International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

The continuous wavelet transform as a stochastic process for damage detection Balafas, Konstantinos; Rajagopal, Ram; Kiremidjian, Anne S. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015The Continuous Wavelet Transform as a Stochastic Process forDamage DetectionKonstantinos BalafasPh.D. Candidate, Dept. of Civil and Environmental Engineering, Stanford University,Stanford, CA, U.S.A.Ram RajagopalAssistant Professor, Dept. of Civil and Environmental Engineering, Stanford University,Stanford, CA, U.S.A.Anne S. KiremidjianProfessor, Dept. of Civil and Environmental Engineering, Stanford University, Stanford,CA, U.S.A.ABSTRACT: This paper presents the formulation of a novel statistical model for the wavelet transform ofthe acceleration response of a structure based on Gaussian Process Theory. The model requires no priorknowledge of the structural properties and all the model parameters are learned directly from the mea-sured data using Maximum Likelihood Estimation. The proposed model is applied to the data obtainedfrom a series of shake table tests and the results are presented. The results, even at a proof-of-conceptlevel, appear to correlate well with the ocurrence of damage, which is an indication of the validity of theunderlying model. The results from the use of a simple metric for the detection of damage are presentedas well.1. INTRODUCTIONThe application of Statistical Pattern Recognition(SPR) in the field of Structural Health Monitor-ing (SHM) has received significant attention byresearchers over the past few decades, especiallyin the context of vibration analysis of structures.There has been considerable research in the appli-cation of various pattern recognition methods fordamage detection (Farrar and Sohn (2000); Sohnet al. (2001); Sohn and Farrar (2001)) while a moreformal presentation of the Statistical Pattern Recog-nition Paradigm can be found in Fugate et al. (2000)and Farrar and Worden (2007). In SPR, damageis detected through changes or outliers in statisticalfeatures that are obtained directly from the acquireddata rather than by changes in estimates of struc-tural properties. As a result, one of the advantagesof SPR is that limited to no knowledge of the struc-tural properties is required. This allows for toolsand algorithms that are modular and eliminate theuncertainty around developing a structural modeland estimating its parameters.A mathematical model that is very widely used inSHM and especially under the Statistical PatternRecognition Paradigm is the Continuous WaveletTransform (CWT). Research on the application ofthe CWT for SHM includes the observation ofchanges in the wavelet coefficients under differ-ent loading conditions (Melhem and Kim (2003);Kim and Melhem (2004)), the extraction of featuresfrom the CWT (Sun and Tang (2002); Robertsonet al. (2003); Noh et al. (2011)) and the combinationof the CWT with other signal processing methodssuch as Empirical Mode Decomposition (EMD) (Liet al. (2007)). The literature on the application ofwavelets in the field of SHM is so rich that hasspurred the publication of several review papers.Comprehensive reviews on the intersection of thewavelet transform and SHM can be found in Pengand Chu (2004), Taha et al. (2006), or Staszewski112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015and Robertson (2007).This paper presents a novel statistical model of thewavelet coefficients at each time sample as a Gaus-sian Process (GP). The scope of the present pa-per is to present the mathematical formulation ofthe statistical model and provide proof-of-conceptfor the efficacy of the model for damage detection.In keeping with the Statistical Pattern RecognitionParadigm, all of the model parameters are estimateddirectly from the acquired data and no knowledgeof the properties of the monitored structure is re-quired, other than an undamaged baseline signal.Only the structural response is required in order todetect damage. The effect of the input excitation isaccounted for through the model parameters.The proposed model is applied to the data acquiredfrom 13 sequential shake table tests on a rein-forced concrete bridge pier conducted at the Uni-versity of Nevada, Reno by Choi et al. (2007). Thisdataset provides an excellent testbed for the pro-posed model due to the large number of experimen-tal runs, the progressive development of structuraldamage and the detailed documentation of the dam-age. An extensive presentation of the experimentalsetup, the testing protocol and the damage docu-mentation can be found in Choi et al. (2007).The present analysis is limited to Single Degree ofFreedom (SDOF) systems under earthquake load-ing. The SDOF system was selected as a startingpoint due to its simplicity and the fact that it canstill approximate well important types of structuressuch as bridge piers. However, the statistical modelproposed is not necessarily limited to SDOF sys-tems as the type of the structure does not affect themodel, which only requires a response data streamas input.The response of a structure to earthquake loading,as a non-stationary signal, is very well suited forwavelet analysis. The high intensity of the earth-quake loading (compared to ambient vibration orwind loading) ensures that any potential occurrenceof damage will be reflected in the recorded responsewhile the relatively short duration of the earthquakeisolates the effects of damage in the signal from en-vironmental effects that can affect the response ofthe structure. As was the case with the type of struc-ture, the proposed model need not be limited to thistype of loading and can be used for other types ofdynamic excitation.2. STATISTICAL MODEL FORMULATIONLet a(t) be the acceleration response of the sys-tem, where t denotes time. The Continuous WaveletTransform (CWT) of the signal a(t) will be denotedas Wa(u,s) and is defined asWa(u,s) =∞∫−∞a(t)1√sψ∗(t−us)dt (1)where u refers to shift (a measure of time), s refersto wavelet scale (a measure of frequency) and the(·)∗ operator is the complex conjugate. Let yt (s)be the wavelet coefficients at shift t which will bereferred to as wavelet “slice” at time (s) =Wa(t,s) (2)Let us define a random process of wavelet scale,Ψ(s) that represents the fundamental shape of thewavelet slices and only depends on the damagestate of the structure. The realizations of Ψ(s). foreach time t are denoted by Ψt (s).Assumption 1. The realizations of Ψ(s) can bewritten as:Ψt (s) = Ψ(s)+ εt (s) (3)where Ψ(s) is an unobservable function of waveletscale that only depends on the damage state of thestructure.Assumption 2. The error terms εt (s) are realiza-tions of a zero-mean Gaussian Process (GP) withcovariance function kε (s,s′).While a different statistical model or distribu-tion could potentially be used, the assumption ofa Gaussian Process is made for simplicity and com-putational efficiency. Testing the validity of this as-sumption is part of the authors’ current work andwill be presented in a future publication.Define the functional F for any function f suchthat:F ( f ;a,b,c)(s) = a · f (b · s+ c) (4)212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Assumption 3. Each wavelet slice, yt (s) can be ex-pressed as:yt (s) =F (Ψ;at ,bt ,ct)(s)+∆y(s)= atΨ(bt · s+ ct)+∆y(s) (5)where at , bt and ct are, generally unobservable,scalar parameters that represent the effect of the in-put motion on the system’s response and ∆y(s) is afunction of scale that represents effects not capturedby the first term.Null Hypothesis: while the structure is undam-aged, the realizations of Ψ(s) are drawn from thesame distribution. When damage occurs, the be-havior of the structure changes and, thus, the shapeof Ψ(s) is assumed to change. Since Ψ(s) is un-observable, that change in shape is manifested as achange in the distribution of Ψ(s).Combining Equations 3 and 5, we can obtain thefollowing expression for the wavelet coefficients attime t:yt (s) = atΨ(bt · s+ ct)+atεt (bt · s+ ct)+∆y(s)=F(Ψ;at ,bt ,ct)(s)+F (εt ;at ,bt ,ct)(s)+∆y(s) (6)Without loss of generality, we can define the slice attime t0 as a reference slice. The reference slice willserve as a baseline for the model and, thus, wouldhave to correspond to the undamaged state of thestructure. In the present analysis, the reference sliceis selected manually so that it is relatively smoothan its general shape is representative of the shape ofthe majority of the rest of the slices. Equation 5 canbe written for the reference slice:y0 (s) =F (Ψ;a0,b0,c0)(s)+∆y(s)= a0Ψ(b0 · s+ c0)+∆y(s)= a0Ψ(b0 · s+ c0)+a0ε0 (b0 · s+ c0)+∆y(s) (7)Solving Equation 6 for Ψ(s) and substituting inEquation 7, we obtain:y0 (s) =a0atyt(b0bts+c0− ctbt)−a0εt (b0s+ c0)−a0at∆y(b0bts+c0− ctbt)+a0ε0 (b0s+ c0)+∆y(s) (8)Define the following:a˜t =a0at(9a)b˜t =b0bt(9b)c˜t =c0− ctbt(9c)ε˜t (s) = a0ε0 (b0s+ c0)−a0εt (b0s+ c0) (9d)∆˜y(s) = ∆y(s)− a˜t∆y(b˜ts+ c˜t)(9e)Equation 8 then becomes:y0 (s) = a˜tyt(b˜ts+ c˜t)+ ε˜t (s)+ ∆˜y(s) (10)The scalar parameters a˜t , b˜t and c˜t can be estimatedfrom the data once the reference slice, y0 (s) hasbeen selected.In order to obtain an estimate for the transformederror term, ε˜t (s), the wavelet slices can be trans-formed as follows:y′t (s) = a˜tyt(b˜ts+ c˜t)(11)As a result, and since the transformed error term,ε˜t (s) is zero-mean, an estimate for the transformed,unmodeled effects term, ∆˜y(s) can be obtained by:∆̂y(s) =t=N∑t=1y0 (s)− y′t (s) (12)n estimate for the transformed noise terms, ε˜t (s)can be obtained as:εˆt (s) = y0 (s)− y′t (s)− ∆̂y(s) (13)In essence, this transformation “fits” each waveletslice to the reference one. While the initial param-eters, a, b and c, are not recovered, all the trans-formed slices refer to the same baseline (the refer-ence slice) in terms of signal energy and bandwidth312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015and, thus, the influence of the amplitude and fre-quency content of the input motion is removed. Thetransformed noise estimates, as well as the param-eters calculated from Equation 9, can then be usedto test whether the slices are indeed drawn from thesame distribution. The case where not all slices aredrawn from the undamaged distribution implies thatdamage has occurred in the structure.3. ESTIMATION OF THE MODEL PARAMETERSThe parameters that need to be estimated are: a˜, b˜and c˜, the bias term ∆˜y and the transformed noisecovariance matrix, Σε˜ . The vectors a˜, b˜ and c˜ havesize N × 1, where N is the number of time sam-ples, the bias term is an M×1 vector and the trans-formed noise covariance matrix is an M×M ma-trix, where M is the number of scales at which thewavelet transform is calculated.The estimation of the model parameters requires theestimation of 3 ·N+M+M ·(M+1)/2 values. Fora typical acceleration response record and a reason-able amount of wavelet scales, the simultaneous es-timation of the parameters is computationally veryexpensive. For that reason, a recursive algorithm isused in the present analysis:Step 1 Initialize ∆y and Σε˜ to an uninformed prior.In the present analysis, ∆y(0) = 0 and Σ(0)ε˜ = I,where I is the M×M identity matrix.Step 2 For each t = 1 . . .N, calculate the param-eters aˆ(k+1)t , bˆ(k+1)t , cˆ(k+1)t using MaximumLikelihood Estimation.Step 3 Calculate the transformed slices from Equa-tion 11.Step 4 Calculate the bias term from Equation 12.Step 5 Calculate the error terms from Equation 13.Step 6 Estimate the covariance matrix from the er-ror terms.Step 7 Repeat Steps 2 through 6 until the bias termand covariance matrix converge.A more extensive presentation of this algorithm,presentation of alternative methods and discussionon the estimation of the model parameters will beprovided in a future paper.It should be noted that this algorithm need only beapplied to a reference signal where the structure isa priori assumed to be undamaged. Once the co-variance matrix and bias terms are estimated, theyessentially describe the structure’s undamaged be-havior since, as mentioned previously, a change inthe damage state of the structure will be reflectedin these two parameters. Then, the learned covari-ance matrix and bias term can be directly appliedto a signal where the structure’s damage state is un-known.The outlined algorithm requires the estimation andinversion of the covariance matrix of a generallyhigh-dimensional random variable. The estimationof the covariance matrix is a well-studied problemand several parametric and non-parametric estima-tion methods exist in the literature (e.g. Ander-son (1973); Andrews (1991); Chen et al. (2010)).The sample covariance matrix was found to be nu-merically unstable, especially after a few iterationsof the presented algorithm and did not converge.To overcome this problem, a parametric covari-ance function was fit to the data and its param-eters were estimated by maximizing the marginallog-likelihood of the data. In the results presentedin subsequent sections, a Matern covariance func-tion was used and the fitting was performed us-ing the Gaussian Processes for Machine Learning(GPML) toolbox, which is based on Rasmussen andWilliams (2006).4. EXPERIMENTAL VALIDATION4.1. Dataset descriptionIn order to evaluate the validity of the proposedmodel and its underlying assumptions, it is appliedto the experimental data obtained from a series ofshake table tests conducted at the University ofNevada, Reno by Choi et al. (2007). This dataset,as mentioned in a previous section, presents an ex-cellent benchmark for this model due to the largenumber of experimental runs and the progressionof damage and documentation thereof. The speci-men used was a 3 : 10 scale model of a reinforcedconcrete bridge pier. The acceleration response thatwas used in the analysis was measured at the top of412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015the pier. The testing sequence consisted of 13 earth-quake runs, each with increasing intensity. The firsttwo runs are damage-free, damage first occurs dur-ing Run 3 and progresses with each subsequent run.A comprehensive presentation of the experimentalset up and damage description can be found in Choiet al. (2007).4.2. Parameter distributionsThe transformed slices, residuals (error terms) andmodel parameters for the 13 experimental runswere calculated. The reference slice was selectedmanually from the first experimental run where thestructure was known to be damage-free. Figure 1shows heatmaps of the transformed residuals for sixdifferent experimental runs. For clarity, only Runs1, 2, 3, 6, 9 and 12 are shown. As mentioned pre-viously, the first two runs are undamaged, Run 3marks the first occurrence of damage and Runs 4through 13 exhibit progressively increasing dam-age. This is consistent with what can be visuallyobserved in Figure 1. The residuals in the first tworuns are approximately zero (the mean of the er-ror term process) for almost all time samples whileclusters of slices with significantly different shapeappear from Run 3 onwards with increasing dura-tion and intensity.Run 1Scale1000 1200 140050100150Run 21000 150050100150Run 31000 150050100150Run 6ScaleTime Sample1000 1500 200050100150Run 9Time Sample500 1500 250050100150Run 12Time Sample500 1500 250050100150Figure 1: Transformed residuals for different experi-mental runsFigures 2 through 4 show histograms of the am-plitude, stretch and shift parameters, respectively,for the same experimental runs shown in Figure 1.It can be observed that the distributions of the pa-rameters are generally stable in the two undamagedcases (Runs 1 and 2) and change when damage oc-curs in the structure, even at low levels of damage(Run 3). This shift in distribution for the modelparameters clearly demonstrates a sensitivity of theproposed model to damage in the structure.0 5 10050100 Run 1Number of observations0 5 100100200300 Run 20 5 100100200300 Run 30 5 100100200300400 Run 6Number of observationsAmplitude parameter 0 50 1000200400600 Run 9Amplitude parameter 0 100 2000200400600 Run 12Amplitude parameterFigure 2: Histograms of the absolute value of the am-plitude parameter0 1 2 3050100 Run 1Number of observations0 1 2 3050100150 Run 20 1 2 3050100150 Run 30 1 2 3050100150 Run 6Number of observationsStretch parameter 0 1 2 30100200300 Run 9Stretch parameter 0 1 2 30100200300 Run 12Stretch parameterFigure 3: Histograms of the stretch parameter5. APPLICATION ON DAMAGE DETECTIONBased on visual inspection of Figure 1, the occur-rence of damage can be correlated with the pres-ence of clusters of outlying slices. For that rea-512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015−10 0 10050100 Run 1Number of observations−10 0 10050100150 Run 2−10 0 10050100150200 Run 3−10 0 100100200300400 Run 6Number of observationsShift parameter −10 0 100200400600800 Run 9Shift parameter −10 0 100200400600800 Run 12Shift parameterFigure 4: Histograms of the shift parameterson, it is reasonable to investigate the temporal pro-gression and statistical distribution of the SquaredMahalanobis Distance (SMD), which, for a randomvariable X , is defined as:d2M = (X−µX)T Σ−1X (X−µX) (14)and is a commonly used metric for the identifica-tion of multivariate outliers. Figure 5 shows time-series plots for the SMD during the experimentalruns shown in previous plots, while Figure 6 showsthe corresponding histograms of the SMD.1000 150000.511.5 Run 1Squared MahalanobisDistance1000 150000.511.5 Run 21000 150000.511.5 Run 31000 1500 200000.511.5 Run 6Squared MahalanobisDistanceTime Sample 1000 200000.511.5 Run 9Time Sample 1000 200000.511.5 Run 12Time SampleFigure 5: Time-series of the Squared MahalanobisDistanceVisually, a differentiation between runs withoutdamage (Runs 1 and 2) and runs with damage (Runs0 0.5 1 1.5050100150 Run 1Number of observations0 0.5 1 1.5050100150 Run 20 0.5 1 1.50100200 Run 30 0.5 1 1.50100200300 Run 6Number of observationsSquaredMahalanobisDistance0 0.5 1 1.50200400 Run 9SquaredMahalanobisDistance0 0.5 1 1.50200400600 Run 12SquaredMahalanobisDistanceFigure 6: Histograms of the Squared MahalanobisDistance3 through 13) can be observed in both Figure 5and 6. Despite the presence of a high-frequencycomponent, the lower envelope of the plots in Fig-ure 5 gives a good indication of the time intervalswhere outliers occur, which can be used for esti-mating the extent of damage.While this is, by no means, a complete damage de-tection scheme, it presents an important validationof the underlying model and its assumptions. TheSMD, while a useful metric, collapses a multivari-ate random variable into just one number and doesnot take into account additional information that theproposed model provides in the form of the ampli-tude, stretch and scale parameters. The informationfrom the residuals and the model parameters willbe combined to develop a robust damage detectionalgorithm. The details of this algorithm will be pre-sented by the authors in a subsequent paper (Bal-afas et al. (2015)).6. CONCLUSIONSThis paper presents the formulation of a novel sta-tistical model for the wavelet transform of the ac-celeration response of a structure. In the proposedmodel, the wavelet coefficients at each moment intime are considered transformed realizations of afundamental Gaussian Process that is only depen-dent upon the presence or not of damage in thestructure. The model only considers the structuralresponse; the input excitation is not required and612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015its effects on the response signal are taken into ac-count by the model using an amplitude parame-ter and two, stretch and shift, parameters in thewavelet scale domain. The model also requires noprior knowledge of the structural properties and allthe model parameters are learned directly from themeasured data. As such, it eliminates the need forcomplex Finite Element Models and the uncertaintyinvolved with material or geometric properties ofthe structure. The parameters of the model are es-timated using Maximum Likelihood Estimation incombination with a recursive algorithm due to theirlarge number of values to be estimated. In calcu-lating the model parameters, it is required to esti-mate the covariance matrix of a multivariate sam-ple. In order to avoid a poorly conditioned matrix,the covariance matrix is estimated through trainingof a Gaussian Process. The proposed model is ap-plied to the data obtained from a series of shaketable tests and the results are presented. The distri-butions of the model parameters are stable while thestructure remains undamaged, and shift with the oc-currence and progression of damage. Furthermore,based on visual observation of the model residuals,there is potential for temporal localization of dam-age. This would provide information on the numberand duration of time intervals where damage oc-curs, information that is potentially important forthe assessment of the extent of damage. The re-sults of the presented analysis, even at a proof-of-concept level, are very encouraging, which is anindication of the validity of the underlying model.This paper shows the results from the use of a sim-ple metric for the detection of damage, also withencouraging results. The derivation of a damagedetection algorithm that would make use of all themodel parameters and learn the parameter distribu-tions in order to detect and classify damage throughhypothesis testing is part of the authors’ work andwill be presented in a future paper.7. ACKNOWLEDGEMENTSThe authors would like to thank the following or-ganizations for their support and funding of thisresearch: The National Science Foundation andthe Network for Earthquake Engineering Simula-tion (NEES) for their funding through Grant No.NEESR 105651, the Stanford Graduate Fellow-ship program for their funding through the Arvan-itidis Fellowship and the John A. Blume Earth-quake Engineering Research Center for their fund-ing through the John A. Blume Research Fellow-ship. They would also like to thank the followingpeople for their help, collaboration and data shar-ing: Dr. Hoon Choi, formerly of the Universityof Nevada, Reno, Professor M. “Saiid” Saiidi andKelly Doyle of the University of Nevada, Reno andAllen Cheung of Yelp.8. REFERENCESAnderson, T. W. (1973). “Asymptotically efficient es-timation of covariance matrices with linear struc-ture.” The Annals of Statistics, 1(1), 135–141.Andrews, D. W. K. (1991). “Heteroskedasticity and au-tocorrelation consistent covariance matrix estima-tion.” Econometrica, 59(3), 817–858.Balafas, K., Rajagopal, R., and Kiremidjian, A. S.(2015). “The wavelet transform as a gaussian pro-cess for damage detection.” Structural Control andHealth Monitoring (in preparation).Chen, Y., Wiesel, A., Eldar, Y. C., and Hero, A. (2010).“Shrinkage algorithms for MMSE covariance esti-mation.” IEEE Transactions on Signal Processing,58(10), 5016–5029.Choi, H., Saiidi, M. S., and Somerville, P. (2007). “Ef-fects of near-fault ground motion and fault-ruptureon the seismic response of reinforced concretebridges.” Report No. CCEER-07-06, University ofNevada, Reno (December).Farrar, C. R. and Sohn, H. (2000). “Pattern recogni-tion for structural health monitoring.” Workshopon Mitigation of Earthquake Disaster by AdvancedTechnologies, Las Vegas, NV, USA.Farrar, C. R. and Worden, K. (2007). “An introduction tostructural health monitoring.” Philosophical Trans-actions of the Royal Society A, 365, 623–632.Fugate, M. L., Sohn, H., and Farrar, C. R. (2000). “Un-supervised learning methods for vibration-baseddamage detection.Kim, H. and Melhem, H. (2004). “Damage detection ofstructures by wavelet analysis.” Engineering Struc-tures, 26(3), 347–362.Li, H., Deng, X., and Dai, H. 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N., Farrar, C. R., and Sohn, H. (2003).“Singularity detection for structural health moni-toring using holder exponents.” Proc. SPIE 5057,Smart Structures and Materials: Smart Systemsand Nondestructive Evaluation for Civil Infras-tructures, 569.Sohn, H. and Farrar, C. R. (2001). “Damage diagno-sis using time series analysis of vibration signals.”Smart Materials and Structures, 10(3), 446.Sohn, H., Farrar, C. R., Hunter, N. F., and Worden,K. (2001). “Structural health monitoring usingstatistical pattern recognition techniques.” Journalof Dynamic Systems, Measurement and Control,123(4), 706–711.Staszewski, W. J. and Robertson, A. N. (2007). “Time-frequency and time-scale analyses for structuralhealth monitoring.” Philosophical Transactions ofthe Royal Society A, 365(1851), 449–477.Sun, Q. and Tang, Y. (2002). “Singularity analysis usingcontinuous wavelet transform for bearing fault di-agnosis.” Mechanical Systems and Signal Process-ing, 16(6), 1025–1041.Taha, M. M. 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