Structural Modal Identification using Data Sets with...

Structural Modal Identification using Data Sets with Missing Observations

Thomas J. Matarazzo1 and Shamim N. Pakzad, Dept. of Civil and Environmental Engineering, Lehigh University, Bethlehem, PA USA

1. ABSTRACT System identification algorithms currently require a full data set, i.e., no missing observations, to estimate

the natural vibration properties of a structural system. These algorithms are often based on parameters estimated from a state-space model. There are circumstances in which a Missing Data Problem can arise during data collection; therefore, it is important to adjust these algorithms to facilitate Structural Modal Identification.

Despite having missing observations, state-space parameters can be estimated for a time series;

subsequently, structural modal properties can be identified. This paper will use the EM algorithm to identify structural modal properties from a data set with missing observations. The end of the paper will focus on the search for a missingness threshold which can be used to assess the probability of extracting useful structural modal properties from a given data set with missing observations. This assessment will be based on the accuracy of modal estimates for data sets with varying magnitudes and patterns of missingness. It is clear that missingness can only reduce the accuracy of modal estimates; however, it is important to establish the associated scale and behavior of the reduction. An example is presented to illustrate the main concepts of this approach.

Keywords: system identification, structural modal identification, state-space, EM algorithm, missing data, missingness threshold

2. INTRODUCTION

2.1. Missing observations in data collection Researchers and scientists often seek opportunities to measure phenomena in the real world to help

explain a governing model(s) of interest. While our expectations for our data may be high, we must admit that we can only control so much of the data collection process. Collected data sets can contain numerous undesirable features – noise, clipped values, missing observations, or sparsity – to name a few. Unfortunately, just about every useful calculation, algorithm, or analysis that one may perform on our data post-collection, requires full data sets, i.e., no missing observations; this especially includes system identification algorithms.

It is often inconvenient or impossible for us to repeat the field experiment in hopes of obtaining a

complete data set. It is usually more practical to make use of the data we do have, which, in the missing observations case, implies using only a subset of the collected data that has full observations. In this case, the user must be especially weary of any biases that may exist in the subset of the collected data. What if every missing observation was the result of a particular phenomenon? Take clipping, for example: let’s say a sensor is unable to measure acceleration over a certain value (say, 1 g). If we assume the clipped observations to be missing, then it is clear that the degree of missingness depends on each acceleration magnitude.

These types of questions are answered once a missingness pattern is recognized. Recognizing the

missingness pattern is perhaps the most important step of any missing data problem. Little and Rubin [1] refer to these missing-data patterns as missing-data mechanisms and have developed three main categories:

1 Email: [email protected]; 610 758-4543

Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems 2013, edited by Jerome Peter Lynch, Chung-Bang Yun, Kon-Well Wang, Proc. of SPIE Vol. 8692,

86920X · © 2013 SPIE · CCC code: 0277-786X/13/$18 · doi: 10.1117/12.2012247

Proc. of SPIE Vol. 8692 86920X-1

missing completely at random (MCAR), missing at random (MAR), and not missing at random (NMAR). Rubin [2,3] discusses methods to work with data sets that have missing observations; however, these methods are geared towards working with survey data, not time-series data.

2.2. Structural modal properties and system identification

The goal of system identification of a structural system (St-Id) is to develop an accurate mathematical model for a physical system by use of experimental data. Once this model is established, unique properties of the physical system can be estimated, e.g., dynamic properties of a structural system. As discussed in the previous section, St-ID requires full finite data sets of finite lengths. These data sets are often the output from a sensor network implemented on a structure. The location of these sensors on the structure determines the DOF under consideration. The sensors collect data at a specified sampling frequency, thus the structure’s vibration information are discretized in time and space. The real infinite DOF structure is now simplified to a finite DOF structure, much like a classical structural dynamics problem.

Figure 1. Basic concept of system identification

There are numerous St-ID algorithms that can determine the natural frequencies (eigenvalues) and mode shapes (eigenvectors) from the sensor network output. The choice of algorithm is dependent on the type of data sets collected, i.e., whether the sensor network is input/output or output only.

These algorithms are often based on state-space or ARMA models. Generally speaking, the goal is to

identify the state matrix in the state-space model or the coefficients in the ARMA model. A few popular algorithms that are based on the state-space model are Eigen Realization Algorithm (ERA) [4], Numerical State-Space Subspace System Identification (N4SID) [5], and System Realization Using Information Matrix (SRIM) [6].

2.3. Mass Discretization

This paper uses the state-space model to represent a structural system discretized in space and time;

however, since measurements will come from the continuous structure, it is important to anticipate the inherent model differences. The modal analysis of an infinite degree of freedom (DOF) system determines the characteristic equation by applying structural boundary conditions to the 4th order partial differential equation of motion. This characteristic equation yields infinitely many eigenvalues and eigenfunctions that correspond to the natural vibration frequencies and mode shapes of the continuous structure.

The modal analysis of an N DOF system solves the eigenvalue problem for the mass and stiffness

matrices to yield N eigenvalues and eigenvectors corresponding to the first N natural frequencies and mode shapes of the structure. It is important to note that the accuracy of a given natural frequency improves as N increases (more DOF are added to the system). For a simple beam, when DOF are added (while maintaining uniform spacing between DOF), the frequencies monotonically converge to the theoretical natural frequency from lower values.

System .. .. ..

.. .. ..

Input Output

Xi Yi


Figure 2. Convergence of N DOF system to continuous for a simple bridge (up to 20 DOF shown)

Figure 1 shows how the Mass Discretization Error (MDE) decreases as N increases for a simply supported beam. MDE is simply the experimental error for the discrete natural frequency estimate compared to the theoretical value obtained from the continuous system. For example, the 4th natural frequency will have MDE = 9% in a 4 DOF system but will have MDE = 3% in a 5 DOF system.

This error is inevitable, but it is feasible to estimate with accuracy because it does not depend on the

mass or stiffness of the system; this error only depends on the mode, n, and total DOF, N. Livesley [7] showed the closed form solution for this relationship using methods derived in [8]. Consider the nth

frequency estimate given by an N DOF system to be

f̂n,N and the theoretical natural frequency given by the continuous

system to be fn. Define MDE as the experimental error of the nth natural frequency estimate from an N DOF system:

MDEn,N =fn − f̂n,N

fn= 1−

sin nπ2(N+1)

⎛⎝⎜

⎞⎠⎟

nπ2(N+1)

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

2

1

1− 23

sin2 nπ2(N+1)

⎛⎝⎜

⎞⎠⎟ (1)

While only a simply supported beam was discussed in detail, the main idea of this section is that

discretized structures, which model continuous systems, will contain more accurate modal estimates when N is large. This is important to keep in mind because as far as data collection is concerned, the measured responses belong to a real continuous system but will be forced to fit a state-space model based on a structure discretized in space. As a general rule of thumb provided by [7], in an N DOF system, frequency estimates will be relatively accurate up to n = N/2, e.g., in a 10 DOF system, the first five frequencies will be reasonably close to the theoretical values.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

2

4

6

8

10

12

14

16

18

20

22

24

26MDE VS DOF System: Uniform Length Segments

MD

En

,N %

(n

th M

od

e)

N DOF System

1st freq

2nd freq

3rd freq

4th freq

5th freq

6th freq

7th freq

8th freq

9th freq

10th freq

11th freq12th freq

13th freq14th freq

15th freq16th freq

17th freq18th freq19th freq


3. MAXIMUM LIKELIHOOD ESTIMATES (MLE) OF A STATE-SPACE MODEL 3.1. State-space model for a second-order linear dynamic system

It is often convenient to express the equation of motion in first order form using the state-space

equations, which are inherently discrete in space. Since the goal in data collection is to obtain a sampled response, the equation of motion is discretized in both space and time.

mut + cd

ut + kut = w t t = 1,2,...,n (2)

For reasons discussed in section 2.3, it should be clear that this discretized model will not describe the behavior of the continuous structure exactly; however, high N DOF models will suffice. The equation of motion represents equilibrium at each DOF and time step, t. For an N DOF system, the mass, m, damping, cd, and the stiffness, k, are N x N matrices, that may be estimated using the finite element method. The sampled response: displacements, ut, velocities, u t accelerations, üt, are N x 1 vectors at each time step. The loading, wt, is a N x 1 vector at each time step, assumed to be uncorrelated and random with covariance matrix Q.

xt =

ut

ut

⎡

⎣⎢⎢

⎤

⎦⎥⎥

xt =ut

ut

⎡

⎣⎢⎢

⎤

⎦⎥⎥

x(t)→ xt

x(t)→ xt+1 − xt (3)

The state variable is chosen to represent the displacements and velocities. Differentiation in time is analogous to the change in state at each time step, t.

xt+1 = Axt + w t t = 1,2,...,n (4)

yt = Ctxt + vt t = 1,2,...,n (5) The discrete-time state equation (4) describes how the states (structural responses) change over time.

The discrete-time observation equation (5) relates our observations to the states. The behaviors of these two equations are best captured by matrices A and C, respectively. In general, the state matrix, A, is a p x p matrix, where p is the model order2 of the system selected by the user. In this case, a model order of 2N will be used so that A is 2N x 2N. The observation matrix, Ct, can vary in size depending on the observed responses, yt. The observations are the response to be measured in the field and will depend on the sensors and sensor network.

yt =

ut

ut

⎡

⎣⎢⎢

⎤

⎦⎥⎥

(6)

It is assumed that displacement and velocity are measured at each sensing location; the observation

vector is 2N x 1 at each time step. In this case, the observation matrix, Ct, will be a 2N x 2N identity matrix that simply maps the observations to the states since yt and xt represent the same responses. To maintain generality, the observation matrix is assumed to change at each time step; this will be especially convenient when missing observations are considered.

x0 N(µ0,V0) wt N(0,Q) vt N(0,R) (7) The observation noise errors, vt, are assumed to be zero-mean uncorrelated normally distributed with N x

N covariance matrix R. The dynamic loading terms, wt, are assumed to be zero-mean uncorrelated normally

2 The model order is an important adjustable parameter in the state-space model. In short, higher model orders are used in more complex systems to force more system poles.


distributed with N x N covariance matrix Q. The initial state x0 is assumed to be a normal random vector with mean vector µ0 and N x N covariance matrix V0.

3.2. Kalman filter and smoother

The Kalman filter equations introduced in [9] are commonly used within the state-space model to obtain

the optimal estimate of the state variable based on observations over a finite time period n. The states, xt, are not observed; however, the Kalman filter estimates these unobserved states for all t using the observations, yt, and other state-space parameters.

x̂t|n = E xt | y1,…,yn( )

(8)

V̂t|n = cov xt | y1,…,yn( ) (9)

V̂t,t−1|n = cov xt,xt-1 | y1,…,yn( )

(10)

The Kalman smoother estimator x̂t|n and mean square error terms

P̂t|n &

P̂t,t−1|n are calculated from the

backwards recursive calculations given in [10] (excluded from this paper). The Kalman smoothers are given in [11] as S11, S10, and S00. The statistical properties of the initial state, input loading and noise, and observation noise are also required to calculate the smoothers. Therefore, initial estimates of the parameters µ0, V0, A, Q, and R, are needed to commence the algorithm. Define a set of parameters, Φi, initially set to Φ0. This set of parameters will update with each iteration, i, of EM until the specified convergence criteria is met at the final iteration, m.

Φ0 = (µ0,0,V0,0,A0,Q0,R0) i = 0,1,...,m (11)

3.3. Expectation maximization (EM) algorithm to compute MLE

Primarily comprised of two steps, the expectation step (E-step) and the maximization step (M-step), the

general EM algorithm iterative procedure was given in [12]. The procedure is given below in a manner similar to [11] where it is described to “simply alternate between the Kalman filtering and smoothing recursions and the multivariate normal maximum likelihood estimators.”

1) Initialize, Φ0. Begin iterations i=0,1,… 2) Compute the incomplete-data likelihood using the innovations form. 3) E-Step: Use the Kalman filter (8-10) and smoother equations (excluded from this paper) to estimate the states

x̂t|n and mean square error terms

P̂t|n &

P̂t,t−1|n for all time steps. Next, use the Kalman

smoothers to compute the new parameter estimates (14-18)

4) M-Step: Update parameter estimates Φi →Φi+1 (19). 5) Repeat 2 – 4 until convergence at i=m. Theses Φm are the MLE for the system. The EM iterations are guaranteed to increase in likelihood as well as converge to a stationary point for an

exponential family [13]. EM is also straightforward in its procedure and is, most importantly, compatible with data sets that have missing observations (more of this is seen in section 4). These are the main reasons why this algorithm was chosen.

In general, the complete data log likelihood depends on the observations and the states; however, the current problem setup contains incomplete data. Not incomplete as in missing – this has yet to be formally introduced for the state-space model. The data is incomplete because the states are unavailable (not measured). The complete data log likelihood formulation has been presented by Shumway and Stoffer in [14]. Since only the observations are known, the EM algorithm is applied to the conditional expectation of the complete data likelihood function.


G Φi |Φi−1( ) = E logLX,Y Φ( ) | y1:n,Φi−1

⎡⎣ ⎤⎦ (12)

The E-step begins by expressing the conditional expectation of the complete data log likelihood function in terms of the observed data, state-space parameters, and Kalman smoothed estimates.

G Φi |Φi−1( ) = − 12

ln V − 12

tr V−1 V0|n + (x̂0|n − µ0)T(x̂0|n − µ0)⎡⎣ ⎤⎦{ }− n

2ln Q − 1

2tr Q−1 S11 −S10A

T − AS10T − AS00A

T⎡⎣ ⎤⎦{ }− n

2ln R − 1

2tr R−1 (yt −Ctx̂ t|n)(yt −Ctx̂ t|n)T +CtV̂t|nCt

Tt=1

n∑{ } (13)

Before this conditional expectation can be maximized, the Kalman smoothers must be determined. These

smoothers generate the minimum mean square error (MSE) smoothed estimator (predictor) of xt and its covariances V through time n. Basically, the Kalman smoother equations “complete” the data by producing values for the state xt. Finally, the conditional expectation of the complete data log likelihood function above in (13) is maximized using equations (14-18) to update the parameters. This is the M-step as shown in [14].

Ai+1 = S10S00−1

(14)

Qi+1 = n−1 S11 −S10S00

−1S10T( )

(15)

Ri+1 = n−1 (yt −Ctx̂t|n)(yt −Ctx̂t|n)T +CtV̂t|nCt

T⎡⎣ ⎤⎦t=1

n

∑

(16)

µ0,i+1 = x̂0|n

(17)

V0,i+1 = V̂0|n

(18)

These parameters that maximize (13) update the previous iteration parameter estimates.

Φi = (µ0,i,V0,i,Ai,Qi,Ri)→Φi+1 = (µ0,i+1,V0,i+1,Ai+1,Qi+1,Ri+1) (19)

Equations (8-10) and (13-18) are to be calculated on the new iteration i→i+1 with Φi+1 . The convergence properties of EM is discussed in detail in [13]. In short, in order to get the iterations to

stop automatically, the user must define at least one of the following properties: maximum number of iterations or threshold for slope of likelihood function. The slope of the likelihood function can be approximated using the present likelihood value and the previous iteration’s value [15]. Once this difference, normalized over the average, falls below the threshold, the algorithm ends (i=m).

converge if LLthreshold >G Φi( ) −G Φi−1( )G Φi( ) +G Φi−1( )

2

⎛

⎝⎜

⎞

⎠⎟

(20)

3.4. Modal properties of a structural system in state space model

Juang [16] shows how the state matrix is formed from the equation of motion in terms of the mass,

stiffness, and damping matrices. He [4] also discusses that the modal properties (natural frequencies, mode shapes, and damping ratios) of the structural system are preserved in this formulation. This is one of the main


reasons why many St-Id algorithms are based on the state-space model. The natural frequencies and mode shapes of the structural system are simply the eigenvalues and eigenvectors of the state matrix. The number of observable modes depends on the size of the state matrix (which is the model order, p).

3.5. Modifications to EM algorithm for missing observations One of the main reasons this algorithm was chosen to estimate state-space model parameters is the

convenience of using data sets with missing observations. Only simple modifications [14] are necessary to use with data sets that contain missing observations. For the case where the observation vectors are partially observed at time step t, one may partition the observation equation at the time-step and replace the rows corresponding with missing observations with zeros. Additionally, the affected measurement error covariances (off-diagonal terms) must also be set to zero. Equation (21) shows the partitioned form of the observation equation (4).

yat

ybt

⎛

⎝⎜

⎞

⎠⎟ =

Cat

Cbt

⎡

⎣⎢⎢

⎤

⎦⎥⎥xt + vt

cov(vt ) =Raa

t Rabt

Rbat Rbb

t

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥ (21)

The missing partitions (assume the “b partition” is missing) are zeroed out as shown below in equation (22).

yat

0

⎛

⎝⎜⎞

⎠⎟=

Cat

0

⎡

⎣⎢⎢

⎤

⎦⎥⎥xt + vt cov(vt ) =

Raat 0

0 Rbbt

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥ (22)

It is important to mention that substitution in the covariance matrix assumes that the observed data and

the missing data are uncorrelated. As discussed in section 2.1, this assumes the missing data mechanism is random and the missing observations would be described as MCAR in [1]. Stoffer [17] shows that these modifications can be carried out through the regular Kalman filter equations, and that the Kalman smoother process can still occur with these “missing data-filtered values”. However, this requires that the M-step is also modified so that equation (16) becomes equation (23) below.

Ri+1 = n−1 Dt

yat

0

⎛

⎝⎜⎞

⎠⎟−

Cat

0

⎡

⎣⎢⎢

⎤

⎦⎥⎥x̂t|n

⎛

⎝⎜

⎞

⎠⎟

yat

0

⎛

⎝⎜⎞

⎠⎟−

Cat

0

⎡

⎣⎢⎢

⎤

⎦⎥⎥x̂t|n

⎛

⎝⎜

⎞

⎠⎟

T

+Ca

t

0

⎡

⎣⎢⎢

⎤

⎦⎥⎥V̂t|n

Cat

0

⎡

⎣⎢⎢

⎤

⎦⎥⎥

T⎡

⎣

⎢⎢

⎤

⎦

⎥⎥+

0 00 Rbb

t,j

⎡

⎣⎢⎢

⎤

⎦⎥⎥Dt

T⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪t=1

n

∑ (23)

Where Dt is an N x N permutation matrix used to organize the partitions at time step t, back to their original positions.

In another approach [18,19], used in the case of full observation “blackouts” all observations are missing at one time, missing observations can be accounted for in a different manner. For each “blackout,” the Kalman filter equations of the E-step are modified so that the filtered state estimates and its variance are set to be the values from the previous step.

x̂t|t = x̂t|t-1 (24)

V̂t|t = V̂t|t−1 (25)

However, the Kalman smoother equations remain unchanged. Additionally, Jones [19] states “this

procedure works for any number of observations missing in a row,” but “running the recursion across a large block of missing data is equivalent to restarting the recursion at the other end.”


4. EXAMPLE 4.1. Setup, assumptions, and computation

In this section, an example is given to demonstrate the implementation of the EM algorithm modal

identification of a structural system when missing data is present. The goal is to use structural response data in the EM algorithm to determine the natural frequencies and mode shapes. An undamped 10 DOF with evenly spaced lumped masses is used in this example; therefore, 10 natural frequencies and 10 mode shapes can be identified.

Figure 3. 10 DOF simple bridge with white noise dynamic loading at each DOF Sampling info: n= 20,000 samples at fs = 1,000 Hz Bridge Properties: ρ = 6000 kg/m3, L= 300 m, A = 70 m2, I = 600 m4, and E = 170 GPa

The initial state matrix was calculated using perturbed versions of the actual mass and stiffness matrices

for the structure shown in Figure 3. Although these matrices were perturbed up to 20% from their true values, they each remained symmetrical. The observation matrix was known at every time-step and the Gaussian observation noise was correctly assumed as 5%. The dynamic load was assumed to be a random white noise excitation; the EM algorithm was used as an output only system identification algorithm.

Ten simulations of the 10 DOF bridge responses were performed using Newmark’s linear acceleration method. Consistent with the terminology introduced in section 3.1, the observations were the measured displacements and velocities at each DOF. In each simulation, 5% noise was added to the responses to mimic real sensor performances. Once structural response data were simulated, observation “blackouts” were applied to the data randomly with 5% missingness. In other words, the measured responses for 1,000 randomly selected samples were removed and labeled as missing. The EM algorithm was applied to each simulated sensor output twice: once for each degree of missingess (0%, 5%).

For comparison purposes, the algorithm’s convergence criteria was selected so that the algorithm would

end after 15 iterations. This number was selected after observing adequate behavior after 10 iterations in preliminary EM trials. More iterations would have provided more accurate MLE in the 0% missingness case; however, due to stability issues with the singularity of the innovation covariance matrix, the EM algorithm would not continue past the 15th iteration in the 5% missingness case. Despite this brick wall limit, the results show comparable performances.

wt = [ ]Twt1 wt2 wt10... ... ... ... ... ... ...


Figure 4. Observed displacement and velocity at DOF 5 (midspan, no missing data)

4.2. Results

With fn,EM as the nth frequency from the final estimated state matrix, AEM, and fn as the nth natural frequency from the 10 DOF structure shown in Figure 3, the frequency error shown in the results is:

en =

fn,EM − fnfn

⋅100%

(26)

Table 1. EM results for no missing data: frequency error for each simulation (15 iterations)

sim. # 1 2 3 4 5 6 7 8 9 10 µ σ log like. 1,911,296 1,921,061 2,011,172 2,010,590 2,025,023 2,019,866 2,122,293 2,036,819 2,004,280 2,001,132 2,006,353 58,958

e1 -0.185 -0.840 5.656 6.070 0.234 5.596 3.180 3.932 -0.699 4.188 2.713 2.807 e2 1.150 0.793 2.289 0.731 1.353 0.105 0.281 1.765 -0.785 -0.090 0.759 0.920 e3 -0.301 0.616 0.059 -0.484 0.380 0.219 0.468 0.030 0.082 0.209 0.128 0.335 e4 2.495 11.361 0.162 1.004 0.428 -0.172 1.488 0.280 0.595 0.066 1.771 3.462 e5 0.571 0.167 0.072 -0.032 1.271 -0.037 -0.005 -0.154 0.449 0.474 0.278 0.429 e6 -0.669 1.221 5.235 0.498 1.537 0.069 -0.098 0.309 -0.050 0.443 0.850 1.668 e7 1.433 1.852 0.544 0.327 -0.170 0.057 0.503 0.375 0.227 0.494 0.564 0.617 e8 0.361 -1.118 0.003 4.495 2.126 1.511 1.030 0.377 0.341 1.146 1.027 1.508 e9 0.037 0.200 0.731 1.603 1.008 0.159 0.395 1.015 1.444 1.633 0.823 0.611 e10 4.450 0.933 1.528 4.977 1.101 0.483 0.382 1.979 2.626 2.033 2.049 1.575 Table 2. EM results for 5% missing data: frequency error for each simulation (15 iterations)

sim. # 1 2 3 4 5 6 7 8 9 10 µ σ log like. 1,523,974 1,527,723 1,631,558 1,696,686 1,641,827 1,623,169 1,725,730 1,643,249 1,610,112 1,624,017 1,624,804 63,104

e1 -0.045 -0.797 5.742 6.679 0.553 5.189 3.397 5.053 -0.481 5.417 3.071 2.940 e2 1.956 -30.851 3.067 0.918 1.577 0.203 0.687 3.113 -0.607 0.046 -1.989 10.216 e3 0.427 -54.248 -31.888 -0.281 1.397 0.586 1.063 1.286 0.688 0.714 -8.026 19.212 e4 17.143 -42.342 -43.394 4.634 -19.594 1.122 7.169 3.208 5.313 0.394 -6.635 21.151 e5 4.139 1.680 -35.417 1.088 -33.847 3.116 1.293 1.485 5.795 4.018 -4.665 15.870 e6 -1.156 6.219 -28.425 1.252 15.154 3.029 1.255 7.923 2.931 3.985 1.217 11.365 e7 9.882 -0.196 3.530 1.499 2.927 1.896 5.734 2.992 2.638 4.698 3.560 2.763 e8 2.150 -7.895 1.723 10.389 10.725 12.315 5.517 2.165 2.389 6.407 4.588 5.924 e9 3.262 3.491 3.380 4.332 4.506 1.628 2.623 4.850 6.597 6.548 4.122 1.595 e10 15.820 4.419 4.927 16.054 -6.847 2.744 2.527 7.532 8.908 6.775 6.286 6.662

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cem

en

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elo

cit

y [

m/s

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Observed response to white noise excitation at DOF 5


Figure 5. EM frequency error (simulation 9, no missing data)

Figure 6. EM frequency error close-up (simulation 9, no missing data)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15í�

í�

í�

í�

í�

0

1

2

3

4

5EM Path to MLE for LL9 = 2004280.3317

% E

rror

e

EM Iteration Number

e10=2.63%

e9=1.44%

e4=0.60%

e5=0.45%e8=0.34%e7=0.23%e3=0.08%e6=-0.05%

e2=-0.79%e1=-0.70%

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Figure 7. 1st four mode shapes from EM algorithm results (simulation 9, 5% missing data)

The modal assurance criterion (MAC) values [20] were used to evaluate the consistency of mode shapes between the EM algorithm results and the 10 DOF system.

Table 3. MAC values for mode shapes for EM algorithm compared with 10 DOF (simulation 9, 5% missing data)

mode # 1 2 3 4 5 6 7 8 9 10 MAC 1.000 1.000 0.620 0.962 0.024 0.164 0.460 0.362 0.740 0.916

4.3. Discussion

Tables 1 and 2 show the frequency results of the EM algorithm after 15 iterations for the cases of 0% and

5% missingness, respectively. Clearly, each simulation yielded different results; the mean and standard deviations of the simulations can be seen at the right side of each table. In general, the EM algorithm estimated each natural frequency within ±6% of the actual 10 DOF values in the case of no missing data. Once missing data was introduced, estimate errors up to ±55% were seen; while some frequency errors increased dramatically, many of them remained within ±6%.

These tables also show the final log likelihood values for the algorithm. The end likelihood values

remained close within each simulation set, but they dropped for every simulation once missing data was introduced. We hypothesize that this decrease is some indication of a confidence reduction in the results or acknowledgement of problematic observations. However, the significance of this final log likelihood value may vary more with each data set, making it difficult to use as an assessment tool.

Figures 1 and 2 show the frequency error with each iteration of the EM algorithm for simulation 9 with no

missing data. The estimates begin to stabilize around the 7th iteration and continue to improve until the last.

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While this stabilization did not occur at the same iteration in each simulation, each simulation did show a distinct transition into a more-stable frequency convergence. Additionally, the lower modes seemed to transition into this stabilization more quickly.

Figure 7 shows the first four mode shapes the final estimated state matrix, AEM plotted with the

eigenvectors of the 10 DOF system. Table 6 lists the MAC values used to determine the consistency of each mode shape; in general, values over 0.90 can demonstrate consistent mode shapes. 4.4. Missingness threshold

After reviewing the performance of the EM algorithm for no missing data and 5% missingness, one may naturally wonder if there exists some missingness threshold for each data set. Most practically, a field engineer would like to know “how many missing observations is too many?” What’s the probability of extracting accurate dynamic properties from a given data set with missing observations?

The answers to these questions are unknown at this time but surely depend on the structure, the

responses observed, sampling rate, the degree of missingness, and the missingness pattern. The results from this example are preliminary and meant for illustration purposes; however, it’s promising to see reasonable modal estimates when 1,000 samples were removed at random. The next step is to implement acceleration data collected from a bridge with known dynamic properties in the EM algorithm and simulate missing observations once again. The missingness pattern may also be adjusted to identify the most undesirable missing-data mechanisms. We hypothesize that full “blackouts” at random are among the most severe patterns.

As mentioned in Section 4.3, there is some concern regarding the stability issues present on higher iterations of the algorithm. The parameter updates were causing singularity problems with the innovation covariance when the likelihood was being computed and essentially limited the algorithm to 15 iterations. While 15 iterations provided good estimates, it was clear that when 30 iterations were used in the 0% missing case, the MLE were more accurate, i.e., the converging trends that begin towards the end of Figures 5 and 6 do continue. Stoffer [21] warns users “another disadvantage is that the EM algorithm may converge slowly in the latter stages of the iterative procedure; one may wish to switch to another algorithm at this stage.” It is clear there is some work that needs to be done to assess how missingness may affect this convergence.

5. CONCLUSIONS In this paper, the EM algorithm was used to identify the dynamic properties of a structural system when

there were missing observations in the data set. Similar algorithms have been used in the past to find MLE in economic time-series [22], traffic data [23], pollution levels [24], and fishing [25], to name a few, when missing data was present; however, this is an unprecedented feat in structural health monitoring. Origins of missing data and the importance of classifying missing data patterns were discussed in Section 2.1. A background of structural modal identification was given in Section 2.2. The accuracy of frequency estimates in lumped mass finite element models was discussed in Section 2.3. The EM algorithm procedure was summarized and a structural dynamics example was used to provide preliminary results to support the feasibility for the use of this algorithm in structural health monitoring. In the example, displacement and velocity were measured at each time-step and then 5% of the samples were removed and denoted as missing. The EM algorithm was used for ten different simulations of the same structural response for two degrees of missingness (0%, 5%). The natural frequencies and mode shapes were compared to the eigenvalues and eigenvectors obtained in the 10 DOF system. In our opinion, these preliminary results are promising enough to implement this algorithm using real bridge data. Different patterns and degrees of missingness can be applied to the acceleration observations to model unfortunate data-collection scenarios. We hope to develop a sophisticated strategy that combines the theory and applications of this algorithm with missingness information to make use of all data collected, allowing useful information to be extracted from data sets that once seemed fruitless.


6. REFERENCES

[1] R. J. A. Little and D. B. Rubin, Statistical Analysis with Missing Data, 2nd ed., p. 381, John Wiley & Sons, Inc., Hoboken, NJ (2002).

[2] D. B. Rubin, “Multiple Imputation After 18 + Years,” Journal of the American Statistical Association 91(434), 473–489 (1996).

[3] D. B. Rubin, Multiple Imputation for Nonresponse in Surveys, p. 258, John Wiley & Sons, Inc., New York (1987).

[4] J.-N. Juang and R. S. Pappa, “An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction,” 620–627, NASA Langley Research Center (1984).

[5] T. W. Flint and R. J. Vaccaro, “Performance Analysis of N4SID State-Space System Identification,” Control(June), 2766–2767 (1998).

[6] J.-N. Juang, “State-Space System Realization With Input- and Output-Data Correlation,” NASA, Hampton, VA (1997).

[7] R. K. Livesley, “The Equivalence of Continuous and Discrete Mass Distributions in Certain Vibration Problems,” University of Manchester (1954).

[8] R. V. Southwell, Theory of Elasticity, 2nd ed., Clarendon Press, Oxford (1941).

[9] R. E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” Transactions of the ASME, Journal of Basic Engineering 8, 35–45 (1960).

[10] A. H. Jazwinksi, Stochastic Processes and Filtering Theory, Academic Press Inc., New York (1970).

[11] R. H. Shumway and D. S. Stoffer, Time Series Analysis and Its Applications – with R Examples, 2nd ed., p. 575, Springer, New York (2006).

[12] D. B. Rubin, A. P. Dempster, and N. M. Laird, “Maximum Likelihood from Incomplete Data via the EM Algorithm,” Journal of the Royal Statistical Society 39(1), 1–38 (1977).

[13] C. F. J. Wu, “On the Convergence Properties of the EM Algorithm,” The Annals of Statistics 11(1), 95–103 (1983).

[14] R. H. Shumway and D. S. Stoffer, “An Approach to Time Series Smoothing and Forecasting Using the EM Algorithm,” Journal of Time Series Analysis 3(4), 253–264 (1982).

[15] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed., Cambridge University Press, New York (1992).

[16] J.-N. Juang and M. Q. Phan, Identification and Control of Mechanical Systems, p. 331, Cambridge University Press, Cambridge (2001).

[17] D. S. Stoffer, “Estimation of Parameters in a Linear Dynamic System with Missing Observations,” University of California, Davis (1982).


[18] V. Digalakis, J. R. Rohlicek, and M. Ostendorf, “ML Estimation of a Stochastic Linear System with the EM Algorithm and its Application to Speech Recognition,” IEEE Transactions on Speech and Audio Processing 1(4), 431–442 (1993) [doi:10.1109/89.242489].

[19] R. H. Jones, “Maximum Likelihood Fitting of Time ARMA Models to Time Series with Missing Observations,” Technometrics 22(3), 389–395 (1980).

[20] R. J. Allemang, “The Modal Assurance Criterion – Twenty Years of Use and Abuse,” Analysis 1(August), 14–21 (2003).

[21] R. H. Shumway and D. S. Stoffer, “An Approach to Time Series Smoothing and Forecasting Using the EM Algorithm,” p. 22 (1981).

[22] A. C. Harvey and R. G. Pierse, “Estimating Missing Observations in Economic Time Series,” Journal of American Statistical Association 79(385), 125–131 (1984).

[23] W. T. Scherer, “Imputation Techniques to Account for Missing Data in Support of Intelligent Transportation Systems Applications” (2003).

[24] P. M. Robinson and W. Dunsmuir, “Estimation of Time Series Models in the Estimation Presence of Missing Data,” American Statistical Association 76(375), 560–568 (1981).

[25] R. Mendelssohn and C. Roy, “Environmental Influences on the French, Ivory Coast, Senegalese and Moroccan Tuna Catches in the Gulf of Guinea,” in ICCAT International Skipjack Year Programme, pp. 170–188 (1986).


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