Bridge Performance Prediction Based on a Novel SHM-Data Assimilation Approach considering Cyclicity

Modern bridges are monitored by an increasing network of sensors that produce massive data for bridge performance prediction. Reasonably and dynamically predicting with monitored data for the time-variant reliability of the existing bridges has become one of the urgent problems in structural health monitoring (SHM). Tis study, taking the dynamic measure of structural stress over time as a time series, proposes a data assimilation approach to predicting reliability based on extreme stress data with cyclicity. To this aim, the objectives ofthisarticlearetopresentthefollowing:(a)a Gaussianmixturemodel-basedBayesiancyclicaldynamiclinearmodel(GMM-BCDLM) based on extreme stress data with cyclicity and (b) a dynamic reliability prediction method in the combination of GMM-BCDLM and SHM data via frst-order second-moment (FOSM) method. An in-service bridge for providing real-time monitored stress data is applied to illustrate the application and feasibility of the proposed method. Ten, the efectiveness and prediction precision of the proposed models are proved to be superior compared to other prediction approaches to extreme stress data with cyclicity.


Introduction
In the past two decades, due to the continuous increase in trafc loads and the uncertainties in both mechanical and environmental conditions, health monitoring concepts for bridge systems have become increasingly necessary for the preventive maintenance of new and existing bridges [1][2][3].Nowadays, although the monitoring technology has developed for a long time, the performance (e.g., stress, defection, and reliability) prediction model, based on the data collected by the structure health monitoring (SHM) systems, is still an urgent demand in the current engineering practice, which could assist in the intervention planning (e.g., maintenance, replacement, and rehabilitation) and the costoptimized aspects on bridges [4][5][6][7][8][9].It is not easy to efciently utilize monitoring data (especially with cyclical trends) for performance prediction.Accordingly, the timevariant SHM data-based dynamic performance prediction, especially involving the monitored cyclical data, is still a grand challenge of the SHM feld.
In recent years, diverse forecasting methods have been developed for predicting the dynamic and static structural responses of bridges, which can generally be categorized into model-based methods and data-driven methods [10].Te model-based models often necessitate a prior knowledge of structural features and emphasize the relationship between external excitation and structural responses [11].For example, Wu et al. [12] proposed an efective method for dynamically predicting long-gauge strains in high-speed railway bridges using fber Bragg grating (FBG) sensors, based on the fnite element approach.Tis method was validated through numerical simulations, experimental comparisons, and feld tests, ofering engineers a valuable tool for monitoring the health conditions of high-speed railway bridges.However, the applicability of modelbased methods may be limited by the complexity of the mechanical environment and uncertain material properties [13].Furthermore, due to the complex external environmental factors during operation, the changes in structural parameters may not directly refect potential structural damage, therefore, afecting the accuracy and reliability of its real-time application [14].
Data-driven models have gained popularity in recent years, and they establish numerical forecasting models based on feld measurement data instead of FE models [15][16][17].Based on data mining, a novel time series prediction model with a combination of Kalman flter and autoregressive integrated moving average-generalized autoregressive conditional heteroskedasticity (ARIMA-GARCH) [18] improves the prediction accuracy of bridge structure deformation, which does not meet requirements for predicting nonstationary performance information with cyclical trends.For early warning, a data-driven method based on an improved variational mode decomposition and conditional kernel density estimation to predict the monitoring deformation data avoids exploring the complex internal mechanism of structural behavior evolution [19].Based on monitoring noncyclical data, bridge structural reliability indices are predicted via a data assimilation method and Bayesian method [20], which is only applicable to nonstationary data without cyclic properties.Besides, another data assimilation method [21] is used to predict longperiodic monitoring information for an in-service bridge.However, the method relies heavily on fxed Fourier function equations, which result in fxed amplitudes for the periods and make it difcult to capture local variations in the real-time monitoring data.As a result, the accuracy of predictions for periodic monitoring data is compromised.Defection data are also used to provide important information about the structural health condition of bridges with the long short-term memory framework [22], but less attention is paid to the periodicity of the data.In addition, several machine learning methods [23][24][25][26][27][28] have also been applied to the prediction and early warning based on bridge monitoring data.For example, based on the pavement temperature data of Sutong Bridge, a prediction model based on the long short-term memory neural network was proposed for the prediction and early warning of the bridge state [23].Te problem with these methods is that they need high computational requirements for model training, and if the data used to train a model are biased or unrepresentative, the model will likely exhibit the same biases.Tese methods mentioned above have a series of limitations and do not provide practical methods for reliability prediction based on cyclical data related to existing monitoring systems.Terefore, fnding an algorithm for efectively and dynamically predicting bridge reliability based on the shortperiodic monitored information for the daily monitoring of extreme stresses, which could assist in daily bridge maintenance and early warning, is the topic of this study.
Based on the above issues, the following problems need to be solved: (1) the dynamic prediction models based on the cyclical monitored extreme stress data need to be established and (2) the probability recursive processes of dynamic prediction models based on cyclical data need to be achieved.
To predict the time-variant reliability indices of an existing bridge based on the monitoring cyclical extreme stress data, we propose a Gaussian mixture model-based Bayesian cyclical dynamic linear model (GMM-BCDLM) that incorporates initial information and corresponding probability recursion processes.Te frst step involves establishing a GMM-CDLM model based on initial information.Next, we use the predicted results to update and correct the parameters of the GMM-CDLM model with the Bayesian method during the dynamic prediction of extreme stress.Finally, we use the predicted extreme stress and the frst-order second-moment (FOSM) method to dynamically predict the main beam's reliability.Te fow plan is shown in Figure 1, and this study is organized as follows: Section 2 proposes the monitored equation and the state equation of CDLM based on extreme stress data with cyclicity.In Section 3, we provide the probabilistic recursion processes of the established CDLM with the Bayesian method based on cyclical data, and then, we provide the recursive processes of the GMM-BCDLM.Section 4 introduces the FOSM method and combines it with the GMM-BCDLM method.Section 5 discusses the application of this method to an existing bridge for performance prediction and compares the efectiveness of the proposed GMM-BCDLM with other methods for monitoring extreme stress data with cyclicity from an actual bridge.Finally, Section 6 concludes this contribution and identifes issues for further examination.

The Cyclical Dynamic Linear Model
Te cyclical dynamic linear model (CDLM), including a linear cyclical state equation, a monitored equation, and the initial state information, meets the basic assumptions for state-space models [29].It primarily includes the following assumptions: (1) A linear approximation of the system is acceptable, and the complete system state can be inferred from the outputs (2) Te process and measurement errors are Gaussiandistributed white noises and are independent and uncorrelated (3) Te state variable θ t   is a Markov chain, namely, Te linear cyclical state equation shows changes in the system with time and refects the level of the monitored cyclical extreme stresses and random disturbances.Te linear monitoring equation reveals the connection between the monitored extreme stresses and the current state parameters of the system.Moreover, based on the historical monitored extreme stress data, the initial state information can be approximately obtained using the cubical smoothing algorithm with a fve-point approximation [30].In this study, an in-service steel bridge is adopted as the research 2 Structural Control and Health Monitoring object, of which the monitored extreme stress is defned as the monitored maximum stress during each hour.
If the monitored extreme stress data, regarded as a time series, show similarity after p time intervals, such as the peak or trough state at the same time, the series is said to be a cyclical time series with a period of p. Tere is a general time series set D t with period p, which includes the data at and before time t.Te state set, also taken as a time series approximately obtained with a cubical smoothing algorithm with fve-point approximation through resampling D t , is denoted by D θ t , the ith cyclical diferenced data set of which is D θ t,j .In general, for the nonstationary general time series set D t with (n − 1)-order cyclical trends, the diference can be carried out until the time series after the diference is stationary, and the number (n − 1) of diference is the order of cyclical trends.Besides, the augmented Dickey-Fuller (ADF) test method [31] can also help ascertain the stationarity of the data before and after cyclical diferences.Let y t be the monitored extreme stress at time t, the cyclical differences about y t are where Te cyclical factor θ j , (j � 0, 1, • • • , p − 1) refers to taking p-state values within a cycle.Te cyclical vector at time t is the vector that is arranged by cyclical factors in proper order, of which the frst element is the cyclical factor at time t, namely, In particular, for arbitrary natural numbers n and k � np, there is where I is a (p − 1) order identity matrix and O means all elements here are zero.Obviously, θ t � Pθ t− 1 .Also, at time t − 2, the cyclicalfactor vector is Generally, the cyclical-factor vector at time t − h can be written as

CDLM Based on Cyclical Extreme Stress Data.
Considering the monitored error v t about y t conditional on the state (or level), transition error from state to monitoring, and the basic assumptions for state-space models [29], assuming that the monitoring noise and process noise are mutually independent and follow a normal distribution, the CDLM is defned as follows.
Monitored equation: State equation: Structural Control and Health Monitoring Initial information: where y t is the monitored data at time t; T are, respectively, the monitored white noise and the state cyclical vector of white noises; V is the monitored error variance; P is a p × p state transfer matrix, namely, equation ( 4); W is state error variance estimated by smoothed data; D t− 1 is the information set at time t − 1 and before time t − 1 [32]; and According to the information provided by the sensor manufacturer, the monitoring error v t for this type of sensor (pressure transducer) is within 3%.Since it is difcult to precisely determine this error in practical work, for ease of computation, we assume it to be white noise following a normal distribution.Given that the monitoring noise is relatively small, this assumption can approximate the sensor's monitoring noise to a certain extent, providing the model with the uncertainty of monitoring.Furthermore, compared to the monitoring error within 3%, the error introduced by this assumption for monitoring noise can be considered a completely negligible secondary error.And state noise ω t,i is a stochastic variable introduced in the dynamic model of the system.Its main purpose is to capture uncertainty and random disturbances in our model.State noise is typically assumed to be a zero-mean Gaussian random process with a specifc covariance.Tis means that over the long term, the infuence of these noises averages out to zero, but in the short term, they can impact the system state.Tis aligns with the stress characteristics observed in the monitoring of in-service bridges.Under normal conditions over longer periods (e.g., months or years), the monitored stresses should be stable (with a noise mean of 0).If there are signifcant fuctuations or trends, it indicates changes in the structural or mechanical environment, requiring further inspection or maintenance.
According to reference [32], W t can be regarded as a fxed proportion of C t .Tis perspective naturally informs our understanding of system variance.Specifcally, when an error, is introduced between observations, this leads to an additive increase in W t from the initial uncertainty C t .It is both convenient and intuitive to adopt a constant rate of uncertainty increase, or information decay, for all time points t.Tus, for a given discount factor δ, choose where δ is the discount factor, which is usually 0.48-0.98 by the author's experience [33].

Main Probability Parameters of CDLM.
Te prior probability parameters of CDLM include V, W t− 1 , m t− 1 , and C t− 1 .Te determination approaches are as follows.
Te relationship between the cyclical factors is only affected by the covariance term of the initial prior and the infuence of this initial prior decay with time [33].Accordingly, for simplicity, suppose where m 0 � m 0 I and C 0 � C 0 I are the initial information, I is a p-order (p is the period, in this paper p � 24) identity matrix, m 0 is the mean value of the initial data, and C 0 is the variance of the initial data, and the initial data are obtained from historical data through the cubical smoothing algorithm with a fve-point approximation [30].Additionally, V is the monitored error variance.V � 1/n − 1 n i�1 (y i − y) 2 and n is the number of historical extreme stress data.Te reason for using n − 1 instead of n is that we are calculating the sample variance rather than the population variance.

Bayesian Recursive Processes of GMM-CDLM
Based on the established CDLM shown in equations ( 7)-( 9), with the Bayesian method [34,35], the probability recursion processes and simplifed Bayesian recursion processes of CDLM can be reached.

Probability Recursion Processes of CDLM.
To calculate the prior probability, we need to express the joint probability density and then expand it based on conditional probability: According to the frst-order Markov property, we obtain By further integration to θ t , we can obtain the marginal probability distribution, i.e., the prior probability distribution: Ten, to derive the posterior probability distribution, we can expand the joint probability density based on the conditional probability equation (12).And according to the conditional independence, we have (16)

Simplifed Recursion Processes of CDLM.
Referring to [34,35], Bayesian probabilistic recursion of CDLM means that dynamic recursive processes about CDLM are achieved with the Bayesian method.Also, Bayesian probabilistic recursion is good at inferring and predicting past and future parameters [36].
Based on the aforementioned recursive relationship, under the criterion of minimum mean-square error, the optimal estimate of the value θ t is the frst moment of the posterior probability density function.Te estimate at time t can be written as follows: Furthermore, it can be expressed further using equation (8).By applying the method of point estimation [36] and assuming the monitored variance is known, we can simplify the recursive relationship as follows.
Assume time t � np, and the current moment is written as M(0).
(1) Te prior PDF about the state at time t can be obtained with where and W t � − C t− p,0 + C t− p,0 /δ are calculated and updated with equation (10), and in this study, δ � 0.8.(2) Te one-step forward prediction PDF at time t can be solved with where f t � Fm t � m t− p,0 is the predicted mean value, F � (1, 0, • • • , 0) T is a p-dimensional state transfer vector, and Q t � R t + V is the predicted variance.According to the defnition of the highest posterior density (HPD) region [37], the predicted interval of the monitored data m t with a 95% confdential interval at time t is (3) With the Bayesian method, the posterior PDF about the state at time t can be calculated with where m t,0 � m t− p,0 + A t e t , C t,0 � A t , A t � R t / (R t + 1), and e t � y t − f t .e t is the one-step forward predicted error and y t is the monitored data at time t.
Similar equations can be obtained for M(1), M(2), . .., and so on, by simply replacing the subscript 0 with the corresponding subscripts 1, 2, 3, and so forth.Te fowchart is shown in Figure 2:

Gaussian Mixed Model-Based Bayesian Recursive
Processes.In practical engineering, the moving loads acting on a bridge and the complexity of the stress environment may cause stress responses to deviate from a normal distribution.In such cases, using only the previously proposed BCDLM may lead to a decrease in prediction accuracy.Considering that Gaussian mixture models (GMMs) can be used to ft and approximate various types of data distributions [38], we have decided to enhance the BCDLM by incorporating a Gaussian mixture model.Tis improvement allows for a more fexible representation of the input information and improves the modeling accuracy.
For the Gaussian mixed model assume the monitored variance V is known.M t (α) can be written as follows: where p t (i) represents the weight of the subdistribution of the mixture Gaussian distribution and N(•) represents the Gaussian distribution.
Monitored equation: State equation: Initial information: where y t is the monitored data at time t; T is the state cyclical vector at time t;

Structural Control and Health Monitoring
Te corresponding recursive processes are as follows: (1) Assume for each , there are Among them, the mean and variance of each distribution can be recursed and corrected by the model where the weight can be calculated with Te initial probability p 0 (i) can be given by the EM algorithm [39] and historical data, and the posterior model probability can be continuously corrected.
(3) Te prediction PDF at time t can be solved with Te predicted mean value at time t is

Components Number Choosing of Gaussian Mixed Model.
For a GMM model, if we set the number of components too low, the model may not have enough fexibility to capture the structure of the data, leading to underftting.On the other hand, if we set the number of Gaussian components too high, the model may become too fexible and lead to overftting.To strike a balance between underftting and overftting, this study employs the Bayesian information criterion (BIC) [40] to determine the optimal number of components in the GMM.Te BIC is defned as follows: where L is the likelihood of the data given the model.Tis represents how well the model fts the data.Higher likelihoods correspond to better fts, so we want to maximize L. k is the number of parameters in the model.Tis represents the complexity of the model.Models with more parameters are considered more complex.n is the number of data points.ln(n) is the natural logarithm of the number of data points, which scales the penalty for the number of parameters.Ten, by minimizing the BIC, the number of components would be calculated.

Evaluation of Prediction Precision.
In this study, the mean square error (MSE) and normalized root mean square deviation (NRMSD) are adopted to measure the prediction precision.Te smaller the MSE and NRMSD, the better the corresponding model.Te formulas are as follows: where N is the total number of the predicted data; f t is the predicted mean value at time t; y t is the monitored data at time ts; and y max and y min are the maximum and minimum monitored data, respectively.

Reliability Prediction Based on GMM-BCDLM and FOSM
4.1.FOSM Method.From a safety perspective, within the design service life of civil infrastructure such as bridge structures, the bearing capacity must be greater than the efects caused by various loads acting on it, which the following formula can express as follows: The posteriori PDF of the state vector at time M (t) The priori PDF about the state vector at time M (t+1) One-step forward prediction PDF

Updated
The posteriori PDF of the state vector at time M (t+1) Real-time monitoring data 6 Structural Control and Health Monitoring R(X) > S(X), (33) where R(X) represents the structure reactance; S(X) is the load efect (including dead load efect and live load efect) caused by both mechanics and environmental conditions on the structure; and X is various random variables that afect the structure reactance.Suppose that there are internally independent and mutually independent random variables R(X) and S(X), and the average and standard deviation are μ A , σ R , μ S , and σ S , respectively.
For the linear limit state functions, the general function of structural performance is According to the central limit theorem, as samples increase, the random variable Z(X) will approach a normal probability distribution.With the FOSM approach [41], the formula of the reliability index can be obtained as

Dynamic Reliability Calculation Formula
Based on FOSM and GMM-BCDLM.For this study, the Ningbo Bund bridge, a cable-stayed bridge with a total length of 337 m [42], is chosen as an example.It is important to note that the stress levels at the consolidation part of the tower and beam are signifcantly higher than in other sections of the structure, requiring special attention.In practical engineering applications, the calculation precision of the FOSM method is deemed sufcient.Under both dead load and live load conditions, the critical section of the main girder experiences static bending failures in both positive and negative bending moments.Te sensors of the SHM system for the Ningbo Bund bridge were installed after the bridge was completed, and the monitoring results do not account for the load efect caused by the dead load component.Terefore, when ignoring the infuence of geometric parameters, the structural performance function for this steel bridge can be expressed as where R represents the steel yield strength which is 345 MPa for this bridge, and the standard deviation is 27.6 MPa; S M is the extreme stresses monitored or predicted with time; and c M � 1.15 is a factor assigned to the data provided by sensors [32].With equations ( 19) and ( 34), the predicted dynamic reliability index is where (μ R , σ R ) is, respectively, the mean and standard deviation of R; (μ M , σ M ) is, respectively, the mean and standard deviation of S M ; and c M is a factor assigned to the data provided by sensors.Especially, when equation ( 37) is used to calculate the reliability indices based on monitored extreme stress data, σ M is the standard deviation of the monitoring information.

Application to an Existing Bridge
5.1.Bridge Introduction.Te Ningbo Bund Bridge, situated in downtown Ningbo, China, is depicted in Figure 3.It is a singlepylon four-plane special-type cable-stayed bridge with a main span of 225 m, primarily composed of a separated steel-box girder [42].Te bridge is predominantly made of steel, with an allowable stress of 345 MPa, and its support conditions are specifed in reference [42].Te health monitoring system is connected to an of-site data acquisition station and utilizes a fber-optic network provided by a telecommunications company to transmit data to the Urban Bridge Monitoring and Management Center in Ningbo City for storage and management.Subsequently, the management center conducts data preprocessing to enhance data quality.Te stress measurement points of this bridge are arranged with two cross sections, respectively, the main beam section at the consolidation part of the tower and beam (measurement points STR1-16) and the root section at the front tower column of the cable tower (measurement points STR17-24).Te sensors used in this study are pressure transducers, which convert pressure signals into electrical signals for transmission and processing.Tese pressure sensors are equipped with temperature compensation functionality and are installed and calibrated by the supplier during bridge construction.Tis allows for efective calibration of the sensor readings and reduces the impact of temperature changes on the measurement accuracy of the sensors.
Te section at the consolidation part of the tower and beam is critical in controlling the safety-based structural performance and is monitored by the stress sensors for the longitudinal stress information.Te pressure transmitters are mounted on the crossbeam reinforcement rib inside the steelbox girder using bolt fasteners.Tese pressure transmitters are equipped with temperature compensation functionality and have been precalibrated by the manufacturer during installation.Tis calibration capability enables the sensors to correct monitoring readings and minimize the impact of temperature fuctuations on measurement accuracy.Te layout of measurement points in this section is shown in Figure 4.It can be seen that these sensors, STR1-16, are located on the same steel beam and are connected in series.As the failure of any monitoring section leads to the collapse of the entire girder, the failure modes of the monitoring sections of the bridge girder are considered a series structure.Tis steel box girder represents an n-dimensional series system with n correlated failure modes.Due to their almost identical Structural Control and Health Monitoring mechanical environment, they have a strong correlation.For the convenience of calculation, it is assumed that they are fully correlated, and the failure probability p f of the main beam system can be simplifed as follows: where Z i,t (i � 1, 2, . .., n) represents the performance function calculated with equation (36) and P(Z i,t ≤ 0) � p f i is the failure probability of sections at the location i � 1, 2, • • • , n.Among these monitoring points, point STR1 stands out as the most critical point and is, therefore, chosen as the focal point of our research.Te average stress levels monitored at this point are notably higher compared to other points.Based on equation (38), the failure probability at monitoring point STR1 is found to be the highest, making its reliability indices indicative of the overall reliability of the main beam.Te hourly monitored extreme stress data collected by the sensor STR1, shown in Figure 5, are utilized to establish the CDLM for Bayesian dynamically predicting the extreme hourly stress at point STR1.
Bridge monitoring information generally exhibits dynamic characteristics, randomness, and periodicity due to the efects of periodic temperature load and peak trafc fow.Based on the real-time monitoring information of the STR1 measurement point of this cable-stayed bridge, the CDLM is established.In May 2020, the point STR1 was dynamically monitored with a 1 Hz sampling frequency of sampling rate, and the extreme hourly stress was utilized to establish the CDLM for Bayesian dynamically predicting the extreme hourly stress at the dangerous point STR1 of the main beam section.While a higher sampling frequency can provide more accurate measurements of instantaneous stress changes, in this study, we focus on the extreme stress values per hour, which can be considered as long-term statistical data relative to 1 Hz sampling data.In this case, collecting a sufcient number of data samples can compensate to some extent for the lower sampling frequency's limitations in capturing instantaneous responses and provide an accurate estimation of extreme stress values per hour.

Predictive Results of GMM-BCDLM.
Our objective is to develop a model that accurately captures the structural response characteristics and exhibits a certain level of generalization to ensure efciency in real-time applications.By utilizing only 240 hours of data for model validation, we can further demonstrate the efcient utilization of response information by the model and the high computational effciency of the model itself.
To dynamically predict the reliability of the main beam section based on monitored extreme stresses, we divided the detailed procedures into three steps: (a) hourly monitored extreme stresses of the former 240 h for sensor STR1 are used to establish GMM-CDLM; (b) the mean value of extreme hourly stresses from the 241st hour to 360th hour is predicted with equations ( 22)- (29); and (c) with equations ( 35)-( 38) and the prediction results, the prediction reliability indices of the main beam are computed.
With (1), the frst-order cyclical diference about y t can be obtained (see Figure 6), which is stationary after the augmented Dickey-Fuller (ADF) test [31].
Trough the cubical smoothing algorithm, hourly extreme stress data of the former 240 h are approximately resampled into initial state data (see Figure 7).Ten, the established cyclical dynamic linear model, with equations (2)-( 9) and ( 24)-( 26), can be obtained as follows.
Te initial state information can be obtained using a cubical smoothing algorithm with fve-point approximation [30] applied to the previous 240 h extreme stress data, and initial parameters are then determined with equations ( 22) and ( 25) and Section 2.3.Finally, the density estimation p 0 (i) is obtained via the expectation-maximization algorithm [39] and initial state data.
Ten, with equations ( 7)-( 29) and updating, the predicted hourly extreme stress mean value about the hourly monitoring extreme stresses is computed and shown in Figure 8, which indicates that the predicted results ft the changing rules of monitored extreme stress data well and remain within the 95% confdence interval.
When establishing a predictive model based on normal historical data, the expected prediction will also refect the structural behavior trends of a bridge's normal operation.By using prediction intervals, the range of potential trends can be further determined.If the monitoring data deviate from this expected trend, particularly beyond a certain confdence interval (typically 95% or 99.7%), it indicates the possibility of sensor or even structural anomalies, necessitating further inspection.Structural Control and Health Monitoring

Analysis of Prediction Results.
In Figure 9, we show diferent prediction results of four existing common approaches for a fair comparison, i.e., GMM-BCDLM, Bayesian Fourier dynamic linear model (BFDLM) [32], seasonal autoregressive and moving average model (SARMA) [43], and long short-term memory (LSTM) [22] to forecast extreme stress data with cyclicity.It indicated the superiority and advantages of the proposed method for monitored data with cyclicity.
To further demonstrate the efectiveness of the proposed model in this paper, two additional sensors, STR2 and STR10, were selected as research objects.Te GMM-CDLM model was established to model and predict their behavior and compared with other commonly used methods.Te predicted results are shown in Figures 10 and 11.
With equations ( 31)-( 32), the computed MSE and NRMSD results for GMM-BCDLM, BFDLM, SARMA, and LSTM are shown in Tables 1 and 2, respectively.Among these, the proposed GMM-BCDLM model yields the smallest MSE and NRMSD, thereby indicating superior predictive performance.Tis further underscores the enhanced accuracy of our proposed model in predicting cyclical bridge structural monitoring information.
It can be observed that in Figure 10, during the period from 280 hours to 300 hours, two measured data points lie far away from the predicted curve.We have included these points in Figure 12 and added a 95% confdence prediction interval.It is important to understand that a 95% confdence prediction interval means that there is a 95% probability of the actual values falling within this interval.Terefore, it is    Structural Control and Health Monitoring still consistent with our expected outcome, considering that only a few data points out of 120 deviate from the predicted interval.
Te choice of a 95% confdence interval is primarily for analytical convenience.In practical engineering applications, higher confdence levels are often employed to determine warning thresholds.A commonly used choice is a 99.7% confdence interval, corresponding to the 3-sigma criterion.By adjusting the confdence level, it is possible to determine reasonable warning thresholds based on engineering considerations.
Here are the prediction interval results with a 99.7% confdence level (Figure 13).It can be observed that as the confdence level increases, the possibility of false alarms is   Structural Control and Health Monitoring reliability of the entire main beam.Comparing the predicted reliability indices with the statistical monitoring values at a 95% confdence level [37], the predicted results, as shown in Figure 15, are in good agreement with the monitored reliability indices.Te predicted reliability indices are all above 5.2, indicating the safety of the main beam.

Conclusions
Tis article proposes a new prediction approach, a Gaussian mixture model-based Bayesian cyclical dynamic linear prediction method, to dynamically predict the performance of steel bridges based on extreme stresses with cyclicity.GMM-CDLM is frst built in the analysis processes, and the corresponding probability recursion method is given in detail with the Bayesian approach.Te monitoring data of the steel bridge are provided to illustrate the efectiveness of the proposed method.Te prediction results illustrate that the proposed method can make dynamic predictions based on monitored data with cyclicity and has excellent prediction precision.Monitored data with cyclicity refer to the type of data collected from the monitoring sensors that exhibit periodic or cyclic patterns over time.In the context of our study, it refers to the monitored data from the bridge's sensors, which demonstrate repetitive variations over a specifc time interval, associated with recurring factors of daily cycles.Usually, the simplest way to identify cyclicity is by plotting the data and visually identifying any apparent patterns, as seen in the monitored data of this study.If there is a need to satisfy this condition mathematically, statistical analysis (autocorrelation analysis to quantitatively assess the presence of periodic patterns) or spectral analysis (like Fourier analysis) methods are commonly employed.
However, some limitations should be noted.Although our method has been validated through critical components of actual bridges, the correlation between sensors has not been studied quantitatively.Future work should include follow-up studies to consider the dependency between sensors.

Figure 1 :
Figure 1: Te fow plan of the dynamic reliability prediction method.

Figure 3 :
Figure 3: A view of Ningbo Bund bridge.

Figure 5 :
Figure 5: Te sensor layout at the critical section.

Figure 6 :
Figure 6: First-order cyclical diference of monitored extreme stress data.

Figure 7 :
Figure 7: Monitored extreme stresses and initial state data of STR1.

Figure 8 :
Figure 8: Monitored data, predicted results, and range of hourly extreme stress of STR1.

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monitored stress data Prediction of GMM-BCDLM Prediction of BFDLM Prediction of SARMA Prediction of LSTM