Bayesian Inference and the Symbolic Dynamics of Deterministic Chaos. Christopher C. Strelioff 1,2 Dr. James P. Crutchfield 2

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1 How Random Bayesian Inference and the Symbolic Dynamics of Deterministic Chaos Christopher C. Strelioff 1,2 Dr. James P. Crutchfield 2 1 Center for Complex Systems Research and Department of Physics University of Illinois at Urbana-Champaign 2 Computational Science and Engineering Center and Department of Physics University of California at Davis NIPS 2006 Workshop on Dynamical Systems, Stochastic Processes and Bayesian Inference December 9th, 2006

2 Sources of Stochastic or Deterministic? Given a data source which appears to be a high-dimensional stochastic process: Is this instead a low-dimensional deterministic system which exhibits a chaotic attractor? Analysis of chaotic data: is very sensitive to the measurement process. This is both a blessing and a curse.

3 Finite-precision instrument: A model of measurement Induces a symbolic representation. Symbolic Dynamics How well does this represent underlying continuous-valued behavior? Symbolic dynamics: Coarse-grained view of continuous dynamics. Use ideas from symbolic dynamics to discuss instrument design.

4 Modeling chaotic time series Continuous-state discrete data - Maximize entropy rate: Project continuous state onto finite set of disjoint regions. Produce maximum information with each measurement. Model inference - Minimize entropy rate: Model discrete data produced by instrument. Search for determinism and structure in data.

5 Our model data source 1-dimensional chaotic map 1-d map with additive noise: x t+1 = f(x t ) + ξ t where x t [0, 1], ξ t N(0,σ 2 ). Why is this a relevant example? We can connect ordinary differential equations with a map.

6 The system of ode s Rössler Attractor How can we treat this ODE as a 1-d map? dx dt dy dt dz dt = y z = x + ay = b + z(x c) Typical chaotic parameters: a = 0.2, b = 0.2, c = 5.7

7 x(t), y(t), z(t) Rössler Attractor Representative chaotic dynamics x(t) y(t) z(t) Time t

8 x(t), x (t) Rössler Attractor - sensitivity to perturbation x(t) x (t) Time t

9 y Rössler Attractor Typical Chaotic Dynamics original perturbed x

10 x(t) Rössler Attractor Consider the sequence of x max x(t) x max Time t

11 Rössler Attractor 1-d map from x max sequence xmax(n + 1) x max (n)

12 Designing an instrument Symbolic dynamics: without noise Assume a discrete time map (state space M): f : M M Partition M into a finite number of non-overlapping regions P = {I i : i I i = M, I i I j =, i j} For one-dimensional maps: Choose decision points at critical points of map.

13 A good partition: 1 Symbolic Dynamics Logistic map: x n+1 = r x n (1 x n) 0.75 xn+1 = f(xn) x n

14 Choose a partition P Create coarse-grained data What if we don t know f(x)? Choose a P and create symbolic sequence: X = x 0 x 1...x N 1 S = s 0 s 1...s N 1, s t A. P Best partition gives true entropy rate h µ = max {P} h µ (P).

15 Piesin s Identity Testing the resulting partition Lyapunov Exponent λ = lim N h µ = i 1 N Lyapunov Exponent λ + i N log 2 f (x t ). t=1

16 Goal is to find posterior: Symbolic data Markov model P(θ k D, M k ) = P(D θ k, M k )P(θ k M k ) P(D M k ) M k : Markov chain order-k θ k : model parameters D: data

17 k-th order Markov Chains P(D θ k, M k ) = p(s s k ) n( s A s k A k n( s k s): counts of word s k s in data D. Assumes: s k s) Finite memory. Stationarity.

18 k-th order Markov Chains is product of Dirichlet distributions { Γ(α( s k )) P(θ k M k ) = s A Γ(α( s k s)) where s k A k δ(1 s A p(s s k )) s A p(s s k ) α( s k s) 1 } α( s k ) = s α( s k s)

19 or marginal likelihood P(D M k ) = = s k A k k-th order Markov Chains dθ k P(D θ k, M k )P(θ k M k ) { Γ(α( s k )) s A Γ(α( s k s)) s A Γ(n( s k s) + α( s k s)) Γ(n( s k ) + α( s k )) } This term is fundamental to model comparison, estimation of h µ.

20 k-th order Markov Chains Combine likelihood, prior, evidence using Bayes theorem to obtain posterior density P(θ k D, M k ) = δ(1 s A s k A k p(s s k )) s A { Γ(n( s k ) + α( s k )) s A Γ(n( s k s) + α( s k s)) p(s s k ) n( s k s)+α( s k s) 1 }

21 Consider set of orders M = {M k } Probability of order k is given by Model comparison How do we select the order k? P(D M k, M) P(M k M) P(M k D, M) = M P(D M k k, M) P(M k M) Penalize for model size P(M k M) = exp( A k ( A 1))

22 becomes partition function Z = Estimating h µ Define a partition function dθ k P(D θ k, M k )P(θ k M k ) Re-write product of likelihood and prior P(D,θ k M k ) 2 β k(d[q P]+h µ[q]) 2 + A k+1 (D[U P]+h µ[u]) where β k = È s k,s n( s k s) + α( s k s). Distributions Q: posterior mean, U: uniform.

23 An average of D[Q P] + h µ [Q] Analytic result Estimating h µ Define a partition function 1 logz = E post [ D[Q P] + h µ [Q]] log 2 β k E post [ D[Q P] + h µ [Q]] = 1 log 2 s k q( [ s k )ψ (0) β k q( ] s k ) 1 q( s k )q(s [ s k )ψ (0) β k q( s k )q(s ] s k ) log 2 s k,s where ψ (0) is the digamma function.

24 An experiment Instrument and Inference We know how to create an instrument - define P. GOAL: Maximize h µ - informative measurements. We know how to infer a Markov chain model of coarse grained data. GOAL: Minimize h µ - find patterns.

25 Our experiment Generate a single time series of length N = 10 4 f(x t ) = r x t (1 x t ) Design an instrument, r = 4.0, σ = 10 3 Choose a decision point d P(d) = { 0 x [0, d), 1 x [d, 1]} Infer a Markov chain and estimate h µ. Consider orders k = 1 8.

26 k: selected order Entropy Rate (bits) Maximize h µ d (Decision Point)

27 Model Inference Minimize h µ P(Mk D, M) d = 0.2 d = 0.3 d = 0.4 d = Entropy Rate (bits) k: Order

28 Bayesian Inference and the Symbolic Dynamics of Deterministic Chaos can be due to low-dimensional chaos. Symbolic dynamics can help analyze this data. For instruments, maximize observed entropy rate. For model inference, minimize entropy rate.