Merging Particle Filter

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1 c 2008 Merging Particle Filter 1,2 1,2 1,2 1, particle filter merging particle filter MPF MPF MPF particle filter merging particle filter 1. data assimilation e.g., Kalnay, 2003;, 2005; Evensen, 2006 Gauss Kalman filter Kalman filter extended Kalman filter

2 e.g., Evensen, 1992 singular evolutive extended Kalman SEEK filter Pham et al., 1998b Ensemble Kalman filter EnKF Evensen, 1994; Burgers et al., 1998 EnKF Kalman filter extended Kalman filter SEEK filter singular evolutive interpolated Kalman SEIK filter Pham et al., 1998a EnKF Nerger et al., 2005 EnKF particle filter PF Gordon et al., 1993; Kitagawa, 1993, 1996; Kitagawa and Gersch, 1996; Higuchi and Kitagawa, 2000; van Leeuwen, 2003 PF Gauss PF EnKF PF Gauss Gauss Kotecha and Djurić, 2003 Gauss kernel filter Hürzeler and Künsch, 1998; Anderson and Anderson, 1999; Musso et al., 2001 PF Gauss PF merging particle filter MPF Nakano et al., 2007 MPF

3 Merging Particle Filter MPF Merging particle filter (2.1a) (2.1b) x k = F k (x k 1, v k ) y k = H k (x k, w k ) x k y k T = t k (k =1,...) v k w k F k H k PF T = t k 1 p(x k 1 y 1:k 1 ) {x (i) k 1 k 1 }N i=1 (2.2) p(x k 1 y 1:k 1 ) 1 N N i=1 ( ) δ x k 1 x (i) k 1 k 1 δ Dirac N T = t k (2.3) p(x k y 1:k 1 ) 1 N N i=1 ( ) δ x k x (i) 1 x (i) 1 {x (i) k 1 k 1 }N i=1 F k (x (i) k 1 k 1,v(i) k ) MPF {x (i) }N i=1 x (i) {x (i) 1 }N i=1 x (i) n N {x (i) }N i=1 n N {ˆx (1,1),...,ˆx(n,1),...,ˆx(1,N),...,ˆx (n,n) } {x (i) 1 }N i=1 x (i) n N {ˆx (i,j) }i,j n {ˆx(1,i) n (2.4) x (i) = α j ˆx (j,i) α j (2.5a) (2.5b) α j =1 α 2 j =1,...,ˆx(n,i) }

4 {x (i) }N i=1 p(x k y 1:k ) Nakano et al., 2007 MPF 2 MPF {x (i) }N i=1 2 3 MPF 2 3 {α j} n x k 1 m (2.6) (x k µ ) m p(x k y 1:k ) dx k (x k µ ) m 1 N = 1 N 1 N N i=1 N N i=1 (x (i) µ ) m = 1 N i=1 α m j α m j ( ˆx (j,i) ( ) δ x k x (i) dx k µ N m i=1 l=1 ) m = n (x k µ ) m p(x k y 1:k ) dx k j l =1 α m j α jl ˆx (j l,i) m (2.7) α m j =1 µ (x k µ ) m 1 N N m ( ) ˆx (j l,i) N µ 0 (unless j 1 = j 2 = = j m) i=1 l=1 N ( ) δ x k ˆx (j,i) dx k 0 {α j} 2.5b 0 < α j < 1 j 2 m (2.8) α j 2 > α j m α j α 2 j > 0 α j > 0 α j < 0 (2.9) α 2 j >α m j. j (2.10) α 2 j > α m j. 0 {α j} 2.5b m> (2.11) α m j < 1 i=1

5 Merging Particle Filter m 2 3 x k 1 x k m n 3 n =1 PF n =2 2.5a 2.5b 1 0 PF n 3 2.5a 2.5b {α 1,...,α n} MPF Lorenz 63 Lorenz Lorenz 63 Lorenz 63 (3.1a) dx = s(x y) dt (3.1b) dy = rx y xz dt (3.1c) dz = xy bz. dt s =10 r =28 b =8/ Lorenz x y z 3 2 Gaussian MPF PF y k x 1 =(x 1,y 1,z 1 ) p(y k x 1 ) w k 0 diag(σ 2,σ 2,σ 2 ) Gauss (3.2) p(y k x 1 )= 1 ( 2πσ) 3 exp [ y k x 1 2 ] σ =3 0 diag(0.01,0.01,0.01) Gaussian T = t 1 x 1 1 diag(16,16,16) Gauss MPF n 3 (3.3) 2σ 2 α 1 = , α2 =, α 3 =

6 Root-mean-square deviations from the true state over 50,000 time steps for an experiment using the Lorenz 63 model. (3.4) α 1 = 19 20, α2 = , α 3 = α 1 1 PF 2 MPF PF 3 N 1 50,000 root-mean-square MPF MPF N PF N PF MPF MPF PF Lorenz 96 Lorenz and Emanuel 1998 Lorenz 96 Lorenz 96 x j(j =1,...,J) (3.5) dx j dt =(xj+1 xj 2)xj 1 xj + f. x 1 = x J 1 x 0 = x J x J+1 = x 1 f =8 J Lorenz 96 (3.6a) (3.6b) x j =8.0 (for j 20) x j =8.008 (for j = 20) 2, x j j x j Gauss

7 Merging Particle Filter Root-mean-square deviations from the true state from time step 3,000 to time step 20,000 for an experiment using the Lorenz 96 model. 0 diag(0.25,..., 0.25) Gauss 1 (3.7) p(y k x 1 )= ( exp [ y k H(x 1 ) 2 ] 2πσ) 20 2σ 2 H H(x 1 )=( x 2, 1 x 4, 1... x 40, 1 ) T σ =3 Lorenz 63 MPF PF 3 2 5,000 20,000 root-mean-square 1 MPF MPF PF N MPF MPF 2 N PF N = 1024 N = Nakano et al MPF PF MPF MPF 1 α PF α PF {ˆx (1,i) }N i=1 ( n j=2 αj ˆx (j,i) )N i=1 kernel filter Hürzeler and Künsch, 1998; Anderson and Anderson, 1999; Musso et al., 2001 MPF MPF 2 α 1 1 kernel filter Lorenz 63 MPF α 1 1

8 Lorenz 63 Gauss Gauss α 1 1 Lorenz 96 N α 1 1 N N α 1 1 α 1 1 PF N α 1 JST/CREST Anderson, J. L. and Anderson, S. L A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts, Monthly Weather Review, 127, Burgers, G., van Leeuwen, P. J. and Evensen, G Analysis scheme in the ensemble Kalman filter, Monthly Weather Review, 126, Evensen, G Using the extended Kalman filter with a multilayer quasi-geostrophic model, Journal of Geophysical Research, 97 C11, Evensen, G Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, Journal of Geophysical Research, 99 C5, Evensen, G Data Assimilation: The Ensemble Kalman Filter, Springer-Verlag, Berlin and Heidelberg. Gordon,N.J.,Salmond,D.J.andSmith,A.F.M Novel approach to nonlinear/non-gaussian Bayesian state estimation, IEE Proceedings F, 140, Higuchi, T. and Kitagawa, G Knowledge discovery and self-organizing state space model, IEICE Transactions on Information and Systems, E83-D, Hürzeler, M. and Künsch,H.R Monte Carlo approximations for general state space models, Journal of Computational Graphical Statistics, 7, Kalnay, E Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, Cambridge. Kitagawa, G A Monte Carlo filtering and smoothing method for non-gaussian nonlinear state space models, Research Memorandum, No. 462, The Institute of Statistical Mathematics,

9 Merging Particle Filter 233 Tokyo. Kitagawa, G Monte Carlo filter and smoother for non-gaussian nonlinear state space models, Journal of Computational Graphical Statistics, 5, Kitagawa, G. and Gersch, W Smoothness Priors Analysis of Time Series, Chapter 6, Springer-Verlag, New York. Kotecha, J. H. and Djurić, P. M Gaussian particle filtering, IEEE Transactions on Signal Processing, 51, Lorenz, E. N Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20, Lorenz, E. N. and Emanuel, K. A Optimal sites for supplementary weather observations: Simulations with a small model, Journal of the Atmospheric Sciences, 55, Musso, C., Oudjane, N. and Le Gland, F Improving regularized particle filters, Sequential Monte Carlo Methods in Practice eds. A. Doucet, N. de Freitas and N. Gordon, Chapter 12, , Springer-Verlag, New York , Nakano, S., Ueno, G. and Higuchi, T Merging particle filter for sequential data assimilation, Nonlinear Processes in Geophysics, 14, Nerger, L., Hiller, W. and Schröter, J A comparison of error subspace Kalman filters, Tellus, 57A, Pham, D. T., Verron, J. and Gourdeau, L. 1998a Filtres de Kalman singuliers évolutifs pour l assimilation de données en océanographie, Comptes rendus de l Académie des sciences, Série 2 : Sciences de la Terre et des Planètes, 326, Pham, D. T., Verron, J. and Roubaud, M. C. 1998b. A singular evolutive extended Kalman filter for data assimilation in oceanography, Journal of Marine Systems, 16, van Leeuwen, P. J A variance-minimizing filter for large-scale applications, Monthly Weather Review, 131,

10 234 Proceedings of the Institute of Statistical Mathematics Vol. 56, No. 2, (2008) Merging Particle Filter and Its Characteristics Shin ya Nakano, Genta Ueno, Kazuyuki Nakamura and Tomoyuki Higuchi The Institute of Statistical Mathematics; Japan Science and Technology Agency A significant problem with the basic particle filter algorithm is degeneration. The merging particle filter algorithm has recently been proposed to overcome this problem at a reasonable computational cost. In an MPF, each member of a filtered ensemble is generated from a weighted sum of multiple samples from the forecast ensemble such that the mean and covariance of the filtered distribution are approximately preserved. In this study, we performed data assimilation experiments using an MPF with two different sets of merging weights. When one merging weight is set to near 1 and the other weights are set small, better estimates are obtained for data assimilation into a low dimensional model. For data assimilation into a relatively large dimensional model, such a weight set requires a large ensemble size. Key words: Data assimilation, particle filter, merging particle filter.

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