Lectures on Stochastic System Analysis and Bayesian Updating

Transcription

1 Lectures o Stochastic System Aalysis ad Bayesia Updatig Jue 29-July James L. Beck, Califoria Istitute of Techology Jiaye Chig, Natioal Taiwa Uiversity of Sciece & Techology Siu-Kui (Iva) Au, Nayag Techological Uiversity, Sigapore

2 Overview of Robust Stochastic System Aalysis 1.1: Stochastic System Modelig Class(es) of probabilistic iput-output models for system to address ucertaities i system modelig (robust aalysis) Termiology: System = real thig; Model =idealized mathematical model of system 1.2: Prior System Aalysis Ucertaities i system iput also addressed Reliability aalysis to compute failure probabilities 1.3: Posterior System Aalysis Bayesia updatig of models i class based o system data Updated reliability aalysis

3 1.1: Stochastic System Modelig Predictive model: Gives probabilistic iput-output relatio for system depedig o model parameters: p( Y U, θ) u (kow) Iput (ukow) N I System U = { u R : k = 0,..., } k N O Y = { y R : k = 1,..., } k Ucertai y Output where iput (if available) ad output time histories:

4 Stochastic System Modelig (Cotiued) Usually have a set of possible predictive probability models to represet system: { py ( U, θ) : θ Θ R } Nomial prior predictive model: Select sigle model, e.g. most plausible model i set But there is ucertaity i which model gives most accurate predictios that should ot be igored N p

5 Stochastic System Modelig (Cotiued) Robust prior predictive model: Select p( θ M) to quatify the plausibility of each model i set, the from Total Probability Theorem: p( Y U, M) = p( Y U, θ) p( θ M) dθ M Here, deotes the class of probability models, i.e. it specifies the fuctioal forms of & p( θ M More about choosig, the prior PDF, later p( θ M) py ( U, θ) )

6 Stochastic System Model: Example 1 Complete system iput kow: Defie determiistic iput-output model for θ Θ R N p u Iput Ucertai predictio error: q ( U, θ) System ( U, θ) Model for predictio-error time history gives Ca take predictio errors as zero-mea Gaussia & idepedet i time (maximum etropy distributio), so Y is Gaussia with mea q ( U, θ) ad covariace matrix Σ(θ) v = y q Ucertai Output y p ( Y U, θ)

7 Stochastic System Model: Example 2 Complete system iput ot kow: Defie state-space dyamic model for system by: Ucertai state: (kow) u Ucertai Iput w System Output y (ukow) ( 1, 1, 1, θ = F x u w ) x y = H( x, u, θ) + Ucertai output: Probability models for missig iformatio (i.e. iitial state x0 ad time histories of ukow iput w ad predictio error v ), defie p( Y U, θ) v

8 First-Excursio Problem: Aalysis Model Dyamical System U ( t; Z ) Stochastic E xcitatio Y Ucertai Respose 1( t; Z ),..., Ym ( t; Z ) Spectrum Liear filter Gaussia White Noise Z Evelope fuctio e(t )

9 1.2: Prior System Aalysis Total Iput Ucertaity: Choose probability model over set of possible system iputs: p( U U) Nomial prior predictive aalysis: Fid the probability that system output lies i specified set F usig omial model: PY ( F U, θ) = PY ( F U, θ) pu ( U) du Y F Reliability problem correspods to defiig failure (= specified uacceptable performace of system) Primary computatioal tools for complex dyamical systems are advaced stochastic simulatio methods (more later) ad Rice s out-crossig theory for simpler systems

10 First-Excursio Problem Ucertai excitatio U Dyamical System Ucertai respose Y (; tu),..., Y (; tu) 1 m m P(Failure) = P( { Y( t; U) > b for some t o [0, T]}) i= 1 i i

11 Prior System Aalysis (Cotiued) Robust prior predictive aalysis: PY ( F UM, ) = PY ( F U, θ) p( θ M) dθ Robust reliability if defies failure Primary computatioal tools: Y F Stochastic simulatio, e.g. importace samplig with ISD at peak(s) of itegrad (eeds optimizatio) Asymptotic approximatio w.r.t. curvature of the peak(s) of itegrad (eeds optimizatio) Huge differeces possible betwee omial ad robust failure probabilities

12 Prior System Aalysis (Cotiued) Asymptotic approximatio itroduced i: Papadimitriou, Beck ad Katafygiotis (1997). Asymptotic expasios for reliability ad momets of ucertai systems. (at website) Au, Papadimitriou ad Beck (1999). Reliability of Ucertai Dyamical Systems with Multiple Desig Poits (at website) Comparisos betwee omial ad robust failure probabilities available i: Papadimitriou, Beck & Katafygiotis (2001). Updatig Robust Reliability usig Structural Test Data. (at website)

13 1.3: Posterior System Aalysis Available System Data: Update by Bayes Theorem: D N N N = { U, Y } p( θ D, M) = cp( Y U, θ) p( θ M) Optimal posterior predictive model: Select most plausible model i class based o data, i.e. that maximizes the posterior PDF (if uique) θˆ N N N Optimal posterior predictive aalysis: PY ( F U, ˆ θ) = PY ( F U, ˆ θ) pu ( U) du Difficulties: No-covex multi-dimesioal optimizatio ( parameter estimatio ); igores model ucertaity

14 Posterior System Aalysis (Cotiued) Robust posterior predictive model: Use all predictive models i class weighted by their updated probability (exact solutio based o probability axioms): p( Y U, D,M) = p( Y U, θ) p( θ D, M) dθ N Robust posterior predictive aalysis: Primary computatioal tools are MCMC simulatio methods ad asymptotic approximatio w.r.t. sample size N PY ( F U, D ) = N,M PY ( F U, θ) p( θ D N,M ) dθ N

15 Posterior System Aalysis (Cotiued) Asymptotic approximatio for large N for robust posterior predictive aalysis (Beck & Katafygiotis 1998; Papadimitriou, Beck & Katafygiotis both at website) P( Y F U, D,M) = P( Y F U, θ) p( θ D,M) dθ N K k= 1 wpy ( F U, ˆ θ ) k k Assumes system is idetifiable based o the data, i.e. fiite umber of MPVs ˆ θ ˆ θ ˆ 1, 2,..., θ K that locally maximize posterior PDF, so eed to do optimizatio; uses Laplace s method for asymptotic approximatio (see later) N

16 Posterior System Aalysis (Cotiued) w k The weights are proportioal to the volume uder the peak of the posterior PDF at ad sum to uity (see Beck & Katafygiotis 1998) Globally idetifiable case (K=1) justifies usig MPV for posterior predictive model whe there is large amouts of data: Gives a rigorous justificatio for doig predictios with MPV model (or MLE, sice θˆ is isesitive to choice of prior) Error i approximatio is O( 1/ N ) θˆk p( Y U, D N,M) p( Y U, ˆ) θ θˆ

17 Posterior System Aalysis (Cotiued) Uidetifiable case correspods to a cotiuum of MPVs lyig o a lower dimesioal maifold i the parameter space Iterest i this case is drive by fiite-elemet model updatig Asymptotic approximatio for posterior predictive model for large amout of data is a itegral over this maifold feasible if it is low dimesio (<4?) (Katafygiotis ad Lam (2002); Papadimitriou, Beck ad Katafygiotis (2001) - both at website) All MPV models give similar predictios at observed DOFs but may be quite differet at uobserved DOFs

18 Posterior System Aalysis (Cotiued) Stochastic Simulatio approaches: Very challegig because most of probability cotet of posterior PDF is cocetrated i a small volume of parameter space (IS does ot work) But potetial of avoidig difficult o-covex multi-dimesioal optimizatio ad hadlig uidetifiable case i higher dimesios Markov Chai Mote Carlo simulatio (e.g. Metropolis-Hastigs algorithm) shows promise (more later)

19 Commets The framework ad computatioal tools give a powerful approach to stochastic system aalysis ad yet it is ot widely used i egieerig why ot? Obstacle: may people are comfortable with p ( Y U, θ) p( θ M) but ot with because they iterpret probability as the relative frequecy of iheretly radom evets i the log ru

20 Commets The two mai themes for the remaiig lectures: Developmet of probability logic which gives a rigorous framework i which probabilities of models makes sese Developmet of a set of computatioal tools to provide efficiet algorithms for hadlig the highdimesioal algorithms eeded for prior ad posterior stochastic predictive system aalysis

21 Probability Logic Primarily due to: R.T. Cox 1946, 1961: The Algebra of Probable Iferece E.T. Jayes 1983, 2003: Probability Theory The Logic of Sciece Major cotributors to developmet of ideas: T. Bayes 1763: A essay towards solvig a problem i the doctrie of chaces P.S. Laplace 1812: Aalytical Theory of Probability H. Jeffreys 1931: Scietific Iferece 1939: Theory of Probability

22 Quote from James Clerk Maxwell (1850): The actual sciece of logic is coversat at preset oly with thigs either certai, impossible or etirely doubtful, oe of which (fortuately) we have to reaso o. Therefore the true logic of this world is the calculus of probabilities, which takes accout of the magitude of the probability which is, or ought to be, i a reasoable ma s mid.

23 Itroductio Features of Probability Logic Probability logic is a quatitative approach to plausible reasoig whe available iformatio is icomplete; it geeralizes biary Boolea logic Framework based o probability axioms ad o other adhoc criteria or cocepts Uses Cox-Jayes iterpretatio of probability as quatifyig plausibility of statemets coditioal o specified iformatio Probability models are used to stad i for missig iformatio; they are (lack of) kowledge models

24 Itroductio (Cotiued) Features of Probability Logic Careful trackig of all coditioig iformatio sice all probabilities are coditioal o probability models ad other specified iformatio Meaigful to talk about probability of probability models, a essetial aspect of Bayesia aalysis Ivolves itegratios over high-dimesioal iput ad model parameter spaces; computatioal tools for this will be give ad are also beig actively developed by may researchers Framework is geeral but our focus is primarily o dyamical systems

25 Decisio Makig uder Ucertaity/Icomplete Iformatio (e.g. Egieerig System Desig) System Models ad Probability Models [Kowledge models for missig iformatio] Iformatio Processig [Axioms for Calculus of Probability] Decisio Makig [Predicted Decisio Variables] Iformatio processig should be doe i such a way that kow iformatio is ot lost ad spurious iformatio is ot added

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