Damage diagnosis algorithm using a sequential change pointdetection method with an unknown distribution for damage

Hae Young Noha, Ram Rajagopala, and Anne S. Kiremidjianb

aStanford Sustainable Systems Laboratory, Department of Civil and EnvironmentalEngineering, Stanford University, Stanford, CA, 94305, USA;

bDepartment of Civil and Environmental Engineering, Stanford University, Stanford, CA,94305, USA

ABSTRACT

This paper introduces a damage diagnosis algorithm for civil structures that uses a sequential change pointdetection method for the cases where the post-damage feature distribution is unknown a priori. This algorithmextracts features from structural vibration data using time-series analysis and then declares damage using thechange point detection method. The change point detection method asymptotically minimizes detection delay fora given false alarm rate. The conventional method uses the known pre- and post-damage feature distributions toperform a sequential hypothesis test. In practice, however, the post-damage distribution is unlikely to be known apriori. Therefore, our algorithm estimates and updates this distribution as data are collected using the maximumlikelihood and the Bayesian methods. We also applied an approximate method to reduce the computation loadand memory requirement associated with the estimation. The algorithm is validated using multiple sets ofsimulated data and a set of experimental data collected from a four-story steel special moment-resisting frame.Our algorithm was able to estimate the post-damage distribution consistently and resulted in detection delaysonly a few seconds longer than the delays from the conventional method that assumes we know the post-damagefeature distribution. We confirmed that the Bayesian method is particularly efficient in declaring damage withminimal memory requirement, but the maximum likelihood method provides an insightful heuristic approach.

Keywords: Damage diagnosis algorithm, sequential change point detection, hypothesis test, false alarm, detec-tion delay, autoregressive model, maximum likelihood estimation, Bayesian updating

1. INTRODUCTION

Structural health monitoring (SHM) systems for civil structures plays an important role in providing safe andfunctional built environment by diagnosing and prognosing structural damage efficiently and reliably. In thepast few decades, there has been a great effort to develop the hardware aspects of SHM;1 however, we still needto develop robust and reliable algorithms that can take full advantage of the collected data. The challenges forproperly analyzing civil structures come from the fact that civil structures are complex systems with variousmaterials and complicated geometry. In addition, they are subjected to significant uncertainties in loadingand environmental conditions. Thus, identifying the features that are sensitive to damage but insensitive toenvironmental and loading conditions is difficult, and the features inherently contain uncertainties because of theaforementioned reasons and measurement errors. Therefore, we need to develop algorithms that appropriatelyconsider the uncertainties of the features.

In this paper, we introduce a vibration-based damage diagnosis algorithm that is computationally efficient,considers uncertainties of individual features and time of damage, and sequentially processes data to make a de-cision. This algorithm uses time-series based damage sensitive features (DSFs) that are computationally efficientto extract from structural responses. As a result, it is particularly suitable to embed in wireless sensing units.In the past decade, various time-series based features have been developed for damage diagnosis algorithms.2–9

Further author information: (Send correspondence to Hae Young Noh.)Hae Young Noh: E-mail: noh@stanford.edu, Telephone: 1 650-862-5453Ram Rajagopal: E-mail: ramr@stanford.edu, Telephone: 1 650-725-4268Anne S. Kiremidjian: E-mail: ask@stanford.edu, Telephone: 1 650-723-4164

Among these numerous options, we use the coefficients of the autoregressive (AR) model as a DSF becauseof their relationship to structural parameters7 and their computational efficiency. The algorithm also uses thechange point detection method,10 which sequentially receives features extracted from structural vibration dataand conducts a hypothesis test. The test involves declaring damage on the basis of the allowable false alarmrate and the posterior distribution of the time of damage. This distribution is computed using the pre- andpost-damage feature distributions and the prior distribution of the time of damage. Consequently, the algorithmconsiders uncertainties of both the feature and the time of damage and systematically combines the informationfrom the previously collected DSFs, instead of relying on decisions based on evaluating a single sample. Thisis important because DSF values are a sequence of weak or noisy damage indicators over time. In addition,unlike the previous damage diagnosis algorithm that uses the change point detection method,11 which assumesthat the pre- and post-damage feature distributions are known a priori, the proposed algorithm estimates thepost-damage distribution using the collected data, which is more applicable in practice.

Our contributions are development of various methods to estimate and update the post-damage feature dis-tribution as we collect data. We also developed an approximate method to reduce computation and memory loadand evaluated their performance. With the estimated distribution, we can represent the damage identificationproblem as the trade-off between the false alarm rate of declaring damage before it happens and the detectiondelay in time incurred to assert damage after it happens.12,13 Sohn et al.4 used sequential probability ratiotests for the standard deviations of the residual of the AR model to classify structural damage, but they didnot consider the uncertainty of the time of damage. Under appropriate assumptions, the change point detectionmethod minimizes the damage detection delay for an allowed probability of falsely declaring damage using aBayesian framework. Therefore, once we choose a false alarm rate, which sets the reliability of the algorithm,the algorithm automatically optimizes the detection delay to locate the damage both in space and time. Ouralgorithm approximates this optimal detection delay closely by updating the estimation of the feature distribu-tion every time a new DSF is collected. Barnard14 proposed the idea to use the maximum likelihood method todetect mean shifts of data with known variance when the amount of shift is unknown, and Siegmund and Venka-traman15 solved the statistical properties of this procedure. Our algorithm incorporates the Bayesian framworkto the problem by assigning a prior distribution to the time of damage and reformulate the procedure usingthe two approaches to estimate the unknown distribution: the maximum likelihood method and the Bayesianmethod.

We verified this algorithm using multiple sets of simulated data and a set of experimental data obtained froma series of shake table tests. The simulation involved the generation of a set of 10,000 time-series that consistof random numbers whose distribution changes during the simulation in order to test the consistency and therobustness of the algorithm. The experiment involved the collection of ambient-vibration response data from afour-story steel moment-resisting frame for multiple damage scenarios. The DSF is extracted from each data,which is sequentially processed using the proposed algorithm to estimate the post-damage feature distributionand then the change point detection mehtod to identify damage with minimal delay within the given false alarmrate. The algorithm resulted in detection delay comparable to the theoretical optimal bound while the falsealarm rate remained below the limit.

This paper is organized as follows. The algorithm section defines the DSF using the AR model and describesthe conventional hypothesis test, sequential change point detection method, and the new method to estimatethe feature distribution for a damaged case. The validation section describes the simulation and the shake tableexperiment and the application results. Finally, the conclusion section summarizes the paper and discusseslimitations of the algorithm and future directions.

2. ALGORITHM

The algorithm consists of three steps: (i) collection of structural responses, (ii) extraction of a damage sensitivefeature (DSF), and (iii) identification and classification of damage. Figure 1 provides the summary of thealgorithm. The first step involves sequentially obtaining structural responses and normalizing them. In thesecond step, we apply the AR model to extract the DSF. In this study, the first AR coefficient is defined as theDSF because they are closely related to structural parameters as derived by Nair et al.7 and sensitive to damageas shown in.11,16 Nair et al. also showed that the distribution of the first AR coefficients follow a Gaussian

Figure 1. Summary of the algorithm.

mixture model.17 The algorithm then defines a score that involves the distributions of the DSF before and afterthe damage. The change point detection method essentially performs a sequential hypothesis test to identifydamage, which is equivalent to a threshold test of the score based on the desired false alarm rate. This thirdstep allows the algorithm to benefit from the relationship of the observed features in time when the currentdamage state is unknown. In this algorithm, we assume that the damage is irreversible; thus, the algorithmcan eventually identify damage if it keeps updating the score long enough. The change point detection method,however, minimizes the detection time delay for a desired false alarm rate. In other words, the method makes atrade-off between the detection time delay and the false alarm rate. Since the post-damage distribution of theDSF is unknown, two methods that use the maximum likelihood estimation and the Bayesian updating schemeare developed to estimate the score, and an approximate method is developed to compute the score efficiently.

Section 2.1 describes the DSF extracted from the collected acceleration responses of the structure in variousdamage states. Section 2.2 then introduces the sequential change point detection method that identify damagewhen the distribution of the DSF is unknown a priori for damaged cases.

2.1 Feature extraction

The AR model is a linear predictor, which predicts the system output as a linear function of previous outputsand captures the linear dynamics of ideal structures. To extract the feature, we first subtract the mean ofeach acceleration recording from the original recording to zero-mean, and then fit the AR model. Let xi(t)(i =1, 2, ..., N) denote the zero-meaned acceleration recordings collected from sensor i, where N is the number ofsensors. The single-variate AR model of order p is given as

xi (t) =

p∑k=1

αkxi(t− k) + εi(t) (1)

where αk is the kth AR coefficient and εi(t) is the residual. We investigated the performance of various combi-nations of these AR coefficients as DSFs, such as {α1}, {α1, α2}, and {α1, α2, α3}. The AR Modeling of thedata includes the removal of trends, obtaining the optimal model order, and checking the assumptions about theresiduals of the AR model. More details are given by Noh et al.16

2.2 Damage identification

We first describe the conventional sequential change point detection method that processes the data sequentiallyconsidering the prior distribution of the time of damage and the pre- and post-damage distributions of the DSFknown a priori. We then describe the method that addresses the cases when the post-damage distribution of theDSF is unknown.

2.2.1 Sequential change point detection method with known feature distributions

The single-fault sequential change point detection method addresses the problem of detecting a fault that happensat a random time λ as quickly as possible under a constraint of the probability of falsely declaring a fault beforeit happens. The sequence of observations {X(t), t ≥ 1} is assumed to consist of independent and identicallydistributed random variables that have the distribution f0 before the fault occurs(i.e., t < λ) and f1 afterthe fault (i.e., t ≥ λ). The distributions f0 and f1 are obtained prior to the analysis from simulations orexperiments. The algorithm that addresses the problem of unknown distribution f1 is explained in the nextsection. The optimal sequential procedure for solving the problem is to perform the following hypothesis test ateach time n based on the observations Fn(X) = {X(t), 1 ≤ t ≤ n}:18

H0 : λ > n,

H1 : λ ≤ n.(2)

This hypothesis test is equivalent to finding the fault detection time ν̄ according to the threshold test on theposterior probability ratio at time n for the false alarm rate P(ν̄ < λ) ≤ α:

Λn(X) =P(λ ≤ n |Fn(X) )

P(λ > n |Fn(X) )≥ Bα, (3)

where the threshold Bα is defined as 1−αα . This posterior probability ratio in Equation (3) can be expanded as

Λn(X) =P (λ ≤ n|Fn(X))

P (λ > n|Fn(X))=

n∑k=0

π(k)

k−1∏r=1

f0(Xr)

n∏r=k

f1(Xr)

∞∑k=n+1

π(k)

n∏r=1

f0(Xr)

=

n∑k=0

π(k)

n∏r=k

f1(Xr)

f0(Xr)

∞∑k=n+1

π(k)

, (4)

where π(n) is the prior distribution of λ (i.e., π(n) = P(λ = n)). The fault detection time ν̄ is then defined asthe earliest time when Λn(X) exceeds the threshold Bα:

ν̄ = inf{n : Λn(X) ≥ Bα}. (5)

The Shiryaev-Roberts-Pollak (SRP) procedure18 achieves the minimum asymptotic detection delay10

Dλm(ν̄) = Eλ [(ν̄ − λ)

m |ν̄ ≥ λ ] =̇

[| logα|

q1(X) + d

]m, (6)

where Eλ denotes expectation with respect to the prior of λ, =̇ denotes asymptotic upper and lower bounds asα→ 0, and typically m = 1 or 2. The delay bound is a function of the false alarm α, the inverse of the mean ofthe prior for λ, denoted by d, and the distance measure q1(X) between the two probability densities f0 and f1:

q1(X) =

∫f1(x) log

f1(x)

f0(x)µ(dx). (7)

This bound is the minimum asymptotic delay achievable by any method with false alarm rate α. Furthermore,if the test statistic in Equation (3) can be computed for n ≥ 1 using the recursion10

Λn(X) =

(Πn−1

ΠnΛn−1 +

π(n)

Πn

)f1(Xn)

f0(Xn),

Λ0 =π(0)

1− π(0),

(8)

where Πn is the prior complementary cumulative distribution of λ (i.e., Πn = P(λ > n)).

2.2.2 Sequential change point detection method with an unknown feature distribution fordamage

In practice, the distribution f1 is most likely to be unknown to us unless we have the means of exactly predictinghow damage will be incurred to the structure. In such cases, f1 can be estimated from the data, and then Λn(X)in Equation (4) can be computed accordingly. We present two methods to estimate f1. First, we estimate bythe maximum likelihood method such that the posterior probability ratio Λn(X) is maximized. For example,let f0 and f1 be N (0, σ) and N (µ, σ), respectively, where N (µ, σ) is the Gaussian distribution with mean µ andstandard deviation σ. These distributions can be substitued into the Equation (4) as follows:

Λn(X) =

n∑k=0

π(k)

n∏r=k

e−(Xr−µ)2

2σ2

e−X2r

2σ2

∞∑k=n+1

π(k)

. (9)

By the Jensen’s inequality, Equation (9) has the following lower bound:

log Λn(X) = logEλ

n∏r=k

e−(Xr−µ)2

2σ2

e−X2r

2σ2

− log

∞∑k=n+1

π(k)

≥ Eλ

log

n∏r=k

e−(Xr−µ)2

2σ2

e−X2r

2σ2

− log

∞∑k=n+1

π(k)

=

n∑k=0

π(k)

(n∑r=k

− (Xr − µ)2

2σ2+X2r

2σ2

)− log

∞∑k=n+1

π(k).

(10)

Because of the complexity of the Equation (9) when differentiated, we will instead find f1 that maximizes thelower bound in Equation (10). Let the lower bound in Equation (10) be denoted as Λ̃n(X). The optimalparameter of f1 can be found as follows:

dΛ̃n(X)

dµ=

d

dµ

(1

2σ2

n∑k=0

π(k)

(2µ

n∑r=k

Xr − µ2 (n− k + 1)

))

=1

σ2

n∑k=0

π(k)

(n∑r=k

Xr − µ (n− k + 1)

)= 0,

∴ µ̂ =

n∑k=0

π(k)

n∑r=k

Xr

n∑k=0

π(k) (n− k + 1)

(11)

If any prior knowledge about µ is available, we can then incorporate the knowledge by using (n− k + 1)µ′′k,n in

place of∑nr=kXr where µ′′k,n is the posterior mean of µ given the data {X(t), k ≤ t ≤ n}. For example, if µ has

the Gaussian prior distribution with mean µ′, and standard deviation σ′, then

µ′′k,n = E [µ|{X(t), k ≤ t ≤ n}, µ′, σ′] =µ′

σ′2 +∑nr=kXrσ2

1σ′2 + n−k+1

σ2

(12)

Another challenge for using the change point detection method with an unknown distribution of DSF afterdamage is that the recursive form in Equation (8) is no longer equivalent to the original form in Equation (4)

computed using the most recent estimation of µ because the recursive form is computed using the entire historyof the estimations of µ that is updated every time new data are collected. We impose a memory limitation toapproximate the original form in Equation (4) while keeping the computation simple. This approach limits theamount of accessible data for computing Λn(X). In other words, only the most recent m data, {X(t), n−m+1 ≤t ≤ n}, are used for computing Λn(X) as follows:

Λn(X) =P(n−m < λ ≤ n |Fn(X) )

1− P(λ ≤ n |Fn(X) )=

n∑k=n−m+1

π(k)

n∏r=k

f1(Xr)

f0(Xr)

∞∑k=n+1

π(k)

. (13)

The second method to estimate f1 involves the Bayesian statistics. Using the Bayes’ rule and the law of totalprobability, the posterior distribution of λ can be represented as

P(λ = k |Fn(X) ) =P(Fn(X)|λ = k)P(λ = k)

P(Fn(X))=

∫P(Fn(X)|λ = k, µ)P(µ)dµP(λ = k)

P(Fn(X))

=

∫ k−1∏r=1

f0(Xr)

n∏r=k

f1(Xr)P(µ)dµπ(k)

P(Fn(X))

=

π(k)

k−1∏r=1

f0(Xr)

∫ n∏r=k

f1(Xr)P(µ)dµ

P(Fn(X))

(14)

If we use the conjugate prior distribution for µ with hyper-parameters µ′ and σ′ for mean and standard deviation,respectively, Equation (14) can be simplified as

P(λ = k |Fn(X) ) =

π(k)

k−1∏r=1

f0(Xr)

n∏r=k

f1(Xr|µ = 0)

P(Fn(X))

σ′′k,nσ′

e

µ′′k,n2

2σ′′k,n

2−µ′2

2σ′2

=

π(k)

k−1∏r=1

f0(Xr)

n∏r=k

f1(Xr|µ = 0)

P(Fn(X))

P(µ = 0)

P(µ = 0|{X(t), k ≤ t ≤ n})

(15)

Equation (15) can be substituted into Equation (3) to compute Λn(X) as follows:

Λn(X) =P (λ ≤ n|Fn(X))

P (λ > n|Fn(X))=

n∑k=0

π(k)

k−1∏r=1

f0(Xr)

n∏r=k

f1(Xr|µ = 0)P(µ = 0)

P(µ = 0|{X(t), k ≤ t ≤ n})∞∑

k=n+1

π(k)

n∏r=1

f0(Xr)

(16)

We can impose the memory limitation as before in Equation (13) to reduce the computation load.

3. VALIDATION

Numerous sets of synthetic data and a set of experimental data are used to validate the change point detectionmethod. Because we want to declare the damage based on the data rather than relying on the prior information,an uninformative or flat prior distribution π(n) for λ is desirable. For mathematical convenience, the geometricdistribution with a large mean (i.e., small d) is selected as π(n). It was found that the results are robust to thevalue of d when d << 1.

Figure 2. Distribution of detection delay.

Figure 3. Distribution of estimated µ as time passes.

3.1 Simulated data

3.1.1 Description of simulation

In order to test the algorithm with multiple sets of data with varying statistical properties, large sets of simulateddata are generated. The data consist of random numbers with the Gaussian distribution. The simulation involved10000 sets of 600 independent random numbers whose mean changes from 0 to 3 after the 100th random number.The variance of the random number remained constant as 1. Using these data sets, the consistency and therobustness of the algorithm were assessed by investigating their false alarm rates and detection delays. For eachof the data sets, we applied the change point detection algorithm using the maximum likelihood method and theBayesian method with various degree of memory limitation. The false alarm rate and the detection delay resultsare compared with the theoretical asymptotic bounds and the case where we know the true f1. In all cases, theGaussian distribution with the mean of 3, and the standard deviation of 3 is used as the prior distribution formu.

3.1.2 Results and discussion

The distribution of the detection time ν̄ is shown in Figure 2, and that of the estimated µ using the maximumlikelihood method is shown in Figure 3. We can observe that the detection delay is distributed symmetrically,and both the detection delay and µ do not have significant outliers nor multiple modes. Thus, the algorithmperformed consistently for each simulation. Moreover, the estimated µ starts approaching the true µ as soon asthe data shifts the mean at n = 101. Note that the estimated µ has the mean near zero for n ≤ 100, and itsvariance decreases as more data is collected. Figure 4 and 5 show the detection delay and the false alarm ratecomputed from the maximum likelihood method and the Bayesian method with various memory limitations,respectively. As expected, the algorithm detects the change faster when we allow a larger false alarm rate α.

Figure 4. Detection delays and false alarm rates using the maximum likelihood method for the simulated data.

Figure 5. Detection delays and false alarm rates using the Bayesian method for the simulated data.

Having a shorter memory helps to reduce false alarm but results in longer detection delay. The detection delayincreases faster as α decreases (i.e., the threshold increases) with a shorter memory. Therefore, a shorter memorymakes the algorithm less sensitive to the changes in the data and respond slower. The memory length of 20 dataperforms almost equivalently to having the full memory in terms of detection delay. This is because having ashorter memory corresponds to making a rougher approximation in exchange of computational load. Knowingthe true µ improves the detection delay by a few unit times and approaches the theoretical asymptotic lowerbound closely. The Bayesian method resulted in the detection delays much closer to knowning the true meancase, which are at most 18% longer with full memory, than the maximum likelihood method, which are at mostabout 250% longer with full memory. The maximum likelihood method, however, provided an insight into howthe post-damage distribution is estimated, by directly showing that the estimated parameter of the unknowndistribution converges to the true value quickly after the change point, as shown in Figure 3.

3.2 Experimental data

3.2.1 Description of experiment

The experiment involved a series of shake table tests of a 1:8 scale model for a four-story steel moment-resistingframe designed according to current seismic provisions (IBC-2003,19 AISC-07-05,20 FEMA-35021). The exper-iment was conducted at the State University of New York at Buffalo.22,23 Figure 6 shows the structure aftercompletion of the erection process on the shake table. The primary focus is on the model frame shown onthe left. A total of 314 channels were used for the instrumentation of the test frame, including strain gauges,accelerometers, and displacement meters. For this analysis, we used the acceleration measurements at each floorincluding the ground motion and the roof response. The sampling rate was 128 Hz. In order to introduce dam-age to the structure, the structure was subjected to a series of the scaled 1994 Northridge earthquake ground

Figure 6. Four-story steel moment-resisting frame after the completion of erection on the shake table.

Figure 7. SDR histories of story 1 and 4.

motions recorded at the Canoga Park, CA, station. The scaling factors were 40%, 100%, 150%, and 190% fora service level earthquake (SLE), a design level earthquake (DLE), a maximum considered earthquake (MCE),and a collapse level earthquake (CLE), respectively. Figure 7 shows the story drift ratio (SDR) of the framefrom elastic behavior (during the SLE) up to collapse for the first and the fourth story, respectively. The secondand the third story had SDR similar to that of the first story. During the SLE the structure remained elastic(undamaged). During the DLE the structure reached a maximum SDR of about 1.6% with the inelastic actionobserved at the column base and first floor beams. Localized damage, such as local buckling of the flange plates,was not noticeable. During the MCE the frame reached a maximum SDR of about 5% with plastic deformationevident from local buckling of the plates that represented plastic hinge elements of the frame.23 During the CLEtest the frame experienced a maximum SDR of about 13% with a full first three story collapse mechanism. Theframe was also subjected to white noise excitations before and after each earthquake loading. For this analysis,we used the white noise excitation responses collected before the SLE and DLE, and after the CLE.

The acceleration response from each white noise excitation was segmented, and the AR coefficients were thenextracted separately from each segment. To obtain the distributions of the DSF before and after the fault (i.e.,f0 and f1, respectively), we assumed Gaussian distributions7 and computed the sample means and variances ofthe DSF. For f0 we used the features extracted from the roof responses before the SLE and the DLE, and for f1we used the response after the CLE.

3.2.2 Results and Discussion

The results of the analysis using the roof acceleration responses are presented in this section. Each accelerationmeasurement was segmented into 50 chunks, each of which includes 100 data points. Thus, each chunk took100/128 seconds to be collected. The model order of 3 was used on the basis of the Akaike Information Criteriatest (see Noh et al.16). 150 values of the DSFs extracted from all 3 white noise excitations were fed into thealgorithm sequentially to compute the test statistic Λn for each roof response.

Figure 8. Damage sensitive feature and their quantile-quantile plot before damage.

Figure 9. Estimated µ as time passes for the experimental data.

Figure 8 shows the DSF values extracted from the white noise response of the frame and their quantile-quantile plot agains the standard Gaussian distribution before damage happens. We can observe that the DSFclosely follow the Gaussian distribution before damage and the damage happened around time 79 second. Thus,the Gaussian assumption for the distribution of the DSF is valid. Figure 9 shows the estimated µ value, as afunction of time, estimated using the maximum likelihood method. The estimate µ quickly converges to themean DSF value for the undamaged structure, -2.12, in the first 20 seconds. It then gradually approaches thetrue mean for the damaged case, -2.54, after the true damage time of 79 second. In this analysis, only the meanof f1 is considered unknown, and the standard devation is assumed to be the same as that of f0. Although theirtrue standard deviations, 0.13 and 0.087, are not identical, the algorithm was able to identify damage as shownbelow.

The detection delay computed using the maximum likelihood method and the Bayesian methods are presentedin Figure 10 and 11, respectively. Similar to the simulation results, the detection delay increases as we set theallowable false alarm rate α small. Also, a shorter memory leads to a longer detection delay for the same reasonexplained in Section 3.1.2. Knowing the true mean case performed nearly as good as the lower asymptoticbound, and having the full memory was about 5 seconds slower than knowing the true mean case for themaximum likelihood method and about 1 second for the Bayesian method. The memory length of 20 resulted inthe equivalent detection delay to the full memory for the maximum likelihood method, and the memory lengthof 10 for the Bayesian method. Overall, the Bayesian method performed much better in terms of minimizing thedetection delay. These results are comparable to the simulation results since the difference between the meansof f0 and f1 normalized by their known or assumed standard deviation σ is 3.23, which is similar to 3 for thesimulation.

Figure 10. Detection delays using the maximum likelihood method for the experimental data.

Figure 11. Detection delays using the Bayesian method for the experimental data.

4. CONCLUSIONS

We have developed a damage diagnosis algorithm based on time-series models and a single fault sequentialchange point detection method that addresses the cases when the feature distribution is unknown when thestructure is damaged. The case of unknown distribution of the DSF is of great interest because in practice weare not likely to have this information. Following the collection of structural responses, the algorithm extractsa damage sensitive feature (DSF) using the autoregressive model and classifies the DSF into a damage stateusing a sequential change point detection method. The classification using the change point detection methodinvolves sequential hypothesis test of the DSF extracted at each time step. The sequential nature of this methodallows us to make a decision using a series of observations sequentially and eliminates the need to wait for a setof features to be collected for statistical analysis. Therefore, it is particularly suitable for a long-term periodicmonitoring of a structure.The sequential test uses the DSF’s pre-damage distribution and the estimated post-damage distribution. The post-damage distribution of the DSF is updated everytime a new set of data is available.This updating scheme is developed using the maximum likelihood estimation and the Bayesian approach. Inaddition, an approximate method with a memory limitation is applied to increase the computational efficiencyof the algorithm.

For validation, we applied the algorithm to multiple sets of simulated data and a set of white noise shaketable test data collected from a four-story steel moment-resisting frame. The results show that the algorithmcan reliably estimate the distribution of the DSF for damaged case using the maximum likelihood method andthe Bayesian method. The simulation results showed that the algorithm can sequentially identify damage withthe fault detection time delay close to the theoretical lower asymtotic bound. The Bayesian method providedfault detection times that are at most 18% longer than the asymptotic bound with a memory length of 10 datapoints. In addition, the emperical false alarm rate was similar to the allowable false alarm rate abound even

with the approximate method with the memory limitation. For the experimental data, the detection was around1 second slower than the aymptotic bound with a memory length of 10 data points. The overall performanceof the Bayesian method was better than that of the maximum likelihood method for both the simulation andthe experimental data, but the maximum likelihood method provided a heuristic approach to the problem bydirectly showing how the estimated parameter of the unknown distribution behaves.

To apply this algorithm in practice, we are currently developing several extensions to the algorithm. Thealgorithm can be expanded using a different set of DSFs to analyze non-stationary responses. Moreover, we arealso investigating the detection of multiple damage states at multiple damage locations simultaneously using thechange point detection method. Last, we need to apply the algorithm to more experimental and field data tofurther validate it.

ACKNOWLEDGMENTS

The research presented in this paper is partially supported by the National Science Foundation CMMI ResearchGrant No. 0800932, the National Science Foundation CIF Research Grant No. 1116377, the Global EarthquakeModel Vulnerability Study Project, Powell Foundation Fellowship, and the John A. Blume Fellowship. Theirsupport is greatly appreciated. The authors would also like to acknowledge Professor Dimitrios G. Lignos andProfessor Helmut Krawinkler for providing the experimental data used in this paper.

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