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Modal frequency is an important indicator reflecting the health status of a structure. Numerous investigations have shown that its fluctuations are related to the changing environmental factors. Thus, modelling the modal frequency– multiple environmental factors relation is essential for making reliable inference in structural health monitoring. In this study, the Bayesian network (BN)-based algorithm is developed for recognizing the pattern between modal frequency and multiple environmental factors. Different candidates of network structure of the BN are proposed to describe the possible statistical relations of different variables. In the BN-based pattern recognition, the learning phase conducts uncertainty quantification in both parameter and model levels; and the prediction phase makes inference under complete and incomplete observed information. Based on the long-term monitoring data, the most plausible network structure is selected, and its associated parameters are identified. The developed algorithm is then utilized for analyzing the long-term monitoring data (modal frequencies, temperature, humidity, wind speed and traffic volume) of the Xinguang Bridge (a 782-m three-span half-through arch bridge). It turns out that the selected network structure properly captures the pattern of modal frequency–multiple environmental factors. Keywords Bayesian network, environmental factor, model class selection, structural health monitoring Introduction Structural health monitoring is to infer structural health status given measurements of structural responses and environmental conditions, which has been developed and utilized for various types of infrastructures such as 2–8 9–16 bridge, building structure, etc. The modal frequency is widely adopted as a global health status indicator in 17,18 SHM. As monitored infrastructures are exposed to environmental conditions, evidence can be observed that the fluctuations of modal frequencies are related to the changing environmental factors such as the temperature, relative humidity, wind speed and traffic loading. Direct ignorance of the environmental effects in SHM possibly leads to misleading inference. Therefore, considerable efforts have been devoted to depicting the pattern between modal parameters and environmental factors. The modal frequency–multiple environmental factors pattern recognition contains the training/learning phase and prediction phase. Based on training SHM data School of Civil Engineering and Transportation, State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou, P.R. China Corresponding author: Cheng Su, School of Civil Engineering and Transportation, State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou 510640, P.R. China. Email: cvchsu@scut.edu.cn; cthqmu@scut.edu.cn Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www. creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 546 Journal of Low Frequency Noise, Vibration and Active Control 39(3) (modal frequencies and environmental factors), the training/learning phase is to select the model class and identify the parameters of modal frequency–multiple environmental factors relation. Once the model is trained, it can be utilized to predict the output data (modal frequencies) based on the given input data (multiple environmental factors) in the prediction phase. In practical SHM applications, the key is how to make accurate and robust predictions when the training input data only comprise a tiny fraction of all possible input data, which is known as generalization in pattern recognition. Cornwell et al. developed a linear model for daily temperature and modal frequencies, and they discovered that the first frequency of the Alamosa Canyon Bridge varied by 4.7% over the 24-h period. Peeters and Roeck utilized the ARX model to capture the relationship between temperatures (air temperature and stuctural temperatures) and modal frequencies by one-year monitoring data of the Z24-Bridge. Xia et al. modelled environmental conditions-modal parameters with a linear regression model by nearly two- year monitoring of a reinforced concrete slab outside the laboratory. It was found that the modal frequencies and damping ratios have strong correlation with temperature and humidity. Hua et al. modelled temperature–fre- quency with combined principal analysis and support vector regression technique. Liu et al. used the linear regression models to evaluate how the temperature variations influence the modal parameters. Zhou et al., Ni 25 26 et al. and Li et al. used the neural network to determine the correlation between the environmental conditions 27,28 and the modal parameters. Yuen and Kuok inspected a set of model class candidates based on Bayesian probabilistic approach. Mu et al. developed a pattern recognition algorithm to select the relevance features in environmental conditions (temperature and relative humidity) and modal frequencies. Moser and Moaveni monitored a steel pedestrian bridge located on Medford. A fourth-order model, out of six regression models, without cross terms was selected as the best representative model for the relationship between the modal fre- quencies and the temperature. Zhang et al. used the Gaussian process regression technique to model the depen- dency between the bridge modal frequencies and the environmental along with operational conditions. 32–35 Since both structural dynamical responses and environmental factors exhibit significant level of uncertainty, uncertainty quantification (UQ) is essential in modal frequency–multiple environmental factors pattern recogni- 36–38 39,40 tion. Bayesian probabilistic framework-based approach has attracted special attention as it provides a 41–44 45–47 rigorous solution to UQ in both parameter level and model level. As the complexity of the pattern between modal frequency–multiple environmental factors grows with the number of the environmental factors in the infer- ence process, a sophisticated graphical model-based tool, the Bayesian network (BN), is explored and developed for pattern recognition purpose. There are three advantages of the BN. First, its inference is based on Bayesian framework, so it possesses the capacity of UQ. Second, it utilizes a graphical interpretation to depict the casual or statistical dependent relationships between different variables and it is capable of directly identifying the conditional and joint probability distributions of the variables. Third, it can make inference under incomplete observed infor- mation (missing data), which is common in SHM. Due to these advantages, it has attracted attention in georisk and 49 50 51 structural engineering, such as ground-motion prediction, reliability analysis, and risk assessment. In this study, the BN-based pattern recognition is performed based on one-year modal frequency and multiple environmental (temperature, relative humidity, wind speed, and traffic) monitoring dataset of a 782-m three-span half-through arch bridge over the Pearl River of Guangzhou City of China. The remaining parts of the paper are organized as follows. Dataset and BN structure candidates are firstly presented. Then, the BN-based pattern recognition is explored and developed. Finally, pattern recognition results of modal frequency–multiple environ- mental factors of the Xinguang bridge are presented and the prediction capability of the BN-based model is validated. Dataset and BN structure candidates The monitored structure is the Xinguang Bridge (shown in Figure 1), which is a three-span half-through arch bridge with the mid span of 428 m, two side spans of 177 m each, and width of 37.62 m, over the Pearl River of Guangzhou City of China. It is the first bridge with a combination of the steel truss arch and the concrete triangular frame in China. The monitoring period is from 1 January to 31 December of year 2014. As the operating time of sensors was set to be uniformly distributed from 00:00 to 23:59, more than one set of data can be achieved within one day. As the monitoring system requires regular maintenance, no record is measured for those maintenance days. Totally four types of data were collected for the environmental dataset: the temperature, the relative humidity, the wind velocity, and the traffic volume. Two weather sensors, located in the mid-span deck and the side-arch crown, measured the temperature and the relative humidity in the sampling time of 1 s. The average temperature T (unit: C) and the average relative humidity H (unit: %) were achieved by averaging the corresponding measured values Mu et al. 547 Figure 1. Elevation of the Xinguang Bridge with the mid span of 428 m, the two side spans of 177 m each, and the width of 37.62 m. Figure 2. The temperature, relative humidity, wind speed and traffic volume of the Xinguang Bridge of year 2014. from the two weather sensors. Three wind sensors, located in the mid-span deck the main-arch crown, and the side-arch crown, measured the wind velocity in the sampling time of 1 s. The average wind velocity W (unit: m/s) was achieved by averaging these three measured values. Two traffic volume cameras, located in the upstream and downstream traffic lanes, measured the number of vehicles passing through the bridge within 5 min. The traffic volume Tr (unit: no. of vehicles) was achieved by summing up the cumulative number of vehicles of the three upstream traffic lanes and the three downstream traffic lanes. Figure 2 shows these four environmental factors of the Xinguang Bridge of year 2014. Forty accelerometers were installed and the corresponding locations of one side 548 Journal of Low Frequency Noise, Vibration and Active Control 39(3) Figure 3. The first and second modal frequencies of the vertical mode of the Xinguang Bridge of year 2014. of the bridge are shown in Figure 1. Note that ‘V’ and ‘H’ represent the vertical and horizontal directions of the sensors, respectively. The measured sampling frequency is 76.8 Hz, while the adjustable measured time spans are 120n s with n ranging from 1 to 10 for different measured times and days. The data-driven random subspace method is used to identify the modal frequency of the structure. In this study, the first and second modal frequencies (f and f , unit: Hz) of the vertical mode of the Xinguang Bridge are achieved. Figure 3 shows the 1 2 fluctuations of these two frequencies. A BN is a directed acyclic graphical (DAG) model, which is a graphical representation of the statistical relation between a set of random variables X ¼ X ; .. . ; X . The structure of the BN is constructed by two types of ðÞ 1 n components: a node and a directed arc. A node represents a random variable X . A directed arc pointing from node X to node X (like X ! X ) represents that X statistically depends on X . In this case, node X is called a j i j i i j j parent of X and the collection of all the parent nodes of X is denoted as pðÞ X . A node without any parent node is i i i called a root node. In modal frequency–multiple environmental factors pattern recognition, six nodes are required for six mea- sured variables: T, H, W, Tr, f , and f . For inference purpose, discretization of data of each node is introduced. 1 2 Over-finer discretization leads to the fluctuation of data probability, while over-coarser discretization leads to the masking of data probability. In this study, considering that there are 7811 points, Table 1 shows discretization of data and Figure 4 shows data histograms of different nodes. Figure 5 shows the proposed candidates of network structure for modal frequency–multiple environmental factors pattern recognition. The candidates are proposed as follows: (1) Candidates M and M treat the four 1 3 environmental factors (T, H, W, Tr) as the root nodes (i.e. the node without any parent node). As the four envi- ronmental factors are the parent nodes of the two modal frequencies (f , f ), the directed arcs are added from the 1 2 environmental factors to the modal frequencies. (2) Candidates M and M consider the correlation effect between 2 4 the temperature T and the relative humidity H, so they add the arc from T to H. Note that adding the arc from H to T is a Markov equivalence to adding the arc from T to H, so only one case of the arc linking between T and H needs to be considered in candidate construction. (3) Candidates M and M consider the correlation effect between f 3 4 1 and f , so they add the arc from f to f . Finally, four candidates of network structure, M ; o ¼ 1; .. . ; 4, of the 2 1 2 o BN are proposed for modal frequency–multiple environmental factors pattern recognition. BN-based pattern recognition Learning phase Let h ¼ PXðÞ ¼ kjpðÞ X ¼ j denote the probability of node X , i ¼ 1; .. . ; n , being in its kth state, k ¼ 1; ... ; r , ijk i i i x i given that its parent node collection pðÞ X is in its jth state, j ¼ 1; ... ; q . It is worth to note that all the inferences i i Mu et al. 549 Table 1. Discretization of data. Number Variable Notation Unit of bins Bin boundaries 1st frequency f Hz 14 0, l 3r , l 2:5r ,.. ., l þ 2:5r , l þ 3r ,þ1 1 x x x x x 1 x 1 x 1 x 1 1 1 1 1 2nd frequency f Hz 14 0, l 3r , l 2:5r ,.. ., l þ 2:5r , l þ 3r ,þ1 2 x x x x x 2 x 2 x 2 x 2 2 2 2 2 Temperature T C10 1; 0; 5; .. . ; 35; 40; þ1 Relative humidity H %10 0; 10; .. . ; 90; 100; Wind speed W m=s11 0; 1; .. . ; 9; 10; þ1 Traffic loading Tr No. of vehicles 19 0; 50; .. . ; 850; 900; þ1 Figure 4. Data histograms of different nodes. Figure 5. Proposed candidates of network structure for modal frequency–multiple environmental factors pattern recognition. 550 Journal of Low Frequency Noise, Vibration and Active Control 39(3) and variables are conditional on the proposed network structure M ; o ¼ 1; ... ; 4, but it is not reflected in some notation due to symbolic simplicity. For an independent and identically distributed complete dataset D¼ðÞ D ; ... ; D , the likelihood function conditional on h ¼ h ; i ¼ 1; .. . ; n ; j ¼ 1; ... ; q ; k ¼ 1; .. . ; r of 1 N ijk x i i a candidate of network structure M ; o ¼ 1; ... ; 4 can be written as q r nx i i YYY ijk PðDjh; M Þ¼ h (1) ijk i¼1 j¼1 k¼1 where N is the cumulative number of data points belonging to the category of X ¼ k and pðÞ X ¼ j, and it can ijk i i be obtained by N ¼ vðÞ i; j; k; D ; M (2) ijk l o l¼1 with the characteristic function vðÞ i; j; k; D ; M l o 1if X ¼ k and pðÞ X ¼ j in D i i l vðÞ i; j; k; D ; M¼ (3) l o 0 otherwise The product Dirichletian distribution, the conjugate prior with global and local independence, is introduced q r nx i i YYY C a ðÞ ij a 1 ijk pðhjM Þ¼ h (4) ijk C a ijk ðÞ i¼1 j¼1 k¼1 where C is the Gamma function; and a ¼ a with a being the hyperparameters, representing the ðÞ ij ijk ijk k¼1 number of virtual samples belonging to the category of X ¼ k and pðÞ X ¼ j. A popular choice for the hyper- i i parameter is a ¼ 1, that is, a uniform prior over the parameters of X for each value combination of pðÞ X . ijk i i Finally, the posterior distribution can be obtained by Bayes’ theorem q r x i i YYY CðÞ N þ a ij ij N þa 1 ijk ijk pðÞ hjD; M / pðÞ hjM PðÞ Djh; M ¼ h (5) o o o ijk C N þ a ðÞ ijk ijk i¼1 j¼1 k¼1 where N ¼ N . It can be observed that the posterior distribution PðhjD; M Þ is again a product ij ijk o k¼1 Dirichletian distribution. The posterior expectation h , mode h and variance VarðÞ h can be derived as ijk ijk ijk N þ a ijk ijk h ¼ (6) ijk ij N þ a 1 ijk ijk h ¼ (7) ijk r r ij i N þ a r N þ a ðÞ½ ðÞ ijk ijk ij ijk ijk Var h ¼ (8) ðÞ ijk rðÞ r þ 1 ij ij r ¼ N þ a (9) ðÞ ij ijk ijk k¼1 In order to select the most plausible network structure, Bayesian probability is utilized as the relative plausi- bility measure of the aforementioned four candidates M ; o ¼ 1; ... ; 4. Given the dataset D, the model o Mu et al. 551 probability PðÞ M jD is evaluated as pðÞ DjM PðÞ M o o PðÞ M jD ¼ ; o ¼ 1; ... ; 4 (10) ðÞ pðÞ DjM P M o o o¼1 where PðÞ M is the prior probability of M and the uniform prior is considered PðÞ M ¼ 1=4. The evidence o o o PðÞ DjM can be evaluated by the theorem of total probability (11) pðÞ DjM ¼ P Djh; M Þpð hjM Þdh logp Djh; M logN; o ¼ 1; ... ; 4 o o o o P P P n q r 1 x i i where h is the posterior expectation of the parameters in equation (6), and H ¼ 1 is the number i¼1 j¼1 k¼1 of effective parameters of M . Here, the Bayesian informative criterion (BIC) is introduced for the integral evaluation as it is accurate in the large samples case. Finally, the optimal network structure is selected as M¼ arg max PðÞ M jD ; o ¼ 1; ... ; 4 (12) Prediction phase In the case of complete observation information, given a full-observed sample D 2 R , the posterior predictive Nþ1 probability PDðÞ jD; M can be obtained by considering the compound distribution of likelihood function Nþ1 o PD ð jh; M Þ, with the uncertain parameters distributed according to posterior PDF pðhjD; M Þ, as follows Nþ1 o o PDðÞ jD; M ¼ PDðÞ jh; D; M pðÞ hjD; M dh Nþ1 o Nþ1 o o q r nx i i Y XX ¼ vðÞ i; j; k; D ; M h pðhjD; M Þdh Nþ1 o ijk o (13) i¼1 j¼1 k¼1 n i r x i YXX ¼ vðÞ i; j; k; D ; M h Nþ1 o ijk i¼1 j¼1 k¼1 In contrast to the traditional inference requiring complete information of observation, the BN is capable to make prediction on the target node with incomplete observed information of other nodes. That is, without uo n o n uo o knowing the unobserved node x ðÞ M 2 R , based on the information of the observed node xðÞ M 2 R , o o p p the predicted value of the target node xðÞ M 2 R can be calculated, where n þ n þ n ¼ n . In this case, the o o uo p x BN makes prediction based on xðÞ M by maximizing the following conditional probability with respect to xðÞ M p p o o bxðÞ M ¼ arg max PðÞ X ¼ x jX ¼ x ; M (14) o o The above conditional probability can be factorized as Table 2. BIC and probability results of different proposed candidates of network structure. M M M M 1 2 3 4 BIC 3.4333Eþ06 23.4323E106 2.5242Eþ07 2.5241Eþ07 PðÞ M jD 0.00 1.00 0.00 0.00 BIC: Bayesian informative criterion. Note: Bold values signifies the most plausible network structure. 552 Journal of Low Frequency Noise, Vibration and Active Control 39(3) Figure 6. Predicted histogram of f with incomplete observed information of the most plausible model M . 1 Mu et al. 553 Figure 7. Predicted histogram of f with incomplete observed information of the most plausible model M . p p o o PðÞ X ¼ x ; X ¼ x ; M p p o o PðÞ X ¼ x jX ¼ x ; M ¼ PðÞ X ¼ x ; M (15) P X ¼ h; M uo uo X ¼x X X P X ¼ h; M p p uo uo X ¼x X ¼x with n q r x i i YYY P X ¼ h; M ¼ h (16) o ijk i¼1 j¼1 k¼1 where h is the posterior expectation of the parameters. ijk 554 Journal of Low Frequency Noise, Vibration and Active Control 39(3) Table 3. Root-mean-square errors (RMSE) on predicted frequencies with different training and test data. Candidate Case RMSE M M M M 1 2 3 4 I. Training data: Full; Test data: Full; Incomplete :No f 3.3826E-03 3.3826E-03 3.5040E-03 3.5109E-03 f 3.3826E-03 3.3826E-03 3.5040E-03 3.5109E-03 II. Training data: Full; Test data: Full; Incomplete: Yes f 3.4124E-03 3.4114E-03 3.5159E-03 3.5134E-03 f 6.5680E-03 6.5674E-03 6.9220E-03 6.9239E-03 III. Training & test data: Leave-one-out cross f 3.9722E-03 3.9721E-03 4.0505E-03 4.0544E-03 validation; Incomplete: Yes f 7.7573E-03 7.7562E-03 7.9677E-03 7.9666E-03 Incomplete: Each test point is randomly selected to be an incomplete point with some probability. An incomplete point means that each component of the four environmental factors is with some probability to be ‘missing’ in prediction. Note: Bold values signifies the most plausible network structure. Figure 8. Predicted to measured ratios and normalized residuals by M of Case I of Table 3. Pattern recognition results Table 2 shows BIC and probability results of different proposed candidates of network structure. The probability of the most plausible network structure M is almost equal to 1; and the probability values of M and M are 2 1 2 higher than those of M and M . Recall Figure 5 of different network structures, the probability results reflect 3 4 that the predictions can be improved by considering the correlation between the temperature and relative humid- ity. On the other hand, the improvement of predictions is not significant by considering the dependency between f and f . In this case, the BN prefers model M as it accounts the dependency between T and H and avoid 2 2 considering the unnecessary dependency between f and f . Figure 6 shows predicted histogram of f with incom- 1 2 1 plete observed information of the most plausible model M . Note that the vertical dash line is the true observed 2 Mu et al. 555 Figure 9. Predicted to measured ratios and normalized residuals by M of Case II of Table 3. value of f , while the grey histogram is the updated probability of different states of f given the available observed 1 1 information. For example, the middle right subplot “Observed: T, H, W” represents that the temperature, relative humidity, and wind speed are observed, while the traffic volume is not observed. With the increasing of the information gain from the environmental effects, the probabilities of different states of f are being updated. It can be seen that the prediction of the BN can be improved with increasing observed information and the accurate prediction can be achieved even though the observed information is not complete (for “Observed: T, H, W,” the traffic volume is not observed). In the same fashion as Figure 6, Figure 7 shows predicted histogram of f with incomplete observed information of the most plausible model M . Again, with increasing observed information, the probability of the optimal predicted state of f is increasing and it approaches to 1 when all the four envi- ronmental factors are observed. These two figures have successfully demonstrated the appealing feature of the BN that it is capable to make prediction under incomplete observed information, and its prediction accuracy can be improved with increasing observed information. In order to compare the prediction capability of different network structure candidates, prediction capability tests are implemented. Table 3 shows root-mean-square errors (RMSEs) on predicted frequencies with different training and test data. Three cases with different training data, test data, and data incompleteness are considered. Case I: The training data and test data are both the full observation dataset (the original dataset with 7811 points). Case II: The training data and test data are both the full observation dataset, but each test point is randomly selected to be an incomplete point with some probability. An incomplete point means that each component of the four environmental factors is with some probability to be ‘missing’, so the ‘missing’ component will not be utilized in frequency prediction. Here, the probability of random incompleteness is 0.05, and for a selected incomplete point, each component of the four environmental factors (T, H, W, Tr) is with 1/3 probability to be ‘missing’. Case III: The training data and test data are constructed using leave-one-out cross validation. That is, for each implementation, one point is selected as the test data, while the remaining points are the training data. Random 556 Journal of Low Frequency Noise, Vibration and Active Control 39(3) Figure 10. Predicted to measured ratios and normalized residuals by M of Case III of Table 3. incompleteness is considered in this case as that in Case II. From Table 3, it is obvious that the optimal network structure M possesses the smallest RMSE in predicating both f and f . This reconfirms the probability results of 2 1 2 the BN in Table 2. Figure 8 shows predicted to measured ratios and normalized residuals by M of Case I of Table 3. Most of the predicted to measured ratios are within 0.99 to 1.01, indicating that the most plausible model M gives very accurate predictions on two frequencies. The normalized residual is within 0.98 to 1.02, reconfirm- ing the high prediction capability of M . Figures 9 and 10 show the predicted to measured ratios and normalized residuals by M of Case II and III, respectively, of Table 3. Again, the accuracy results by M for frequency 2 2 prediction can be observed. Finally, it can be concluded that the optimal network structure M is capable to precisely recognize the pattern of modal frequency–multiple environmental factors. Conclusion In this study, the BN-based algorithm is developed for recognizing the pattern between modal frequency–multiple environmental factors of the Xinguang Bridge based on long-term monitoring data (model frequencies, temper- ature, humidity, wind speed, and traffic volume). Taking the advantages of the BN approach, the uncertainty is quantified in both parameter and model levels in learning phase; and the inference is made under both complete and incomplete observed information in the prediction phase. Based on the monitoring data, the results of the most plausible network structure indicate that consideration of the correlation between the temperature and relative humidity can improve prediction, while consideration of the correlation between two frequencies cannot. The appealing feature of the BN for making prediction under incomplete observed information is dem- onstrated. The performances of different network structure are evaluated by the full training dataset along with full test dataset, the full training dataset along with full test dataset considering random incompleteness of test points, and the leave-on-out cross validation considering random incompleteness of test points. The positive evaluation results of the most plausible network structure confirm that the BN-based approach is capable to precisely recognize the pattern of modal frequency–multiple environmental factors. The proposed algorithm can Mu et al. 557 be utilized in structural condition assessment. The predicted residuals of modal frequencies can be calculated and compared with a prescribed threshold. The predicted residuals being larger than the threshold can be treated as an alert for structural health, and the corresponding actions (further data analysis, engineering judgement and/or safety inspection) need to be taken. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (51508201, 51678252), the Natural Science Foundation of Guangdong Province, China (2017A030313262), Pearl River S&T Nova Program of Guangzhou (201806010172), and Science and Technology Program of Guangzhou (201804020069). This generous support is gratefully acknowledged. References 1. Farrar CR and Worden K. An introduction to structural health monitoring. Phil Transac R Soc Lond A 2007; 365: 303–315. 2. Alampalli S and Fu G. Remote bridge monitoring systems for bridge condition. J Low Frequency Noise Vib Active Control 1997; 16: 43–56. 3. Jang S, Jo H, Cho S, et al. 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"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE
Published: Jul 24, 2018
Keywords: Bayesian network; environmental factor; model class selection; structural health monitoring
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