterior distribution p(f|y,y new ,θ new )make Maximize and obtain the new optimal variational free energy bound and the new optimal approximate posterior distribution q new,vfe (u new |θ new );

[0051] It should be noted that in step 5, based on the old true posterior distribution p(f|y,θ) and the old optimal approximate posterior distribution q(f|θ), the new true posterior distribution p(f|y,y new ,θ new ) According to formula (5.1), we get:

[0052]

[0053] Where: q new (f|θ new ) is the new approximate posterior distribution after acquiring new monitoring data; p(f|y,y new ,θ new ) is the new true posterior distribution of the training function f after obtaining new monitoring data; is dependent on the hyperparameter θ new The standardization constant of new ) is the new true prior distribution of the training function f after obtaining the new monitoring data; p(y|f) is the conditional distribution of the old output vector y in the new data given the training function f after obtaining the new monitoring data; p(y new |f) is the new output vector y in the new data given the training function f after obtaining the new monitoring data new The conditional distribution of

[0054] Among them, p(y|f) is obtained according to formula (5.2):

[0055]

[0056] Where: is a normalization constant that depends on the hyperparameter θ; q vfe (f|θ) is the optimal approximate posterior distribution before acquiring new monitoring data; To obtain the true prior distribution of the training function f before new monitoring data.

[0057] It should be noted that in step 5, the new variational free energy bound According to formula (5.3), we can get:

[0058]

[0059] Where: logp(y new |y,θ new ) is the new output vector y new The log-marginal likelihood function conditioned on the old output vector y; KL[q new (f|θ new )||p(f|y,y new ,θ new )] is the new approximate posterior distribution q new (f|θ new ) and the new true posterior distribution p(f|y,y new ,θ new )

[0060] Among them, KL[q new (f|θ new )||p(f|y,y new ,θ new )] According to formula (5.4), we get:

[0061]

[0062] Where: p(f ≠u,new |u new ,θ new ) is the training function value set u except the new pseudo point new The remaining potential function value f ≠u,new The prior distribution of q new (u new |θ new ) is the new pseudo-point training function value set u new The posterior distribution of p(u new |θ new ) New pseudo-point training function value set u new The prior distribution of ≠u,new =f new -u new ;f new is the set of training function values ​​for the new historical monitoring data;

[0063] In which, q(f|θ) is assumed to obey the normal distribution, and q(f|θ) is determined according to formula (5.5):

[0064]

[0065] Where: and are the mean and variance of the normal distribution respectively;

[0066] Among them, the new approximate posterior distribution q new (f|θ new ) The expansion of u is obtained according to formula (5.6):

[0067] q new (f|θ new )=q new (u new |θ new )p(f ≠u,new |u new ,θ new ) (5.6)

[0068] Where: p(f ≠u,new |u new ,θ new ) is the training function value set u except the new pseudo point new The remaining potential function value f ≠u,new The prior distribution of q new (u new |θ new ) is the new pseudo point u new The training function value set u new The posterior distribution of .

[0069] It should be noted that in step 5, the new optimal approximate posterior distribution q new,vfe (f|θ new ) and the new optimal variational free energy bound According to formula (5.7) and (5.8) respectively, we can get:

[0070]

[0071] Where: p new (u new |θ new ) is the new pseudo point P new The training function value set u new The prior distribution of K fu,new f and u new The covariance matrix between uu,new for u and u new The covariance matrix between ; K uu,new The inverse matrix of

[0072] Step 6: According to Get θ new Value, according to q new,vfe (u new |θ new ) and θ new It is worth getting the new specific expression and variance of the training function f Substitute the new specific values ​​into step 1 to obtain a new specific expression for the benchmark probability prediction model.

[0073] Compared with the prior art, the present invention has the following beneficial effects:

[0074] 1. The flow sparse Gaussian process model described in the present invention is a standard Gaussian process sparse approximate inference framework based on variational free energy. It uses pseudo-points to update the approximate prior distribution to obtain an approximation of the true posterior distribution, which is consistent with the standard Gaussian process model, can provide a predicted mean, and explain the uncertainty caused by measurement and model errors.

[0075] 2. The time complexity of training and prediction of the flow sparse Gaussian process model described in the present invention is lower than that of the standard Gaussian process model, and avoids the situation where the standard Gaussian process model has low efficiency or is unsolvable when processing massive bridge health monitoring data, significantly improving the computational efficiency and accuracy of model training and prediction.

[0076] 3. The present invention establishes an online learning framework of stream sparse Gaussian process approximation based on variational inference, which can perform online posterior update and hyperparameter learning in a data stream environment where monitoring data arrives sequentially, ensuring the model prediction accuracy under bearing aging and changes in the service environment.

[0077] 4. The present invention only needs to store a small amount of model posterior parameters and new monitoring data, which significantly reduces the storage requirements of the model and its dependence on data. At the same time, the present invention summarizes historical old data through a small number of pseudo points, which can avoid the problem of computational complexity explosion caused by processing massive accumulated historical old data. BRIEF DESCRIPTION OF THE DRAWINGS

[0078] Figure 1 The present invention is a flow chart of the on-line probability prediction method for bridge support displacement. DETAILED DESCRIPTION

[0079] The technical solution of the present invention will be clearly and completely described below in conjunction with the accompanying drawings.

[0080] like Figure 1 , Figure 1 The present invention is a flow chart of the on-line probability prediction method for bridge support displacement.

[0081] The present invention provides an online probability prediction method for bridge bearing displacement, comprising the following steps:

[0082] Step 1: Use the standard Gaussian process model as the baseline probability prediction model: y n =f(x n )+ε n , where f is the training function, ε n is the noise term, is the variance, assuming that the training function f obeys a Gaussian process prior distribution with a mean of 0 and a covariance of k: Where θ is the hyperparameter on which the covariance k depends and The set of hyperparameters composed of

[0083] Definition: Given a training function f and variance After that, when the main beam temperature input vector is X, the support displacement output vector of the probability prediction model is y.

[0084] The standard Gaussian process model is a conventional technology. In order to introduce symbols to facilitate the derivation of the flow sparse Gaussian process model, this patent only simply derives the standard Gaussian process model:

[0085] Given N input and output pairs hour( Corresponding to the historical monitoring data of main beam temperature and support displacement respectively), the standard Gaussian process regression model is:

[0086] y n =f(x n )+ε n

[0087] Where f is the unknown training function; is the noise term, is the variance.

[0088] It is usually assumed that the training function f follows a Gaussian process prior distribution with mean 0 and covariance k(·,·):

[0089]

[0090] Where k(·,·) is the covariance function, k(·,·) is an unknown function, which contains unknown hyperparameters; θ is the hyperparameter and variance on which the covariance function k(·,·) depends The set of hyperparameters.

[0091] Therefore, the joint distribution of the training function f and the output vector y is:

[0092]

[0093] in, is the true joint distribution of the training function f and the output vector y; is the true prior distribution of the training function f is the original likelihood function of the output vector y.

[0094] Step 2: Obtain the main beam temperature historical monitoring data through the bridge health monitoring system And the corresponding support displacement historical monitoring data according to Introducing a set of pseudo points in u i =f(z i ), M<<N, M is the number of pseudo points, N is the number of samples; Definition: The set of training function values ​​of pseudo point P The training function value set of historical monitoring data The remaining training function value set f ≠u =fu;

[0095] According to u and f, f ≠u The relationship between the two functions is obtained by using the product rule to obtain the true prior distribution of the training function f. The expansion of u, and according to Get the expansion of the approximate prior distribution p(f|θ) of the training function f with respect to u, and get the expansion of the approximate joint distribution p(f,y|θ) of the training function f and the output vector y with respect to u based on the original likelihood function of p(f|θ) and the output vector y;

[0096] Define the true prior distribution of u under the output vector y as p(u|y,θ). According to p(u|y,θ) and the relationship between u and f, f ≠u The product rule is used to obtain the expansion of the true posterior distribution p(f|y,θ) of the training function f under the output vector y with respect to u, and the expansion of the approximate posterior distribution q(f|θ) of the training function f with respect to u is obtained based on p(f|y,θ);

[0097] It should be noted that in step 2, u is f={f ≠u ,u}, satisfying u=ff ≠u , according to u and f, f ≠u The relationship between the true prior distribution of the training function f The expansion of u is obtained by using the product rule according to formula (2.1):

[0098]

[0099] Where: p(u|θ) is the prior distribution of u; p(f|u,θ) is the prior distribution of f under u; p(f ≠u |f,u,θ) is f ≠u Prior distribution under the conditions of f and u;

[0100] Among them, p(u|θ) is determined according to formula (2.2):

[0101]

[0102] Where: K uu is the covariance matrix between u and u;

[0103] Among them, p(f|u,θ) is determined according to formula (2.3):

[0104]

[0105] Where: K fu is the covariance matrix between f and u; K ff is the covariance matrix between f and f; K uf is the covariance matrix between u and f; K uu The inverse matrix of

[0106] To reduce K fu and K ff To reduce the complexity of the derivation process, we define the approximate posterior distribution q(f|u,θ) of the training function value set f under the condition of the training function value set u as the approximation of p(f|u,θ), that is, q(f|u,θ)≈p(f|u,θ); since u and f ≠u are independent of each other, and u is related to f and f ≠u u=ff ≠u The relationship, that is, p(f ≠u |u,θ)≈p(f ≠u |f,u,θ); q(f|u,θ), p(f ≠u |u,θ) replace p(f|u,θ) and p(f ≠u |f,u,θ), the expansion of the approximate prior distribution p(f|θ) of the training function f with respect to u is as follows:

[0107] p(f|θ)=p(u|θ)q(f|u,θ)p(f ≠u |u,θ)(2.4)

[0108] Where: p(f|θ) is the approximate prior distribution of the training function f; q(f|u,θ) is the approximate posterior distribution of the training function value f under condition u; p(f ≠u|u,θ) is the prior distribution p(f ≠u |f,u,θ) is simplified.

[0109] It should be noted that in step 2, based on the approximate prior distribution p(f|θ) of the training function f and the original likelihood function of the output vector y, the approximate joint distribution p(y,f|θ) of the training function f and the output vector y is obtained according to formula (2.5):

[0110]

[0111] Where: p(y,f|θ) is the approximate joint distribution of the training function f and the output vector y; is the original likelihood function.

[0112] Among them, combining equations (2.4) and (2.5) can rewrite the approximate joint distribution p(y,f|θ):

[0113] p(y,f|θ)=p(y|f,θ)p(f ≠u |u,θ)p(u|θ)(2.6)

[0114] Where p(y|f,θ) is the prior distribution (likelihood function) of the output vector y under the training function f.

[0115] It should be noted that in step 2, according to p(u|y,θ) and u and f, f ≠u The relationship between the true posterior distribution p(f|y,θ) and u is obtained by formula (2.7):

[0116] p(f|y,θ)=p(u|y,θ)p(f ≠u |y,u,θ)(2.7)

[0117] Where p(f|y,θ) is the true posterior distribution of f under the condition of y; p(u|y,θ) is the prior distribution of u under the condition of output vector y; p(f ≠u |y,u,θ) is the remaining training function value set f ≠u The prior distribution conditional on u and the output vector y;

[0118] Since p(f ≠u |u,θ)≈p(f ≠u |f,u,θ), p(u|y,θ)≈q(u|θ), so the expansion of the approximate posterior distribution q(f|θ) of the training function f with respect to u is obtained according to formula (2.8):

[0119] q(f|θ)=p(f ≠u |u,θ)q(u|θ)(2.8)

[0120] Where: q(u|θ) is the posterior distribution of u; p(f ≠u |u,θ) is the prior distribution p(f ≠u |f,u,θ) is simplified.

[0121] Due to historical monitoring data The number of samples is large, and the training time complexity of the standard Gaussian process model is The prediction time complexity of the test set is in: is a time complexity function, N is the number of samples, and the model training time is relatively long. To characterize the entire historical monitoring data The approximate prior distribution p(f|θ) of the training function f is constructed to replace its true prior distribution. The probability prediction model based on the approximate prior distribution p(f|θ) is the benchmark model of the stream sparse Gaussian process model established by the present invention, and its time complexity is The prediction time complexity of the test set is M is the number of pseudo points; usually M<<N, so the use of the sparse Gaussian process regression model described in this application can significantly improve the computational efficiency.

[0122] It should be noted that in step 1, a probability prediction model based on a standard Gaussian process is constructed. After approximation, the approximate joint distribution p(f,y|θ) of the training function f and the output vector y is obtained. p(f,y|θ) includes the approximate process. The probability prediction model obtained based on f in p(f,y|θ) is the approximate probability prediction model based on the flow sparse Gaussian process.

[0123] Step 3: Define the marginal likelihood function p(y|θ) of the output vector y, and construct the lower bound of the logarithmic marginal likelihood function logp(y|θ) based on the approximate posterior distribution q(f|θ) and the approximate joint distribution p(y,f|θ), combined with the principle that p(y|θ) has an analytical solution; define the KL divergence between the approximate posterior distribution q(f|θ) and the true posterior distribution p(f|y,θ), and construct the variational free energy bound based on the difference between the logarithmic marginal likelihood function logp(y|θ) and the KL divergence according to the variational inference principle. As a lower bound for logp(y|θ), we derive Expression about u; according to Select different pseudo points P and use numerical iteration method to iteratively calculate u, so that the approximate posterior distribution q(f|θ) approaches the true posterior distribution p(f|y,θ) so that Maximize and obtain the optimal variational free energy bound and the best approximate posterior distribution q vfe (f|θ);

[0124] It should be noted that in step 3, based on the principle that the marginal likelihood function p(y|θ) of the output vector y has an analytical solution, the hyperparameter set θ can be trained by optimizing the marginal likelihood function. According to the approximate posterior distribution q(f|θ) and the approximate joint distribution p(y,f|θ), the lower bound of the logarithmic marginal likelihood function logp(y|θ) is constructed according to formula (3.1):

[0125]

[0126] Where: logp(y|θ) is the logarithmic marginal likelihood function of the output vector y; q(f|θ) is the approximate posterior distribution of the training function f; is the variational free energy bound that depends on the approximate posterior distribution q(f|θ); p(y,f|θ) is the approximate joint distribution of the training function f and the output vector y;

[0127] Among them, the variational free energy bound It can be written as the difference between the logarithmic marginal likelihood function logp(y|θ) and the KL divergence between the approximate posterior distribution q(f|θ) and the true posterior distribution p(f|y,θ) of the training function f under the output vector y, that is, Determine according to formula (3.2):

[0128]

[0129] Where: KL[q(f|θ)||p(f|y,θ)] is the KL divergence between the approximate posterior distribution q(f|θ) and the true posterior distribution p(f|y,θ); p(f|y,θ) is the true posterior distribution of the training function f under the condition of the output vector y.

[0130] It should be noted that in step 3, substituting equations (2.6) and (2.8) into equation (3.1) can eliminate a key term p(f ≠u |u,θ), a computationally tractable variational free energy bound can be derived Right now The expression of u is obtained according to formula (3.3):

[0131]

[0132] Where: f n =f(x n ) is x n The implicit function value at x n That is the historical monitoring data of the main beam temperature The nth sample data in ; To eliminate items.

[0133] It should be noted that in step 3, the main beam temperature z at the pseudo point Pi Define the historical monitoring data of beam temperature In the main range, that is However, when selecting the pseudo point P, it is not possible to minimize the KL divergence or the variational free energy bound at one time. The maximum value is obtained by using numerical iteration method to determine whether the KL divergence is the minimum after selecting a pseudo point P. If the KL divergence is not the minimum, the temperature z of the main beam at the pseudo point P needs to be i Update and iterate u to make the approximate posterior distribution q(f|θ) approach the true prior distribution p(f|y,θ) to minimize the KL divergence. Minimizing the KL divergence is equivalent to maximizing the variational free energy bound, that is, the optimal variational free energy bound and the best approximate posterior distribution q vfe (f|θ) is determined according to formula (3.4) and (3.5) respectively:

[0134]

[0135] Where: I is the identity matrix; trace(·) is the regularized tracking term to prevent the model from overfitting.

[0136] Conventional Bayesian theorem normalizes the likelihood function and the prior distribution to derive the posterior distribution, which has high computational complexity, is dependent on the prior distribution, and has small data volume, which leads to unstable results. This patent obtains the optimal approximate posterior distribution by minimizing the KL divergence between the approximate posterior distribution and the true posterior distribution, and strictly approximates the exact GP model by minimizing the distance between the sparse model and the exact model.

[0137] The standard Gaussian process is a non-parametric supervised learning model with a specific probability interpretation under the Bayesian framework, and provides model interpretation capabilities with statistical knowledge. The stream sparse Gaussian process model described in the present invention is a standard Gaussian process approximate reasoning framework based on variational free energy. It uses pseudo-points to update the approximate prior distribution to obtain an approximation of the true posterior distribution, which can not only provide a predicted mean, but also explain the uncertainty caused by measurement and model errors.

[0138] Step 4: According to the optimal variational free energy bound Get the value of the hyperparameter set θ, according to the optimal approximate posterior distribution q vfe (f|θ) and the hyperparameter set θ get the specific expression and variance of the training function f Specific values ​​of , substitute the two into step 1 to get the specific expression of the probability prediction model;

[0139] Step 5: Obtain new monitoring data of the main beam temperature and support displacement through the bridge health monitoring system, and compare the new monitoring data with the old historical monitoring data in step 1. Composing new historical monitoring data according to Introduce a new pseudo point P according to the method in step 2 new and P new The training function value set u new ; Based on the old true posterior distribution p(f|y,θ), use Bayes’ theorem to derive the new true posterior distribution p(f|y,y new ,θ new );Define the new approximate posterior distribution of the training function f as q new (f|θ new ), define a new approximate posterior distribution q new (f|θ new ) and the new true posterior distribution p(f|y,y new ,θ new ) and construct a new variational free energy bound based on the new KL divergence according to the variational inference principle According to u and u new The relationship between About u, u new The expression of Select multiple different pseudo points P new , using numerical iteration method to calculate u new Perform iterative calculations to make the new approximate posterior distribution q new (f|θ new ) approximates the new true posterior distribution p(f|y,y new ,θ new )make Maximize and obtain the new optimal variational free energy bound and the new optimal approximate posterior distribution q new,vfe (u new |θ new );

[0140] It should be noted that in step 5, based on the old true posterior distribution p(f|y,θ) and the old optimal approximate posterior distribution q(f|θ), the new true posterior distribution p(f|y,y new ,θ new ) According to formula (5.1), we get:

[0141]

[0142] Where: q new (f|θ new ) is the new approximate posterior distribution after acquiring new monitoring data; p(f|y,y new ,θ new ) is the new true posterior distribution of the training function f after obtaining new monitoring data; is dependent on the hyperparameter θnew The standardization constant of ; is the new true prior distribution of the training function f after obtaining new monitoring data; p(y|f) is the conditional distribution of the old output vector y in the new data given the training function f after obtaining new monitoring data; p(y new |f) is the new output vector y in the new data given the training function f after obtaining the new monitoring data new The conditional distribution of

[0143] Among them, p(y|f) is obtained according to formula (5.2):

[0144]

[0145] Where: is a normalization constant that depends on the hyperparameter θ; q vfe (f|θ) is the optimal approximate posterior distribution before acquiring new monitoring data; To obtain the true prior distribution of the training function f before new monitoring data.

[0146] It should be noted that in step 5, the new variational free energy bound According to formula (5.3), we can get:

[0147]

[0148] Where: logp(y new |y,θ new ) is the new output vector y new The log-marginal likelihood function conditioned on the old output vector y; KL[q new (f|θ new )||p(f|y,y new ,θ new )] is the new approximate posterior distribution q new (f|θ new ) and the new true posterior distribution p(f|y,y new ,θ new )

[0149] Among them, KL[q new (f|θ new )||p(f|y,y new ,θ new )] According to formula (5.4), we get:

[0150]

[0151] Where: p(f ≠u,new |u new ,θ new ) is the training function value set u except the new pseudo pointnew The remaining potential function value f ≠u,new The prior distribution of q new (u new |θ new ) is the new pseudo-point training function value set u new The posterior distribution of p(u new |θ new ) New pseudo-point training function value set u new The prior distribution of ≠u,new =f new -u new ;f new is the set of training function values ​​for the new historical monitoring data;

[0152] In which, it is assumed that q(f|θ) obeys a normal distribution, that is, q(f|θ) is determined according to formula (5.5):

[0153]

[0154] Where: and are the mean and variance of the normal distribution respectively;

[0155] Among them, the new approximate posterior distribution q new (f|θ new ) According to formula (5.6), we can get:

[0156] q new (f|θ new )=q new (u new |θ new )p(f ≠u,new |u new ,θ new )(5.6)

[0157] Where: p(f ≠u,new |u new ,θ new ) is the training function value set u except the new pseudo point new The remaining potential function value f ≠u,new The prior distribution of q new (u new |θ new ) is the new pseudo-point training function value set u new The posterior distribution of .

[0158] It should be noted that in step 5, the new optimal approximate posterior distribution q new,vfe (f|θ new ) and the new optimal variational free energy bound According to formula (5.7) and (5.8) respectively, we can get:

[0159]

[0160] Where: p new (u new |θ new ) is the new pseudo point P new The training function value set u new The prior distribution of K fu,new f and u new The covariance matrix between uu,new for u and u new The covariance matrix between ; K uu,new The inverse matrix of

[0161] Step 6: According to Get θ new Value, according to q new,vfe (u new |θ new ) and θ new It is worth getting the new specific expression and variance of the training function f Substitute the new specific values ​​into step 1 to obtain a new specific expression of the probability prediction model.

[0162] In the first step of the present invention, the standard Gaussian process model is used as the benchmark probability prediction model. In the second step, the monitoring data set is simplified by introducing pseudo points to construct an approximate probability prediction model based on the flow sparse Gaussian process. In the third step, the variational free energy bound is constructed by variational inference, and the optimal variational free energy bound is solved by numerical iteration. and the best approximate posterior distribution q vfe (f|θ), after solving the hyperparameter θ and the training function f, the two are used as the true solution of the benchmark probability prediction model, so that the specific expression of the probability prediction model can be obtained. Based on the specific expression, the support displacement in the future period can be predicted, and the predicted support displacement is compared with the measured support displacement. According to the difference between the two, the bearing operation status can be evaluated; however, with the passage of time, the model error will gradually increase, and the accuracy of the bearing operation status evaluation will decrease. Therefore, in step five of the present invention, the probability prediction model is updated based on the newly acquired monitoring data based on the new monitoring data, and a new specific expression of the probability prediction model based on the new historical monitoring data is established. The support displacement in the next time period is predicted based on the new expression. Compared with the old expression of the probability prediction model, the new expression of the probability prediction model contains the characteristics of the new monitoring data, so its prediction result is more accurate; and so on, new data is continuously acquired, and the prediction result of each stage is more reliable.

[0163] The present invention deploys a stream sparse Gaussian process approximation framework based on variational reasoning in a data stream environment where monitoring data arrives sequentially. According to the approximate posterior distribution of the training function f obtained based on the old historical monitoring data of the previous time period and the new approximate posterior distribution of the training function f obtained by updating the new historical monitoring data of the new time period, the storage of historical monitoring data in the stream sparse Gaussian process model, the updating of the stream sparse Gaussian process model and the real-time probability prediction of bearing displacement are realized, thus overcoming the problem of time-varying prediction model parameters caused by bearing aging and changes in the service environment.

[0164] The present invention only needs to call the approximate posterior distribution of the training function f in the previous time period, and can propagate the approximate posterior distribution forward without calling massive old historical monitoring data, which significantly reduces the memory requirement for storing massive data. At the same time, a small number of pseudo points can be used to characterize the inherent laws of historical monitoring data, thereby further improving the online computing efficiency of the model while ensuring the prediction accuracy of the model.

[0165] The applicant further declares that the present invention illustrates the implementation method of the present invention through the above-mentioned embodiments, but the present invention is not limited to the above-mentioned implementation methods, that is, it does not mean that the present invention must rely on the above-mentioned methods and structures to be implemented. Those skilled in the art should understand that any improvement of the present invention, equivalent replacement of the implementation method selected by the present invention, addition of steps, selection of specific methods, etc., all fall within the protection scope and disclosure scope of the present invention.

[0166] The present invention is not limited to the above-mentioned embodiments, and all embodiments that use structures and methods similar to the present invention to achieve the purpose of the present invention are within the protection scope of the present invention.

Claims

1. An online probability prediction method for bridge bearing displacement, characterized by: The steps include: Step 1: Use the standard Gaussian process model as the baseline probability prediction model: y n =f(x n )+ε n , where ε n is the noise term, is the variance, f is the training function, and the training function f obeys the Gaussian process prior distribution with mean 0 and covariance k: Where θ is the hyperparameter on which the covariance k depends and The set of hyperparameters composed of Assume that the training function f and variance are given After that, when the main beam temperature input vector is X, the support displacement output vector of the probability prediction model is y; Step 2: Obtain historical monitoring data of main beam temperature And support displacement historical monitoring data according to Introducing a set of pseudo points in u i =f(z i ), M<N, M is the number of pseudo points, N is the number of samples; Definition: The set of training function values ​​of pseudo point P The training function value set of historical monitoring data The remaining training function value set f ≠u =fu; According to u and f, f ≠u The relationship between the two functions is obtained by using the product rule to obtain the true prior distribution of the training function f. The expansion of u, and according to Get the expansion of the approximate prior distribution p(f|θ) of the training function f with respect to u, and get the expansion of the approximate joint distribution p(f,y|θ) of the training function f and the output vector y with respect to u based on the original likelihood function of p(f|θ) and the output vector y; Define the true prior distribution of u under the output vector y as p(u|y,θ). According to p(u|y,θ) and the relationship between u and f, f ≠u The product rule is used to obtain the expansion of the true posterior distribution p(f|y,θ) of the training function f under the output vector y with respect to u, and the expansion of the approximate posterior distribution q(f|θ) of the training function f with respect to u is obtained based on p(f|y,θ); Step 3: Define the marginal likelihood function of the output vector y as p(y|θ), and construct the lower bound of the logarithmic marginal likelihood function logp(y|θ) based on the approximate posterior distribution q(f|θ) and the approximate joint distribution p(y,f|θ); define the KL divergence between the approximate posterior distribution q(f|θ) and the true posterior distribution p(f|y,θ), and construct the variational free energy bound based on the difference between the logarithmic marginal likelihood function logp(y|θ) and the KL divergence according to the variational inference principle. As a lower bound for logp(y|θ), we get Expressions about u; according to Select different pseudo points P and use numerical iteration method to iterate u, so that the approximate posterior distribution q(f|θ) approaches the true posterior distribution p(f|y,θ) so that Maximize and obtain the optimal variational free energy bound and the best approximate posterior distribution q vfe (f|θ); Step 4: According to Get the value of θ, according to q vfe (f|θ) and θ values ​​to obtain the specific expression of f and Specific values ​​of , substitute the two into step 1 to obtain the specific expression of the benchmark probability prediction model; Step 5: Obtain new monitoring data of the main beam temperature and the new monitoring data of the support displacement, and compare the new monitoring data with the old historical monitoring data in step 2. Composing new historical monitoring data according to Introduce a new pseudo point P according to the method in step 2 new and P new The training function value set u new ; Based on the old true posterior distribution p(f|y,θ), use Bayes’ theorem to derive the new true posterior distribution p(f|y,y new ,θ new ); Define the new approximate posterior distribution of the training function f as q new (f|θ new ), define a new approximate posterior distribution q new (f|θ new ) and the new true posterior distribution p(f|y,y new ,θ new ) and construct a new variational free energy bound based on the new KL divergence according to the variational inference principle According to u and u new The relationship between About u, u new Expression of according to Select multiple different pseudo points P new , using numerical iteration method to calculate u new Perform iterative calculations to make the new approximate posterior distribution q new (f|θ new ) approximates the new true posterior distribution p(f|y,y new ,θ new )make Maximize and obtain the new optimal variational free energy bound and the new optimal approximate posterior distribution q new,vfe (u new |θ new ); Step 6: According to Get θ new Value, according to q new,vfe (u new |θ new ) and θ new The value is f's new specific expression and variance Substitute the new specific values ​​into step 1 to obtain a new specific expression for the benchmark probability prediction model.

2. The method according to claim 1, characterized in that: In step 2, The true prior distribution of the training function f The expansion of u is obtained by using the product rule according to formula (2.1): Where: p(u|θ) is the prior distribution of u; p(f|u,θ) is the prior distribution of f under u; p(f ≠u |f,u,θ) is f ≠u Prior distribution under the conditions of f and u; Among them, p(u|θ) is determined according to formula (2.2): Where: K uu is the covariance matrix between u and u; Among them, p(f|u,θ) is determined according to formula (2.3): Where: K fu is the covariance matrix between f and u; K ff is the covariance matrix between f and f; K uf is the covariance matrix between u and f; K uu The inverse matrix of .

3. The method according to claim 2, characterized in that: In step 2, The expansion of the approximate prior distribution p(f|θ) of the training function f with respect to u is obtained according to formula (2.4): p(f|θ)=p(u|θ)q(f|u,θ)p(f ≠u |u,θ) (2.4) Where: q(f|u,θ) is the approximate posterior distribution of f under u, which is an approximation of p(f|u,θ); p(f ≠u |u,θ) is the prior distribution p(f ≠u |f,u,θ) is simplified.

4. The method according to claim 3, characterized in that: The approximate joint distribution p(y,f|θ) of the training function f and the output vector y is obtained according to formula (2.5): Where: is the original likelihood function; Substituting equation (2.4) into equation (2.5), the expansion of the approximate joint distribution p(y,f|θ) with respect to u is obtained according to equation (2.6): p(y,f|θ)=p(y|f,θ)p(f ≠u |u,θ)p(u|θ) (2.6) Where p(y|f,θ) is the prior distribution of y under the condition of f.

5. The method according to claim 4, characterized in that: In step 2, The expansion of the true posterior distribution p(f|y,θ) of the training function f under the output vector y with respect to u is obtained according to formula (2.7): p(f|y,θ)=p(u|y,θ)p(f ≠u |y,u,θ) (2.7) Where p(u|y,θ) is the prior distribution of u under the condition of y; p(f ≠u |y,u,θ) is f ≠u Prior distribution under u and y conditions; The expansion of the approximate posterior distribution q(f|θ) of the training function f with respect to u is obtained according to formula (2.8): q(f|θ)=q(u|θ)p(f ≠u |u,θ) (2.8) Where: q(u|θ) is the posterior distribution of u.

6. The method according to claim 5, characterized in that: In step 3, the lower bound of the logarithmic marginal likelihood function logp(y|θ) is constructed according to formula (3.1): Where: logp(y|θ) is the logarithmic marginal likelihood function of y; is the variational free energy bound that depends on q(f|θ); Among them, the variational free energy bound Determine according to formula (3.2): Where: KL[q(f|θ)||p(f|y,θ)] is the KL divergence between q(f|θ) and p(f|y,θ); p(f|y,θ) is the true posterior distribution of f under the condition of y; Substituting equations (2.6) and (2.8) into equation (3.1), The expression of u is obtained according to formula (3.3): Where: f n =f(x n ) is x n The implicit function value at ; To eliminate items.

7. The method according to claim 6, characterized in that: In step 3, the optimal variational free energy bound F vfe (θ) and the best approximate posterior distribution q vfe (f|θ) is determined according to formula (3.4) and (3.5) respectively: Where: I is the unit matrix; trace(·) is the regularized trace term.

8. The method according to claim 7, characterized in that: In step five, The new true posterior distribution p(f|y,y new ,θ new ) According to formula (5.1), we get: Where: q new (f|θ new ) is the new approximate posterior distribution of f after acquiring new monitoring data; p(f|y,y new ,θ new ) is the new true posterior distribution of f after obtaining new monitoring data; is dependent on the hyperparameter θ new The standardization constant of new ) is the new true prior distribution of f after obtaining the new monitoring data; p(y|f) is the conditional distribution of the old output vector y in the new data given f after obtaining the new monitoring data; p(y new |f) is the new output vector y in the new data given f after obtaining the new monitoring data new The conditional distribution of Among them, p(y|f) is obtained according to formula (5.2): Where: is a normalization constant that depends on the hyperparameter θ; q vfe (f|θ) is the optimal approximate posterior distribution of f before acquiring new monitoring data; To obtain the true prior distribution of f before new monitoring data.

9. The method according to claim 8, characterized in that: In step 5, the new variational free energy bound According to formula (5.3), we get: Where: logp(y new |y,θ new ) is the new output vector y new The log-marginal likelihood function conditioned on the old output vector y; KL[q new (f|θ new )||p(f|y,y new ,θ new )] is q new (f|θ new ) and p(f|y,y new ,θ new ) Among them, KL[q new (f|θ new )||p(f|y,y new ,θ new )] According to formula (5.4), we get: Where: p(f ≠u,new |u new ,θ new ) is the remaining training function value f ≠u,new The prior distribution of q new (u new |θ new ) is the new pseudo point P new The training function value set u new The posterior distribution of p(u new |θ new ) is u new The prior distribution of ≠u,new =f new -u new ;f new is the set of training function values ​​for the new historical monitoring data; Among them, q(f|θ) is determined according to formula (5.5): Where: and are the mean and variance of the normal distribution respectively; Among them, the new approximate posterior distribution q new (f|θ new ) The expansion of u is obtained according to formula (5.6): q new (f|θ new )=q new (u new |θ new )p(f ≠u,new |u new ,i new (5.6) Where: q new (u new |θ new ) is u new The posterior distribution of .

10. The method according to claim 9, characterized in that: In step 5, the new optimal approximate posterior distribution q new,vfe (f|θ new ) and the new optimal variational free energy bound According to formula (5.7) and (5.8) respectively, we can get: Where: p new (u new |θ new ) is the new pseudo point P new The training function value set u new The prior distribution of K fu,new f and u new The covariance matrix between uu,new for u and u new The covariance matrix between ; K uu,new The inverse matrix of

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