EnKF-based particle filters - PDF Free Download

EnKF-based particle filters Jana de Wiljes, Sebastian Reich, Wilhelm Stannat, Walter Acevedo June 20, 2017

Filtering Problem Signal dx t = f (X t )dt + 2CdW t Observations dy t = h(x t )dt + R 1/2 dv t. 1 0.5 0 Signal Obs 0.5 time Goal: determine π(x Y 0:t )

State of the art Linear Model: Kalman- Bucy Filter d x M t = A x M t dt + bdt P M t H T R 1 (H x M t dt dy t ) Non-linear Model: approximate with empirical measure, i.e., π(x Y 0:t ) 1 M M δ(x Xt i ). i=1 Ansatz: define modified evolution equation for particles X i t

Ensemble Kalman Filter (EnKF) P(x Y 0:t ) P(x) time t dx i t = f (X i t )dt + CdW i t 1 2 PM t H T R 1 ( HX i t dt + H x M t dt 2dY t ) x M t = 1 M M i=1 X i t P M t = 1 M 1 M (Xt i x t M )(Xt i x t M ) T i=1 Works remarkably well in practice: meteorology, oil reservoir exploration But: theoretical understanding is largely missing

Recent accuracy results for EnKF Interacting particle representation of the model error: dx i t = f (X i t )dt+cc (P M t ) 1 (X i t x M t )dt 1 2 PM t H T R 1 ( HX i t dt + H x M t dt 2dY t ) Setting: dy t = X t dt + R 1/2 dw t with R = εi Results: ([dwrs16]) Control of spectrum of covariance matrix Pt M over t [0, ). Control of estimation error e t = Xt ref x t M 2 in expectation and pathwise over fixed interval t [0, T ]. E[E t ] = O(ɛ 1/2 ) (1) P[sup E t c 1 ɛ q ] O(ɛ 1/2 η q ) (2) t T

Numerical confirmation for L63 10 1 time averaged mean squared error 10 0 error theoretical order 10 1 10 2 10 4 10 3 10 2 10 1 measurement noise variance

EnKF-based particle filters Ansatz: modified evolution equation for the particles e.g., of the form dxt i = f (Xt i )dt + CdWt i Xt j dst ji j Aims: achieve first/second order accuracy to go beyond Gaussian assumption consistency for M hybrid formulation to combine different interacting particle filters ([CRR16])

Linear ensemble transform filters (LETF) Given: M samples x f i π(x) (prior) Analysis step: x a j = i x f i d ij (3) with transformation matrix D = {d ij } subject to M d ij = 1 (4) i=1 Examples: ENKF, ESRF, NETF, ETPF (see [RC15])

Ensemble transform Particle Filter (ETPF) Given: M samples xi f π(x) (prior ensemble) normalized importance weights w i π(y xi f ) (likelihood) Desired: M samples xi a π(x y) Ansatz: replace resampling step with linear transformation by interpreting it as discrete Markov chain given by transition matrix s.t. d ij 0 and D R M M d ij = 1, i 1 M d ij = w i. j Benefits: localization, hybrid

Ensemble transform Particle Filter (ETPF) Then the posterior ensemble members are distributed according to the columns of the transformation d 1j d 2j X a j. d Mj and x j a = E[ X a j ] = i x a i d ij (5) Example. Monomial resampling w 1 w 1 w 1 w 2 w 2 w 2 D Mono := w 1 =...... w M w M w M.

Ensemble transform Particle Filter (ETPF) Solve optimization problem D ETPF = arg min M M i,j=1 t ij x f i z f j 2 to find transformation matrix D ETPF that increases correlation ([Rei13]). Remarks: consistent for M first-order accurate for finite M, i.e, x a = 1 M M i=1 x a i = M i=1 w i x f i (6) But: not second-order accurate

Second-order accuracy The analysis covariance matrix P a = 1 M M (xi a i=1 x a )(x a i x a ) T is equal to the covariance matrix defined by the importance weights, i.e. P a = M i=1 w i (t k )(x f i x a )(x f i x a ) T.

First-order accurate LETFs Notation: and analogously X f = (x1 f,..., xm), f X a = (x1 a,..., xm) a Nx M R w = (w 1,..., w M ) T R M 1 and W = diag (w) LETF is first-order accurate if This holds if D satisfies 1 M Xa 1 = X f w. 1 D1 = w. M

First-order accurate LETFs Class of first-order accurate LETFs D 1 = {D R M M D T 1 = 1, D1 = Mw } Examples: D EnKF / D 1 D ESRF / D 1 D ETPF D 1 D Mono D 1

Second-order accurate LETF Analysis covariance matrix: P a = 1 M Xf (D w1 T )(D w1 T ) T (X f ) T (7) Importance sampling covariance matrix: P a = X f (W ww T )(X f ) T. (8) Thus the following equation has to hold for second-order accuracy: (D w1 T )(D w1 T ) T = W ww T (9) Class of second-order accurate LETFs D 2 = {D D 1 (D w1 T )(D w1 T ) T = W ww T }. (10)

Second-order corrected LETF Given: D D 1 Goal: correct transformation D D 2 Ansatz: D = D + with R M M such that 1 = 0, T 1 = 0 ([dwar17]). Need to solve algebraic Riccati equation: M(W ww T ) (D w1 T )(D w1 T ) T =(D w1 T ) T + (D w1 T ) T + T.

Numerical example I Gaussian prior, non-gaussian likelihood: Figure: Prior and posterior distribution for the single Bayesian inference step

Numerical example I Figure: Absolute errors in the first four moments of the posterior distribution.

Numerical example II Lorenz-63 model, first component observed infrequently ( t = 0.12) and with large measurement noise (R = 8): Figure: RMSEs for various second-order accurate LETFs compared to the ETPF, the ESRF, and the SIR PF as a function of the sample size, M.

Numerical example II Hybrid filter: D := D ESRF D ETPF. Figure: RMSEs for hybrid ESRF (α = 0) and 2nd-order corrected LETF/ETPF (α = 1) as a function of the sample size, M.

Data Assimilation Center Potsdam I PhD and Postdoc positions in 11 projects I Mathematical foundation and applications in geophysics, biologie and cognitive sciences I Support for external projects and long term visitors

References I Nawinda Chustagulprom, Sebastian Reich, and Maria Reinhardt. A hybrid ensemble transform filter for nonlinear and spatially extended dynamical systems. SIAM/ASA J Uncertainty Quantification, 4:592 608, 2016. J. de Wiljes, W. Acevedo, and S. Reich. A second-order accurate ensemble transform particle filter. accepted in SIAM Journal on Scientific Computing, (ArXiv:1608.08179), 2017. J. de Wiljes, S. Reich, and W. Stannat. Long-time stability and accuracy of the ensemble Kalman-Bucy filter for fully observed processes and small measurement noise. Technical Report https://arxiv.org/abs/1612.06065, 2016. S. Reich and C. Cotter. Probabilistic Forecasting and Bayesian Data Assimilation. Cambridge University Press, 2015.

References II Sebastian Reich. A non-parametric ensemble transform method for Bayesian inference. SIAM J Sci Comput, 35:A2013 A2014, 2013.