A temperature-driven MPCA method for structural anomaly detection

Engineering Structures 190 (2019) 447–458

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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct

A temperature-driven MPCA method for structural anomaly detection a,⁎

b

c

d

Yanjie Zhu , Yi-Qing Ni , Hui Jin , Daniele Inaudi , Irwanda Laory

a,⁎

T

a

Civil Research Group, School of Engineering, University of Warwick, Coventry CV4 7AL, UK Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong Jiangsu Key Laboratory of Engineering Mechanics, School of Civil Engineering, Southeast University, Nanjing, China d CTO, SMARTEC SA, Manno, Switzerland b c

A R T I C LE I N FO

A B S T R A C T

Keywords: Anomaly detection Blind source separation Moving principal component analysis Structural health monitoring Temperature effects

An important issue in structural health monitoring (SHM) is to develop appropriate algorithms that can explicitly extract meaningful changes in measurements due to structural anomalies, especially damage. However, the effects due to environmental factors, especially temperature variations may produce significant misinterpretations. Consequently, developing solutions to identify the structural anomaly, accounting for temperature influence, from measurements, is crucial and highly anticipated. This paper presents a Temperature-driven Moving Principal Component Analysis method, designated as Td-MPCA, for anomaly detection. The Td-MPCA introduces the idea of blind source separation (BSS) for thermal identification with intent to enhance the performance of Moving Principal Component Analysis (MPCA) for anomaly detection. To achieve this target, temperature-induced strain variations are first investigated and revealed by employing Independent Component Analysis based on maximization non-Gaussianity, also known as Fast ICA. Afterwards, the MPCA is adopted for anomaly detection on the separated temperature-related response. Three case studies are provided in this paper to evaluate the proposed method. The first one is a numerical truss bridge with a simulated 5% stiffness reduction. The results confirm that Td-MPCA is more sensitive than MPCA in detecting anomalies, where the simulated stiffness loss fails to be detected by MPCA. The second case study is on an experimental truss bridge where two damage scenarios are introduced and interpreted. The detection results show that Td-MPCA outperforms MPCA since the damage is identified at the expected time by Td-MPCA but not by MPCA. The third case study is an in-situ curved viaduct in Switzerland. Data acquired during both construction period and normal service period has been used for interpretation. Results demonstrate that Td-MPCA is able to identify the date of change in construction process without any delay when compared with the application of MPCA only.

1. Introduction The degradation of civil structures is inevitable because of the cyclic loading in service condition, such as thermal loading, traffic loading and others due to environmental or operational variations. To avoid sudden structural failure, diagnostic procedures are developed to detect and trace such changes before failure happens. Structural health monitoring (SHM) techniques are emerging to complement the traditional non-destructive evaluation methods, e.g., the visual-based inspection. By employing sensor-based SHM techniques, the real condition of an existing structure can be determined, and the data can be used for structural identification. Results from the identification have fundamental significance in informing the maintenance plan or other asset management [1]. In addition, a monitoring procedure is indispensable for new bridges starting with the construction period, since real-time

⁎

structural behaviour can be obtained to validate the design assumptions and to support the subsequent construction work. Early warning of structural degradation and timely alarm of imminent structural damage are challenging goals within the SHM discipline. The real-time assessment of structural condition can be established from knowing changes in structural responses that reflect an adverse effect on structural integrity [2]. Commonly, the structural condition is inferred and assessed from changes between physical measurements and numerical predictions from finite element (FE) model [3]. In general, the methods for anomaly detection can be divided into two streams: physics-based models and data-driven approaches. The detailed distinction can refer to [4]. For physics-based structural identification to succeed, an accurate FE model is critical to the whole process. The inevitable weakness of physics-based methods is model uncertainties [5]. For example, an FE model constructed from

Corresponding authors. E-mail addresses: [email protected] (Y. Zhu), [email protected] (I. Laory).

https://doi.org/10.1016/j.engstruct.2019.04.004 Received 9 September 2018; Received in revised form 1 April 2019; Accepted 1 April 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.

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detection if the temperature effect can be separated from the measurements [19]. Later, neural network models for correlation between modal frequencies and environmental temperatures were formulated with the optimal network configuration being obtained by applying the early stopping technique and the Bayesian regularization technique in the training process [20]. Yarnold et al. proposed to seek for a threedimensional best-fit plane among the local strain, global displacements and local temperature, which is a reliable response surface as the representation of structural condition [11,21,22]. This unique correlation can be employed for model calibration and behaviour prediction; and by comparing this correlation at certain time intervals, the error due to property changes can be detected. The apparent shortcoming is the ignorance of temperature distribution among the structural cross section. Nguyen et al. adopted the curving fitting method to formulate a linear relation between displacement and temperature. Temperature compensation was subsequently completed by setting a reference temperature and re-calculating all the displacement values under fixed temperature condition according to the linear relation they formulated [23]. However, the assumption of linear relationship is not always valid, and the reliability of baseline data is critical to this method. Su et al. studied the temperature-induced displacement and stresses and obtained the temperature distribution model by applying temperature loads on reduced-order finite element model. Subsequently they subtracted this temperature-induced effect prior to studying the typhooninduced responses of a supertall structure [24]. However, the model uncertainty is inevitable since assumptions exist due to the limitation of temperature measuring points. In this paper, the idea of extracting temperature-induced responses blindly and directly is employed. As an extension of the authors’ previous work [25,26], a temperature-driven anomaly detection method is subsequently proposed and presented. In general, the temperature related structural response can be separated blindly from sensor measurements by utilizing Independent Component Analysis (ICA). Since the scale of monitoring system varies depending on the structure scale, a minimum number of input sensors for ICA separation will be determined by applying Principal Component Analysis (PCA). After doing so, the ICA-estimated thermal response is manipulated by Moving Principal Component Analysis (MPCA) to reveal abnormal variations for anomaly alarm as appropriate. Since the MPCA has been applied for damage detection directly [27–29], this study will also compare the performance of the proposed Temperature-driven MPCA (Td-MPCA) with the original MPCA for anomaly detection. The first step in the proposed temperature-driven method is employing PCA as a pre-indicator. PCA was first formulated by Peason in 1901 and Hottelling in 1933 [30]. The detailed description of PCA can be found in [31] and mathematical background information related to this research will be given in the next section. As one of proper orthogonal decomposition techniques, the analysis of the behaviour in terms of eigenvalues and eigenfunctions of the covariance matrix of the data set gives a good indication of the damage initiation and provides information about the severity of the damage [32–34]. The utilization of PCA for structural health monitoring is mainly divided into data pattern recognition and data reduction [35]. For the purpose of data pattern recognition, researchers have used reference data across a certain period obtained from health structure to construct a statistical model to represent the structure’s normal condition. For example, Mujica et al. utilized PCA to obtain a baseline pattern of the structure as an undamaged state and then combined hypothesis testing for damage detection [36,37]. PCA has also been examined by Cross et al. [38]. They found that the mapped principal components (PCs) with less variance were damage-sensitive but environment-insensitive, which therefore could be used for further damage detection. However, the damage could be ignored if the damage information manifests as one of the PCs. One more concern addressed is environmental trends which may not be sufficiently removed due to the orthogonality constriction among all principal components. Yan et al. proposed PCA-based

engineering drawings cannot truly reflect the actual structure’s performance. One difference is that the assumed constant mechanical properties cannot be implemented and guaranteed for the in-situ structure due to inherent material uncertainties and local variability. A second difference is that the assumed perfect joints and connections as given in design drawings are not matched well with the real physical situation. The discrepancy between the real structure and the designed one is also due to the practicalities of on-site construction [3]. Furthermore, it is challenging to predict with confidence the evolving patterns of real actions that might happen once the bridge is put into use [1]. As regards the data-driven approaches, an issue of great concern for anomaly detection within this category is the temperature impacts, which can induce significant cyclic or abnormal changes in measurements and also distinct influence on system reliability [6,7]. A number of works have focused on the evaluation of temperature distribution in, e.g., composite box girder bridges [8] and steel box girder bridges [9]. However, characterizing temperature distribution, especially in a longspan bridge, is a complicated procedure due to different thermal properties of various structural portions in a bridge, such as deck, tower/pier and cable [10]. Since the temperature variations can induce distinct thermal effect [11], numerous studies have focused on identifying this thermal effect [6,12–14]. Of particular significance in anomaly detection when considering temperature distribution’s effect is that the structural damage may produce smaller variations compared with the thermal effect. Consequently, the paramount target is enabling the identification of the meaningful changes due to structural abnormal behaviour in the presence of temperature fluctuations. Two mainstream solutions to avoid temperature effects have been developed, in simple terms, elimination or utilization. In some cases, e.g. short period data, the thermal effect can be eliminated directly without any side effect on final detection. For example, Chen et al. analysed the transient alterations in vibration characteristics from the concrete deck with cracks [15]. The deviation due to crack reflected in the observed data is sufficiently manifest within several minutes. Therefore, the temperature effects can be eliminated in this short-lived data interpretation. Deraemaeker et al. have tried to find structural features that are insensitive to temperature variations but only sensitive to structural damage. They have demonstrated that the appearance of spurious peaks in the outputs of modal filters is more sensitive to damage instead of environment impacts and proposed to utilize these peak indicators, obtained by performing Fourier transform on the output of modal filters, for damage detection [16]. Zhou et al. proposed a parametric approach for eliminating the temperature effect in vibration-based damage detection when both dynamic properties and temperature are measured [17]. Yang et al. eliminated temperature effect indirectly by separating monitoring data, i.e. frequency, into various clusters with the same probability distribution [18]. The subsequent detection was carried out on each independent cluster, in which temperature has a similar effect on structural properties. Thus, the temperature effect can be eliminated indirectly for short period data. In other cases, the temperature-induced fluctuation is treated as substantial noise in the observed signals. Increasing the signal-to-noise ratio by operating Wavelet and Hilbert-Huang transform helps to eschew temperature effects [3]. However, the non-negligible conditions are the noise level, i.e. temperature-induced deviation must be within 20% of the signal and the damage should occur during the monitoring period. In recent years, an increasing attention of directly utilizing temperature related to structural responses for structural identification has been drawn in a number of studies. To utilize the temperature effect, the first crucial step is to identify it. Correlating temperature with the structural properties of interest is the most popular way to compensate for temperature effects. For example, Ni et al. utilized support vector machine (SVM) to identify the correlation pattern of temperature with modal frequencies and indicated the further potential of anomaly 448

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Fig. 1. Flowchart of MPCA and proposed Td-MPCA for anomaly detection.

matrix; and the mixing matrix was treated as the representation of structural vibration features. The two parts of ICA output were subsequently utilized to build a neural network model as an indicator for detecting damage. Poncelet et al. applied ICA to estimate the damping ratios and modal frequencies in mechanical systems [53]. Yang and Nagarajaiah combined wavelet transform with ICA to obtain the recovered mixing matrix, which contains interesting damage information [54]. More applications of ICA for modal identification can be found in the investigations conducted by Chang et al. [55] and Yang and Nagarajaiah [56,57]. Spiridonakos et al. utilized ICA to merge and find small set of operational sensor records from the whole sensor system [58]. However, applications mentioned above did not take temperature influence in consideration. In this study, ICA will be leveraged for extracting mechanical strain induced by temperature varations. MPCA is the abbreviation of moving principal component analysis [27,59,60] which is based on the classical principal component analysis. The method of MPCA is designed to figure out the characteristics of a certain period of time series measurements. This period of records is named as the initialization phase, in which the structure is supposed to be in healthy condition. After that, anomalous behaviours can be identified according to this initial phase. This period is also denominated as window size. The covariance matrix of data inside an active window is calculated and then moving in time; more details can be found in [59–61]. Cavadas et al. examined the performance of MPCA and found that MPCA could give an early detection of anomalous behaviours [62]. With the moving window, the computational cost is lower for each step and detection of the presence of new situations is timelier because old measurements do not buffer results. The window size should be sufficiently large, so that the periodic variability, i.e. the seasonal temperature cycles, can be exposed; while rapidity of computation can be guaranteed at the same time. Therefore, the window

damage detection methods for both linear [39] and nonlinear conditions [40] under various environmental conditions. Mojtahedi et al. used PCA to discern the nonlinearity of measured data of an offshore structure [41]. Reynders et al. proposed kernel PCA based technique to eliminate environmental influence for damage detection relying on the output only measurements [42,43]. Shokrani et al. evaluated the performance of PCA on modal frequency and modal shape curvature data accounting for the temperature variation [44]. For the purpose of data reduction, PCA maps the target sources into a different space to find a low-dimensional representation. Loh et al. applied PCA to compress frequency response function data collected from a twin-tower steel structure [45]. Datteo et al. combined PCA with auto regression model techniques to obtain structural representation in a concise way [46]. In this study, the main function of PCA can be classified as data dimension reduction, but the number of principal components is of interest instead of the PCs themselves. The independent component analysis, or known as ICA, is the most popular solution for blind separation problem which is also employed in this study. ICA is to separate a set of non-Gaussian data into a new collection of variables, which are statistically independent. The original idea of ICA was first proposed by Comon [47], aiming to maximize the statistical independence among the separated components by performing a linear transformation on the target data. There are some applications that leverage ICA along with other techniques for structural identification or damage detection. For instance, Jutten and Herault proposed a source separation method, combing ICA and a recursive interconnected neural network in the early 1990s [48,49]. The other early applications of ICA can be found in [50] and [51]. In the recent decade, ICA was employed in conjunction with artificial neural network for damage diagnosis [52]. The time history records were decomposed into a set of independent components and the mixing 449

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Target signal X Step 1: Normalization process Setp 2: Covariance process Step 3: Eigen process Step 4: Sorting process

•Column ( ): measurements of each sensor •Row ( ): measurements from all sensors at time t •Zero-mean process for each sensor records, i.e. each column. •Calulating covariance matrix •Calulating eigenvectors of covariance matrix

of

Fig. 3. Schematic graph of moving window of MPCA.

and relative eigenvalues

•The eigenvectors will be sorted followed by the value of relative eigenvalues

Fig. 2. Basic theory of principal component analysis.

size should be theoretically multiple of periodic variability. In the numerical simulation of this study, one-year window size is chosen to achieve a lower computational cost, instead of two-year window size used in [27], because integrated and continuous measurements can be obtained. After selecting the window size, the first principal component, i.e. the eigenvector related to the main eigenvalues, is analysed at each step. The standard deviation of eigenvectors from the first set of data within the fixed window is recorded as σ , which is subsequently used for threshold definition. According to the previous studies [27,28], the confidence interval is defined as 3σ off the initial data’s eigenvectors, however, this limitation can be narrowed to ± 2σ in this paper. The next section is devoted to describing the mathematical background of the algorithms required in the proposed method, i.e., fast ICA, PCA, and MPCA. The case study on a numerical truss bridge is given subsequently in Section 3 and followed by the experimental truss bridge case study in Section 4. In Section 5, the in-situ Ricciolo viaduct is also examined. The evaluation of the proposed Td-MPCA for anomaly detection will be summarized at the end of this paper.

Fig. 4. Case study 1: aluminium truss bridge model with six sensors and four fixed ends at A, B, C, and D.

Gaussianity is measured by kurtosis. Taking two independent components as an example, as expressed in Eq. (1). Matrix X is composed of two vectors, x1 and x2 , which are timehistory measurements. The vectors x1 and x2 are linear combination of s1 and s2 through mixing matrix A . The matrix A contains weight factors a11, a12, a21, and a22 .

2. Temperature-driven MPCA

2

X = AS → x i =

The proposed Td-MPCA method is operating MPCA on ICA-estimated thermal responses, with a pre-processing of PCA on input measurements. The flowchart in Fig. 1 gives a visualized procedure, whose details are given as follows. Basically, Td-MPCA can be divided into three parts: pre-indication, blind separation, and anomaly detection. The principal component analysis (PCA) is first applied on all target strain measurements, e.g. mchannel signals, to find the number of intrinsic components, notated as n. The number n, which is of interest, represents the minimum essential number of input channels for ICA separation. Hence, n-channel measurements will be selected from all target signals; as a result, Cnm various collections arise in total. The ICA is then executed on these various combinations individually. For each collection, the thermal-related strain is selected from ICA-estimators according to its correlation with an available temperature record nearby. The one which is highest correlated with temperature fluctuation is saved as the thermal-related strain for next step. Finally, the MPCA will be applied on the previously saved thermal-related strain for anomaly detection. In addition, the MPCA is also performed on all target strains in three case studies, seeing the flowchart in Fig. 1. The results will be compared with the performance of Td-MPCA, and the improvement will be summarized. This section is mainly interpreting the theoretical background of the involved methodologies in following order. The introduction of fast ICA will be given first, followed by a description of MPCA along with PCA. For the sake of brevity, the bold capital letter represents matrix, i.e. X, and the small bold letter denotes vector, i.e. x, throughout the text.

∑ aij s j, i = 1, 2 (1)

j=1

Theoretically, the estimator of the independent components, s1 and s2 , can be obtained by finding the decomposing matrix BT , to let Eq. (2) hold.

Sest = BTX

(2)

BT

supposes to be the inverse of the matrix The decomposing matrix A . However, in blind source separation case, the mixing matrix A is unknown, therefore the challenge is to find the unknown decomposing matrix BT to separate observed sources, x1 and x2 , into independent components, sest2 and sest1. Since both matrices A and S are unknown, one assumption in ICA estimation is introduced as follows: the ICA estimating components, i.e., Sest , have the unit variance. This is due to that the ambiguity of magnitudes of ICA estimation is inevitable. In other words, the magnitudes of independent components cannot be estimated precisely, since the scalar multiplier or division in estimated components, s , and weight factors, a , can be cancelled each other. Taking the following Eq. (3) as an example, the scalars α and β in s1 and s2 can be canceled due to the same division in mixing weights a11 and a12 respectively.

x1 =

1 1 a11α s1 + a12β s2 α β

(3)

To simplify ICA estimation, the pre-processing contains centring and whitening procedure. The vector x in Eq. (1) is the zero-mean vector obtained from its original sensor measurements, x org , by subtracting their row mean, x = x org − E{x org} . This centring process will result in zero mean of independent components too, that’s why all the estimators in case study are zero-mean sources. The second process is whitening the records. The purpose of the whitening process is to reduce the

2.1. Fast ICA The fast ICA algorithm is using the criteria of maximizing nonGaussianity to estimate the independent components, where the non450

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(a). Sensed strain at C71, C75 and C82

(b). Sensed strain at C73, C80 and C81

Fig. 5. Case study 1: overall strain of six sensors.

order cumulant and is a classic measure of non-Gaussianity. The kurtosis value is defined in Eq. (7) for zero-mean variables [63].

Table 1 Pre-indication of Td-MPCA by PCA.

Cumulative percentage of total variation

1st principal component

2nd principal component

94.5%

99.9%

kurt(bTx w) = E{(bTx w ) 4} − 3(E{(bTx w )2})2

Taking a Gaussian variable y as an example, the kurtosis is zero, because the fourth moment of a Gaussian variable equals to 3(E{y 2})2 . Thus, a non-Gaussian random variable should have a nonzero kurtosis. As mentioned before, the independent components are assumed with unit convenience, hence, the above Eq. (7) can be simplified into Eq. (8) and the absolute value of kurtosis is chosen since it can be positive or negative.

estimated parameters in the mixing matrix by half. The proof can be found in the previous paper [25]. Eq. (4) describes the whitening process, where matrix Λ is the whitening matrix.

kurt(bTx w) = E{(bTx w ) 4} − 3

(4)

X w = ΛX

(5)

According to the central limit theorem, the distribution of a sum of independent random variables tends toward a Gaussian distribution under certain conditions, it is known that any vector x w is closer to Gaussian distribution than vector s . The fast fixed-point algorithm is then employed herein as an iteration scheme to find a unit vector bT that maximizes the non-Gaussianity of bTx w , which is the estimated sources of s , as shown in Eq. (6) updated from Eq. (2).

Sest = BTX w

(8)

The weight vector bT with unit norm starts from a random vector. The iterative computation is to find the direction where the gradient of kurtosis of bTx w is the extrema. In this study, Eq. (9) is utilized for fixed point iteration in fast ICA, followed by the normalization (unit norm) in Eq. (10) [64].

X w will be utilized for next ICA estimation procedure and Eqs. (1) and (3) can then be updated to Eq. (5) as follows. X w = ΛAS

(7)

b ← E {x w f (bTx w)} − E {f ' (bTx w)} b

b←

b = ||b||

(9)

b ∑i b2i

(10)

where the functions f and f ' of bTx w are defined in Eqs. (11) and (12).

f = 4 × (bTx w )3

(11)

f ' = 12 × (bTx w )2

(12)

(6)

The non-Gaussianity of bTx w for ICA estimation can be measured by minimizing or maximizing kurtosis, which is also known as fourth-

The iteration will be stopped if the absolute value of previous and

(a) Estimated source 1

(b) Estimated source 2

Fig. 6. Case study 1: blind separation of Td-MPCA by ICA from C71, C73 and C75 (Normalized strain means zero-mean and unit norm). 451

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(a) Td-MPCA: three inputs for fast ICA

(b) Td-MPCA: four inputs for fast ICA

(c) Td-MPCA: five inputs for fast ICA

(d) MPCA only

Fig. 7. Case study 1: anomaly detection by Td-MPCA (with various inputs for ICA) and MPCA only (window size: 365 days, and threshold: ± 2σ over 135 days reference period).

Fig. 9. Case study 2: experimental truss bridge model with sensor and damage position.

Fig. 8. Case study 2: experimental aluminium truss bridge set up description.

2.2. PCA and MPCA

new weight vector b shows convergence tend, which means the value closes to 1. In consequence, the decomposing matrix, BT , can be obtained and the estimated independent components, Sest , can be calculated according to Eq. (6). It has to be emphasized again that the ICA estimators of strain are all normalized strain with zero-mean (due to the centring process) and unit norms (due to the ambiguity of magnitudes).

As previously introduced in the first section, MPCA is the abbreviation of Moving Principal Component Analysis which is proposed on the basis of classical Principal Component Analysis (PCA). Hence, the basic theory of PCA is given first. As a quantitatively rigorous method for data dimensionality reduction, the core rationale behind PCA is the orthogonal decomposition of the covariance matrix of target variables, which can generate a new and smaller set of uncorrelated variables, called principal components, 452

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(a). Damage scenario A (DS-A)

(b). Damage scenario B (DS-B)

Fig. 10. Case study 2: measurements from SG2, 4, 6, 8 and 10.

(a). Td-MPCA detection

(b). MPCA detection

Fig. 11. Case study 2: damage detection results under damage scenario A (DS-A).

(a). Td-MPCA detection

(b). MPCA detection

Fig. 12. Case study 2: damage detection results under damage scenario B (DS-B).

value decomposition (SVD), since SVD might be the most computationally efficient solution to find principal components for PCA [31]. The basic algebraic theory of SVD is decomposing X∗ (step 1 in Fig. 2) as in Eqs. (13) and (14).

from the target variables. Those principal components are a linear combination of the original variables and orthogonal to each other without any redundant information. The basic procedures of PCA are summarized in Fig. 2, where the matrix X contains structural measurements from all sensors. Each column represents an individual sensor’s time series record, while each row shows the collected data from all sensors at a specific time step, seeing Fig. 2. In this study, steps 2 and 3 in Fig. 2 are achieved by using singular

X∗ = ULAT

(13)

r

x ij∗ =

∑ uik lk1/2 ajk k=1

453

(14)

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(a). Measurement system layout

(b). Sensor placement at section A Fig. 13. Case study 3: data acquisition layout of Ricciolo curved viaduct, reprinted from [68].

k th eigenvalue of the covariance matrix C is lk /(m − 1) (X∗)T X . Hence, the variance of matrix U is 1/(m − 1) . The target of SVD is to find out a new m × n matrix, Xnew , with the first p PCs ( p < r ) which minimize the Euclidean norm of difference between X  and Xnew , ‖Xnew − X∗‖, as displayed in Eq. (16).

Table 2 Data acquisition periods and viaduct condition summary. Start Time

End Time

Frequency

Duration

Condition

02/04/2005 00:03 02/05/2005 16:20

02/05/2005 12:03 01/06/2005 16:20

Every 4 h

31 days

Construction + Clear

Every 6 h

31 days

Clear

m

‖x new, ij − x ij∗ ‖ =

n

∑ ∑ (xnew,ij − xij )2 i=1 j=1

(16)

*Clear means no construction work is recorded.

Thus, the information that SVD can provide is not only the coefficients and variances for the PCs, but also the PC scores, which will be used for visualized PCs’ plot [65]. The first principal component is the eigenvector with the largest eigenvalue, which means that the projected original variables on this direction have the maximum variances among all eigenvector-eigenvalue choices [31,66]. Therefore, the first few principal components contain most characteristics of the whole observations since their variance together can exceed 80% or 90% of the total variances of observations. In this study, the cumulative percentage of selected principal components is set to be over 95% of the total variation. In this study, PCA is employed as a tool to indicate the intrinsic variables of the observations, or in other words, the number of main driving forces that contribute to the overall strains. This is utilized as a guide for fast ICA separation, as shown in Fig. 1. As previously

The matrix X∗ contains m sensor channels with n independent observations. The m × r orthonormal matrix A is eigenvectors of (X∗)T X , and L is a r × r diagonal matrix that contains root square roots of eigenvalues of (X∗)T X , with its elements lk being the k th eigenvalue of (X∗)T X . The extra information from SVD separation is the n × r orthonormal matrix U which relates to PC scores. The PC scores represent the values of X∗ in new principal component space, i.e. each independent observation on each principal component space. The definition of PC scores, notated as Z in Eq. (15), is given by Jolliffe [30], whose kth column represents the kth PC scores.

Z = X∗A = ULATA = UL

(15)

Therefore, the eigenvalue of the kth PC score is lk /(m − 1) , where k = 1, 2, ⋯, m , and lk is the k th eigenvalue of (X∗)T X ; consequently the 454

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(a) Strian measuremetns

(b) Temperature measurements

Fig. 14. Case study 3: axial strain and temperature measurements of section A.

(a) Td-MPCA detection

(b) MPCA detection

Fig. 15. Case study 3: anomaly detection results. (window size: 48 h, and threshold: ± 2σ over 120 h reference period).

the longest periodic variability, that is, a one-year window is enough for accounting for seasonal thermal variability [61]. In this paper, a 365day window is utilized for truss bridge case study and 48 h for Ricciolo viaduct case study, as the target data duration is 730 days for the former case and one month for the latter case respectively. Since the inconsequent behaviour due to structural damage will be reflected in the mean values of target data, as well as its covariance matrix and relative eigenvector-eigenvalue, the anomaly detection is to catch those abnormal changes in eigenvectors from each time step. Thus, a reference state is required to define the threshold. In this study, the threshold is defined and narrowed as two times of standard deviation of the reference period, designated as ± 2σ , instead of the ± 3σ in the previous study [61]. The detail on the reference period for each case study will be stated in the next section.

mentioned, the plots are based on the singular value decomposition (SVD). As a graphical representation, it will be utilized in the case study section to visualize the correlation between target signals and principal components, i.e., the magnitude and sign of target signal’s contribution to the components. The next part of this subsection will give the details of MPCA. Since the decomposed covariance matrix, i.e., eigenvectors and the corresponding variances, are sensitive to anomalous behaviours [66], the eigenvector has the potential to be a damage indicator for anomaly detection. However, the unsatisfactory performance of classic PCA has been observed as follows [27,60]. First, the computation cost is increased when the number of observations increases. Another issue is an evident delay in the time domain. Therefore, an improved statistical method, named as moving principal component analysis, is proposed [27,59,60]. The difference between MPCA and PCA is that MPCA calculates the covariance matrix within a pre-selected window size instead of the whole time series, as shown in Fig. 3. The box with red dash line is the first active window of MPCA, where variables inside are utilized for PCA interpretation, i.e. steps 1 to 3 given in Fig. 2. The first eigenvector with maximum eigenvalue from this active window is saved as the eigenvector of this period at this step, i.e. t1, which is the starting time of this active window. By analogy, for the other active window, whose starting time is ti , the first principal component of the data within that active window will be saved as the eigenvector at time ti . According to Posenato et al. [27], the window size should be a multiple of the periodic variability. A two-year window was therefore suggested to avoid temperature cycles’ effect on threshold estimation. Furthermore, Laory et al. limited the window size to the same length of

3. Case study 1: Numerical simulation of a truss bridge This simulation study is on an aluminium truss bridge, which is only affected by temperature and traffic loads. The bridge is made by aluminium, whose Young’s modulus is 70 GPa , density 2.7 g/cm3 , Passion ration 0.35, and thermal expansion coefficient 23.1 μm·m−1K−1. The damage is introduced on the bottom chord C74, which is adjacent to the floor chord C75 and opposite to the bottom chord C82, as shown in Fig. 4. The length of floor chords, i.e. C71, 73 and 75, is 382 mm and the length of bottom chords C74, 80, 81 and 82 is 406 mm . The Young’s modulus of the bottom chord C74 is reduced by 5% from 70 GPa to 66.5 GPa since the 500th day in the simulation. Other features of this truss bridge can be referred to Fig. 4. 455

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As a result, the outcomes of this numerical case study can be summarized as follows: First of all, the fast ICA is able to separate thermalrelated strain from the influence of traffic and others, as shown in Fig. 6. Secondly, the proposed Td-MPCA is capable to uncover slight damage, as shown in Fig. 7(a)–(c), while the original MPCA fails to detect the damage as shown in Fig. 7(d).

The sensors are placed in chords 71, 73, 75, 80, 81, and 82, which will be abbreviated as C71, C73, C75, C80, C81, and C82 respectively. The overall strain from all six sensors, which means the strain caused by all loading conditions, is shown in Fig. 5. The temperature loading is simulated as 9.7 °C average value with maximum 4 °C daily variation and maximum 7 °C seasonal fluctuation. The simulated duration is 2 years, while the sampling rate is approximately 2.4 h per day, but only 3-day variations are simulated for each month. Hence, the 730-day measurements only contain 72 daily cycles. The traffic load is simulated with double peaks within 24 h and applied on all bottom nodes of the bridge model. The daily traffic load is varying from 0 KN to 5 KN and decreasing to approximately 2.5 KN at noon before increasing back to the second peak value. After that, the recession period leads the value to 0 KN again to complete the daily variation. As described in Fig. 1, the first step in implementing Td-MPCA is pre-indication. The principal component analysis (PCA) is first utilized to estimate the minimum components among all sensor records, represented as the overall strain in Fig. 5. This minimum number will guide the subsequent ICA estimation. As indicated before, the cumulative variance of the selected principal components should be over 95% of the total variation. Therefore, the first two components are satisfactory according to Table 1, which contribute almost 99.9% variance, indicating that the recommended input sources for the second procedure, blind separation, is over two but less than the total number of sensors. Hence the option can be three, four or five. Taking three as the first example. The clustering process within pre-indication is selecting 3-channel sources from 6-channel measurements. Hence, C63 = 20 collections can be obtained. In the second step of Td-MPCA, which is blind separation, the 20 collections have been investigated. The collection of C71, C73, and C75, shown in Fig. 5, is selected as an example. The ICA-estimated strain components, with zero mean and unit norm, are given in Fig. 6. It is apparent that the expected temperature-related strain (Fig. 6(a)) and traffic-relevant strain (Fig. 6(b)) are separated by fast ICA from the overall strain of C71, C73, and C75. In addition, the abnormal shift at the 500th day is much more visible in Fig. 6 than in Fig. 5. This is because the 5% reduction of stiffness is a negligible-level damage. It is hard to be reflected in any single mixed measurements, as shown in Fig. 5. However, the aggregation of three sensors can enlarge and deepen the impact of stiffness loss during ICA estimation process to uncover it. The apparent shift in residual sources can also demonstrate that the temperature-induced strain, as ‘substantial noise’ [3], is possible to cover real damage-induced variation in measurements. The other combinations of three strain records are also examined, among which all temperature-related strain, similar to Fig. 6(a), is of interest and saved for the final step of Td-MPCA. The final process is anomaly detection, where MPCA is employed and applied on ICA-separated results. The detection results of Td-MPCA are displayed in Fig. 7(a). The window size of 365 days and a reference period of 135 days are considered to establish the threshold, i.e. ± 2σ over the reference period [67]. The non-negligible shift at the 500th day is obvious. As mentioned previously, the recommended number of inputs is over two. Hence, the ICA estimation process on four and five should also be able to reveal the concealed temperature-related and trafficinduced strain. This expectation is verified in Fig. 7(b) and (c). The six inputs for ICA estimation will not be considered within Td-MPCA, because the assumption of MPCA is that the input sources must be correlated, at least not independent. However, the ICA estimators from six sensors are independent from each other, therefore, it will be meaningless. As shown in Fig. 7(b) and (c), the evident shift at the 500th day could be clearly observed. To have a visible comparison of Td-MPCA and MCPA, the MPCA is also applied to six overall strain measurements. The detection results are shown in Fig. 7(d). The failure of detection is apparent due to the small damage level of only 5% stiffness reduction.

4. Case study 2: Experimental truss bridge The experimental truss bridge, with the same dimension and material as the numerical model in Section 3, is studied in this section. As illustrated in Fig. 8, temperature load is simulated as the daily variations by controlling the heating lamps, while the moving load is applied on the bridge by using dumbbells and two wood tracks. Five strain gauges, selected from the monitoring system, are interpreted in this study to demonstrate the ability of temperature-driven damage detection method. Those five sensors are located on the bottom surface, denoted as SG2, 4, 6, 8 and 10, as shown in Fig. 9. Two damage scenarios are created as shown in Fig. 9, denominated as DS-A and DSB.

• DS-A: damage is created by removing one bottom chord, which is in the middle span and opposite to SG6. • DS-B: damage is created by removing the connections of top chord. The anomaly detection ability of Td-MPCA is investigated under these two damage scenarios and compared with MPCA. The window size for either MPCA or Td-MPCA is slightly adjusted for each test, considering the uncertainty of temperature variations during the lab test. It usually equals to 1–1.5 times of a complete temperature cycle period. Fig. 10 displays the strain measurements from target sensors under two damage scenarios. More specifically, DS-A is incurred into the structure manually during 1220–1302 s, as shown in Fig. 10(a), while DS-B is generated manually during 627–657 s, as shown in Fig. 10(b). Therefore, the successful detection should be within 1220–1302 s and 627–657 s for DS-A and DS-B, respectively. The detection outcome under DS-A by Td-MPCA is given in Fig. 11(a), while Fig. 11(b) shows the result by employing MPCA directly. Apparently, the Td-MPCA could detect DS-A at the expected time, between 1220–1302 s, but abnormal variations could not be observed by applying the original MPCA. Fig. 12 provides the outcomes for the second damage scenario, DS-B. The Td-MPCA could uncover anomalies during the expected period, i.e. 627–657 s as shown in Fig. 12(a); however, the detection by MPCA is 39 s delay, seeing Fig. 12(b). As shown in Figs. 11 and 12, unexpected variations can be observed close to the end of the observation period, this is because the bridge is in an unstable state with uncertainties in loading condition. Firstly, the structural response induced by temperature and moving load is unexpected, because the damage is introduced into the bridge system. Secondly, the simulated loading conditions cannot be the same as the original condition, especially the manual-controlled moving load. Therefore, large variations can be observed after the damage is introduced. This experimental truss bridge study is to evaluate the performance of Td-MPCA in detecting damage. The results can confirm the conclusion made in the previous numerical simulation. First of all, the anomalies cannot be reflected in strain measurements directly, as shown in Fig. 10, because of the thermal variations. Secondly, the proposed Td-MPCA is capable to uncover anomalies in both damage scenarios, i.e. DS-A and DS-B; however, applying MPCA directly fails to detect DS-A and causes a 39 s delay in the case of DS-B, as evidenced in Figs. 11 and 12.

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5. Case study 3: Ricciolo curved viaduct

Acknowledgement

The Ricciolo curved viaduct is part of Swiss motorway A2 and has been monitored since its construction in 2005. The monitoring system is installed at the longest span, which is 35 m long. The measurement system layout can be found in Fig. 13(a), while the cross-section view of sensor placement in section A refers to Fig. 13(b). In this study, two months data from 00:03 in 02/04/2005 to 16:20 in 01/06/2005 are selected for anomaly detection, as listed in Table 2. The construction events that happened during this period are described as follows: (a) Construction of lateral protection walls from 02/04/ 2005 to 22/04/2005; (b) Post-tensioning, cast of left side wing and removal of external formworks from 25/04/2005 to 26/04/2005. The bridge works properly after 27/04/2005, because of no construction work since then. To detect the anomalous behaviour, the time scale will be inverted from 01/06/2005 to 02/04/2005, since MPCA requires a training period as the reference state of the bridge. Therefore, the reference period, i.e. healthy condition, ranges from 01/06/2005 to 28/04/2005. The abnormal changes should be detected at 27/04/2005. This time reversal procedure utilizes change structural behaviour evolving with construction progress as a damage scenario when looking backwards in time. The measurements from section A are interpreted in this case study. The axial strain and corresponding temperature from the extremities of the span are displayed in Fig. 14, where the maximal negative vertical bending occurs. The detection results by applying Td-MPCA are shown in Fig. 15(a). The detection is successful without any delay. The performance of MPCA on all four sensor records is given in Fig. 15(b), where the delay of detection is obvious. Overall, several points can be concluded from the Ricciolo curved viaduct case study. First of all, the ICA-separated thermal strain is enabling subsequent anomaly detection as there is hidden information in temperature-induced strain. Moreover, by comparing Fig. 15(a) and (b), the delay in anomaly alarm by applying MPCA is well overcome when Td-MPCA is implemented. It corroborates that the proposed temperature-driven anomaly detection method is more efficient for anomaly detection.

This work was supported by the British Council (Grant ID: 217544274) and China Scholarship Council. References [1] Inaudi D. Long-term static structural health monitoring. Struct Congress 2010:566–77. https://doi.org/10.1061/41130(369)52. [2] Van Buren K, Reilly J, Neal K, Edwards H, Hemez F. Guaranteeing robustness of structural condition monitoring to environmental variability. J Sound Vib 2017;386:134–48. https://doi.org/10.1016/j.jsv.2016.08.038. [3] Chang PC, Flatau A, Liu SC. Review paper: Health monitoring of civil infrastructure. Struct Heal Monit 2003;2:257–67. https://doi.org/10.1177/145792103036169. [4] American Society of Civil Engineers. Structural Identification of Constructed Systems. Reston, VA: American Society of Civil Engineers; 2013. [5] Döhler M, Hille F, Mevel L, Rücker W. Structural health monitoring with statistical methods during progressive damage test of S101 Bridge. Eng Struct 2014;69:183–93. https://doi.org/10.1016/j.engstruct.2014.03.010. [6] Xia Y, Chen B, Zhou X, Xu Y. Field monitoring and numerical analysis of Tsing Ma Suspension Bridge temperature behavior. Struct Control Heal Monit 2013;20:560–75. https://doi.org/10.1002/stc.515. [7] Catbas FN, Susoy M, Frangopol DM. Structural health monitoring and reliability estimation: Long span truss bridge application with environmental monitoring data. Eng Struct 2008;30:2347–59. https://doi.org/10.1016/j.engstruct.2008.01.013. [8] Dilger Walter H, Ghali Amin, Chan Mathew, Cheung Mo S, Maes Marc A. Temperature Stresses in Composite Box Girder Bridges. J Struct Eng 1983;109(6):1460–78. https://doi.org/10.1061/(ASCE)0733-9445(1983) 109:6(1460). [9] Ding Y-L, Wang G-X. Estimating extreme temperature differences in steel box girder using long-term measurement data. J Cent South Univ 2013:20. https://doi.org/10. 1007/s11771-013-1766-6. [10] Zhou L, Xia Y, Brownjohn JMW, Koo KY. Temperature analysis of a long-span suspension bridge based on field monitoring and numerical simulation. J Bridg Eng 2016;21:04015027. https://doi.org/10.1061/(ASCE)BE.1943-5592.0000786. [11] Yarnold M, Moon F, Dubbs N, Aktan A. Evaluation of a long-span steel tied arch bridge using temperature-based structural identification. International Association for Bridge Management and Safety 2012. https://doi.org/10.1201/b12352-361. [12] Xu YL, Chen B, Ng CL, Wong KY, Chan WY. Monitoring temperature effect on a long suspension bridge. Struct Control Heal Monit 2010;17:632–53. https://doi.org/10. 1002/stc.340. [13] de Battista N, Brownjohn JMW, Tan HP, Koo K-Y. Measuring and modelling the thermal performance of the Tamar Suspension Bridge using a wireless sensor network. Struct Infrastruct Eng 2015;11:176–93. https://doi.org/10.1080/15732479. 2013.862727. [14] Westgate RJ. Environmental Effects on a Suspension Bridge’s performance. The University of Sheffield; 2012. [15] Chen G, Asce M, Yang X, Alkhrdaji T, Wu J, Nanni A. Condition assessment of concrete structures by dynamic signature tests. Eng Mech Spec Conf 1999. [16] Deraemaeker A, Reynders E, De Roeck G, Kullaa J. Vibration-based structural health monitoring using output-only measurements under changing environment. Mech Syst Signal Process 2008;22:34–56. https://doi.org/10.1016/j.ymssp.2007. 07.004. [17] Zhou HF, Ni YQ, Ko JM. Eliminating temperature effect in vibration-based structural damage detection. J Eng Mech 2011;137:785–96. https://doi.org/10.1061/ (ASCE)EM.1943-7889.0000273. [18] Yang C, Liu Y, Sun Y. Damage detection of bridges considering environmental temperature effect by using cluster analysis. Proc Eng 2016;161:577–82. https:// doi.org/10.1016/j.proeng.2016.08.695. [19] Ni YQ, Hua XG, Fan KQ, Ko JM. Correlating modal properties with temperature using long-term monitoring data and support vector machine technique. Eng Struct 2005;27:1762–73. https://doi.org/10.1016/j.engstruct.2005.02.020. [20] Ni YQ, Hua XG, Wong KY, Ko JM. Assessment of bridge expansion joints using longterm displacement and temperature measurement. J Perform Constr Facil 2007;21:143–51. https://doi.org/10.1061/(ASCE)0887-3828(2007) 21:2(143). [21] Yarnold MT. Temperature-based structural identification and health monitoring for long span bridge. Drexel University; 2013. [22] Yarnold MT, Moon FL. Temperature-based structural health monitoring baseline for long-span bridges. Eng Struct 2015;86:157–67. https://doi.org/10.1016/j. engstruct.2014.12.042. [23] Nguyen VH, Schommer S, Maas S, Zürbes A. Static load testing with temperature compensation for structural health monitoring of bridges. Eng Struct 2016;127:700–18. https://doi.org/10.1016/j.engstruct.2016.09.018. [24] Su JZ, Xia Y, Zhu LD, Zhu HP, Ni YQ. Typhoon- and temperature-induced quasistatic responses of a supertall structure. Eng Struct 2017;143:91–100. https://doi. org/10.1016/j.engstruct.2017.04.007. [25] Zhu Y, Jesus A, Laory I. Thermal effect identification and bridge damage disclosure by using blind source separation method. Struct Heal Monit 2017:527–34. https:// doi.org/10.12783/shm2017/13907. [26] Zhu Y, Ni Y-Q, Jesus A, Liu J, Laory I. Thermal strain extraction methodologies for bridge structural condition assessment. Smart Mater Struct 2018. https://doi.org/ 10.1088/1361-665X/aad5fb. [27] Posenato D, Lanata F, Inaudi D, Smith IFC. Model-free data interpretation for continuous monitoring of complex structures. Adv Eng Informatics

6. Conclusions In this paper, the Td-MPCA method is presented to separate the thermal-induced response and detect structural anomalies. The evaluation and assessment of the proposed method are conducted on three case studies. The first case study is the numerical simulation of a truss bridge with 5% stiffness loss on a chord that is away from sensor position. The second case study is on an experimental truss bridge with two damage scenarios. The last case study is on two months monitoring data including the construction period which was collected from the Ricciolo viaduct in Switzerland. According to the outcomes of the case studies, the following conclusions can be drawn.

• The apparent anomalous changes can be observed after ICA se•

paration in the first case study. This demonstrates that the temperature-induced variations in measurements can mask structural damage-induced effects; The proposed Td-MPCA is more sensitive than the original MPCA in detecting structural anomalies. In the first case study, MPCA fails to detect the simulated 5% stiffness loss, but Td-MPCA succeeds. Meanwhile, Td-MPCA is able to uncover both damage scenarios in the experimental case study, while MPCA fails or has some delay to disclose the damage. Moreover, the alarm delay is also apparent in the third case study by applying MCPA. In contrast, Td-MPCA performs well without such drawback, i.e. the punctual anomaly alarm can be obtained as expected even in the case of small-level damage, as evidenced in both second and third case studies. 457

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2008;22:135–44. https://doi.org/10.1016/j.aei.2007.02.002. [28] Posenato D, Kripakaran P, Inaudi D, Smith IFC. Methodologies for model-free data interpretation of civil engineering structures. Comput Struct 2010;88:467–82. https://doi.org/10.1016/j.compstruc.2010.01.001. [29] Laory I. Model-free methodologies for data-interpretation during continuous monitoring of structures. EPFL 2013. https://doi.org/10.5075/epfl-thesis-5518. [30] Hotelling H. Analysis of a complex of statistical variables into principal components. J Educ Psychol 1933;24:417–41. https://doi.org/10.1037/h0071325. [31] Jolliffe I. Principal component analysis. Int. Encycl. Stat. Sci., 2011:1094–6. [32] Liang YC, Lee HP, Lim SP, Lin WZ, Lee KH, Wu CG. Proper orthogonal decomposition and its applications—part I: Theory. J Sound Vib 2002;252:527–44. https://doi.org/10.1006/jsvi.2001.4041. [33] Liang YC, Lin WZ, Lee HP, Lim SP, Lee KH, Sun H. Proper orthogonal decomposition and its applications – Part II: Model reduction for MEMS dynamical analysis. J Sound Vib 2002;256:515–32. https://doi.org/10.1006/jsvi.2002.5007. [34] Lanata F, Del Grosso A. Damage detection and localization for continuous static monitoring of structures using a proper orthogonal decomposition of signals. Smart Mater Struct 2006;15:1811–29. https://doi.org/10.1088/0964-1726/15/6/036. [35] Figueiredo E, Park G, Figueiras J, Farrar C, Keith W. Structural health monitoring algorithm comparisons using standard data sets. Los Alamos Natl Lab 2009. [36] Mujica L, Rodellar J, Fernández A, Güemes A. Q-statistic and T2-statistic PCA-based measures for damage assessment in structures. Struct Heal Monit An Int J 2011;10:539–53. https://doi.org/10.1177/1475921710388972. [37] Mujica LE, Ruiz M, Pozo F, Rodellar J, Güemes A. A structural damage detection indicator based on principal component analysis and statistical hypothesis testing. Smart Mater Struct 2014;23:025014. https://doi.org/10.1088/0964-1726/23/2/ 025014. [38] Cross EJ, Manson G, Worden K, Pierce SG. Features for damage detection with insensitivity to environmental and operational variations. Proc R Soc A Math Phys Eng Sci 2012;468:4098–122. https://doi.org/10.1098/rspa.2012.0031. [39] Yan A-M, Kerschen G, De Boe P, Golinval J-C. Structural damage diagnosis under varying environmental conditions—Part I: A linear analysis. Mech Syst Signal Process 2005;19:847–64. https://doi.org/10.1016/j.ymssp.2004.12.002. [40] Yan A-M, Kerschen G, De Boe P, Golinval J-C. Structural damage diagnosis under varying environmental conditions—part II: local PCA for non-linear cases. Mech Syst Signal Process 2005;19:865–80. https://doi.org/10.1016/j.ymssp.2004.12. 003. [41] Mojtahedi A, Lotfollahi Yaghin MA, Ettefagh MM, Hassanzadeh Y, Fujikubo M. Detection of nonlinearity effects in structural integrity monitoring methods for offshore jacket-type structures based on principal component analysis. Mar Struct 2013;33:100–19. https://doi.org/10.1016/j.marstruc.2013.04.007. [42] Reynders E, Wursten G, De Roeck G. Output-only fault detection in structural engineering based on kernel PCA. BIL2014 Work Data-driven Model. methods Appl., Leuven, Belgium. 2014. [43] Reynders E, Wursten G, De Roeck G. Output-only structural health monitoring in changing environmental conditions by means of nonlinear system identification. Struct Heal Monit An Int J 2014;13:82–93. https://doi.org/10.1177/ 1475921713502836. [44] Shokrani Y, Dertimanis VK, Chatzi EN, Savoia M. Structural damage localization under varying. 11th HSTAM Int Congr Mech. 2016. [45] Loh C-H, Huang Y-T, Hsueh W, Chen J-D, Lin P-Y. Visualization and dimension reduction of high dimension data for structural damage detection. Proc Eng 2017;188:17–24. https://doi.org/10.1016/j.proeng.2017.04.452. [46] Datteo A, Lucà F, Busca G. Statistical pattern recognition approach for long-time monitoring of the G.Meazza stadium by means of AR models and PCA. Eng Struct 2017;153:317–33. https://doi.org/10.1016/j.engstruct.2017.10.022. [47] Comon P. Independent component analysis, a new concept? Signal Process 1994;36:287–314. https://doi.org/10.1016/0165-1684(94)90029-9. [48] Jutten C, Herault J. Blind separation of sources, part I: an adaptive algorithm based on neuromimetic architecture. Signal Process 1991;24:1–10. https://doi.org/10. 1016/0165-1684(91)90079-X. [49] Comon P, Jutten C, Herault J. Blind separation of sources, part II: problems

[50]

[51]

[52]

[53]

[54]

[55]

[56]

[57]

[58]

[59]

[60]

[61]

[62]

[63] [64]

[65]

[66]

[67]

[68]

458

statement. Signal Process 1991;24:11–20. https://doi.org/10.1016/0165-1684(91) 90080-3. Chaumette E, Comon P, Muller D. ICA-based technique for radiating sources estimation: application to airport surveillance. IEE Proc F Radar Signal Process 1993;140:395. https://doi.org/10.1049/ip-f-2.1993.0058. Bell AJ, Sejnowski TJ. An information-maximization approach to blind separation and blind deconvolution. Neural Comput 1995;7:1129–59. https://doi.org/10. 1162/neco.1995.7.6.1129. Zang C, Friswell MI, Imregun M. Structural damage detection using independent component analysis. Struct Heal Monit An Int J 2004;3:69–83. https://doi.org/10. 1177/1475921704041876. Poncelet F, Kerschen G, Golinval J-C, Verhelst D. Output-only modal analysis using blind source separation techniques. Mech Syst Signal Process 2007;21:2335–58. https://doi.org/10.1016/j.ymssp.2006.12.005. Yang Y, Nagarajaiah S. Time-frequency blind source separation using independent component analysis for output-only modal identification of highly damped structures. J Struct Eng 2013;139:1780–93. https://doi.org/10.1061/(ASCE)ST.1943541X.0000621. Chang J, Liu W, Hu H, Nagarajaiah S. Improved independent component analysis based modal identification of higher damping structures. Measurement 2016;88:402–16. https://doi.org/10.1016/j.measurement.2016.03.021. Yang Y, Nagarajaiah S. Dynamic imaging: real-time detection of local structural damage with blind separation of low-rank background and sparse innovation. J Struct Eng 2015;142:1–9. https://doi.org/10.1061/(ASCE)ST.1943-541X.0001334. Yang Y, Dorn C, Mancini T, Talken Z, Nagarajaiah S, Kenyon G, et al. Blind identification of full-field vibration modes of output-only structures from uniformlysampled, possibly temporally-aliased (sub-Nyquist), video measurements. J Sound Vib 2017;390:232–56. https://doi.org/10.1016/j.jsv.2016.11.034. Spiridonakos MD, Chatzi EN, Sudret B. Polynomial chaos expansion models for the monitoring of structures under operational variability. ASCE-ASME J Risk Uncertain Eng Syst Part A Civ Eng 2016;2:B4016003. https://doi.org/10.1061/ AJRUA6.0000872. Lanata F, Posenato D. A multi-algorithm procedure for damage location and quantification in the field of continuous static monitoring of structures. Key Eng Mater 2007;347:89–94. https://doi.org/10.4028/www.scientific.net/KEM.347.89. Lanata F, Posenato D, Inaudi D. Data anomaly identification in complex structures using model free data statistical analysis. 3rd Int. Conf. Struct. Heal. Monit. Intell. Infrastruct. Canada: Vancouver, British Columbia; 2007. Laory I, Trinh TN, Smith IFC. Evaluating two model-free data interpretation methods for measurements that are influenced by temperature. Adv Eng Informatics 2011;25:495–506. https://doi.org/10.1016/j.aei.2011.01.001. Cavadas F, Smith IF, Figueiras J. Damage detection using data-driven methods applied to moving load responses. Mech Syst Signal Process 2013;39:409–25. https://doi.org/10.1016/j.ymssp.2013.02.019. Hyvärinen A, Karhunen J, Oja E. Independent component analysis. John Wiley & Sons; 2004. Hyvärinen A, Oja E. Independent component analysis: algorithms and applications. Neural Networks 2000;13:411–30. https://doi.org/10.1016/S0893-6080(00) 00026-5. Gabriel KR. The biplot graphic display of matrices with application to principal component analysis. Biometrika 1971;58:453–67. https://doi.org/10.1093/ biomet/58.3.453. Hubert M, Rousseeuw PJ, Vanden Branden K. ROBPCA: A New Approach to Robust Principal Component Analysis. Technometrics 2005;47:64–79. https://doi.org/10. 1198/004017004000000563. Laory I, Trinh TN, Posenato D, Smith IFC. Combined model-free data-interpretation methodologies for damage detection during continuous monitoring of structures. J Comput Civ Eng 2013;27:657–66. https://doi.org/10.1061/(ASCE)CP.1943-5487. 0000289. Glišić B, Posenato D, Inaudi D, Figini A. Structural health monitoring method for curved concrete bridge box girders. Sensors Smart Struct Technol Civil, Mech Aerosp Syst 2008. https://doi.org/10.1117/12.778643.