Prediction Model for Long-Term Bridge Bearing Displacement Using Artificial Neural Network and Bayesian Optimization

. Bridge bearings are critical components in bridge structures because they ensure the normal functioning of bridges by accommodating the long-term horizontal movements caused by changing environmental conditions. However, abnormal structural behaviors in long-term horizontal displacement are observed when the structural integrity of bridge structures is degraded. Tis study aims to construct an accurate prediction model for long-term horizontal displacement under varying external environmental conditions to support the reliable assessment of bridge structures which has not been fully explored in previous studies. Te main challenge in developing an accurate prediction model lies in modeling the infuencing factors that accurately simulate the efect of external environmental conditions on long-term horizontal displacement. To enhance the prediction accuracy in the proposed study, the surrounding environmental efects by considering the relationship between the current and past displacements in addition to air temperature, thermal inertia, and solar radiation are modeled as critical infuencing factors. In addition, a data-driven method based on an artifcial neural network (ANN) integrated with Bayesian optimization (BO) is employed to model and predict long-term horizontal displacement with the adopted critical infuencing factors. An overpass bridge equipped with bearing displacement monitoring and temperature sensors is used to validate the robustness and efectiveness of the proposed method. Te analysis of the results concludes that the proposed method can generate an accurate and robust long-term horizontal displacement prediction model that supports a reliable anomaly detection approach for early warning systems of bridge structures.


Introduction
Bearings are considered a key component for maintaining the normal operation of bridge structures because they help transmit loads from the superstructure to the substructure while accommodating horizontal movements [1]. Bridge structures exhibit abnormal structural behavior when their serviceability and structural integrity are compromised. Te horizontal displacement of bridge bearings, which refects the overall bridge behavior, is a prominent indicator for assessing the structural condition of bridge structures [2,3]. Terefore, changes in the horizontal displacement of bridge bearings, regarded as anomalous behavior, are a crucial issue that raises concerns regarding bridge maintenance and structural safety.
Structural health monitoring (SHM) is an efective approach to the maintenance and safety evaluation of bridge structures [4]. SHM systems for bridges measure data related to the structural response and surrounding environmental variations to evaluate their structural performance [5][6][7]. Prior studies in the SHM feld that focused on bridge monitoring highlighted the signifcance of temperature effects on the long-term structural behavior of bridge components [3,[8][9][10][11], along with an emphasis on systematic approaches to account for temperature efects in the data interpretation technique to ensure efective assessments of their structural condition. Tus, temperature-based data interpretation (TBDI) techniques in SHM have received signifcant attention in recent years for evaluating the structural condition of critical bridge components [2,[12][13][14][15]. As the horizontal displacement of bridge bearings is mostly governed by diurnal and annual temperature variations [3,11], the characterization of long-term horizontal displacement in response to temperature fuctuations can yield substantial information for assessing their structural performance using the TBDI approach.
Te TBDI technique in SHM detects damage and provides a warning about anomalous structural behavior by analyzing and separating temperature-induced responses from long-term measurements [15]. Two diferent methods can be used to deal with the temperature efects for anomaly detection using the TBDI technique. Te frst method relies only on the structural response (i.e., the output-only method) and treats the temperature efect as undesirable noise or an embedded variable that is discarded from the measured response using numerical techniques for anomaly detection [16,17]. However, this method ofers limited success as the TBDI technique for SHM. Te efect of daily and annual temperature variations on the measured response is substantial compared with that of other external and environmental loads [11,13,18]. Terefore, there it is necessary to consider the temperature-induced response rather than ignoring it to efectively detect the anomalous structural behavior using the TBDI approach [14,15]. Te second method explicitly accounts for the temperature efect on the measured response by modeling the relationship between the temperature variation and temperatureinduced response (i.e., the input-output method) and thus ofers more promise. Te residuals obtained as a discrepancy between the measured and predicted temperature-induced responses (TIR) can be analyzed for anomaly detection [19][20][21]. Accordingly, the efectiveness of the TBDI technique for detecting anomalous structural behavior and its gradual trend, analyzed based on the time histories of the residuals, seems to depend on the reliability of the predicted horizontal response based on temperature fuctuations. Tus, it is necessary to develop an appropriate TBDI approach that can predict the temperature-induced bearing response to ensure bridge maintenance and structural safety.
Existing TBDI approaches for SHM can be broadly divided into two categories: model-based and data-driven approaches. Model-based approaches are based on numerical fnite element (FE) models to identify intrinsic changes in measured TIR [3,22,23]. Te initial FE models should undergo a calibration process to obtain a better prediction of TIR, which is currently undergoing exhaustive research [3,24,25]. However, the application of FE models to TBDI from long-term SHM remains limited because incorporating FE models with varying environmental conditions (most importantly temperature) having a signifcant infuence on the long-term structural behavior is challenging, and thus, the prediction accuracy can be undermined [15,20,26]. In addition, it is difcult to assess the accuracy of the computed FE models because of modeling uncertainties and assumptions (e.g., boundary conditions, chosen model behavior and geometry, and simplifed structural elements) [27,28]. Data-driven approaches ofer promise for addressing the shortcomings of model-based approaches by adopting measured responses instead of FE models to understand structural behavior. Data-driven approaches for TBDI utilize available long-term measurements to establish baseline conditions for normal structural behavior. Subsequently, new measurements are analyzed against the measurements predicted by data-driven methods to detect deviations from the normal baseline condition [20]. Terefore, data-driven methods are robust and ofer great promise for long-term SHM, which can be utilized to detect and identify changes in structural behavior using the TBDI technique.
Data-driven methods can efciently learn the complicated input-output relationships of the system using longterm measurements without requiring extensive prior structural knowledge. Terefore, relevant studies have focused on predicting the TIR of bridge structures by explicitly modeling the relationship with temperature variations in bridges using data-driven methods [12,21,[29][30][31][32][33][34][35]. Te prediction of TIR using the established temperaturedisplacement relationship (data-driven) model can then be utilized to assess the bridge's structural condition. However, bridge structures can have complex nonlinear temperature distributions [15], and therefore, not all temperature measurements are strongly correlated to the thermal response at a specifc location [33,36]. Consequently, the prediction accuracy of data-driven TBDI is inevitably infuenced by the location of the temperature sensors installed on the bridge structure [30,33]. Air temperature afects bridge structures globally and is considered a major factor that causes bridge temperature variations [37], thus showing a strong correlation with the TIR. Terefore, air temperature is utilized as an infuencing factor in data-driven TBDI to efciently predict the TIR such as strain, girder defection, and horizontal displacement [38][39][40]. However, previous studies [38][39][40] could not model the thermal inertial and seasonal solar radiation variation efects, and this can afect the prediction accuracy of the TIR, which in turn afects the reliability of the TBDI approach. Te diference between the bridge and air temperature exists [41] because changes in air temperature are not immediately refected in the bridge temperature, owing to the thermal inertial efect, and follows the seasonal variation attributed to the direct solar radiation and local heat island (irradiation from the ground) efect [38,42]. Te strong and intense solar radiation and the irradiation from the ground during the summer season increase the bridge temperature signifcantly above the air temperature compared to the relatively weak sunlight and cold ground in the other seasons [38]. Tus, seasonal solar radiation variation afects the bridge temperature in addition to air temperature, which signifcantly infuences the TIR and should be considered to enhance the prediction accuracy of data-driven methods for reliable TBDI. Furthermore, the bridge temperature in addition to air temperature and seasonal solar radiation is infuenced by 2 Structural Control and Health Monitoring other surrounding environmental factors and the residuals attributed to such external environmental efects in datadriven methods should be considered for an accurate and reliable TBDI. Considering the limitations in previous studies to comprehensively model the external environmental efects for thermal response modeling, this research focuses on developing an accurate data-driven method that solves a regression problem by employing critical infuencing factors to efciently model and predict the long-term horizontal displacement of bridge bearings. Te critical infuencing factors for efectively modeling the bearing horizontal displacement were selected based on the potential sources of heat exchange between the bridge structures and the external surrounding environment. Te relationship between the critical infuencing factors and bearing horizontal displacement may not be always linear in practice but rather can be complex and nonlinear depending upon the bridge's structural characteristic and bearing type. Terefore, an artifcial neural network (ANN) is employed as a data-driven model because of its ability to efciently model the linear and nonlinear relationships between the infuencing factors and structural response compared to traditional statistical regression models while dealing with highdimensional data-mapping problems [43][44][45][46][47]. Several related research studies owing to the capability of ANN to efectively model and predict the bridge responses are reported in [30,39,[48][49][50][51][52]. However, the ANN model requires tuning of hyperparameters, which is critical for increasing modeling fexibility and enhancing prediction performance [47,48,50,53]. It is challenging to determine the optimum ranges of the hyperparameters quickly and robustly via conventional trial-and-error methods because many hyperparameters of the ANN model have coupling efects with a vast search space owing to their wide range of numerical values [54,55]. To overcome this issue, Bayesian optimization (BO), which emerged as a powerful tool for hyperparameter optimization [56][57][58][59], was integrated with the ANN model to solely search for the optimal ANN hyperparameters in the proposed study. Long-term bearing horizontal displacement and air temperature data collected from a sensor network installed on a bridge with a total span length of 6345 m were utilized to explore and validate the efectiveness of the proposed method.

Artifcial Neural Network. Te ANN, introduced by
McCulloch and Pitts [60], is a data-driven model that is extensively used in the felds of machine learning and data mining. Te ANN model was developed based on the functioning of the biological nervous system of the human brain. Te learning process is the main idea of the ANN model, and the model involves a network of interconnected neurons that work together to discover complex relationships in the data to be analyzed. Research studies have been conducted in the feld of SHM to predict the structural response of bridge structures owing to the capability of the ANN model to efectively learn and simulate the behavior of complex structural systems [30,39,[48][49][50][51][52]. During ANN modeling, no hypotheses or preconstraints are necessary, which allows the ANN model to have a signifcant advantage over traditional computational models [47]. Furthermore, the ANN model is robust against outliers [46] and is a powerful tool for high-dimensional data-mapping regression problems, exploiting the advantages of parallel processing [43,48,50]. ANN was employed as a data-driven model to predict the horizontal displacement in this study owing to its capability to accurately model the linear and nonlinear relationships between the infuencing factors and structural response compared to traditional statistical regression models. Figure 1 shows that the feedforward multilayer perceptron (MLP) structure comprises three layers connected by artifcial neurons and is considered a shallow ANN model that is widely used in many applications. An input layer receives the infuencing factors for a specifc problem and propagates them to hidden layers. Te hidden layers use an activation function to process the weighted sum of incoming signals and output them to the next layer, which can be either another hidden layer or output layer [48]. For the sake of simplicity, the following is a description of a single hidden layer MLP: where X k are the number of infuencing factors at the input layer, H l are the outputs of hidden layer, Y is the predicted response variable, f 0 , W 0 , and b 0 are the activation function, weight matrix, and bias vector of the hidden layer, respectively, while f 1 , W 1 , and b 1 are the activation function, weight matrix, and bias vector of the output layer, respectively. Te ANN model is trained to compute weight matrices and bias vectors using the gradient descent method. During the training phase, the ANN model iteratively reduces the mean square error (MSE) of the dataset by modifying the collection of known input-output pairings until the output value falls below a certain threshold.

Bayesian Optimization for ANN Hyperparameter Tuning.
Te efectiveness and prediction accuracy of the ANN model depend on its architecture and training parameters, regarded as ANN hyperparameters, which should be selected carefully to enhance the ANN performance [46-48, 50, 53-55]. However, there is no consensus regarding the appropriate method for selecting the ANN hyperparameters. Trial-anderror-based optimization is commonly adopted to fne-tune the hyperparameters of the ANN model. In this approach, diferent ANN models are generated by varying the parameters and evaluating the prediction performance of each model. Te ANN model with the best prediction accuracy is then selected for further assessment [46][47][48]. However, the trial-and-error method is computationally expensive and labor-intensive because many ANN hyperparameters have coupling efects with a vast search space, which makes it difcult to quickly determine their optimal ranges, thus necessitating the use of robust optimization approaches.

Structural Control and Health Monitoring
BO internally retains a Gaussian process (GP) model (i.e., probabilistic surrogate model) of the objective function,which makes it particularly ideal for the global optimization of black-box objective functions that are difcult to evaluate. In BO, the loss is modeled as an objective function f(θ) of the hyperparameters θ, and it searches for its global minimum value drawn from the GP prior which can be expressed as where A represents the search space of θ, and μ(θ) and k(θ, θ ′ ) are the mean function and covariance function of the GP respectively. Te μ(θ) captures the expected value of the objective function f(θ) at a given hyperparameter setting, and the k(θ, θ ′ ) captures the similarity or correlation between the objective function f(θ) values at diferent hyperparameter settings. Te BO explores the posterior distribution of the objective function f(θ) using the prior distribution of the objective function f(θ) with the sample information as evidence. Te posterior distribution at each new evaluation of the objective function f(θ) can be defned as follows: where μ(θ + ) and σ 2 (θ + ) are the posterior mean function and variance function, respectively. Te posterior information is then utilized to identify where the objective function f(θ) is minimized, based on a criterion represented by the acquisition function α(θ). Te role of the acquisition function α(θ) is to measure the expected improvement EI(θ) in the objective function f(θ) while discarding the values that would increase it. Te expected improvement EI(θ) can be calculated as follows: where μ(θ + best ) is the lowest observed value of the posterior mean function, Θ is the standard normal cumulative density function, φ is the standard normal probability density function, and z � (μ(θ + ) − μ(θ))/(σ(θ)). Te acquisition function is employed iteratively in an exploration (sampling from the areas of high uncertainty) and exploitation (sampling from that with high values) manner to determine the next hyperparameter confguration by maximizing the acquisition function over the GP. More detailed information on BO can be found in [56,57]. Te BO method can determine the optimal parameter confguration with relatively fewer iterations and is often signifcantly faster than the trialand-error method [57][58][59]. Accordingly, in the present study, the ANN was integrated with BO, as illustrated in Table 1, to solely search for optimal ANN hyperparameters that generate a robust and accurate data-driven prediction model for long-term horizontal displacement.

Bearing Horizontal Response Prediction Methodology
Tis section describes the proposed modeling and prediction method for the horizontal displacement of bridge bearings, with a comprehensive consideration of the external surrounding environment. Previous studies have not comprehensively explored the use of environmental factors in combination as input information to data-driven models for thermal response modeling. To predict the long-term bearing responses accurately, the proposed prediction model employs critical infuencing factors to efectively model the external environmental efects on the horizontal displacement to support a reliable early warning system.

Horizontal Displacement Modeling of Bridge Bearings.
Tis study aimed to construct a prediction model for the horizontal displacement of bridge bearings. However, the main challenge lies in the selection of infuencing factors to accurately model horizontal displacement under changing external environmental conditions. In addition to air temperature and solar radiation as the major infuencing factors, other surrounding environmental factors (e.g., wind) [15] can infuence the horizontal displacement, as shown in Figure 2. Te prediction accuracy can be improved by appropriately selecting the infuencing factors for the horizontal displacement modeling. Terefore, considering the potential sources of heat exchange between the bridge structure and the external surrounding environment, the infuencing factors listed in Table 2 were adopted in the proposed study to model the horizontal displacement and were examined based on the prediction accuracy.
To demonstrate the feasibility of the proposed method, an ANN was employed to model the relationship between the bearing horizontal displacement and the adopted infuencing factors that refect the external environmental efects, as listed in Table 2. Variable set S 1 considers only the air temperature T 0 as an efective factor for modeling the horizontal displacement, where T 0 represents the air temperature at the current displacement measurement time. Te segmented air temperature T p-q along with T 0 as proposed in [61][62][63]   considering the thermal inertia (lag between the displacement response and air temperature) efect. Variable T p-q represents the average air temperature from p to q observations before the current horizontal displacement measurement. According to previous studies [61][62][63], the segmented air temperature T p-q efectively refects the infuence of the air temperature with a lag efect on the displacement variation. Terefore, the segmented air temperature T p-q variables were adopted as the infuencing factors for horizontal displacement modeling. In addition, in variable set S 2 , the thermal efect attributed to the seasonal variation in solar radiation on the horizontal displacement was modeled with variable d, which represents the day of the year from the beginning of the measurement. Variable d simultaneously captures the time and season of the year [46]. Furthermore, the surrounding environmental factors afecting the horizontal displacement, in addition to air temperature and seasonal solar radiation, which are complex to model explicitly, were refected by the lagged displacement variable D n in the variable sets S 3 -S 8 . Te lagged displacement variable D n captures the relationship between current and n previous displacement observations. Te maximum lagged observations for the T p-q and D n variables considered in the present study that can have a signifcant infuence on the current displacement measurement is six than going beyond further in time.

ANN Model Integrated with Bayesian Optimization.
BO is utilized in the proposed study to solely determine the optimal ANN hyperparameters, and a hybrid model is introduced to enhance the prediction accuracy. Te search space for the ANN hyperparameters selected in this study is presented in Table 3.
(1) Te number of hidden layers determines the computational speed and learning efciency of the ANN model. According to previous studies [30,[46][47][48], a single hidden layer can solve any complex function approximation problem. Tus, the number of hidden layers was fxed as one in the proposed study, which considers the computational cost. (2) Te number of hidden neurons in a hidden layer during ANN modeling is generally selected between the number of input and output variables to avoid overftting or underftting problems owing to unnecessarily many or insufcient hidden neurons, respectively [64]. Considering this along with the computational cost, the search range for hidden neurons was set accordingly in the interval [1, 2 n+1 ], where n represents the maximum number of input variables in the proposed study [54,55]. (3) Commonly used activation functions and backpropagation training algorithms that provide acceptable accuracy for function approximation problems [30,47,48,50,65] are selected to construct the search space for BO. Te activation function search space includes linear, tangent-sigmoid, and log-sigmoid activation functions, whereas the backpropagation training algorithm search space includes gradient descent, Levenberg-Marquardt, and Bayesian regularization. Te linear transfer function was fxed for the output layer [55]. (4) Training parameters such as learning rate (lr), momentum constant (mc), and Marquardt parameter (μ) used in their respective backpropagation algorithms can have any value between 0 and 1 depending on the complexity of the data, and therefore, the typical ranges for training parameters were selected accordingly [53,64].
Te ANN prediction performance, as aforementioned, depends on the optimal parameters to train the ANN model determined within the defned search space using BO in the Objective function f(θ) to be optimized: θ * � argmin f(θ), θ ∈ A where A � search space of ANN hyperparameters and θ � θ 1 , θ 2 , ..., θ n � vector of ANN hyperparameters from defned search space. Algorithm: ANN with BO 1: For j � 1,2,3, ... 2: Find θ j by maximizing the acquisition function α(θ) over the GP: θ j � argmaxα(θ | D 1: j−1 ) 3: Calculate the prediction error E(θ j ) through ANN with θ j determined in step 2 4: Augment the data D 1:j � D 1:j−1 , (θ j , E(θ j )) and update the posterior distribution of GP 5: End for   Input variable sets based on adopted environmental infuencing factors present study. For this purpose, the dataset used to train the ANN model was subdivided into training and validation datasets. To avoid overftting (i.e., to obtain better ftting and generalization capability), the ANN model was trained with an early stopping technique [43], and its performance was based on the weighted root mean square error (RMSE) defned in equation (7) over the training and validation datasets was used as the objective function f(θ) for BO. In the weighted RMSE as the objective function to minimize, the weights assigned to the training and validation errors were 0.4 and 0.6, respectively. Te reason can be justifed as the validation dataset is kept unseen while training the ANN model to assess the generalization performance of the ANN model, more weightage value can be assigned to validation error as compared to training error in the f(θ) to be minimized to avoid overftting. To evaluate the prediction capability of the optimized ANN model, the training and validation datasets were again combined into one dataset to retrain the ANN model with the obtained optimal parameters.
where RMSE train and RMSE valid represent the root mean square error for the training and validation datasets, respectively; w 1 and w 2 represents the weights of training and validation errors, respectively, such that w 1 + w 2 � 1.

Model Assessment Metrics.
Once the optimally trained ANN model was generated through BO, its prediction performance was assessed to determine the efective environmental factors adopted for accurate horizontal displacement modeling. Four statistical assessment metrics commonly used for function approximation problems were employed in this study: RMSE, mean absolute error (MAE), Akaike information criterion (AIC), and coefcient of determination (R 2 ).
where n represents the number of observations, k represents the number of infuencing factors, and y i , y i , and y i represent the measured, predicted, and mean bearing displacement values, respectively. Regarding the assessment metrics, the RMSE and MAE represent the overall error distribution as model precision and accuracy, respectively; AIC corresponds to the robustness of the model considering the number of infuencing factors while penalizing the most complex model; and R 2 indicates the goodness of ft.

Procedure for the Bearing Horizontal Response Prediction.
Te procedure for the bearing horizontal displacement prediction with the optimized ANN model is illustrated in Figure 3 and can be described as follows.

Dataset Pre-processing.
Te long-term air temperature and bearing horizontal displacement measurements collected from the safety monitoring system underwent the dataset pre-processing to make the ANN more efcient and to improve its estimation performance, which involves the following steps: (i) Te long-term monitored datasets were checked for constant, corrupt (unknown), and incorrect (outliers) values, and they were removed from the monitored datasets (ii) After checking for suspicious and improper data, the infuencing factors rearranged as input variable sets according to the dataset confguration, as described in Section 3.

Test Bed Bridge.
Te proposed prediction methodology was applied to an overpass bridge in South Korea, as shown in Figure 4. Te target bridge constructed in 1999 was a steel-concrete composite bridge with a total number of 72 spans and a total length of 6345 m. Te deck-pier connection was built using pot bearings, which allow for translation in the horizontal direction owing to the thermal efects in the target bridge structure. However, the relative horizontal displacement required between the deck and piers can be afected by anomalous structural behavior. Tus, this study intended to build a horizontal displacement prediction model for the implementation of a robust early warning decision-making system for bridge structures.

Description of SHM System
Installed on the Test Bed. Figure 5(a) shows the SHM system installed in the test bed in 2020 to ensure the serviceability and structural safety of the target bridge [66]. Figure 5(b) shows the layout of the sensor network installed on the test bed of the target bridge. A computer vision system was used to measure the horizontal displacement of pot bearings. A CCTV camera was mounted underneath the test bed girder using a magnetic base at a distance of approximately 50 cm from the main target on the bearing support, as shown in Figure 6. A CCTV camera travels horizontally with the main girder while the target remains stationary and measures the relative displacement between the girder and bearing support, which is identical to that of the bridge bearing. CCTV cameras with image sensors having progressive scans (RGB CMOS 1/2.7″) with full HD resolution (i.e., 1920 × 1080) and a lens with a focal length of 4 mm were used when the horizontal feld of view was 90.2°. Te air temperature near the monitored bearings was recorded using a platinum resistance temperature detector with a PT1000 sensor, which can measure the temperature with excellent accuracy over a wide range from −30 to 60°C. Te CCTV image sensors measuring the bearing horizontal displacement considered in the proposed study are referred to as B 1 -B 4 (B: bearing), as depicted in Figure 5(b).

Long-Term Measurement.
Te horizontal bearing displacement and air temperature were recorded with a sampling period of 1 h for approximately 18 months from September 1, 2020, to February 13, 2022, using the SHM system. Tere were disconnections in the monitored datasets at the beginning of the measurements, which can occur in the long-term monitoring of in-situ structures because of a variety of issues, such as sensor malfunction, electricity problems, and visual interruption by light refection and backlight [66]. Te bearings' horizontal displacement caused by the dynamic live load was fltered out by recording static measurements with a sampling period of 1 h. Te reason can be justifed as the efect of the trafc load on the bearings' horizontal displacement is signifcantly small when compared to air temperature [11]. Figure 7 shows the overall trends in the monitored longterm horizontal displacement of the selected bearings (B 1 -B 4 ) on the respective piers. Te average air temperature and bearing (B 1 ) horizontal displacement monitored on Pier #1 were normalized by the range of each measurement to examine the relationship between the long-term measurements more closely, and the normalized measurements are plotted in Figures 8(a)-8(b). Te efects of diurnal (daily) and annual (seasonal) air temperature variations can be refected in the horizontal displacement over time by showing similar variation patterns. Tis confrms that the air temperature variations considerably infuence the bearing horizontal response, having a strong correlation between them. Figures 8(c)-8(d) show the efects of thermal inertia and seasonal solar radiation on the monitored horizontal displacement. Te horizontal displacement shows a small magnitude variation during the winter season compared to the summer season, and it also lags behind the air temperature. As discussed, although the seasonal solar radiation being the critical infuencing factor in addition to air temperature for bearing horizontal displacement, the seasonal solar radiation variation was not monitored using the SHM systems installed at the test bed and was explicitly modeled with variable d as day of the year information. Furthermore, the horizontal displacement variation caused by the surrounding environmental efects at the current measurement time can be refected in the prior displacement information. Terefore, it is necessary to select the infuencing factors that efectively model external environmental efects for an accurate horizontal displacement prediction model.

Evaluation of Proposed Method and Discussion of the Results
Te major contribution of this study is the modeling of external environmental efects with the adopted critical infuencing factors for predicting the long-term horizontal displacement of bridge bearings. Te average air temperature and horizontal displacement of bearings B 1 -B 4 on the respective piers measured between September 1, 2020, and February 13, 2022, were utilized to evaluate the proposed methodology. Note that the displacement sensors of the selected bearings exhibited the least data loss. Incorrect (outliers) and missing data points from the beginning of the long-term measurements were removed from the monitored datasets to obtain a continuous and compatible measurement time history for a robust and efcient prediction model. Te optimal parameters were determined within the defned search space, based on the weighted RMSE defned in equation (7) as an objective function for the BO. Figure 9 (a) (b) Figure 4: Test bed: overpass bridge. 8 Structural Control and Health Monitoring shows the performance of the optimized ANN models for each adopted set of infuencing factors. Te input variable sets S 3 -S 8 with the proposed infuencing factors refecting the external environmental efects on bearings (B 1 -B 4 ) horizontal displacement resulted in a lower prediction error (RMSE) over the training and testing periods compared to infuencing factors adopted in the S 1 and S 2 sets. However, the best prediction performance (least RMSE over the testing period as unseen data) with the optimally trained ANN model among the input variable sets S 3 -S 8 is obtained with the critical infuencing factors adopted in sets S 4 and S 5 for bearings B 1 -B 4 as shown in Figures 9(a)-9(d). Furthermore, Figure 10 shows the convergence of the weighted RMSE as an objective function f(θ) obtained using the optimal hyperparameters confguration after employing 30 iterations, as recommended by [59,67]. Note that Figures 10(a)  summarizes the optimal parameters determined via BO only for the ANN model with critical infuencing factors that resulted in the least prediction error for the horizontal displacement of the bridge bearings (B 1 -B 4 ).
Te BO was compared with the random search optimization approach in terms of prediction performance to further evaluate the efectiveness and robustness of BO for parameter tuning. Te random search algorithm is commonly adopted to tune the hyperparameters of data-driven machine-learning models [58,68]. For a fair comparison, a random search optimization technique was implemented on the same search space and datasets for the same number of iterations (i.e., 30) as those used for BO. Figures 11(a)-11(d) illustrate the performance of the optimized ANN model via BO and the random search algorithm, respectively, with critical infuencing factors S 4 and S 5 for the horizontal displacement modeling of bearings B 1 -B 4 . As shown in Figures 11(a)-11(d), for the overall comparative computational cost (time in seconds), the ANN model optimized with BO exhibited signifcant superiority over the random search in terms of prediction accuracy for the testing dataset (RMSE test ) as our major concern. Tis can be Input variable sets   justifed as the major drawback of the random search algorithm is that it ignores prior information regarding hyperparameter combinations in each iteration, which causes the risk of losing the optimal hyperparameters [58].

Model Assessment and Results Visualization.
Te results of the model application phase, in which the prediction performance of the optimized ANN model determined using BO was assessed and visualized, are presented in this section. Te feasibility of the critical infuencing factors S 4 and S 5 which efectively models the external environmental efects (as discussed in Section 5.1) on the horizontal displacement of bearings B 1 -B 4 is compared with the infuencing factors S 1 and S 2 in terms of assessment metrics. Note that S 1 simulates the air temperature efect only, whereas S 2 models the air temperature while considering the thermal inertia in addition to seasonal solar radiation efects on the horizontal displacement. Figure 12 presents the assessment metrics of the testing dataset with the optimal ANN models for bearings B 1 -B 4 . As shown in Figures 12(a)   As an aspect of the result visualization, Figures 13-14 illustrate the prediction and amplitude of the prediction residual with critical infuencing factors S 5 compared to S 1 for the horizontal displacement of bearings B 1 on Pier#1 and B 3 on Pier#27, respectively. Note that bearing B 3 has the largest prediction error among the considered bearings. Te optimized ANN-S 5 model, in contrast to the ANN-S 1 model, results in predictions that closely ft the measured horizontal displacement specifcally at the daily extreme values and hence shows a small amplitude variation of the prediction error. Furthermore, when examining the overall prediction accuracy of the bearings' daily extreme (minimum and   Table 5. Based on the aforementioned discussion regarding the model performance assessment and result visualization, the optimized ANN model led to an accurate and precise horizontal displacement prediction model for a robust and reliable early warning system for bridge structures. Tis can be attributed to the fact that the adopted critical infuencing factors efciently model the surrounding environmental efects (D n ) by capturing the relationship between the current and previous displacements, in addition to the air temperature (T 0 ) with thermal inertia (T p-q ) and seasonal solar radiation (d) efects on the long-term horizontal displacement of bridge bearings.

Performance Comparison with Regression Models.
Te prediction performance of the optimized ANN model was compared with that of multiple linear regression (MLR) and non-linear regression (MNLR) models. MLR and MNLR are the traditional statistical modeling approaches for linear and non-linear regression analyses, respectively, and are commonly used to predict the TIR, such as strain, displacement, and tilt of bridge structures [12,21,[29][30][31]40]. Te MNLR model is a polynomial with a degree of measured temperature greater than one when compared to the MLR model [12,21], and the temperature regression coefcients to predict the TIR with the MLR and MNLR models are computed based on the leastsquare method.
For a comparative study, the MLR and MLNR models were trained and evaluated using assessment metrics over the same training and testing periods as for the ANN model.  Figure 15 illustrates the performance comparison between the MLR, MNLR, and optimized ANN models with input variable sets S 1 -S 8 for the horizontal displacement modeling of bearings B 1 -B 4 . Note that the temperature terms with a polynomial of degree two are adopted together with the infuencing factors in the input variable sets S 1 -S 8 (as listed in Table 2   16 Structural Control and Health Monitoring d}. In addition, when temperature terms greater than polynomial degree two are adopted, it can result in a singular modeling error as the temperature regression coefcients cannot be estimated correctly [21]. As shown in Figures 15(a)  TIR. Tus, it is challenging to predict the long-term horizontal displacement of bridge bearings accurately using the MLR and MNLR models. A direct performance comparison of the test dataset in terms of the assessment metrics was also performed between the MLR, MNLR, and optimized ANN models with the critical infuencing factors S 4 and S 5 for selected bearings B 1 -B 4 as shown in Figures 16(a)-16(d). Te optimized ANN-S 4 and ANN-S 5 models resulted in lower prediction errors (RMSE, MAE, and AIC) and a better goodness of ft (R 2 ). For illustration purposes, the horizontal displacement prediction and amplitude of residuals by the MLR-S 5 , MNLR-S 5, and optimal ANN-S 5 models for bearings B 1   18 Structural Control and Health Monitoring optimal ANN-S 5 model illustrates a low prediction error, which resulted in a prediction that closely fts the measured horizontal displacement. To further confrm this, Table 6 illustrates the overall prediction accuracy in terms of RMSE of the prediction errors of daily extreme horizontal displacement of the bridge bearings shown in Figures 17-18. Te optimal ANN-S 5 model resulted in lower RMSE values for the daily extreme prediction errors of the bearings' horizontal displacement. Hence, the optimal ANN model, in contrast to the MLR and MNLR models, efciently modeled the linear and nonlinear relationships between the critical infuencing factors and long-term horizontal displacement while dealing with high-dimensional data-mapping problems. Terefore, the optimized ANN model with the adopted critical infuencing factors provides an accurate and robust prediction model to support reliable early warning systems for bridge structures.

. Conclusion
Tis study proposes a modeling and prediction method for long-term horizontal displacement attributed to gradual temperature variations in a bridge structure, which is considered an essential aspect of the TBDI approach for anomaly detection in SHM. In contrast to previous studies, the proposed method employs critical infuencing factors to efciently model and predict the long-term horizontal displacement, thereby exploring the potential sources of heat exchange between the bridge structure and the external surrounding environment. A hybrid model based on an ANN with BO was utilized to determine the relationship between the critical infuencing factors and long-term horizontal displacement. Te viability of the presented methodology was demonstrated using long-term air temperature and bearing horizontal displacement data collected from an overpass bridge in Seoul, South Korea. Te assessment metrics in terms of RMSE, MAE, AIC, and R 2 were used to analyze the prediction performance of the hybrid model with the critical infuencing factors. Te key fndings of this study can be summarized as follows.
(1) Te long-term horizontal displacement can be modeled accurately with the proposed comprehensive consideration of the surrounding environmental conditions by capturing the relationship between the current and past displacement information including the air temperature with thermal inertia and seasonal solar radiation efects (2) BO is a computationally efcient method for ANN hyperparameter tuning, yielding robust optimal parameters that enhanced the ANN prediction performance for long-term horizontal displacement (3) Te optimized ANN model with the adopted critical infuencing factors that efectively refected the external environmental efects in combination resulted in a small prediction error, and generated an accurate, precise, and robust horizontal displacement prediction model (4) Compared with the MLR and MNLR models, which are traditional statistical modeling approaches for the TIR of bridge structures, the optimized ANN model can efciently simulate the linear and nonlinear relationships between the adopted critical infuencing factors of external environmental conditions and long-term horizontal displacement, and thus exhibits the better prediction performance In future research, the proposed method for accurate modeling and prediction of bearing long-term horizontal displacement will be investigated by integrating it with the TBDI approach for anomaly detection, which will support the implementation of a reliable early warning decisionmaking system for bridge structures. Furthermore, the ANN model developed with the proposed method represents a potential approach to accurately predict the bearing longterm horizontal displacement and generally can be applicable to any structural bridge and bearing types. Te reason can be attributed to the capability of the developed ANN model to efectively learn and simulate the linear and complex non-linear correlations between the critical infuencing factors and bearing horizontal displacement. Hence, the implementation of the proposed method will be explored on the long-term monitoring data from the other bridge structures in the future study.

Data Availability
Some or all data, models, or code generated or used during the study are proprietary or confdential in nature and may only be provided with restrictions.

Conflicts of Interest
Te authors declare that they have no conficts of interest.