Bayesian Updating and Model Class Selection for Hysteretic Structural Models Using Stochastic Simulation

TLDR: It is shown here that Bayesian updating and model class selection provide a powerful and rigorous approach to tackle the problem of hysteretic system identification when implemented using a recently developed stochastic simulation algorithm called Transitional Markov Chain Monte Carlo.

Abstract: System identification of structures using their measured earthquake response can play a key role in structural health monitoring, structural control and improving performance-based design. Implementation using data from strong seismic shaking is complicated by the nonlinear hysteretic response of structures. Furthermore, this inverse problem is ill-conditioned for example, even if some components in the structure show substantial yielding, others will exhibit nearly elastic response, producing no information about their yielding behavior. Classical least-squares or maximum likelihood estimation will not work with a realistic class of hysteretic models because it will be unidentifiable based on the data. It is shown here that Bayesian updating and model class selection provide a powerful and rigorous approach to tackle this problem when implemented using a recently developed stochastic simulation algorithm called Transitional Markov Chain Monte Carlo. The updating and model class selection is performed on a previously-developed class of Masing hysteretic structural models that are relatively simple yet can give realistic responses to seismic loading. The theory for the Masing hysteretic models, and the theory used to perform the updating and model class selection, are presented and discussed. An illustrative example is given that uses simulated dynamic response data and shows the ability of the algorithm to identify hysteretic systems even when the class of models is unidentifiable based on the data.

Figures

Figure 2. Hysteresis loop for transient loading of the extended Masing model (from Jayakumar, 1987).

Figure 5. Simulated acceleration time histories. The solid plots are the actual response of the model, and the dashed plots represent the “measured” response with added white noise.

Figure 6. Simulated inter-story shear forces plotted against inter-story drifts.

Table 2. Bayesian model class selection results for drift data. The information gain is the difference between the average data fit and the log of the evidence, which are estimated using stochastic simulation.

Figure 12. Aggregate samples from the final level of three runs of the TMCMC algorithm when updating model class 4 with drift data, plotted in the ru i i space for all three stories.

Table 3. The most probable parameter values for model updating with acceleration data obtained from both a lognormal fit to the samples generated from stochastic simulation, shown with standard deviations in parentheses, and numerical optimization of the un– normalized posterior PDFs, shown with the difference from the simulation estimate in brackets.

Full text

Bayesian Updating and Model Class Selection for
Hysteretic Structural Models Using Stochastic
Simulation
MATTHEW MUTO
JAMES L. BECK
Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125 USA
(jimbeck@its.caltech.edu)
(Received 21 March 20061 accepted 9 August 2006)
Abstract: System identification of structures using their measured earthquake response can play a key role
in structural health monitoring, structural control and improving performance-based design. Implementation
using data from strong seismic shaking is complicated by the nonlinear hysteretic response of structures.
Furthermore, this inverse problem is ill-conditioned1 for example, eve n if some components in the structure
show substantial yielding, others will exhibit nearly elastic response, producing no information about their
yielding behavior. Classical least-squares or maximum likelihood estimation will not work with a realistic
class of hysteretic models because it will be unidentifiable based on the data. It is shown here that Bayesian
updating and model class selection provide a powerful and rigorous approach to tackle this problem when
implemented using a recently develop ed stochastic simulation algorithm called Transitional Markov Chain
Monte Carlo. The updating and model class selection is performed on a previously-dev eloped class of Masing
hysteretic structural models that are relatively simple yet can give realistic responses to seismic loading.
The theory for the Masing hysteretic models, and the theory used to perform the updating and model class
selection, are presented and discussed. An illustrative example is given that uses simulated dynamic response
data and shows the ability of the algorithm to identify hysteretic systems even when the class of models is
unidentifiable based on the data.
Key words: Bayesian methods, Masing hysteretic models, system identification, Markov Chain Monte Carlo simu-
lation, model class selection
1. INTRODUCTION
Current methods for developing finite-element models can produce structural responses that
are consistent qualitatively with behavior observed during strong earthquake shaking, but
there has long been an interest in using system identification methods for quantitative as-
sessment of structural models using recorded seismic response (see Beck, 1978 for early
work and Beck, 1996 for a review). The objective may be to improve the predictive capabili-
ties of structural models for dynamic design or for the design of structural control systems, or
to implement structural health monitoring. System identification based on updating of finite-
element models using measured seismic response is challenging, however, because the large
number of uncertain parameters associated with these models makes the inverse problem
extremely ill-conditioned.
Journal of Vibration and Control, 14(1–2): 7–34, 2008 DOI: 10.1177/1077546307079400
1
12008 SAGE Publications Los Angeles, London, New Delhi, Singapore
© 2008 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.
at University of Thessaly on June 18, 2008 http://jvc.sagepub.comDownloaded from

8 M. MUTO and J. L. BECK
Simplified models can be used in the identification procedure but the selection of an
appropriate class of models to employ is complicated by the nonlinear response of structures
under strong seismic loading1 in particular, the structural restoring forces are hysteretic, de-
pending on the previous time history of the structural response rather than on an instanta-
neous finite-dimensional state. Although some research into the identification of hysteretic
systems has been carried out (Jayakumar, 19871 Jayakumar and Beck, 19881 Cifuentes and
Iwan, 19891 Benedettini et al., 1995, Ashrafi and Smyth, 2005), this pre vious work did not
quantify the modeling uncertainties and did not properly deal with the ill-conditioning in-
herent in this inverse problem. Howe ver, the uncertainty associated with structural model
predictions can have a significant impact on the decision-making process in structural de-
sign, control and health monitoring. Furthermore, classical estimation techniques such as
least-squares and maximum likelihood do not usually work properly when applied to hys-
teretic model classes because they are nearly always unidentifiable based on the available
data.
The Bayesian updating approach treats the probability of all models within a set of can-
didate models for a system, and consequently has the advantage of being able to quantify all
of the uncertainties associated with modeling of a system and to handle ill-conditioned iden-
tification problems. Note that the probability of a model will not make sense if one interprets
probability as a long-run frequency of an event, but it does when probability is interpreted as
a multi-valued logic that expresses the degree of plausibility of a proposition conditioned on
the given information (an interpretation given a rigorous foundation by Cox, 19611 see also
Jaynes, 2003). Although Bayesian methods are widely used in many fields, their application
to identification of dynamic hysteretic models seems to be very limited.
Beck (1989) and Beck and Katafygiotis (1991, 1998) presented a Bayesian statistical
framework for model updating and predictions for linear or nonlinear dynamic systems that
explicitly treats prediction-error and model uncertainties. This earlier work utilized Laplace’s
method for asymptotic approximation to evaluate the Bayesian predictive inte grals. An intro-
duction to this theory is giv en in Papadimitriou and Katafygiotis (2005). A basic concept is
that any set of possible deterministic dynamic models for a system can be embedded in a set
of predictive probability models for the system by specifying a probability distribution (usu-
ally Gaussian) for the uncertain prediction error, which is the difference between the actual
system output and the deterministic model output. Each predictive probability model is as-
sumed to be uniquely specified by assigning a value to a model parameter vector. Therefore,
a probability distribution over the set of possible predictive models that specifies the plausi-
bility of each such model is equivalent to a probability distribution over a corresponding set
of possible values for the model parameter vector. When dynamic data is available from the
system, a chosen initial (prior) probability distribution over the parameters can be updated
using Bayes’ Theorem to give a posterior probability distribution.
It is useful to characterize the topology of this posterior as a function of the model pa-
rameter vector by whether it has a global maximum at a single most probable parameter
value, at a finite number of them, or at a continuum of most probable parameter values ly-
ing on some manifold in the parameter vector space. These three cases may be described as
globally identifiable, locally identifiable,andunidentifiable model classes based on giv en
dynamic data from the system. The Laplace asymptotic approximation is most useful when
there is a large amount of data and the model class is globally identifiable, although it can
© 2008 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.
at University of Thessaly on June 18, 2008 http://jvc.sagepub.comDownloaded from

BAYESIAN UPDATING AND MODEL CLASS SELECTION 9
be applied in the locally identifiable case (Beck and Katafygiotis, 1991, 1998) and even
unidentifiable cases when the manifold of most probable values of the parameter vector is
of very low dimension (Papadimitriou et al., 20011 Katafygiotis and Lam, 2002). However,
it requires a non-convex high-dimensional optimization to find the most probable parameter
vectors, which can be computationally challenging. To avoid these difficult optimizations
and to more readily treat cases where the model class is not globally identifiable (which
will occur in finite-element model updating because of the large number of uncertain para-
meters that occur in realistic structural models), in recent years attention has been focused
on stochastic simulation methods for Bayesian updating and prediction, especially Markov
Chain Monte Carlo methods, such as the Metropolis–Hastings, Gibbs Sampler and Hybrid
Monte Carlo algorithms. The emergence of these stochastic simulation methods has led to
a renaissance in Bayesian methods across all disciplines in science and engineering because
the high-dimensional inte grations that are involved can now be readily evaluated.
The goal of the stochastic simulation methods is to generate samples which are distrib-
uted according to the posterior probability density function (PDF). The posterior PDF gives
the plausibility of each of the candidate models in the model class (specified by a correspond-
ing vector of model parameters), based on the data. While a variety of stochastic simulation
methods are available, many of them are not useful for Bayesian updating. In this work, we
focus specifically on Markov Chain Monte Carlo (MCMC) methods (see Neal, 19931 Gilks
et al., 19961 MacKay, 20031 or Robert and Casella, 2004 for more comprehensive overviews
of this topic). One advantage of these methods is that non-normalized PDFs can be sampled,
so that samples may be drawn from the posterior PDF without evaluating the normalizing
constant in Bayes’ Theorem (called the evidence or marginal likelihood), which can be a
difficult procedure because it usually requires eva luating a high-dimensional integral over
the parameter space. A remaining challenge associated with model updating by stochastic
simulation is the fact that, unless the data is very sparse, the posterior PDF occupies a much
smaller volume in the parameter space than the prior PDF over the parameters. This fact
makes it difficult to draw samples from the posterior PDF.
One commonly-implemented MCMC method is the Gibbs sampler (Geman and Geman,
1984). When applicable, the Gibbs sampler is a powerful method for generating samples
from high-dimensional posterior PDFs1 for example, Ching et al. (2006) apply it to the prob-
lem of using modal data to update a stochastic linear structural model that has 312 para-
meters. However, the Gibbs sampler is only readily applied to model classes that produce
posterior PDFs that have a special structure to them1 and for this reason, it is difficult to
apply to the updating of hysteretic models.
Another commonly-implemented MCMC method is the Metropolis–Hastings (M-H) al-
gorithm (Metropolis et al., 19531 Hastings, 1970), which can be used to create samples
from a Markov Chain whose stationary distribution is any specified target PDF, ev en a non-
normalized one. Although in theory the M-H algorithm can generate samples from any pos-
terior PDF, for higher-dimensional parameter spaces it may still be very dif ficult to draw
samples that cover all the regions of high-probability content. For this reason, Beck and Au
(2000, 2002) proposed gradual updating of the model by using the M-H algorithm to sample
from a sequence of target PDFs, where each target PDF is the posterior PDF based on an
increasing fraction of the available data. In this manner, the target PDF gradually converges
from the broad prior PDF to the final concentrated posterior PDF. The samples from each
© 2008 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.
at University of Thessaly on June 18, 2008 http://jvc.sagepub.comDownloaded from

10 M.MUTOandJ.L.BECK
intermediate PDF are used to form a kernel density, which is used as a global proposal PDF
in the M-H algorithm for the next “level” of sampling.
Ching and Chen (2007) modified this approach to develop what they call the Transi-
tional Markov Chain Monte Carlo (TMCMC) method. This technique also uses a sequence
of intermediate PDFs. However, rather than applying updating with part of the av a ilable data,
the entire data set is used but its full effect is diluted by taking the target PDF for the mth
level of the sampler to be proportional to p
1
1223 2
4
5
m
p
1
222
4
,where03 5
m
3 11 here,
5
0
4 0 gives the initial target distribution proportional to the prior PDF and 5
M
4 1forthe
final le vel of the sampler gives a target distribution proportional to the posterior PDF. Due
to the conceptual similarities between this approach and the simulated annealing approach
(Fishman, 19961 Neal, 1993), 5
m
will be referred to as the tempering parameter. In TMCMC,
re-sampling is used between levels to improve the rate of convergence.
Another difference between the TMCMC algorithm and the approach of Beck and Au
(2002) is in the application of the Metropolis–Hastings algorithm. Rather than using a global
proposal PDF based on a kernel density constructed from the samples from the previous
level, a local proposal PDF is used in what is essentially a local random walk in the parameter
space.
This work focuses on the application of the TMCMC algorithm to Bayesian updating of
the model parameters and to Bayesian model class selection between competing sub-classes
of models, for a class of Masing hysteretic models that are believed to be well-suited to
realistic modeling of the seismic behavior of structures. These models are described in the
next section.
2. MASING HYSTERETIC MODELS
One fundamental approach to constructing hysteretic force–deformation relations for struc-
tural members and assemblages of members is to build them up from constitutive equations
(“plasticity models”) which govern material behavior at a point. How ever, factors such as
complex stress distributions, material inhomogeneities and the large number of structural el-
ements make this approach impractical. Also, there is no general consensus on the choice of
models for cyclic plasticity under arbitrary loading.
An alternative approach is to develop simplified models that capture the essential fea-
tures of the hysteretic force-deformation relationship but then, lacking a fundamental theo-
retical basis, these models should be validated against the observed behavior of structures.
This has been done by Jayakumar (1987) for the well-known Bouc–Wen model (Wen, 1976),
which is in essence a planar version of the early endochronic model (Valanis, 1971)1 these
models are mathematically convenient, especially for random vibration studies using equi v-
alent linearization, but when they are subjected to asymmetric cyclic loading, these models
can exhibit an unphysical “drifting” behavior (Jayakumar, 1987). This behavior makes them
unsuitable as a class of identification models for strong seismic response where this type of
irregular loading occurs.
A simplified hysteretic model with a physical basis was presented by Masing (1926),
which is based on the hypothesis that a one-dimensional hysteretic system may be viewed
as a collection of ideal elasto-plastic elements (a linear spring in series with a Coulomb
© 2008 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.
at University of Thessaly on June 18, 2008 http://jvc.sagepub.comDownloaded from

BAYESIAN UPDATING AND MODEL CLASS SELECTION 11
Figure 1. Conceptual sketch of the Distributed Element Model (from Chiang, 1992).
damper) with the same elastic stiffness but with a distribution of different yield strengths.
This idea was used in structural dynamics by Iwan to form the Distributed Element Model
(DEM), which consists of a collection of N ideal elasto-plastic elements connected in parallel
(Iwan, 1966, 1967) with a common stiffness k6N for the springs but different yield strengths
r
i
*6N3 i 4 13777N , as shown in Figure 1. The restoring force r for a single-degree of
freedom DEM subjected to a displacement x is given by:
r 4
n
1
i41
r
i
*
N
5 kx
N 6 n
N
(1)
where n is the number of elements which have yielded. Infinite collections of elasto-plastic
elements can considered by introducing a yield strength distribution function 8
1
r*
4
,such
that restoring force r
1
x
4
during initial loading is:
r
1
x
4
4
kx
2
0
r*8
1
r*
4
dr * 5 kx
7
2
kx
8
1
r*
4
dr *7 (2)
Because there is an underlying physical basis for the model, DEMs with a finite number
of elements have been shown to give good representations of the hysteretic behavior of some
structures, and do not exhibit the previously-discussed drifting behavior (Cifuentes, 19831
Thyagarajan, 1989). However, DEMs with an infinite number of elements are difficult to
implement directly, in contrast to the finite case where the state of each element is tracked.
Fortunately, there are two hysteretic rules that exactly describe the behavior of DEMs without
needing to keep track of the internal behavior of the elements, which we now present.
Masing (1926) postulated that the steady-state behavior of such a hysteretic system sub-
jected to cyclic loading could be described as follows. If the initial or “virgin” loading curve
is described by the implicit relationship:
f
1
x3 r
4
4 0(3)
© 2008 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.
at University of Thessaly on June 18, 2008 http://jvc.sagepub.comDownloaded from

Frequently asked questions

Q1. What have the authors contributed in "Bayesian updating and model class selection for hysteretic structural models using stochastic simulation" ?

In this paper, Bayesian updating and model class selection are used to identify hysteretic structural models. 

Q2. What are the future works mentioned in the paper "Bayesian updating and model class selection for hysteretic structural models using stochastic simulation" ?

This suggests that future research might study the development of initial loading curves for Masing models which are based on ultimate strength distributions with a larger number of elements yielding at lower force levels than in the sub-class of Masing models considered in this work, in order to create a bi-modal distribution for r * in equation ( 2 ). Other plans for future work in this area include a study of more complicated hysteretic systems for generating data, particularly system models that are not contained in any of the candidate model classes and, ultimately, the application of stochastic simulation methods and Masing models to real-world data from structures that have experienced significant yielding during earthquakes. In this case, since substantial yielding is often associated with damage, application of stochastic simulation techniques to the class of degrading Distributed-Element models, or the equivalent Masing models, with a finite ( Cifuentes, 1983 ) and infinite ( Chiang, 1992 ) number of internal elements, may be important for more realistic modeling of hysteretic structural behavior, while still employing models simple enough to be used in design, monitoring and control applications. 

Q3. What is the main reason for the renaissance in Bayesian methods?

The emergence of these stochastic simulation methods has led to a renaissance in Bayesian methods across all disciplines in science and engineering because the high-dimensional integrations that are involved can now be readily evaluated. 

Q4. What is the potential for large spurious forces in viscous damping models?

linear viscous-damping models, particularly Rayleigh damping, have the potential to introduce large spurious forces into the calculated structural response (Hall, 2006). 

Q5. What is the common method for generating posterior PDFs?

When applicable, the Gibbs sampler is a powerful method for generating samples from high-dimensional posterior PDFs for example, Ching et al. (2006) apply it to the problem of using modal data to update a stochastic linear structural model that has 312 parameters. 

Q6. What is the main challenge associated with the posterior PDF?

A remaining challenge associated with model updating by stochastic simulation is the fact that, unless the data is very sparse, the posterior PDF occupies a much smaller volume in the parameter space than the prior PDF over the parameters. 

Q7. What is the impact of the uncertainty associated with structural model predictions?

the uncertainty associated with structural model predictions can have a significant impact on the decision-making process in structural design, control and health monitoring. 

Q8. What is the way to characterize the topology of the posterior?

It is useful to characterize the topology of this posterior as a function of the model parameter vector by whether it has a global maximum at a single most probable parameter value, at a finite number of them, or at a continuum of most probable parameter values lying on some manifold in the parameter vector space. 

Q9. What is the common method for generating samples from a posterior PDF?

Another commonly-implemented MCMC method is the Metropolis–Hastings (M-H) algorithm (Metropolis et al., 1953 Hastings, 1970), which can be used to create samples from a Markov Chain whose stationary distribution is any specified target PDF, even a nonnormalized one. 

Q10. What is the straightforward approach to evaluating the evidence?

Using the Theorem of Total Probability, the evidence can be expressed as:p j p j j p j j d j (15)The most straightforward approach to evaluating the evidence, in the typical case where the integral in equation (15) is too complex to be analytically integrated and of too high a dimension for numerical quadrature, is to use stochastic simulation with samples drawn from the prior PDF pj j. 

Q11. How can the authors obtain the equation of any hysteretic force-deformation curve?

The equation of any hysteretic force-deformation curve can be obtained by applying the original Masing rule to the virgin loading curve using the latest point of load reversal. 

Q12. What is the manifold of probable models?

For model classes 1 and 2, the manifold of most probable (or plausible) models is essentially constrained to move along a curve in the parameter space where only ru 3 varies, because the value of i is pinned down for all three stories by the yielding in the first story. 

Q13. What is the probability distribution for the small-amplitude stiffnesses?

The prior probability distributions for the small-amplitude stiffnesses Ki are taken to be independent lognormal distributions with the logarithmic mean equal to log 2 5 108 and a logarithmic standard deviation of 0.5. 

Q14. What is the definition of the prediction error?

The prediction error, which is defined to be the difference between the uncertain system output and the identification model output, is taken as Gaussian (based© 2008 SAGE Publications. 

Q15. What is the prediction error variance for identification with acceleration records?

the prediction-error variance for identification with acceleration records, 2acc, is equal for all the stories and is uniformly distributed between 0 and 3, which again is approximately one-half the mean-square of the “measured” acceleration time histories. 

Q16. How many different models are used to generate the predicted response?

All of them use the Masing shear-building model in equations (5)–(7) to generate the predicted response, q i t i 1 Nd , t 1 Nt , for Nd = 3 channels of Nt = 500 time-points at a time-step of 0.02 s. 

Q17. What is the restoring force for a single-degree of freedom DEM?

The restoring force r for a single-degree of freedom DEM subjected to a displacement x is given by:r ni 1ri * N kx N n N (1)where n is the number of elements which have yielded.