Bayesian Dynamic Mode Decomposition Naoya Takeishi § , Yoshinobu Kawahara † , ‡ , Yasuo Tabei ‡ , Takehisa Yairi § § Dept. of Aeronautics & Astronautics, The University of Tokyo † The Institute of Scientific and Industrial Research, Osaka University ‡ RIKEN Center for Advanced Intelligence Project (AIP) 22 August 2017, IJCAI @ Melbourne
Motivation: Analysis of dynamical systems ◮ Various types of complex phenomena can be described in terms of (nonlinear) dynamical systems . x ∈ M (state space) x t +1 = f ( x t ) , � When f is nonlinear, analysis based on trajectories of x is difficult. N. Takeishi (The Univ. of Tokyo) Bayesian DMD 2 of 14
Operator-theoretic view of dynamical systems Definition (Koopman operator [Koopman ’31, Mezi´ c ’05] ) Koopman operator (composition operator) K represents time-evolution of observables (i.e., observation function) g : M → R or C . K g ( x ) = g ( f ( x )) , g ∈ F (function space) ◮ K describes temporal evolution of function (infinite-dimensional vector) instead of the finite-dimensional state vector. ◮ Defining K , we can lift the analysis of nonlinear dynamical systems into a linear (but infinite-dimensional) regime! � Since K is linear, we can analyze dynamics using the spectra of K . N. Takeishi (The Univ. of Tokyo) Bayesian DMD 3 of 14
Koopman mode decomposition (KMD) ◮ Eigenvalues and eigenfunctions of K : K ϕ i ( x ) = λ i ϕ i ( x ) for i = 1 , 2 , . . . . ◮ Projection of g ( x ) to span { ϕ 1 ( x ) , ϕ 2 ( x ) , . . . } (i.e., transformation to a ⇒ Coefficients are called Koopman modes . canonical form). ∞ � g ( x ) = ϕ i ( x ) v i i =1 ◮ Since ϕ is eigenfunction, ∞ � λ t g ( x t ) = i ϕ i ( x 0 ) v i , (KMD) � �� � i =1 w i where | λ i | = decay rate of w i , ∠ λ i = frequency of w i . ◮ A numerical realization of KMD is dynamic mode decomposition (DMD) [Rowley+ ’09, Schmid ’10, Tu+ ’14] . N. Takeishi (The Univ. of Tokyo) Bayesian DMD 4 of 14
Dynamic mode decomposition (DMD) Assumption ( K -invariant subspace [Budiši´ c+ ’12] ) Dataset is generated with a set of observables � � T g ( x ) = g 1 ( x ) g 2 ( x ) · · · g n ( x ) that spans (approximately) K -invariant subspace. ⇒ Then, KMD can be (approximately) realized by DMD. Algorithm (DMD [Tu+ ’14] ) Input time-series ( y 0 , . . . , y m ) s.t. y t = g ( x t ) eigenvalues { λ }, eigenfunctions { ϕ } , and modes { w } Output 1. Estimate a linear model y t +1 ≈ Ay t . 2. On A , compute eigenvalues λ i and right-/left-eigenvectors w i , z H i . 3. Compute ϕ i,t = z H i y t . N. Takeishi (The Univ. of Tokyo) Bayesian DMD 5 of 14
Quasi-periodic modes extraction by KMD/DMD ◮ Review: KMD/DMD computes the decomposition of time-series into modes w i that evolve with frequency ∠ λ i and decay rate | λ i | . ◮ w i is termed dynamic modes . ∞ � λ t g ( x t ) ≈ i w i i =1 Example (2D fluid flow past a cylinder) Flow past a cylinder is universal in many natural/engineering situations. DMD − − − → + + w 1 w 2 w 3 + + + · · · · · · w 4 w 5 N. Takeishi (The Univ. of Tokyo) Bayesian DMD 6 of 14
Other applications of KMD/DMD ◮ Lots of applications in a wide range of domains ◮ fluid mechanics [Rowley+ ’09, Schmid ’10, & many more] , ◮ neuroscience [Brunton+ ’16] , ◮ image processing [Kutz+ ’16, Takeishi+ ’17] , ◮ analysis of power systems [Raak+ ’16, Susuki+ ’16] , ◮ epidemiology [Proctor&Eckhoff ’15] , ◮ optimal control [Mauroy&Goncalves ’16] , ◮ finance [Mann&Kutz ’16] , ◮ medical care [Bourantas+ ’14] , ◮ robotics [Berger+ ’15] , etc. N. Takeishi (The Univ. of Tokyo) Bayesian DMD 7 of 14
Issue ◮ DMD relies on linear modeling g ( x t +1 ) ≈ Ag ( x t ) and eigendecomposition of A . ◮ So it lacks an associated probabilistic/Bayesian framework , by which we can ◮ consider observation noise explicitly, ◮ perform a posterior inference , ◮ consider DMD extensions in a unified manner, etc. ◮ Let’s do it! ◮ analogously to PCA’s formulation as probabilistic/Bayesian PCA [Tipping&Bishop ’99, Bishop ’99] N. Takeishi (The Univ. of Tokyo) Bayesian DMD 8 of 14
Proposed method (1/2): Probabilistic DMD ◮ Dataset: snapshot pairs with observation noise � � D = ( y 0 , 1 , y 1 , 1 ) , . . . , ( y 0 ,t , y 1 ,t ) , . . . , ( y 0 ,m , y 1 ,m ) , where y 0 ,t = g ( x t ) + e 0 ,t and y 1 ,t = g ( x t +∆ t ) + e 1 ,t , Definition (Generative model of probabilistic DMD) �� k � i =1 ϕ t,i w i , σ 2 I y 0 ,t ∼ CN �� k � i =1 λ i ϕ t,i w i , σ 2 I y 1 ,t ∼ CN ϕ t,i ∼ CN (0 , 1) � If k = n and σ 2 → 0 , the MLE of ( λ, w ) coincides with DMD’s solution. N. Takeishi (The Univ. of Tokyo) Bayesian DMD 9 of 14
Proposed method (2/2): Bayesian DMD Definition (Prior on parameters for Bayesian DMD) � � �� w i | v 2 v 2 i, 1 , . . . , v 2 v 2 i, 1: n ∼ CN 0 , diag , i,d ∼ InvGamma ( α v , β v ) i,n λ i ∼ CN (0 , 1) σ 2 ∼ InvGamma ( α σ , β σ ) v 2 σ 2 y 0 , t w i i, 1: n λ i ϕ t k y 1 , t m � For a posterior inference , a Gibbs sampler can be constructed easily. N. Takeishi (The Univ. of Tokyo) Bayesian DMD 10 of 14
Extension example: Sparse Bayesian DMD Definition (Prior on parameters for sparse Bayesian DMD) � � �� w i | v 2 0 , σ 2 diag v 2 i, 1 , . . . , v 2 v 2 i,d ∼ Exponential( γ 2 i, 1: n ∼ CN , i / 2) i,n λ i ∼ CN (0 , 1) σ 2 ∼ InvGamma ( α σ , β σ ) v 2 σ 2 y 0 , t w i i, 1: n λ i ϕ t k y 1 , t m � We can extend the model in a unified Bayesian manner. N. Takeishi (The Univ. of Tokyo) Bayesian DMD 11 of 14
Numerical example (1) Example (Fixed-point attractor) Generate data by � � T � � T y t = λ t + λ t 2 2 2 − 2 + e t , 1 2 where e is Gaussian observation noise. True eigenvalues are λ 1 = 0 . 9 and λ 2 = 0 . 8 . 1.3 1.2 truth 1.2 1.1 DMD 1.1 TLS-DMD 1 6 1 ) 1 6 2 ) 0.9 Re(~ Re(~ 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 < < N. Takeishi (The Univ. of Tokyo) Bayesian DMD 12 of 14
Numerical example (2) Example (Limit-cycle attractor) Generate data from Stuart–Landau equation r t +1 = r t + ∆ t ( µr t − r 3 t ) , θ t +1 = θ t + ∆ t ( γ − βr 2 t ) , and Gaussian observation noise. True (continuous-time) eigenvalues lie on the imaginary axis. DMD TLS-DMD BDMD (average) Re(log( 6 ) = " t ) 0.5 0 -0.5 -1 -1.5 -3 -2 -1 0 1 2 3 Im(log( 6 ) = " t ) N. Takeishi (The Univ. of Tokyo) Bayesian DMD 13 of 14
K g ( x ) = g ( f ( x )) Analysis of dynamical systems based on Koopman operator is a useful tool. Dynamic mode decomposition (DMD) is a numerical method for Koopman analysis. In this work, we developed probabilistic & Bayesian DMDs to ◮ consider observation noise, v 2 σ 2 w i y 0 , t i, 1: n ◮ infer posterior distribution, λ i ϕ t ◮ extend DMD in a unified manner, etc. k y 1 , t m Implementation available at https://github.com/thetak11/bayesiandmd N. Takeishi (The Univ. of Tokyo) Bayesian DMD 14 of 14
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