The Temperature-Induced Deflection Data Missing Recovery of a Cable-Stayed Bridge Based on Bayesian Robust Tensor Learning

Abstract

Changes in the deflection of cable-stayed bridges due to thermal effects may adversely affect the bridge structure and reflect the degradation of bridge performance. Therefore, complete deflection field data are important for bridge health monitoring. A strong linear correlation has been found between temperature-induced deflections in different positions of the same span of a cable-stayed bridge in many studies, which make the deflection data matrix/tensor have a low-rank structure. Therefore, it is appropriate to use a low-rank matrix/tensor learning to model the temperature–deflection field of a cable-stayed bridge. Moreover, to avoid disturbing the recovery results via abnormal data (e.g., baseline shift and outliers), a Bayesian robust tensor learning method is proposed to extract the spatio-temporal characteristics of the bridge temperature–deflection field. The missing data recovery and abnormal data cleaning are achieved simultaneously in the process of reconstructing the temperature-induced field via tensor learning. The performance of the method is verified with actual continuous monitoring data from a cable-stayed bridge. The experimental results show that low-order tensor (i.e., matrix) learning has a good recovery and cleaning performance. The extension to higher-order tensor learning is proposed to extract the spatial symmetry of the sensor locations, which is experimentally proven to have better missing recovery and abnormal data cleaning performance.

1. Introduction

Infrastructure construction around the world, especially large bridges, are constantly growing, as is the economy. The form of cable-stayed bridge is favored in the construction of long-span bridges due to their long span and flexibility [1]. Many new bridges built since the 1970s have been cable-stayed bridges [2]. However, damage to the main girder and cables of cable-stayed bridges can cause serious damage. For instance, a stay cable of the Maracaibo Lake Bridge in Venezuela suddenly broke due to severe corrosion of the anchor box, causing the bridge to partially collapse [3]. Therefore, the health monitoring of cable-stayed bridges is extremely important. The evolution rule of temperature-induced deflection in the main girder is an important indicator of the condition assessment of cable-stayed bridges [4], which directly reflects the deterioration of the bridge structural or the nonstructural components [5].

Structural health monitoring systems (SHMS) have been widely installed on large span bridges for 40 years [6]. Based on the monitoring data, many studies can be conducted on bridge temperature-induced deflections. Yang et al. [5] found a significant linear correlation existing between temperature-induced deflections at different positions of the same span of a cable-stayed bridge, which is an important feature of the deflection field of cable-stayed bridges. In addition, a significant correlation was found between the temperature-induced deflections and temperature of bridges [7,8]. Many scholars have also conducted research on structural condition assessments based on monitoring data and machine learning [9,10,11,12].

Although the SHMS can collect a large amount of data, there are inevitably missing data in the monitoring data due to the harsh environment where the SHMS operates, which can hinder the condition assessments of bridges [13]. Therefore, high-quality bridge temperature-induced deflection data are the key to using the evolution rule of temperature-induced deflections for bridge condition assessments. There are many machine learning methods currently being used for missing data recovery. Chen et al. [14] used functional data analysis techniques to analyze and model the inter-sensor relationship between two strain sensors located at different locations. The correlations were modeled in explicit ways and were used to recover the missing data. However, the model selection is time-consuming work in this method. Zhang et al. [15] recovered missing stress data by analyzing spatial correlations of different strain sensors. Deep learning is a new branch of machine learning, which has also been widely used for missing data recovery. Liu et al. [16] used LSTM networks to recover the missing structural temperature data from bridge SHM systems. Nevertheless, deep learning performs better in handling highly nonlinear data [17], which may not be the most suitable way to process existing significant linear correlations of the deflection dataset. Deep learning methods require high-quality data as a training dataset, which is not easily available in practice. However, a simple linear regression model cannot meet the accuracy requirements of many application scenarios (e.g., missing recovery). Thus, a more suitable and accurate method is needed to process temperature-induced deflections.

Tensor learning is a branch of machine learning used to process data in the format of a tensor, which has contributed to image processing, recommender systems and traffic data analysis [18,19,20,21,22]. Temperature-induced deflection data collected from sensors at different locations are a kind of spatio-temporal data, which can be organized in the format of a matrix (e.g., sensor × time) or a tensor. Moreover, the low-rank structure of the matrix/tensor of temperature-induced deflection data is easy to implement due to the strong linear correlation between the temperature-induced deflections at different locations. Therefore, low-rank tensor learning is an appropriate method to handle temperature-induced deflection field data. In addition, the actual monitoring data inevitably contain abnormal data (e.g., outliers and baseline shifts) except missing data, which can obscure the true information contained in the data and lead to false alerts [23]. The abnormal data contained in the data may interfere with the results of missing recovery. However, most current studies on missing data recovery do not simultaneously consider the presence of abnormal data. Therefore, there is an urgent need for a more robust data recovery method, namely recovering the missing data of a dataset containing abnormal data. To this end, this research inherits the modeling idea for Bayesian robust matrix factorization (BRMF) of Naiyan Wang and Dit-Yan Yeung [24]. The recovery of missing data and the cleaning of abnormal data can be achieved simultaneously via BRMF. Because of the full Bayesian framework, there are no problems of overfitting and manual parameter tuning existing in the optimization-based methods. However, BRMF cannot handle data with more than two modes of variation because of the low dimensionality of the matrix. To solve this problem, the BRMF model is extended to a high-order tensor so that the symmetry of the deflection field data due to the symmetry of the bridge structure and sensor locations can be captured to increase recovery accuracy.

2. Methodology

2.1. Problem Description

The bridge temperature-induced deflection data measured with multiple sensors can be organized in the format of a matrix or tensor [25]. For instance, the matrix $Y\in R^{\mathrm{M}\times \mathrm{T}}$ can be used to represent temperature-induced deflection data containing missing and abnormal data, which are collected from M sensors on T time stamps. Data can be denoted by a tensor $Y\in R^{\mathrm{M}\times \mathrm{N}\times \mathrm{T}}$, which is a more flexible organized form that is able to represent “sensor types × sensors × time stamps”, “sensors in the same span×spans×time stamps” or ‘‘sensors × days × time stamps per day” [26]. The process of recovering missing data involves factorizing the original matrix/tensor under a Bayesian framework to obtain its low-rank approximation that does not contain missing or abnormal data [27].

2.2. Bayesian Robust Matrix Factorization

2.2.1. Full Bayesian Model

A partially observed spatio-temporal matrix $Y\in R^{\mathrm{M}\times \mathrm{T}}$ can be decomposed into a spatial factor matrix $U\in R^{\mathrm{K}\times \mathrm{M}}$ and a temporal factor matrix $X\in R^{\mathrm{K}\times \mathrm{T}}$, viz.,

$$Y\approx U^{\mathrm{T}}X,$$

(1)

where K denotes the rank of matrix $Y$. The column vector of $U$ and $X$ is $u_{\mathrm{i}}\in R^{\mathrm{K}}$ and $x_{\mathrm{t}}\in R^{\mathrm{K}}$, respectively. Therefore, Equation (1) can be written at the elemental level as

$$y_{i,t}\approx u_{\mathrm{i}}^{\mathrm{T}}{x}_{\mathrm{t}},$$

(2)

An indicator matrix $\Omega =\left\{\left(i,t\right)|y_{i,t} is observed\right\}$ is defined to locate the position of the missing data. The conceptual diagram of the basic matrix factorization model is shown in Figure 1.

The standard matrix factorization model is a good method to recover missing data. However, when the data matrix is contaminated with abnormal data (e.g., outliers), the estimations may not match the ground truth significantly due to the sensitivity of the $l_2$ norm to the abnormal data [28]. A common solution is to use the $l_1$ norm instead of the $l_2$ norm [29,30]. However, the model with the $l_1$ loss function may still perform not robustly enough when dealing with massive abnormal data. To this end, a Laplace mixture with the generalized inverse Gaussian (GIG) distribution is considered to model the noise to enhance model robustness. Following the above ideas, the graphical model for the BRMF framework is built as shown in Figure 2. The model is entirely built on observed data in $\Omega$, which means it can be trained on datasets with missing and abnormal values. Each component in this probabilistic graphic model is introduced below.

To perform the matrix factorization under the $l_1$ loss function, a Laplace distribution is assumed as the prior distribution of entries of the matrix $Y$, i.e.,

$$y_{i,t}~\mathcal{L}\left(u_i^T{x}_t,{\eta }_{i,t}\right),\left(i,t\right)\in \Omega ,$$

(3)

where ${\eta }_{i,t}$ is a spatially varying and temporally varying precision parameter modeling the noise of $y_{i,t}$. To make the model more robust, the noise is modeled by a Laplace mixture with the generalized inverse Gaussian (GIG) distribution, which has been demonstrated in [31] and performs better than the $l_1$ norm when it is used to define a regularizer for variable selection. We expect it to be better when it is used as a loss function as well. ${\eta }_{i,t}$ is assumed to follow a generalized inverse Gaussian distribution, i.e.,

$${\eta }_{i,t}~GIG\left(p,a,b\right),$$

(4)

where $GIG(·)$ represents the generalized inverse Gaussian distribution, and $p,a,b$ are hyperparameters. However, there is a computational efficiency problem because the Laplace distribution is not easy to sample. An important property of Laplace distributions is used to solve this problem. When a random variable $\epsilon$ follows the Laplace distribution $\mathcal{L}\left(\epsilon |\mu ,{\lambda }^2\right)$, the probability density function of $\epsilon$ is

$$p(\epsilon |\mu ,{\lambda }^2)=\frac{{\lambda }^2}{2}\exp \left(-{\lambda }^2\left|\epsilon -\mu \right|\right).$$

(5)

Then, the Laplace distribution can be equivalently written as an infinite Gaussian mixture with the exponential distribution, i.e.,

$$\mathcal{L}\left(\epsilon |\mu ,{\lambda }^2\right)={{\displaystyle \int }}_0^{\infty }\mathcal{L}\left(\epsilon |\mu ,\tau \right)E\left(\tau |{\lambda }^2\right)d\tau ,$$

(6)

where $E\left(\tau |{\beta }^2\right)$ denotes an exponential distribution. An important property is called the hierarchical view of the Laplace distribution. A matrix $\mathbf{T}\in {\mathbf{R}}^{M\times T}$ whose entry ${\tau }_{i,t}$ is the precision parameter of $y_{i,t}$ is defined to apply the hierarchical view of the Laplace distribution. The exponential prior is placed on ${\tau }_{i,t}$.

$${\tau }_{i,t}~E\left(\frac{{\eta }_{i,t}}{2}\right).$$

(7)

Thus, the prior distribution of $y_{i,t}$ should be changed from a Laplace distribution in Equation (3) to the following Gaussian distribution:

$$y_{i,t}~𝒩\left(u_i^{\mathrm{T}}{x}_t,{\tau }_{i,t}\right).$$

(8)

Due to the introduction of a hierarchical view, sampling from the Laplace distribution is converted to sampling from the Gaussian and exponential distributions, where they are much easier to sample than from the Laplace distribution. The computational efficiency is greatly improved by the hierarchical view of the Laplace distribution.

Then, the spatial factors are modeled. Each column vector $u_{\mathrm{i}}$ of spatial factor $U$ follows an i.i.d. multivariate Gaussian distribution:

$$u_{\mathrm{i}}~𝒩\left({\mu }_{\mathrm{u}},{\Lambda }_u^{-1}\right),$$

(9)

where ${\mu }_{\mathrm{u}}\in R^{\mathrm{K}}$ is a mean vector and ${\Lambda }_u\in R^{\mathrm{K}\times \mathrm{K}}$ is the precision matrix. We further place a conjugate Gaussian–Wishart prior on the parameters of $u_{\mathrm{i}}$, viz.,

$${\mu }_{\mathrm{u}}~𝒩\left({\mu }_0,{\left({\beta }_0{\Lambda }_{\mathrm{u}}\right)}^{-1}\right),\hspace{1em}\hspace{1em}{\Lambda }_{\mathrm{u}}~𝒲\left(W_0,v_0\right),$$

(10)

where ${\mu }_0,{\beta }_0,W_0,v_0$ are hyperparameters, $𝒲(·)$ denotes the Wishart distribution with degrees of freedom $v_0$ and scale matrix $W_0\in R^{\mathrm{K}\times \mathrm{K}}$.

The modeling process of temporal features is like that of spatial features. A multivariate Gaussian distribution prior is placed on the temporal feature vector $x_{\mathrm{t}}$:

$$x_{\mathrm{t}}~𝒩\left({\mu }_{\mathrm{x}},{\Lambda }_x^{-1}\right).$$

(11)

Similarly, the conjugate Gaussian–Wishart distribution is placed on the parameters of the temporal feature, namely

$${\mu }_{\mathrm{x}}~𝒩\left({\mu }_0,{\left({\beta }_0{\Lambda }_{\mathrm{x}}\right)}^{-1}\right),\hspace{1em}\hspace{1em}{\Lambda }_{\mathrm{x}}~𝒲\left(W_0,v_0\right).$$

(12)

where the mean vector ${\mu }_x\in R^{\mathrm{K}}$ and the precision vector ${\Lambda }_x\in R^{\mathrm{K}\times \mathrm{K}}$.

Note that the column vectors of both U and X can be sampled in parallel due to their independence from each other. This potential for parallelization is important for large-scale applications and computational efficiency.

2.2.2. Model Inference

Monte Carlo–Markov chain (MCMC) sampling is adopted by deriving the full conditional distributions for all parameters and hyperparameters to perform Bayesian inference. Due to the use of conjugate priors in Section 2.2.1, especially the corresponding posterior distribution of the mixture of exponential power distributions with a generalized inverse Gaussian also following a generalized inverse Gaussian distribution [30], the analytical solutions of all conditional distributions can be obtained. The detailed process of model inference is given below.

Sample spatial feature parameters ${\mu }_{\mathrm{u}} \mathbf{and} {\Lambda }_{\mathrm{u}}$: Based on the property of conjugate priors, the joint posterior distribution of ${\mu }_{\mathrm{u}} \mathrm{and} {\Lambda }_{\mathrm{u}}$ is the Gaussian–Wishart distribution as well:

$$p\left({\mu }_{\mathrm{u}},{\Lambda }_{\mathrm{u}}|U,W_0,{\mu }_0,v_0,{\beta }_0\right)=𝒩\left({\mu }_{\mathrm{u}}|{\mu }_0^{\prime },{\left({\beta }_0^{\prime }{\Lambda }_{\mathrm{u}}\right)}^{-1}\right)𝒲\left({\Lambda }_{\mathrm{u}}|W_0^{\prime },v_0^{\prime }\right),$$

(13)

where

$${\mu }_0^{\prime }=\frac{{\mathsf{\beta }}_0{\mu }_0+\mathrm{M}\overline{u}}{{\mathsf{\beta }}_0+\mathrm{M}},{\beta }_0^{\prime }={\beta }_0+\mathrm{M}, v_0^{\prime }=v_0+\mathrm{M}, {\left(W_0^{\prime }\right)}^{-1}=W_0^{-1}+{\overline{\Sigma }}_u+\frac{{\beta }_0\mathrm{M}}{{\beta }_0+\mathrm{M}}\left(\overline{u}-{\mu }_0\right){\left(\overline{u}-{\mu }_0\right)}^{\mathrm{T}}.$$

(14)

$\overline{x}$ and $\overline{\Sigma }$ in Equation (14) are two parameters:

$$\overline{u}=\frac{1}{\mathrm{M}}{\sum }_{i=1}^{\mathrm{M}}{u}_i,{\overline{\Sigma }}_u={\sum }_{i=1}^{\mathrm{M}}\left(u_{\mathrm{i}}-\overline{u}\right){\left(u_{\mathrm{i}}-\overline{u}\right)}^{\mathrm{T}}.$$

(15)

The sampling process for ${\mu }_{\mathrm{x}}$ and ${\Lambda }_{\mathrm{x}}$ is similar.

Sample spatial feature $u_i$: According to Bayes’ rule, the conditional distribution of $u_i$ related to the temporal factor X, the partially observed Y, the accuracy parameter ${\tau }_{i,t}$ and all relevant hyperparameters is given as follows:

$$p\left(u_{\mathrm{i}}|Y,X,{\mu }_{\mathrm{u}},{\Lambda }_{\mathrm{u}},T\right)={\displaystyle \prod }_{\mathrm{t}=1}^{\mathrm{T}}𝒩\left(y_{i,t}|u_i^{\mathrm{T}}{x}_t,{\tau }_{i,t}\right)𝒩\left(u_i|{\mu }_{\mathrm{u}},{\left({\Lambda }_{\mathrm{u}}\right)}^{-1}\right),$$

(16)

where

$${\Lambda }_u^{\prime }={\Lambda }_{\mathrm{u}}+{\sum }_{t=1}^{\mathrm{T}}\frac{x_t{x}_t^{\mathrm{T}}}{{\tau }_{i,t}}, {\mu }_{\mathrm{u}}^{\prime }={({\Lambda }_{\mathrm{u}}^{\prime })}^{-1}\left({\sum }_{t=1}^{\mathrm{T}}\frac{y_{i,t}{x}_{\mathrm{t}}}{{\tau }_{i,t}}+{\Lambda }_{\mathrm{u}}{\mu }_{\mathrm{u}}\right), \left(i,t\right)ϵ\Omega .$$

(17)

The sampling process for ${\mathbf{X}}_t$ is similar.

Sample ${\tau }_{i,t}$: Based on Bayes’ rule, the posterior distribution of ${\tau }_{i,t}$ is

$$p\left(\frac{1}{{\tau }_{i,t}}|Y,U,X,{\eta }_{i,t}\right)=IG\left(\frac{1}{{\tau }_{i,t}}|\frac{\sqrt{{\eta }_{i,t}}}{\left|r_{i,t}\right|},{\eta }_{i,t}\right)\propto {\displaystyle \prod }_{i=1}^{\mathrm{M}}{\displaystyle \prod }_{t=1}^{\mathrm{T}}𝒩\left(y_{i,t}|u_i^{\mathrm{T}}{x}_t,{\tau }_{i,t}\right)E\left({\tau }_{i,t}|\frac{{\eta }_{i,t}}{2}\right),$$

(18)

where

$$r_{i,t}=y_{i,t}-u_i^{\mathrm{T}}{x}_t, \left(i,t\right)ϵ\Omega .$$

(19)

Sample ${\eta }_{i,t}$: Based on the corresponding posterior distribution of the mixture of exponential power distributions with a generalized inverse Gaussian also follows a generalized inverse Gaussian distribution:

$$p\left({\eta }_{i,t}|{\tau }_{i,t},p,a,b\right)=GIG\left({\eta }_{i,t}|p^{\prime },a^{\prime },b^{\prime }\right)\propto E\left({\tau }_{i,t}|\frac{{\eta }_{i,t}}{2}\right)GIG\left({\eta }_{i,t}|p,a,b\right),$$

(20)

where

$$p^{\prime }=p+1, a^{\prime }={\tau }_{i,t}+a, b^{\prime }=b.$$

(21)

However, it is not convenient to sample directly from the generalized inverse Gaussian distribution. Thus, a form of the generalized inverse Gaussian distribution was chosen to sample. Specifically, we let p = −1/2, and the posterior distribution was converted to the inverse Gaussian distribution, which was more efficient to sample:

$$\frac{1}{{\eta }_{i,t}}~IG\left(\sqrt{\frac{{\tau }_{i,t}+a}{b},}{\tau }_{i,t}+a\right),$$

(22)

2.2.3. High-Order Extension to Bayesian Tensor Learning

Limited by the low dimensionality of the matrix, there is only one form of data organization. However, it has been shown that data often have more than two modes of variation [31,32,33]. As a result, the best form of data organization is tensor. In practice, the locations of sensor networks always have a certain spatial feature, such as symmetry. To reflect this spatial feature in the form of data organization, we extended BRMF to Bayesian robust tensor factorization (BRTF) by organizing the data as a a high-order tensor $Y\in R^{\mathrm{M}\times \mathrm{N}\times \mathrm{T}}$. The complete expansion process to BRTF is described below.

The Tucker decomposition and CP decomposition both can be used for tensor factorization. Because the result of Tucker decomposition may not be unique when the rank is fixed, the CP decomposition is adopted here, which is one of the most commonly used tensor factorizations that captures multi-linear structures [33]. CP decomposition represents a high-order tensor as a sum of several rank 1 tensors. As shown in Figure 3, the partially observed data $Y\approx {\Sigma }_{p=1}^{\mathrm{p}}{u}_p\circ v_p\circ x_p$, where $u_p\in R^{\mathrm{M}},v_p\in R^{\mathrm{N}},x_p\in R^{\mathrm{T}}$, are the rank 1 tensors of the pth row factor matrices $U,V$ and $X$, respectively. P is the rank of the tensor. The entries in $Y$ can be expressed as

$$y_{i,j,t}~𝒩\left(\sum _{p=1}^{\mathrm{P}}{u}_{i,p}{v}_{j,p}{x}_{t,p},{\tau }_{i,j,t}\right),\hspace{1em}\left(i,j,t\right)ϵ\Omega ,$$

(23)

where $\Omega =\left\{\left(i,j,t\right)|y_{i,j,t} is observed\right\}$ is a third-order indicator tensor. Consistent with BRMF, the column vectors $u_{\mathrm{i}}, x_{\mathrm{t}}$ and $v_{\mathrm{j}}$ of three factor matrices in BRTF follow a multivariate Gaussian distribution, i.e.,

$$u_{\mathrm{i}}~𝒩\left({\mu }_{\mathrm{u}},{\Lambda }_u^{-1}\right), v_{\mathrm{j}}~𝒩\left({\mu }_{\mathrm{v}},{\Lambda }_v^{-1}\right), x_{\mathrm{t}}~𝒩\left({\mu }_{\mathrm{x}},{\Lambda }_x^{-1}\right).$$

(24)

Regarding the parameters of $u_{\mathrm{i}},x_{\mathrm{t}}$ and $v_{\mathrm{j}}$, the conjugate Gaussian–Wishart priors are placed on them. Because there is only an addition factor matrix $V$ in BRTF compared to BRMF, only the prior of the parameters of $V$ is written here.

$${\mu }_{\mathrm{v}}~𝒩\left({\mu }_0,{\left({\beta }_0{\Lambda }_{\mathrm{v}}\right)}^{-1}\right),\hspace{1em}\hspace{1em}{\Lambda }_{\mathrm{v}}~𝒲\left(W_0,v_0\right).$$

(25)

The posterior distribution of ${\mu }_{\mathrm{v}}$ and ${\Lambda }_{\mathrm{v}}$ is

$$p\left({\mu }_{\mathrm{v}},{\Lambda }_{\mathrm{v}}|V,{\mu }_{\mathrm{v}},{\Lambda }_{\mathrm{v}},T\right)={\displaystyle \prod }_{n=1}^{\mathrm{N}}𝒩\left(v_{\mathrm{n}}|{\mu }_{\mathrm{v}},{\Lambda }_{\mathrm{v}}^{-1}\right)𝒩\left({\mu }_{\mathrm{v}}|{\mu }_0,{\left({\beta }_0^{\prime }{\Lambda }_{\mathrm{v}}\right)}^{-1}\right)𝒲\left({\Lambda }_{\mathrm{v}}|W_0,v_0\right),$$

(26)

where

$${\mu }_0^{\prime }=\frac{{\beta }_0{\mu }_0+\mathrm{N}\overline{v}}{{\beta }_0+\mathrm{N}},{\beta }_0^{\prime }={\beta }_0+\mathrm{N}, v_0^{\prime }=v_0+\mathrm{N}, {\left(W_0^{\prime }\right)}^{-1}=W_0^{-1}+{\overline{\Sigma }}_v+\frac{{\beta }_0\mathrm{N}}{{\beta }_0+\mathrm{N}}\left(\overline{v}-{\mu }_0\right){\left(\overline{v}-{\mu }_0\right)}^{\mathrm{T}}.$$

(27)

Similarly, $\overline{v}$ and ${\overline{\Sigma }}_v$ are defined as follows:

$$\overline{v}=\frac{1}{\mathrm{N}}{\sum }_{j=1}^{\mathrm{N}}{v}_j,{\overline{\Sigma }}_v={\sum }_{j=1}^{\mathrm{N}}\left(v_j-\overline{v}\right){\left(v_j-\overline{v}\right)}^{\mathrm{T}}.$$

(28)

The priors of the parameters of $u_{\mathrm{i}}$ and $x_{\mathrm{t}}$ are the same as those in Equations (10) and (12), respectively. In addition, the posterior distribution of the parameters of $u_{\mathrm{i}}$ is the same as those in Equations (13) and (14), respectively.

Because the partially observed data $Y$ are organized as a third-order tensor rather than a matrix in BRMF, the posterior distributions of $u_{\mathrm{i}}$, $v_{\mathrm{j}}$ and $x_t$ in BRTF are totally different. The posterior distribution of $u_{\mathrm{i}}$ is given by $u_{\mathrm{i}}~𝒩\left({\mu }_{\mathrm{u}}^{\prime },{({\Lambda }_u^{\prime })}^{-1}\right)$, where

$${\Lambda }_u^{\prime }={\Lambda }_u+{\sum }_{j,t}\frac{{\mathit{W}}_{j,t}{\mathit{W}}_{j,t}^{\mathrm{T}}}{{\tau }_{i,j,t}}, {\mu }_u^{\prime }={({\Lambda }_u^{\prime })}^{-1}\left({\sum }_{j,t}\frac{y_{i,j,t}{\mathit{W}}_{j,t}}{{\tau }_{i,j,t}}+{\Lambda }_u{\mu }_u\right),{\mathit{W}}_{j,t}=v_j\odot x_t,\left(i,j,t\right)ϵ\Omega ,$$

(29)

⨀ denotes the Hadamard product. The posterior distributions over $v_{\mathrm{j}}$ and $x_t$ have the same form as that in Equation (29). The posterior distribution of ${\eta }_{i,j,t}$ and ${\tau }_{i,j,t}$ can be obtained by simply replacing the two-dimensional subscripts (i, t) with three-dimensional subscripts (i, j, t) in Equations (18), (19) and (22), i.e.,

$$\frac{1}{{\eta }_{i,j,t}}~IG\left(\sqrt{\frac{{\tau }_{i,j,t}+a}{b},}{\tau }_{i,j,t}+a\right),$$

(30)

$$\frac{1}{{\tau }_{i,j,t}}~IG\left(\frac{\sqrt{{\eta }_{i,j,t}}}{\left|r_{i,j,t}\right|},{\eta }_{i,j,t}\right),$$

(31)

where

$$r_{i,j,t}=y_{i,j,t}-{\sum }_{p=1}^p{u}_{i,p}{v}_{j,p}{x}_{t,p},\left(i,j,t\right)ϵ\Omega .$$

(32)

2.3. Bridge and SHM System Description

The Tongling highway–railway bridge is a cable-stayed bridge that connects the two sides of the Yangtze River in Anhui Province. As shown in Figure 4, the bridge has a symmetric span configuration with a midspan of 630 m, a side span of 240 m, and an auxiliary span of 91.8 m. The main truss is an N-shaped truss structure. The cross section of the bridge is shown in Figure 5. The three main trusses are 17.1 m apart at the center with a 34.2 m width and 15.5 m height. The steel orthogonal anisotropic plate is adopted in the highway bridge; the railroad bridge deck is an orthogonal anisotropic integral deck with a ballast track structure.

The locations of the deflection sensors are shown in Figure 4. There are 12 deflection sensors placed in 10 sections of the bridge. As shown in Figure 5, Sections 1–5 and 7–11 are each installed with one deflection sensor; Section 6 is installed with two deflection sensors. The sampling frequency of the deflection sensors is 5 Hz. The raw data are shown in Figure 6a. The bridge girder deflection caused by vehicle loads is a dynamic response, whereas the temperature-induced deflection of the bridge girder is a quasi-static response. Therefore, the vehicle loads should be removed from the raw data. The raw data can be resampled at a rate of 5 min intervals to obtain the temperature-induced deflection, as shown in Figure 6b. The SHM system recorded monitoring data from March 2019 to August 2019. The temperature-induced deflection data measured by each sensor are shown in Figure 7. To prevent confusion, the temperature-induced deflections of the main span and side spans are plotted separately, as shown in Figure 7a,b.

3. Results

3.1. The Correlation between Temperature-Induced Deflections of Different Sensors

In this section, the correlation between temperature-induced deflections of different sensors is studied. The sensors can be divided into main span sensors (i.e., ND3~10) and side span sensors (i.e., ND1~2 and ND11~12). The correlation coefficients of the different deflections are shown in Figure 8 to quantitatively demonstrate the correlation between the deflections of different position. As shown in Figure 8, all the correlation coefficients of the deflections of the same span approximate to 1, which shows that there is a significant linear correlation between the deflections of the same span. In contrast, all the correlation coefficients of the deflections of different spans are significantly less than 1, which indicates that the correlation between the temperature-induced deflection of the main span and the side spans is weak.

Although there is a significant linear correlation between the temperature-induced deflections of different positions of the same span, the accuracy of simple linear regression is still not enough for some scenarios (e.g., missing recovery). For example, the temperature-induced deflections of ND3 are used to linearly fit the temperature-induced deflection of ND4. As shown in Figure 9a, the linear regression equation is given by $y=2.0950x+3.6987$. The fitted temperature-induced deflection data from ND4 can be given by using the linear regression equation. As shown in Figure 9b, the fitted temperature–deflection data from ND4 deviate significantly from the ground truth. To calculate the accuracy of the fitted results, the recovery loss rate ${\rho }_r$ is defined as the mean absolute error (MAE) between the estimations and ground truth, normalized by the absolute mean of the target value:

$${\rho }_r=\frac{\frac{1}{n}{\sum }_{i=1}^n\left|y_i-y_i^{\ast }\right|}{\frac{1}{n}{\sum }_{i=1}^n\left|y_i\right|},$$

(33)

where $y_i$ and $y_i^{\ast }$ represent the ground truth and estimated values at the same position I, respectively, and n is the total number of missing values. The ${\rho }_r$ of linear regression’s fitted results for ND4 temperature–deflection data is as high as 19.39%, which indicates that the accuracy of linear regression cannot meet the requirements of missing recovery.

3.2. Experimental Verification

To present the details in pictures, the main span deflection data of July 2019 are used to test the performance of the proposed method. The deflection data of July 2019 are shown in Figure 10, where there exists an outlier in the data of ND6.

3.2.1. Abnormal Data Types of Cable-Stayed Bridge Deflection Field Data

As shown in Figure 11, the common types of abnormal data in structural health monitoring data are outliers, baseline shifts and noise. Outliers are points that deviate significantly from the overall trend from the perspective of statistical probability, as shown in Figure 11a. The specific characteristics of noise are shown in Figure 11b, which is a section of noise in the deflection data with the original sampling frequency of 5 Hz. As shown in Figure 11c, a baseline shift is a phenomenon in which part of the monitoring data significantly deviates from the overall trend in the time domain. However, the abnormally deviated part of the data retains a normal component in terms of pattern and trend.

The above three types of abnormal data are common in raw data. However, abnormal data types in temperature-induced deflection data are slightly different. Resampling is used to smooth raw data to obtain temperature-induced deflection data, which results in outliers and noise in the raw data being cleaned. However, the small segment baseline shift is transformed into an outlier by resampling. The data with the sampling frequency of 5 Hz in Figure 11 is resampled at a rate of 5 min intervals. The corresponding results are shown in Figure 12. A large segment (e.g., one day or a few days) baseline shift is still a baseline shift after resampling, which is easy to understand. In summary, there are only two common types of abnormal data in bridge temperature-induced deflection data, namely outliers and baseline shifts. Therefore, only outliers and baseline shifts are added to the data in subsequent experiments. Finally, the abnormal rate ${\gamma }_a$ is introduced into the experiments, which is the ratio of the amount of abnormal data to the total amount of data.

The cleaning loss rate ${\rho }_c$ is calculated using the mean absolute error (MAE) and is normalized by the absolute mean of the target values, as follows:

$${\rho }_c=\frac{\frac{1}{m}{\sum }_{j=1}^m\left|y_j-y_j^{\ast }\right|}{\frac{1}{m}{\sum }_{j=1}^m\left|y_j\right|},$$

(34)

where $y_i^{\ast }$ and $y_i$ represent estimated values and the ground truth at position i, respectively.

3.2.2. Missing Recovery Based on Low-Order Tensor Learning

In the framework of low-order tensor (i.e., matrix) learning, deflection data containing missing and abnormal data can be organized as a matrix $Y\in R^{\mathrm{M}\times \mathrm{T}}$, where $\mathrm{M}$ is the number of the deflection sensors and T is the number of timestamps during a continuous monitoring period. In our experiments, the main span deflection data of July 2019 are organized as the matrix $Y\in R^{8\times 8928}$ In our experiments, the hyperparameters are initialized as follows: ${\mathit{\mu }}_0=0,W_0=I, {\beta }_0=2$, $v_0=\mathrm{K}$, $a={10}^{-4}$, $b=10$.

The missing data should be set in temperature-induced deflection data to test the performance of missing recovery. The missing rate ${\gamma }_m$ is introduced in the missing setting, which means the ratio of the amount of the missing data to the total number of the entries of the matrix/tensor. In the experimental design of [34], three types of missing data were set up, namely random missing, structural missing and mixed missing data. Random missing is a scenario in which each entry in the data matrix is set as missing randomly. Structural missing data are more common in actual monitoring data, where data are missing continuously due to sensor failure or other reasons. Mixed missing data consist of random missing and structural missing data. Considering that it is more common and challenging that large proportions of structural missing data occur to multiple sensors at the same time in real situations, only scenarios of large proportions of structural missing data are set in experiments. Specifically, multiple sensors are chosen to add large proportions of structural missing data at the same time.

To demonstrate that the proposed method can recover missing data if the data contain abnormal data, abnormal data should be set in the dataset. According to the analysis in Section 3.2.1, only outliers and baseline shifts are set in the bridge temperature–deflection data. Abnormal data are randomly set at any entry (except for missing entries) of the matrix. The specific scenarios are shown in

Table 1. Scenarios of large proportions of structural missing data.

Scenario	${\mathit{\gamma }}_{\mathit{a}}$	${\mathit{\gamma }}_{\mathit{m}}$	Sensors Containing Missing Data
1	10%	50%	ND4, ND5, ND7, ND9
2	10%	50%	ND4, ND6, ND8, ND9
3	10%	50%	ND4 ND5, ND7, ND9, ND10
4	15%	50%	ND4 ND5, ND7, ND9, ND10

. To leverage the low-rank structure of the deflection data matrix’s low-rank structure, rank K is set as 5 in the experiments. The results of ND4 are selected to present missing recovery results in each experimental scenario. Figure 13 shows the results of missing recovery in different scenarios, where the shaded area presents missing data.

As shown in Figure 13, the estimations match well with the ground truth in all scenarios despite the observed data containing a lot of abnormal data. It is worth mentioning that a large percentage of missing data can greatly decrease the number of observations, which makes the actual abnormal rate larger than the set abnormal rate, namely ${\gamma }_a$. The loss rate in the four scenarios is 3.03%, 3.07%, 3.16% and 3.35%, respectively, which indicates that the high accuracy of the proposed method does not decrease significantly as the missing rate and the abnormal rate increase. In addition, the convergence of the model is very fast. The model converges after about 50 iterations in settled scenarios.

3.2.3. Missing Recovery Based on High-Order Tensor Learning

Effectively modeling and exploiting complex correlations across spatial and temporal dimensions is key to recovering missing data [35]. However, the spatial correlation may not be leveraged when the deflection data of different sections is organized in the format of a matrix. Note that the structure of the Tongling Bridge and the deflection sensor locations are symmetrical, which leads to symmetry between the deflection data measured by the symmetrical sensors. First, the symmetric sensor pair is introduced in the experiments, which refers to a pair of sensors arranged in a symmetrical position (e.g., ND3 and ND10). Then, a three-order tensor $Y$ is used to characterize this spatial symmetry, which represents the “sensors in the same span × spans × time stamps” with a size of 8 × 2 × 8928. The horizontal slices of tensor $Y$ are shown in Figure 14a, which are two matrices where white boxes represent missing data and red boxes represent abnormal data. Each vertical slice of tensor $Y$ is a matrix formed by the data measured by a symmetric sensor pair, as shown in Figure 14b. To test the performance of tensor learning, we compare the ${\rho }_r$ and ${\rho }_c$ of the tensor learning results with those of the matrix learning results in the same missing scenarios. In the experiments, data from one or several symmetric sensor pairs are set as missing, and the effect of missing data in symmetric sensor pairs on the results of tensor learning and matrix learning is compared. The deflection data of the main span from July 2019 is used as the test dataset in this section. The rank of the matrix is set as 5, and the rank of the tensor is set as 10. The specific test scenarios are shown in

Table 2. Missing scenarios for the comparison experiments of tensor learning and matrix learning.

Scenario	${\mathit{\gamma }}_{\mathit{a}}$	${\mathit{\gamma }}_{\mathit{m}}$	Sensors Containing Missing Data	Number of Symmetric Sensor Pairs Containing Missing Data
1	10%	50%	ND4, ND5, ND6, ND7	1
2	10%	50%	ND6, ND7, ND8, ND9	1
3	10%	50%	ND4, ND6, ND8, ND9	1
4	10%	50%	ND4, ND6, ND8, ND9, ND10	1
5	10%	50%	ND4, ND5, ND7, ND9	1
6	10%	50%	ND4, ND5, ND7, ND9, ND10	1
7	10%	50%	ND4, ND5, ND8, ND9	2
8	10%	50%	ND4, ND5, ND6, ND8, ND9	2

. The relative recovery accuracy rate ${\lambda }_r$ and relative cleaning accuracy rate ${\lambda }_c$ are introduced to quantitatively compare the performance of tensor learning and matrix learning, which is defined by the following equation:

$${\lambda }_r=\frac{{\rho }_{r,m}-{\rho }_{r,t}}{{\rho }_{r,m}}\times 100\%,$$

(35)

$${\lambda }_c=\frac{{\rho }_{c,m}-{\rho }_{c,t}}{{\rho }_{c,m}}\times 100\%,$$

(36)

where ${\rho }_{r,m}$ is the recovery loss rate of matrix learning, ${\rho }_{r,t}$ is the recovery loss rate of tensor learning, ${\rho }_{c,m}$ is the cleaning loss rate of matrix learning and ${\rho }_{c,t}$ is the cleaning loss rate of tensor learning.

The results of tensor learning and matrix learning are shown in Figure 15. As shown in Figure 15a, it can be seen that the recovery accuracy of tensor learning is higher than that of matrix learning in all missing scenarios. However, the improvement of the accuracy of tensor learning (i.e., ${\lambda }_r$) decreases as the number of symmetric sensor pairs containing the missing data increases, as shown in Figure 15b. This is because the latent spatial symmetry in the data is broken severely when the number of symmetric sensor pairs containing missing data is too large. In conclusion, the experimental results show that tensor learning has significant advantages when the symmetric structure of the data is not badly broken. On the other hand, the cleaning accuracy of tensor learning is much higher than that of matrix learning, as shown in Figure 15c. As shown in Figure 15d, ${\lambda }_c$ does not decrease as the number of symmetric sensor pairs containing missing data increases like ${\lambda }_r$, which shows the great advantage of high-order tensor learning.

4. Discussion and Conclusions

The Bayesian matrix/tensor learning for the missing recovery of temperature–deflection field data of cable-stayed bridge girders is presented in this paper. First, the correlation between temperature-induced deflections at different locations is studied. The results show that there is a strong linear correlation between temperature-induced deflections at different locations of the same span of a cable-stayed bridge, which makes the low-rank matrix/tensor learning method ideal for the missing recovery of temperature–deflection data. A Laplace mixture with a generalized inverse Gaussian distribution is used to enhance the robustness of the model to recover missing data while cleaning the abnormal data inevitably existing in real monitoring data. The experimental results show that the proposed method has satisfactory missing recovery and abnormal data cleaning performance. The extension of high-order tensor learning is proposed to characterize spatial symmetry in bridge structure and sensor arrangements. The experimental results show that tensor learning outperforms matrix learning for missing recovery in many scenarios.

The highlights of this paper are described below. First, low-rank tensor learning is proposed to model the temperature–deflection data of cable-stayed bridges with a high linear correlation, which is proven to be a very effective method. Second, a Laplace mixture with a generalized inverse Gaussian distribution is used to improve model robustness, which allows the model to recover missing data while cleaning abnormal data. Third, the Bayesian framework of the model avoids overfitting and manual tuning of the parameters. Last, the high-order extension to Bayesian tensor learning is proposed to characterize the symmetry of deflection data, which shows higher missing recovery accuracy and abnormal data cleaning accuracy than those of low-order tensor learning.

This study indicates that tensor learning has great potential for SHM applications. Some future research directions are proposed: first, the application of tensor learning to other bridge types can be investigated; second, the variational inference method with computational efficiency should not be ignored; and finally, applications of tensor learning in other areas of SHM, such as anomaly detection, should also be investigated.