Quantification of Aleatory Uncertainty in Modal Updating Problems using a New Hierarchical Bayesian Framework

Abstract

Identification of structural damage requires reliable assessments of damage-sensitive quantities, including natural frequencies, mode shapes, and damping ratios. Lack of knowledge about the correct value of these parameters introduces a particular sort of uncertainty often referred to as epistemic uncertainty. This class of uncertainty is reducible in a sense that it can be decreased by enhancing the modeling accuracy and collecting new information. On the contrary, such damage-sensitive parameters might also have intrinsic randomness arising from unknown phenomena and effects, which gives rise to an irreducible category of uncertainty often referred to as aleatory uncertainty. The present Bayesian modal updating methodologies can produce reasonable quantification of the epistemic uncertainties, while they often fail to account for the aleatory uncertainties. In this paper, a new multilevel (hierarchical) probabilistic modeling framework is proposed to bridge this significant gap in uncertainty quantification and propagation of structural dynamics inverse problems. Since multilevel model calibration schemes establish a complicated model structure associated with additional parameters and variables, their computational costs are often considerable, if not prohibitive. To reduce the computational costs, the modal updating procedure is simplified using a second-order Taylor expansion approximation. This approximation is combined with a Markov chain Monte-Carlo (MCMC) sampling method to compute marginal posterior distributions of quantities of interest. The proposed framework is illustrated using one simple experimental example. As a result, it is demonstrated that the proposed framework surpasses the present Bayesian modal updating methods as it accounts for both the aleatory and epistemic uncertainties.

Full text

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
1
Quantification of Aleatory Uncertainty in Modal Updating
Problems using a New Hierarchical Bayesian Framework
Omid Sedehi
PhD Candidate, Dept. of Civil and Environmental Engineering, The Hong Kong University of Science
and Technology, Hong Kong, China; Department of Civil Engineering, Sharif University of
Technology, Tehran, Iran
Daniz Teymouri
MPhil Student, Dept. of Civil and Environmental Engineering, The Hong Kong University of Science
and Technology, Hong Kong, China
Lambros S. Katafygiotis
Professor, Dept. of Civil and Environmental Engineering, The Hong Kong University of Science and
Technology, Hong Kong, China
Costas Papadimitriou
Professor, Dept. of Mechanical Engineering, University of Thessaly, Volos, Greece
ABSTRACT: Identification of structural damage requires reliable assessments of damage-sensitive
quantities, including natural frequencies, mode shapes, and damping ratios. Lack of knowledge about
the correct value of these parameters introduces a particular sort of uncertainty often referred to as
epistemic uncertainty. This class of uncertainty is reducible in a sense that it can be decreased by
enhancing the modeling accuracy and collecting new information. On the contrary, such damage-
sensitive parameters might also have intrinsic randomness arising from unknown phenomena and
effects, which gives rise to an irreducible category of uncertainty often referred to as aleatory
uncertainty. The present Bayesian modal updating methodologies can produce reasonable
quantification of the epistemic uncertainties, while they often fail to account for the aleatory
uncertainties. In this paper, a new multilevel (hierarchical) probabilistic modeling framework is
proposed to bridge this significant gap in uncertainty quantification and propagation of structural
dynamics inverse problems. Since multilevel model calibration schemes establish a complicated model
structure associated with additional parameters and variables, their computational costs are often
considerable, if not prohibitive. To reduce the computational costs, the modal updating procedure is
simplified using a second-order Taylor expansion approximation. This approximation is combined with
a Markov chain Monte-Carlo (MCMC) sampling method to compute marginal posterior distributions of
quantities of interest. The proposed framework is illustrated using one simple experimental example.
As a result, it is demonstrated that the proposed framework surpasses the present Bayesian modal
updating methods as it accounts for both the aleatory and epistemic uncertainties.
1. INTRODUCTION
Bayesian operational modal analysis (BOMA) is
originally developed by Katafygiotis and Yuen
(2001 and 2003). Despite great novelty in the
original formulations proposed to identify modal
parameters, they are computationally demanding.
Among different variants of the BOMA, fast
Fourier transform (FFT) methodologies have
gained greater publicity and attention. Au (2011)
developed a new FFT method to dramatically
reduce the computational costs involved with this
approach, while theoretical, computational, and
practical issues involved with applying this

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
2
method to ambient vibration tests are discussed
in Au et al. (2013). Yan and Katafygiotis (2015b)
have continued this line of research to develop a
method to separate the identification of mode
frequencies and damping ratios from the
estimation of mode shapes.
In general, the main advantage of using the
BOMA over deterministic methods is its
capability to estimate the involved uncertainties.
The uncertainty considered by the present
BOMA methods is solely due to the lack of
knowledge about the modal parameters. This
category of uncertainty is often referred to as
epistemic uncertainty, which can be reduced
when additional observations are obtained
(Kiureghian and Ditlevsen 2009). However,
there is a prospect that the unknown parameters
can also be subjected to inherent randomness
when different data sets are used for the
Bayesian inference. This category of uncertainty
is irreducible, often originates from modeling
errors, and cannot be accounted for by the
present Bayesian methods. In this paper, a new
multilevel probabilistic framework is proposed to
bridge this significant gap in modal identification
problems. Subsequently, a basic experimental
example is used to demonstrate the proposed
framework.
2. PROPOSED BAYESIAN FRAMEWORK
2.1. Vibration data
Let
 
1
ˆ, 0,..., 1
n
j
D j N

   y
denote a data set
comprising discrete-time response of a
dynamical system measured at
n
degrees-of-
freedom (DOF), where the index
j
corresponds
to the discrete-time
j
t j t
and
t
is the
sampling interval. The scaled FFT of the
response can be computed as (Yuen and
Katafygiotis 2003):
12/
0
ˆˆe
Nijk N
kj
j
t
N





Fy
(1)
where
1
ˆkn
F
is the Fourier transform of the
response at the frequency
/ ( )
k
f k N t
.
2.2. Frequency-domain model
To construct a parametric model in frequency-
domain, the Fourier transform of the theoretical
response can be used. When the resonance bands
are well-separated, the modal inference can be
performed over each individual band.
Considering the system to be linear having clear
resonance peaks, the Scaled FFT of the response
over a frequency band containing only one
dynamical mode can be expressed as (Au 2017):
k i ik ik
hpFφ
(2)
where
1
in
φ
is the ith mode shape,
ik
p
is the
scaled FFT of the ith modal force corresponding
to
k
f
, and
ik
h
is the transfer function at
k
f
given
by (Au 2017):
2
(2 )
12
q
k
ik ik i ik
if
hi

  


(3)
Here,
q
takes on 0, 1, and 2 for acceleration,
velocity, and displacement response
measurements, respectively;
ik

is the frequency
ratio,
/
ki
ff
;
i
f
and
i

are the modal frequency
and damping ratio corresponding to the ith
dynamical mode. Therefore, the free parameters
can be collected into
{ , , }
i i i
f

φ
, which
should be calibrated based on the vibration data
described in the next section.
2.3. Modal identification
Over a particular frequency band, which contains
only the resonance peak corresponding to the ith
dynamical mode, the scaled FFT of the response
can be predicted as:
ˆk k k
FFε
(4)
where
1
kn
ε
is prediction errors assumed to
have constant power spectral density (PSD) over
the frequency band of interest. Considering the
prediction errors to be statistically independent
and identically distributed (i.i.d.) and describing
them by using a Gaussian distribution leads to:
...
~ (0, )
k e n
i d d NSεI
(5)

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
3
Here,
e
S
is prediction error variance assumed to
be constant over the entire frequency band of
interest and across all DOF. Given this
assumption, the FFT response can be described
by a complex Gaussian distribution expressed as
(Au 2017):
   
*1
1/2
ˆ ˆ ˆ
| exp ( )
n
k k k k
k
p



Fθ F E θ F
E
(6)
and
*
( ) | T
k k k k i i e n
E SD S

  

Eθ F F θ φ φ I
(7)
where
()
k
Eθ
is the theoretical PSD,
S
is the
PSD of the modal force
ik
p
, and
θ
denotes the
parameters of this probabilistic model given by:
{ , , , , }
i i i e
f S S

θφ
(8)
Considering
ˆk
F
’s to be statistically
independent over the entire frequency band
allows constructing the likelihood function as
follows:
 
 
ˆˆ
||
kk
k
pp
Fθ F θ
(9)
By using the Bayes’ rule the posterior
distribution of the parameters can be computed
readily. Given a uniform prior distribution for the
modal parameters, the negative log-likelihood
function should be minimized to yield the MAP
estimations:
 
*1
ln ln ( )
ˆˆ
( )
fk
k
k k k
k
L nN






θ E θ
FEθF
(10)
where
f
N
is the number of data points over the
frequency band of interest. This optimization
problem yields the most probable values (MPV)
of the parameters. An asymptotic approximation
can simplify the posterior distribution giving
(Papadimitriou et al. 1997):
 
 
ˆˆ
| | , p D Nθ
θ θ θ Σ
(11)
and
 
1
ˆ
ˆTL


  

θ θ θ θθ
Σθ
(12)
In the latest equations,
ˆ
θ
and
ˆθ
Σ
denotes
the MPV and covariance matrix of
θ
,
respectively, where
ˆθ
Σ
is approximated as the
inverse of the Hessian matrix of
 
Lθ
evaluated
at
ˆ
θ
. The Hessian can be computed both
analytically and numerically as addressed in Au
(2017).
2.4. Hierarchical Bayesian Approach
The posterior distribution computed earlier only
accounts for the epistemic sources of uncertainty.
To account for the aleatory uncertainty involved
with the modal parameters, multiple sets of
vibration data should be combined under a
hierarchical probabilistic modeling.
Let
 
, 1,...,
rD
D r ND
be the full data set
comprising
D
N
independent sets of vibration
data. Corresponding to each data set, the
foregoing Bayesian inference can be applied, and
D
N
independent realizations for the modal
parameters are thus obtained. Based on Eq. (11),
one can write:
 
 
,
ˆˆ
| | ,
r r r r r
p D Nθ
θ θ θ Σ
(13)
where
r
θ
denotes the modal parameters inferred
from the data set
r
D
. To account for the aleatory
uncertainties, we assume that the dataset-specific
parameters,
r
θ
’s, follow a Gaussian distribution
with the unknown mean
θ
μ
and covariance
matrix
θ
Σ
. These parameters are often referred
to as hyper-parameters, and this probabilistic
model is called the hierarchical modeling
technique (Nagel and Sudret 2016). Given these
assumptions, Sedehi et al. (2019) have recently
shown that the marginal distribution of the
hyper-parameters updated based on multiple data
sets can be computed from:

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
4
   
 
,
1
, | ,
ˆˆ
| ,
D
N
rr
r
pp
N




θ θ θ θ
θ θ θ
μ Σ D μ Σ
μ θ Σ Σ
(14)
Moreover, the posterior predictive
distribution of the modal parameters can be
computed using a Markov chain Monte-carlo
(MCMC) sampling method giving (Sedehi et al.
2019):
   
( ) ( )
1
1
| | ,
s
N
new new m m
m
s
pN
N
θθ
θ D θ μ Σ
(15)
where
()m
θ
μ
and
()m
θ
Σ
denote samples of the
hyper-parameters drawn from
 
,|pθθ
μ Σ D
given by Eq. (14). The latest equation provides a
new formulation to combine multiple data sets
accounting for the inherent randomness of the
modal parameters observed over different data
sets. Sedehi et al. (2018) have proven that the
second-moment statistics of
 
|
new
pθD
can
easily be computed from:
 
()
1
1s
N
new m
m
s
EN
θ
θμ
(16)
   
( ) ( ) ( )
1
( ) ( )
11
1
CoV
11
s
ss
N
new m m T m
m
s
T
NN
mm
mm
ss
N
NN



  
  
  


θ θ θ
θθ
θ μ μ Σ
μμ
(17)
where
 
new
Eθ
and
 
CoV new
θ
denote the
expected value and covariance matrix of
new
θ
,
respectively. In the next section, the proposed
hierarchical Bayesian approach is demonstrated
using an experimental example.
3. EXPERIMENTAL EXAMPLE
Figure 1 shows a three-story shear building
prototype structure tested on a Shaking table at
the Hong Kong University of Science and
Technology (HKUST). The acceleration
responses of the three stories were measured
when the prototype was subjected to
20
D
N
independent Gaussian White noise (GWN) input
excitations. Each set of time-history response
measurements is 120s long, sampled at 0.005s
time intervals.
Figure 1: Structure prototype tested subjected to
white Gaussian noise base excitations.
The FFT of discrete-time acceleration
responses corresponding to a particular data set
is shown in Figure 2. The three well-separated
peaks appearing on this plot correspond to the
three dynamical modes of the structure. Thus, the
presented BOMA approach can simply be
applied to compute the posterior distribution of
dynamical properties from each individual data
set. The proposed hierarchical Bayesian
approach should next be applied to combine
multiple data sets. For the sake of simplicity, we
neglect the correlation between the modal
parameters and assign only one pair of hyper-
parameters to each quantity of interest, the mean
and standard deviation. By using Eq. (14), we
computed the marginal posterior distribution of
the hyper-parameters. Figure 3 shows the
marginal posterior distribution of the modal
frequencies. As can be seen, the MPV of the
mean and standard deviation of the three mode
frequencies are estimated as (4.21Hz, 0.022Hz),
(13.04Hz, 0.045Hz), and (18.82Hz, 0.018Hz).
Figure 4 shows the marginal posterior
distribution of the modal damping ratios. The
MPV of the mean and standard deviations of the

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
5
damping ratios are obtained as (0.038, 0.01),
(0.013, 0.002), and (0.0036, 0.0075). The
validity of this results can be confirmed when
compared with the past studies (Zhouquan 2013).
Figure 2: Frequency response function of
measured output accelerations.
Figure 3: Posterior distributions of the hyper-
distributions chosen to represent mode frequencies.
Figure 4: Posterior distributions of the hyper-
distributions chosen to represent damping ratios.
Transitional MCMC method (Ching and
Chen 2007) is next used to draw the samples
from the marginal posterior distribution of the
hyper-parameters. Eqs. (16) and (17) are used to
compute the mean and standard deviation of the
modal parameters according to multiple data
sets. Figures 5 and 6 compare the data set
specific posterior distributions, shown by the
blue error bars, with the mean and standard
deviations computed by the hierarchical method
indicated by the shaded areas plotted along with
the red line. As shown, the uncertainty bounds
computed by the hierarchical Bayesian approach
are in good agreement with the dataset-specific
uncertainties shown by the blue error bars. In
other words, the uncertainty computed by using
one single data set is fairly consistent with the
uncertainty estimated by using one single data
set. Thus, the hierarchical Bayesian approach is
not only a tool to combine different data sets
reliably, but also it can be used test the
robustness of estimated uncertainties obtained
from the classical BOMA approaches.

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
6
The example demonstrated herein was
rather simple, the modeling assumptions were
flawless, and the full acceleration response
measurements were fed into the algorithm.
However, these results confirm the robustness of
the present BOMA methods to quantify the
involved uncertainty with reasonable accuracy
when there are not considerable modeling errors.
Nevertheless, the prospect that this nice accuracy
cannot be maintained in the presence of drastic
modeling errors still remains open.
Figure 5: Posterior distribution of mode frequencies
and damping ratios (The shaded area and the red
lines show the results of hierarchical Bayesian
approach and the blue error bars show posterior
distributions estimated using a particular data set)
Figure 6: Posterior distribution of mode shapes (The
shaded area and the red lines show the results of
hierarchical Bayesian approach and the blue error
bars show posterior distributions estimated using a
particular data set)
4. CONCLUSIONS
A hierarchical Bayesian formulation is presented
to account for both aleatory and epistemic
uncertainties involved with the operational
modal analysis problems. An experimental
example is adopted to demonstrate the efficacy
of the present BOMA methods, when the models
are sufficiently accurate. Although the
hierarchical method is used herein to combine
different data sets, it is powerful to test and
verify the robustness of estimations obtained
from the classical BOMA methods.
5. ACKNOWLEDGEMENTS
Financial support from the Hong Kong research
grants councils under grant numbers 16234816
and 16212918 is gratefully appreciated. The last
author gratefully acknowledges the European
Commission for its support of the Marie
Sklodowska Curie program through the ETN
DyVirt project (GA 764547).
This paper is completed as a part of the
second author’s PhD dissertation conducted
jointly at Sharif University of Technology and
the Hong Kong University of Science and
Technology. The second author would like to
gratefully appreciate kind support and
supervision of Professor Fayaz R. Rofooei at
Sharif University of Technology.
We would also like to express our sincere
appreciation to Professor Chih-chen Chang for
generously sharing sensors, prototypes, and
laboratory facilities.
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13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
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