Model Updating Using Bayesian Estimation
C. Mares, B. Dratz 1, J.E. Mottershead2, M. I. Friswell3
Brunel University, School of Engineering and Design, Uxbridge, Middlesex, UB8 3PH, UK
1
Ecole Centrale de Lille, Mechanical Engineering, France
2
University of Liverpool, Dept. of Engineering, Brownlow Hill, Liverpool L69 3GH, UK
3
University of Bristol, Department of Aerospace Engineering, Queen’s Building, University Walk,Bristol
BS8 ITR, UK
Abstract
Variability in real structures, which could arise from manufacturing processes, and the modelling
assumptions and limitations require the creation of a statistical model of the relationship between
experimental and model predictions and the quantification of the uncertainty of this estimate. In this paper
Markov-Chain Monte Carlo theory (MCMC) is discussed and applied to model updating in the case of
multiple sets of experimental results by using frequency responses functions. The MCMC method allows
the solution of complex problems in a unifying framework, by integrating over high dimensional
probability distributions in order to make inferences about the model parameters. A simulated three
degree-of-freedom system is used to illustrate some aspects of the method, allowing for practical
assumptions to be tested on a simple example within the WINBUGS environment (Bayesian inference
Using Gibbs Sampling).
1
Introduction
Model updating is usually performed by analysing the degree to which a finite element model adequately
represents a single set of experimental data [1], [2]. The updated parameters should be justified physically
and the quality of the final model should be assessed within the operating range. Robustness-touncertainty, fidelity-to-data and confidence-in-prediction are aspects which give a measure of credibility
for the updated structural model [3], [4].
Variability in real structures, which could arise from manufacturing processes, and the modelling
assumptions and limitations have an important effect when complicated joints and interfaces, such as
welds and adhesively connected surfaces, are present in the actual structure. For such complex structures,
predictions based on a single calibration of the model parameters cannot give a clear measure of
confidence in the capability of an analytical model to represent the actual structure.
The possibility of correction of a set of analytical models with randomized parameters based on a set of
experimental results from a collection of nominally identical test pieces was presented in [5],[6] where a
stochastic Monte-Carlo correlation and inverse uncertainty propagation was carried out on a simulated
example and a benchmark structure with spot-welds.
The three goals presented above are antagonistic and an estimate of the uncertainty leads to the necessity
of creating a model of the relationship between experimental and model predictions. In this paper the
Markov-Chain Monte Carlo theory (MCMC) [7],[8], is discussed and applied to model updating in the
case of multiple sets of experimental results by using frequency response functions. The MCMC method
allows the solution of complex problems in a unifying framework, by integrating over high dimensional
probability distributions in order to make inference about the model parameters.
A simulated three degree-of-freedom system is used to illustrate some aspects of the method, allowing for
practical assumptions to be tested on a simple example within the WINBUGS environment (Bayesian
inference Using Gibbs Sampling) [10]. This software uses Gibbs sampling [11],[12],[13] and the
Metropolis algorithm [14] to generate a Markov chain by sampling from the full conditional distributions.
Different practical assumptions are analysed and an assessment of the consequences of wrongly chosen
2607
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P ROCEEDINGS OF ISMA2006
updating parameters, model structure errors, noise effects, observability of the model errors through
measurements locations and optimal frequency measurement points for the minimisation of model errors
is carried out.
2
Theory
2.1
Bayesian Analysis
Bayesian analysis provides a rational approach for the solution of complex problems, in particular to
parameter estimation problems with an assumed correct model or in model selection problems. The
learning process, based on data, is performed using Bayes’ theorem in a hypothesis space, where
probabilities are assigned to each competing hypothesis regarding the problem under study. The parameter
estimation problem is then solved by estimating the distribution of a random parameter within an
ensemble of data sets and the prior information.
The usual form of Bayes’ theorem is given by:
p ( H i D, I ) =
p( H i I ) p( D H i , I )
(1)
p( D I )
where Hi – proposition asserting the truth of a hypothesis of interest
I - proposition representing our prior information
D - proposition representing data
p ( D H i , I ) - probability of obtaining data D, if Hi and I are true (called the likelihood function )
p ( H i I ) - prior probability hypothesis
p ( H i D, I ) - posterior probability of Hi
p ( D I ) = ∑ p ( H i I ) p ( D H i , I ) - normalisation factor ensuring that
∑ p( H
i
D, I ) = 1
Equation (1) shows how the prior probability of a hypothesis H i is updated to a posterior probability
p ( H i D, I ) which includes all the information provided by the data D . The updating factor is the ratio
of two terms and only the likelihood function (or sampling distribution) p ( D H i , I ) depends explicitly
on H i , the denominator p ( D I ) , called the prior predictive probability or the global likelihood, being
independent of H i .
2.2
Markov Chain Monte-Carlo Bayesian Inference
Quantification of the uncertainty in model parameters when experimental data and model predictions in
some output variables of interest, can be performed by using the Bayesian inference. The Bayesian
solution requires integrals over the model parameter space and Markov Chain Monte Carlo (MCMC)
algorithms form the basis of practical computation for probability distributions in several or many
unknowns. The MCMC method estimates the required posterior distributions for high dimensional models
by using sampler algorithms such as the Metropolis–Hastings or Gibbs approaches, which concentrate the
sampling in regions with significant probability performing an efficient stochastic search in the model
space.
The models are constructed to assess the relationship between the response variables yi and other
characteristics expressed in variables θ j usually called covariates. The explanatory variables are linked
M ODEL UPDATING AND CORRELATION
2609
with the response variables via a deterministic function. For independent model responses following a
probabilistic rule f ( y i θ ) the joint distribution contains all the available information provided by the
sample:
f ( y θ ) = ∏ f ( yi θ )
(2)
i
A
f (θ
Markov
( t +1)
chain
is
θ ,Lθ ) = f (θ
(t )
(1)
( t +1)
a
stochastic
process
{θ
(1)
, θ ( 2) , L θ (t )
}
such
that
θ ) the parameter distribution θ at time t + 1 given all the preceding θ
(t )
depends only on θ (t ) and f (θ (t +1) θ ( t ) ) is independent of the time step t . It can be shown that under
certain conditions the distribution θ (t ) tends to an equilibrium independent of the initial estimates θ ( 0 ) [8]
converging on a posterior distribution called the stationary distribution of the Markov chain. Assuming
that the equilibrium distribution is the target distribution f ( y θ ) , after initialization θ ( 0 ) , the Gibbs
sampler algorithm introduced by Geman and Geman [11] generates samples of θ with a probability
density which converges on the desired posterior target. The Gibbs sampler algorithm is a special case of a
single component Metropolis-Hastings algorithm [12],[13],[14].
2.3
Stochastic Search Variable Selection
In building a model for the estimation problem, a key aspect is that of the variables to be included. The
Bayesian model selection using MCMC techniques explores the model space, tracing the best models and
estimating their posterior probability based on the observed data and the prior probabilities. A possible
two-step solution for a subset of predictors was developed by George and McCulloch [14]. The Stochastic
Search Variable Selection (SSVS) method assigns a probability distribution on a set of models such that
the most appropriate models are given the highest probability distribution and the approach is effective
even for a large number of predictors and small number of observations [14]. The main feature of this
approach is the assignment of a latent inclusion variable δ i to every predictor. Suppose the adopted model
is described by a set of predictors θ 1* , θ 2* , θ 3* Kθ q* from a larger set θ 1 , θ 2 , θ 3 Kθ n . The components
θ i are modelled as a normal mixture:
π (θ i δ i ) = (1 − δ i ) N (0,τ i2 ) + δ i N (0, ciτ i2 )
P (δ i = 1) = 1 − P(δ i = 0) with τ i2 << ciτ i2
(3)
If δ i = 1 then θ i is distributed as a normal with mean zero and variance ciτ i2 where ci is a large constant
pre-multiplying a small default variance τ i2 . If the inclusion of θ i is not supported by the data, then the
prior with the default variance N (0,τ i2 ) will be selected more often. The choice δ i = 1 corresponds to
retaining the predictor θ i while for δ i = 0 the predictor is distributed around zero with a small variance,
meaning that the data does not support the appearance of the predictor among the defining variables for
the model.
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3
P ROCEEDINGS OF ISMA2006
Example
If the model is described by using a finite element model, the global matrices are expressed as linear
combinations of constant element or substructure matrices multiplied by the variable parameters and
affecting all the terms in the substructure stiffness or mass matrix. For example a general parameterisation
for the stiffness matrix may be written as,
K (x) = ∑ x j K ej
(4)
with similar decompositions for the mass or damping matrices. The choice of updating parameters and
their type of variation (bounds and probability distributions), must be justified physically and requires
experimental data that are not always accessible.
Different measures for goodness-of-fit used in test-analysis correlation can be used for Bayesian
estimation [15],[16]. In this paper the likelihood function is built using a root mean square error (RMS)
between experimental and predicted frequency response functions:
Nfreq
L( y θ) =
∑
Nfreq
ε k2
k =1
∑
=
k =1
α∈{measurement point set}
β∈{excitation point set}
(h
exp
αβ
( ωk ) − hαβ ( ωk ; θ ) )
2
(5)
where the sum the measured FRFs for a chosen set Nfreq of the frequency lines, at the excitation and
measurement locations used in test. For variability studies, the sum in equation (5) is augmented to
Nsamples Nfreq
L( y θ) =
∑ ∑
j =1
Nsamples
ε 2k =
k =1
∑
j =1
Nfreq
∑
k =1
α∈{measurement point set}
β∈{excitation point set}
(h
exp
αβ, j
( ωk ) − hαβ, j ( ωk ; θ ) )
2
(6)
When the information about the system is uncertain, the model prediction includes error components due
to the measurement and statistical uncertainty in the model, generally these being uncorrelated. In the
example discussed in this paper, only the errors due to the analytical model are discussed in order to assess
different possible aspects in model updating. Each updating variable in each sample, varies according to a
Gaussian law, expressed by:
ki ~ N (ki ,τ i ); τ i =
1
σ i2
(7)
The prior information for each mean and variance parameter ki , σ i is encoded in a probability distribution
to be used with the likelihood term, equation (6) for Bayesian estimation using MCMC:
k i = Normal (0.0,1.0e − 6);
τ i = Gamma(0.001,0.001)
The example considered is the 3 degree-of-freedom system, shown in Figure 1.
(8)
M ODEL UPDATING AND CORRELATION
2611
Figure 1. Three Degree-of-Freedom Mass-Spring Example.
The nominal values of the parameters for the experimental system are: mi = 1.0 kg ( i = 1,2 ,3 ) k i = 1.0
N/m ( i = 3,4, ) and k 6 = 3.0 N/m. The analysis random parameters have Gaussian distributions with
mean values, k1 = k 2 = k5 = 1.0 and standard deviations, σ 1 = σ 2 = σ 5 = 0.20 N/m. For the analytical
model, the erroneous parameters have the initial values k1 = k 2 = k5 = 1.5 N/m. A set of 10 systems are
used for statistical inference.
Experimental Model
2
10
Set A
Set B
Set C
1
10
0
H11(m/N)
10
−1
10
−2
10
−3
10
0
0.5
1
1.5
2
Frequency(rad/s)
2.5
3
3.5
4
Figure 2. Frequency point sets in Case 1.
The following cases are simulated:
Case 1: Only the erroneous parameters are updated as in the ideal case of perfectly localized errors.
Case 2: In the experimental model a spring connection: k 6 is not present, the analytical model presenting
a connectivity error. The updating parameters remain the same as in Case 1.
Case 3: The same as Case 1 except that all the springs are Gaussian random variables with standard
deviations σ i = 0.20 N/m for i = 1, K 5 and σ 6 = 0.6 N/m.
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P ROCEEDINGS OF ISMA2006
Case 1:
This is an ideal case with all the errors contained in the analytical model located correctly and it will be
used to evaluate the method’s performance. In order to obtain the most accurate prediction, an optimum
subset of frequency lines should be determined within the acquired frequency range. The computational
costs, and many factors influencing the relative accuracy with which the response can be measured,
precludes the use of all the frequency range in the updating algorithm. In the case of real data, the noise
impacts upon the results and different frequencies can dramatically alter the results, and each algorithm is
affected by a combination of measurement and processing errors.
Figure 3. Mean parameter variation and pdf in Case 1.
Parameter
k1
k2
k5
Experimental
Mean
Std
1.000
0.180
1.046
0.176
1.005
0.238
Updated Model
Mean
Std
1.004(-0.36)
0.194(-7.77)
1.069(-2.17)
0.181(-2.84)
0.991(1.39)
0.257(-7.98)
Table 1. Parameter Estimate in Case 1: h11 and h22 measured, frequency set A.
M ODEL UPDATING AND CORRELATION
2613
The finite element model is updated using the likelihood function, equation (1):
• h1,1 , h2, 2 evaluated for set A consisting of 25 equally spaced frequency lines in the interval
[0.0,4.0] ( rad ) ;
•
s
h1,1 for set A;
•
h1,1 , h2, 2 evaluated for set B consisting of 14 frequency lines avoiding the areas close to the
•
resonances or anti-resonances;
h1,1 , h2, 2 evaluated for set C with 16 frequency lines in the areas close to the resonances.
The location for each frequency set is shown in Fig. 2 for a nominal experimental model. The results for
the mean and standard deviation obtained after updating in each case after 6000 iterations, are presented in
Tables 1-4. The convergence history for each updated parameter and the final probability density
functions when using set A and h1,1 , h2, 2 are presented in Fig. 3.In this case, the estimated mean and
standard deviation of the experimental sample are not exactly the same as the distribution that was used to
produce them. The updated parameters converge upon the statistics of the experimental sample rather than
the underlying distribution. The results are improved when using two frequency response functions due to
increased information contained in the response functions. The errors for the mean values are reduced
when using frequency lines avoiding the resonances (set B) as opposed to a equally spaced selected
frequencies (set A) but both sets present standard deviations errors (increased for set B). Set C presents big
errors for all the estimates showing that in general it is better to avoid frequencies located in the resonance
or anti-resonance areas. The experimental sample is small, as encountered usually in industrial
applications and it can be expected that if more experimental samples were available, the MCMC process
would estimate better the model parameters.
Experimental
Parameter ean
Std
k1
1.000
0.180
k2
1.046
0.176
k5
1.005
0.238
Updated Model
Mean
Std
0.965(3.5)
0.207(-15.0)
1.104(-5.54)
0.158(10.22)
0.982(2.28)
0.259(-8.82)
Table 2. Parameter Estimate in Case 1: h11 measured, frequency set A.
Parameter
k1
k2
k5
Experimental
Mean
Std
1.000
0.180
1.046
0.176
1.005
0.238
Updated Model
Mean
Std
0.999(0.23)
0.197(-9.44)
1.044(0.19)
0.187(-6.25)
1.006(-0.10)
0.263(-10.50)
Table 3. Parameter Estimate in Case 1: h11 and h22 measured, frequency set B.
Parameter
k1
k2
k5
Experimental
Mean
Std
1.000
0.180
1.046
0.176
1.005
0.238
Updated Model
Mean
Std
1.384(-38.40)
0.398(-121.11)
1.261(-21.55)
0.202(-14.77)
0.878(12.63)
0.294(-23.52)
Table 4. Parameter Estimate in Case 1: h11 and h22 measured, frequency set C.
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P ROCEEDINGS OF ISMA2006
Case 2
The model correction is carried out in two phases, first by analyzing the structure of the analytical model
and then by an estimation phase similar to that discussed for Case 1. The possibility of having a different
structure of the analytical model is investigated by using the model selection procedure proposed by
George and McCulloch [14]. In the Stochastic Search Variable Selection (SSVS) method, the relationship
between the output variables and a set of predictor variables is described by a latent inclusion variable.
Three different models are considered for comparison, each having a different structure by adding a spring
connection k5, k6 or both, to a baseline system as shown in Table 5. The initial spring stiffnesses
are k1 = k 2 = k5 = 1.5 N/m.
Model
1
2
3
Elements
k1,k2,k3,k4,k5
k1,k2,k3,k4,k6
k1,k2,k3,k4,k5, k6
Table 5. Models used in SSVS.
As previously, the frequency responses h11 and h22 are used in the likelihood function, equation (4). The
probability of appearance for each model, named p, is defined as a categorical distribution, p = (1,...,3)
3
and
∑ P( p) = 1 . A prior distribution for p is given by the inverse of the number of models containing
k =1
each k. A link is created between the models and variables as described by (9):
model 1
model 2
model 3
→
→
→
⎛1
⎜
⎜0
⎜1
⎝
↑
k5
0⎞
⎟
1⎟
1 ⎟⎠
(9)
↑
k6
The variable parameters are defined by equation (3) as a normal mixture and the stochastic search will
determine a posterior distribution for the inclusion variable which will define the model with the best
structure.
The results for a simulation with 5000 updates are presented in Figure 4, where the probability of
occurrence for the first model, ranks it as having the most appropriate for the analysed data. In this case
the connection represented by the spring k6 should not exist in the final analytical model ( model 1 is the
most probable).
Case 3
Table 6 shows the results in this case. All of the stiffnesses are randomised and the estimated parameters
incorporate all the variability presented by the experimental set. The two models come from different
families with different statistical properties and both the mean and standard deviations are in error. The
same type of result can be obtained when a localised error is not located prior to the updating process and
consequently not included in the updating loop.
M ODEL UPDATING AND CORRELATION
2615
a)
b)
c)
Figure 4. Probability of appearance for each model: a) convergence of the latent variable;
b) detail of the convergence process; c) final probabilities, model 1 being the most probable one.
Parameter
k1
k2
k5
Experimental
Mean
Std
1.045
0.171
0.989
0.151
0.906
0.183
Updated Model
Mean
Std
1.072(-2.58)
0.267(-56.14)
0.989(0.0)
0.166(-9.93)
0.885(2.31)
0.211(15.30)
Table 6. Parameter Estimate in Case 3.
4
Conclusions
In this paper a non-linear least squares analysis seen from a Bayesian perspective is used for model
updating. The uncertainty about some model parameter or about the model structure is encoded based on
prior information and used in a Bayesian inference for the determination of a posterior probability density
function based on measured frequency response data. A simulated example is used to demonstrate the
potential of the method for applications related to real life applications. It is accepted nowadays that the
structure of the model (the parameterisation used to describe the model dynamics) is the most important
aspect related to the updating process. A two-stage methodology for variable selection and an analysis of
measurement locations and frequency lines distribution are presented together with the effects on
parameter estimation.
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P ROCEEDINGS OF ISMA2006
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