JHU Workshop: 13-14 November 2003 Elements of Predictability Robust System Analysis, Model Updating...

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


K

kknk

nn

YPw

dpYPYP

1)ˆ,|(

)|(),|(),|(

US

M,DUSM,DUS NN

K ˆ,...,ˆ,ˆ21


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


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


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


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


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( | ) [ ( | )]


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


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.

JHU Workshop: 13-14 November 2003 Elements of Predictability Robust System Analysis, Model Updating...

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