JHU Workshop: 13-14 November 2003 Elements of Predictability
Robust System Analysis, Model Updating and Model Class Selection
James L. Beck
Applied Mechanics & Civil Engineering
California Institute of Technology
Introduction
Part I: Present probabilistic framework for robust system analysis and its updating based on system data Robust means uncertainties in system
modeling explicitly addressed Part II: Present probabilistic framework
for model class selection based on system data
Introduction Features
Framework based on probability axioms and no other ad-hoc criteria or concepts. (Repeated use of Total Prob. Theorem and Bayes Theorem).
Employs probability of models - uses Cox-Jaynes interpretation of probability as a multi-valued logic quantifying plausibility of statements conditional on specified information
Careful tracking of all conditioning information since all probabilities are conditional on probability models and other specified information.
Introduction Features
Involves integrations over high dimensional input and model parameter spaces but computational tools are available and are improving with research
Framework is general but our focus is on dynamical systems in engineering so much prior information is available
Part I: Robust System Analysis
Stochastic System Modeling Probabilistic input-output model for system
with uncertainties in system modeling addressed
Prior System Analysis Uncertainties in system excitation addressed Reliability analysis
Posterior System Analysis Bayesian updating based on system data Updated reliability analysis
Stochastic System Modeling Predictive model: Gives probabilistic
input-output relation for system :
),|( nn UYp SystemInput
Output
(unknown)
(known)
nynu
},...,0:{ nkRuU ONkn
},...,1:{ nkRyY INkn
where input (if available) and output time histories:
Usually have a set of possible predictive models for system:
Nominal prior predictive model: Select single model, e.g. most plausible model in set
But there is uncertainty in which model gives most accurate predictions
:),|( { nn UYpM }pNR
Stochastic System Modeling (Continued)
Robust prior predictive model: Select to quantify the plausibility
of each model in set, then from Total Prob. Theorem:
Stochastic System Modeling (Continued)
)|( Mp
dpUYpUYp nnnn )|(),|(),|( MM
Stochastic System Model: Example 1 Complete system input known: Define
deterministic input-output model for
SystemInput
Output nynu
nY
PDF for prediction-error time history gives
pNR
),( nn Uq
),( nnnn Uqyv Prediction error:
),|( nn UYpCan take prediction errors as zero-mean Gaussian & independent in time (max. entropy distribution), so is Gaussian with mean and covariance matrix
),( nn Uq)(
Stochastic System Model: Example 2 Complete system input not known: Define
state-space dynamic model for system by:
),|( nn UYp
System Input
Output
(unknown)
(known)
nynu
),,,( 111 nnnn wuxFx
nnnn vuxHy ),,(
nw nx
Probability models for initial state and the time histories of and define
0xnw nv
Excitation Uncertainty: Choose probability model over set of possible system inputs:
Nominal prior predictive analysis: Find the probability that system output lies in specified set using nominal model:
Reliability problem corresponds to defining ‘failure’ (= specified unacceptable performance of system)
Primary computational tools for complex dynamical systems are advanced simulation methods (examples later) and Rice’s out-crossing theory for simple systems
) |( UnUp
Prior System Analysis
nnnnn dUUpUYPYP ) |(),|(),|( USUS
S
SYn
Robust prior predictive analysis:
Robust reliability if defines failure Primary computational tools:
Simulation e.g. importance sampling with ISD at peak(s) of integrand (needs optimization)
Asymptotic approximation w.r.t. curvature of the peak(s) of integrand (needs optimization)
Huge differences possible between nominal and robust failure probabilities
Prior System Analysis (Continued)
dpYPYP nn ) |(),|() ,|( MUSMUSSYn
Comparisons between nominal and robust failure probabilities available in:Papadimitriou, Beck & Katafygiotis, “Updating Robust Reliability using Structural Test Data”, Prob. Eng. Mech., April 2001 Au, Papadimitriou and Beck, “Reliability of Uncertain Dynamical Systems with Multiple Design Points” Structural Safety, 1999
Prior System Analysis (Continued)
Available System Data: Update by Bayes Theorem: Optimal posterior predictive model:
Select most plausible model in set based on data
i.e. that maximizes the posterior PDF (if unique) Optimal posterior predictive analysis:
Difficulties: Non-convex multi-dimensional optimization (‘parameter estimation’); ignores model uncertainty
},{ NN YUND
Posterior System Analysis
nnnnn dUUpUYPYP ) |()ˆ,|()ˆ,|( USUS
M)MDN |(),|(),|( pUYcpp NN
Robust posterior predictive model: Use all predictive models weighted by their updated probability (exact solution based on prob. axioms):
Robust posterior predictive analysis:
Primary computational tools are advanced simulation methods and asymptotic approximation w.r.t. sample size
Posterior System Analysis (Continued)
dpUYpUYp nnnn )|(),|(),|( M,DM,D NN
dpYPYP nn ) |(),|() ,|( M,DUSM,DUS NN
Asymptotic approximation for large N for robust posterior predictive analysis (Beck & Katafygiotis, J. Eng. Mech., April 1998; Papadimitriou, Beck & Katafygiotis, Prob. Eng. Mech., April 2001)
Assumes system is identifiable based on the data, i.e. finite number of MPVs that locally maximize posterior PDF, so needs optimization. Uses Laplace’s method for asymptotic approximation
Posterior System Analysis (Continued)
K
kknk
nn
YPw
dpYPYP
1)ˆ,|(
)|(),|(),|(
US
M,DUSM,DUS NN
K ˆ,...,ˆ,ˆ21
Posterior System Analysis (Continued)
The weights are proportional to the volume under the peak of the posterior PDF at
Globally identifiable case justifies using MPV for posterior predictive model when there is large amounts of data:
Gives a rigorous justification for doing predictions with MPV or MLE model
Error in approx. is
kw
k
)ˆ,|(),|( nnnn UYpUYp M,DN
)1( K
)/1( NO
Posterior System Analysis (Continued)
Unidentifiable case corresponds to a continuum of MPVs lying on a lower dimensional manifold in the parameter space
Our interest in this case is driven by finite-element model updating
Asymptotic approximation for posterior predictive model for large amount of data is an integral over this manifold – feasible if it is low dimension (<4?) (Katafygiotis, Papadimitriou and Lam, Soil Dyn. Eq. Eng., 1998; Papadimitriou, Beck and Katafygiotis, Prob. Eng. Mech., April 2001)
All MPV models give similar predictions at observed DOFs but may be quite different at unobserved DOFs
Posterior System Analysis (Continued)
Simulation approaches: Very challenging because most of probability
content of posterior PDF concentrated in a small volume of parameter space (IS does not work)
Potential of avoiding difficult non-convex multi-dimensional optimization and handling unidentifiable case in higher dimensions
Markov Chain Monte Carlo simulation using Metropolis-Hastings algorithm shows promise. (Beck & Au, J. Eng. Mech., April 2002: Adaptive method that works OK for up to 10 or so model parameters)
Recent Simulation Tools for Dynamic Reliability Calculations
Dynamic Reliability: First-Excursion Problem
What is the probability that any response quantity of interest exceeds some limits within a given duration of interest?
+b
-b
“Failure”
Advanced Simulation Methods
Goal is to achieve accurate estimates of small failure probabilities using much fewer samples than Standard Monte Carlo Simulation which is not efficient for small failure probabilities
General philosophy of improving simulation is to gain more information about failure region from samples or prior analysis
Recent Advanced Simulation Methods
ISEE (IImportance SSampling using EElementary EEvents):
For linear dynamical systems with uncertain excitation Requires less that 50 system analyses for any first-
excursion failure probability to get c.o.v. 20% (Au and Beck, Prob. Eng. Mech., July 2001)
Subset Simulation: Applicable to general dynamical systems with both
excitation and modeling uncertainties Substantial improvement in efficiency over standard
MCS for small failure probabilities(Au and Beck, Prob. Eng. Mech., October 2001
Au and Beck, J. Eng. Mech., August 2003)
For simulation, formulate failure probability as a multi-dimensional integral over input variable space:
First-Excursion Problem: Discrete Form
functiondensity
y Probabilit
)(zp)(FP
safe if 0
failed if 1 functionIndicator
)(
zdzF
I
Challenge: High dimension of input variable space
density sampling importance thePDF,chosen a is )( where
)()(
)()(
zf
zdzfzf
zFIzp
)(
)()(
zf
zIzpE F
f
Importance Sampling Simulation
Estimation of failure probability:
zdzFIzpFP )()()(
)( from generated i.i.d. are ,..., where
)(
)()(1)(
1
1
zfzz
zf
zIzp
NFP
N
N
k k
kFk
Importance Sampling Density Goal is to choose importance sampling density (ISD)
to reduce variance of the estimator of the failure probability Theoretical optimal choice for ISD is the conditional PDF given failure
)|()(
)()()(opt Fzp
FP
zFIzpzf
But not feasible since the normalizing constant is the unknown failure probability!
F)|zp(
The ISD should be chosen to approximate the conditional PDF, which requires information about failure region Usual choice is weighted sum of Gaussian PDFs located at “design points”, i.e. maxima of
)( zf
Solution of First-Excursion Problem by Importance Sampling
Challenges: The dimension of the uncertain input
parameter space is extremely large (e.g. 1000)
Calculation of design points is a difficult non-convex optimization problem, in general
Geometry of failure region is difficult to visualizeSolving first-excursion problem by importance
sampling for general dynamical systems is not yet successful
ISEEISEE: Key Aspects
ISEE focuses on linear dynamical systems, so Information about failure region can
be obtained analytically
Design points can be readily obtained using unit impulse response functions
ISEEISEE: Elementary failure events/regions
tn
kk bZtYF
1
|);(| event, Failure
1
tn
kkF
k
t
bktYkF
at time fails response
event failure elementary |)(|
region failure elementary |);(:| bzktYRz n
Consider one output response:
ISEEISEE: Elementary failure regions
Analytical study of elementary failure region in standard Gaussian input space
bzktYRzF nk
|);(:|
Linear dynamics, so failure surface is two parallel hyperplanes
ISEEISEE: Elementary failure region
How to obtain each design point?
Elementary failure region
)(th
kt
kt
kt
k
*kz
Failure Region (Up-crossing)
*kz
*kz
Failure Region (Down-crossing)
Standard Gaussian
Space
Design point is ‘critical excitation’ (Drenick 1970), i.e. the excitation with least ‘energy’ that drives the response to level at time
Readily obtained from unit impulse response functions
Euclidean norm:
ISEEISEE: Design point for each elementary failure region
ktb
)(
* ||||ktY
k
bz
*kz
We know a lot about each elementary failure region:
Design point
Probability content
Efficient generation of random samples according to conditional PDF
*kz
||)||(2)( *kk zFP
)(/)()()|( kFk FPzIzpFzpk
ISEEISEE: Elementary failure regions
ISEEISEE: Union of elementary failure regions
Complexity of first excursion stems from union of elementary failure events
In addition to the ‘global’ design point, a large number of neighboring design points are important in accounting for the failure probability, and hence should be used in constructing the importance sampling density
1F *1z
2F *2z
3F
*3z
4F
*4z
ISEEISEE: Conventional choice of ISD
)()( 1
*
tn
iii zzwzf
1F *1z
2F *2z
3F
*3z
4F
*4z
ISEEISEE: Optimal choice of ISD
)|()( 1
tn
kkk Fzpwzf
ISEEISEE: Failure Probability Estimator
N
r r
rrF Zf
ZZI
NFP
1 )()(
)(1
)(
N
rm
i
n
k
rF
rFF t
ikZI
ZIN
PFP1
1 1
)(
1)(
1ˆ)( )(
11ˆ)(1
1 1
N
rm
i
n
k
rF
F t
ikZI
NPFP
Counting the no. of time instants
at which response fails
Estimate assuming elementary failure
events are mutually exclusive
(analytical)
Correction term to be estimated by simulation
=1 if elementary failure events are mutually
exclusive
ISEEISEE: Applicability
Linear dynamics with Gaussian input
Non-stationary input, e.g., white noise modulated by envelope function
Multiple input excitations
Multiple output responses
Non-white input, e.g., filtered white noise
ISEEISEE: Linear 3-bay-by-6-story steel frame
P e a k I n t e r s t o r y d r i f t r a t i o
),,;( rMZtY
Gaussian White Noise Z
),,;( rMZta
Envelope function ),;( rMte
Spectrum (M,r)
Atkinson-Silva Model for Ground Motion
0 20 40 60 80 10010
-5
10-4
10-3
10-2
10-1
100
No. of samples N
b = 1%
b = 0.75%
b = 0.5%
ISEEISEE: Linear 3-bay-by-6-story steel frame Failure Probability estimates
)km 307,|bIDRPeak ()( rMPFP
ISEEISEE: Concluding Remarks
ISEE is built upon the following observations:
Analysis of failure region as a union of elementary failure regions corresponding to failure at different time instants
For linear systems, the design points can be obtained in terms of unit impulse response functions
In addition to the global design point, a large number of design points are important in contributing to the failure probability
The importance sampling density is constructed as a weighted sum of conditional failure PDFs
ISEEISEE: Concluding Remarks
Continued:
It’s remarkably efficient for first-excursion problem with linear dynamics and Gaussian excitation Failure probability estimator for N samples has c.o.v.
where decreases slightly with decreasing failure probability, in contrast to Monte Carlo simulation where
N
)1(O
)(/1 FP
SSubset SSimulation: Basic Idea
For c.o.v. of 30%:
Carlo Monteby samples 10,000 requires 3
10)(
FP
Carlo Monteby samples 1,000 requires 2
10)(
FP
Carlo Monteby samples 100 requires 1
10)(
FP
Can we avoid simulation of rare events?
300
100? 100?
100?
SSubset SSimulation: Basic Idea Express failure probability as a product of
conditional failure probabilities, e.g.
P(IDR>1.5%)= P(IDR>1.5%|IDR>1%)
P(IDR>1%)P(IDR>1%|IDR>0.5%)
P(IDR>0.5%)
0.001 0.1 0.10.1 =
10,000# samples:
How do we efficiently simulate conditional samples to estimate conditional probabilities?
SSubset SSimulation with Markov Chain Monte Carlo (MCMC)
Generate conditional samples by MCMC simulation for each nested failure region:
Use specially designed Markov chain so that at each level conditional samples are distributed as
MH algorithm: Metropolis et al. (1954), Hastings (1971) Samples are dependent but same failure
probability estimator as MCS can be used
1 2 ... mF F F F
)iF|zp(
Subset Simulation: Concluding Remarks
Efficient failure probability estimation for general dynamical systems
Expresses a small failure probability as a product of larger conditional failure probabilities
Substantial improvement over standard Monte Carlo for small failure probabilities (1,400 samples at 0.001 compared with 10,000 for same c.o.v).
Applicable in high dimensions (e.g. 1000 or more)
Continued:
Failure probability estimator for N samples has c.o.v.
where and
whereas Monte Carlo simulation has
N
|)(log| FP
)(/1 FP
Subset Simulation: Concluding Remarks
)1(O
Appendix to Part I:
Foundations of plausibility logic and probability
Plausibility Logic and Probability Axioms
Axiomatic Foundations:[R. T. Cox: 1946, 1961: “The Algebra of Probable Inference”, Johns Hopkins Press; E. T. Jaynes: 1957, 2003: “Probability Theory: The Logic of Science”, C.U.P.]
Consider extension of binary Boolean logic:
if a is more plausible than b, given c What is the calculus of plausibility logic? How
do we get
( | )a c ameasure of how plausible statement is,given the information in statement c
( | ) ( | )a c b c
)|(~),| (),|&( cacboracba
Plausibility Logic and Probability Axioms
Impose conventions:
Impose consistency with Boolean logic:
0 1 ( | )a c
( | )a c 1 if c logically implies a is true ( )c a( | )a c 0 c a ~if c logically implies a is false ( )
( &( & )| ) ( & )& |
(~ ~ | ) ( | )
a b d c a b d c
a c a c
Plausibility Logic and Probability Axioms
Theorem (Cox 1946 + Aczel 1966): Under a strict monotonicity condition on :
Scale of plausibility is arbitrary, so set
[ ( | )] [ (~ | )]
[ ( & | )] [ ( | & )] [ ( | )]
a c a c
a b c a b c b c
1
where any continuous monotonically increasingfunction with
( )( ) , , ( )
0 0 0 1 1
)|&( cba
p a c a c( | ) [ ( | )]
Plausibility Logic and Probability Axioms
Gives axioms for plausibility measure:
De Morgan’s Law: , so
Plausibility logic axioms=Probability logic axioms!
0 1 p a c( | )p a c c ap a c c a( | )( | ) ~
10
ifif
p a b c p a b c p b cp a c p a c
( & | ) ( | & ) ( | )( | ) (~ | )
1
a b a b or ~ (~ & ~ )p a b c p a c p b c p a b c( | ) ( | ) ( | ) ( & | ) or
Plausibility Logic and Probability Axioms
Conclusion: Interpret probability based on PDF as a measure of the plausibility of the model specified by given the information
dp )|( I
I
References:
R. T. Cox: 1946, 1961: “The Algebra of Probable Inference”, Johns Hopkins Press;
E. T. Jaynes: 1957, 2003: “Probability Theory: The Logic of Science”, C.U.P.
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