Introduction to Particle Filters for Data Assimilation

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1 Introduction to Particle Filters for Data Assimilation Mike Dowd Dept of Mathematics & Statistics (and Dept of Oceanography Dalhousie University, Halifax, Canada STATMOS Summer School in Data Assimila5on, Boulder, Colorado, May

2 Problem Statement Noisy 'me series observa'ons on the system state y 1:T = (y 1,..., y T a discrete 'me stochas'c dynamic model describing the 'me evolu'on of the system state x t = θx t 1 + n t e.g. Bivariate AR(1 process We want to es'mate the system state (and some'mes parameters

3 Noisy Observations of System State Observations state time

4 Stochastic Dynamics governing System State 10 Realizations of System State state time

5 Estimation of System State by Particle Filters Particle Filter Estimate of the State state time

6 Target: Evolution of the Probability Distribution Of the State through Time 'me state

A General Framework for DA Hierarchical Bayesian Model p(x 1:T,θ y 1:T p(y 1:T x 1:T,θ p(x 1:T θ p(θ target (posterior measurement dynamics prior Special Cases in

7 A General Framework for DA Hierarchical Bayesian Model p(x 1:T,θ y 1:T p(y 1:T x 1:T,θ p(x 1:T θ p(θ target (posterior measurement dynamics prior Special Cases in DA Parameter es'ma'on (no model error Filtering solu'on for state * Smoothing solu'on for state Joint state & parameter es'ma'on Inference on dynamic model structure

8 Data Assimilation: A Statistical Framework State Space Model: x t = d(x t 1,θ,e t y t = h(x t,φ,ν t or x t ~ p(x t,θ x t 1 y t x t ~ p(y t,φ x t The general goal is to make inferences about the state x t and some'mes the sta'c parameters* θ and ϕ**. We use explicit priors on process noise & observa'on errors (i.e. parametric distribu'ons for e t and v t. Assume y t condi'onally independent given x t, and that e t and v t are independent. For DA, we want to do space- 'me problems: space is incorporated in x t è Solu'ons involve sequen'al Monte Carlo based for nonlinear and nongaussian state space models * Dynamic parameters are part of the state ** drop φ hereater, and defer θ t = 1,,T

9 Problem Classes We want to es'mate the system state at 'me t, or given the observa'ons up to 'me s, or y 1:S = (y 1, y 2,.., y S x t A Filtering (Nowcast B Forecasting C Smoothing (Hindcast s s s s=t è Filtering s>t è Predic'on s<t è Smoothing Time data available no data available estimation time

10 Sequential Estimation Single stage transition of system from time t-1 to time t prediction x t = d(x t 1,θ,e t measurement y t nowcast p(x t 1,θ y 1:t 1 forecast nowcast p(x t,θ y 1:t 1 p(x t,θ y 1:t time = t-1 time = t time - We ll drop parameters θ for now

11 Probabilistic Solution for Filtering A single stage transition of the system for time t-1 to t involves: Prediction: p(x t y 1:t 1 = p(x t x t 1 p(x t 1 y 1:t 1 dx t 1 Measurement update: p(x t y 1:t = p(y t x t p(x t y 1:t 1 p(y 1:t è General Filtering Solution: p(x t y 1:t p( y t x t p(x t x t 1 p(x t 1 y 1:t 1 dx t 1 do for t=1,, T, given p(x 0 and y 1:t = (y 1, y 2,.., y T

12 Sampling Based Estimation Let {x (i t t,w (i t }, i = 1,..., n be a random sample* from p(x t y 1:t drawn from the target distribu'on (the filter density (i where: x t t w t (i are the support points are the normalized weights The filter density can then be approximated as p(x t y 1:t n i=1 w (i t δ (x t t x (i t t *referred to an ensemble, made up of a collec'on of par'cles (which are sample members

13 Sampling Based Estimation (a unweighted ensemble, n= (b weighted ensemble, n=25 probability 0.2 probability x (c unweighted ensemble, n= x (d weighted ensemble, n= probability 0.2 probability x x

14 Sequential Importance Sampling (SIS (i At time t-1 we have: i {x t 1 t 1,w (i t 1 }, draws from p(x t 1 y 1:t 1 i y t, an observation If a candidate x (i t t is drawn from a proposal q(x t t then w (i t p(x (i t t y t (i y t q(x t t or w (i (i t w t 1 p(y t x (i t t p(x (i (i t t x t 1 t 1 q(x (i t t A draw from p(x t y 1:t is then available as {x (i t t,w (i t } see Arulampalam et al. 2002

15 SIS Remarks SIS solves the filtering problems for the nonlinear, non- Gaussian state space model via an algorithm for sequen'al online es'ma'on of the system state relying on a proposal and weight update. A common simplifica'on is to let the proposal be the the prior (the predic've density q(x t t = p(x t x t 1 leading to: w (i (i t w t 1 p(y t x (i t 1 The main prac'cal problem with SIS is weight collapse wherein ater a few 'me steps, all the weights become concentrated on a few par'cles. How to fix?

16 Sequential Importance Resampling (SIR The major breakthrough in sequential Monte Carlo methods (Gordon et al was to incorporate a resampling step to obviate weight collapse. (i i Specifically, for each time t, {x t t 1,w (i t } is a sample from p(x t y t 1 i We can resample (with replacement the x (i t t with a probability proportional to w t (i (i.e. a weighted bootstrap i This yields a modified sample {x (i t t }, wherein particles were both killed and replicated, and the weights are now equal, i.e. w t (i = 1 i. Remaining Issue: sample impoverishment - some particles replicated often in the unweighted sample {x (i t t }. Less severe than weight collapse (but of course related.

17 Basic Particle Filter Algorithm Given: (1 dynamical model, (2 observa'ons, (3 ini'al condi'ons for t = 1 to T (i (a Prediction: {x t 1 t 1 (i } {x t t 1 } as (i x t t 1 (i = d(x t 1 t 1,e (i t for i = 1,...,n (i (b Measurement: {x t t 1 } {x (i t t } as end (for t w (i (i t p(y t x t t 1 for i = 1,...,n (i (i resample with replacement {x t t 1 } using weights w t yields {x (i t t }

18 Particle Filter Schematic

19 Interpretation: Particle History state 'me

20 R-code for a Basic Particle Filter! for (t in 2:T! predic'on step {! from t- 1 to t! xf=t(apply(xa,1,dynamicmodel!! extract measurement yobs=matrix(y[t,],1,ncol(y! at 'me t! w=apply(matrix(xf[,1],np,1,1,dmvnorm,x=yobs,sigma=r!! assign weights m=sample(1:np,size=np,prob=w,replace=t;! xa=xf[m,]!! }! weighted resampling

21 Element 1. Prediction Step { (i x } t 1 t 1 { (i x } t t 1 draw from p(x t 1 y 1:t 1 filter density at time t-1 draw from p(x t y 1:t 1 predictive density at time t Role: acts as proposal distribution for particle filter (i x t t 1 (i = d(x t 1 t 1,e (i t for i = 1,...,n Remarks: - computa'onally expensive to generate large samples for big GFD models - hard to effec'vely populate a large dimension state space with par'cles

22 Element 2. Measurement Update { (i x } t t 1 draw from p(x t y 1:t 1 predictive density at time t-1 { (i x } t t draw from p(x t y 1:t filter density at time t uses p(x t t y 1:t p(y t x t t 1 p(x t t 1 (Bayes Formula Approaches: (i i PF/SIR: weighted resample of x t t 1 y t { (i,w } (i t x t t { } Metropolis Hastings MCMC Ensemble Kalman filter (approximate plus many others (auxiliary, unscented,. that modify proposal distributions

23 Sequential Metropolis-Hastings Construct (independence chain to yield sample from target filter distribu5on. for k = 1 to K 1. Generate candidate from predictive density: * draw x t t 1 2. Evaluate acceptance probability from predictive density p(x t y 1:t 1 α = min 1, p(y x * t t t 1 (k p(y t x t t 3. Let x * t t enter the posterior sample {x t t (k otherwise retain x t t end (for k yields (i } with probability α, { (i x t t }, a sample from p(x t y 1:t flexible, configurable and efficient (+ a parallel with SIR can alleviate sample impoverishment (resample- move

24 Limitations Theory: convergence to target marginal distribu'on holds under weak assump'ons (e.g. Kunsch 2005 p(x t y 1:t Prac7cal Problem: The standard par'cle filtering (SIR when used in DA suffers from sample impoverishment (recall path degeneracy Typically, the predic'on ensemble far away from the observed state (due to small ensemble size, and high dimensional state space. è likelihood is poorly represented and es'ma'on compromised. What modifica5ons are made to improve par5cle filters?

25 Some Fixes Add Ji;er : overdisperse the sample of predic've density (inflate process error Make a Gaussian approxima5on, or transforma5on/anamorphosis: Can Use Kalman filter upda'ng equa'on Approximate the Likelihood func5on: change its func'onal form, or inflate or alter measurement error. Error Subspace: confine stochas'city in parameters only. Dimension reduc'on. Use Fixed lag smoother, Batch processing incorporate observa'ons from mul'ple 'mes into observa'on update. Clever Proposal Distribu5ons and look- ahead filters: move beyond using prior (predic've density as proposal.

26 What about Parameter Estimation? 1. State Augmentation: append parameters to the state x t = x t θ t Can use particle filter machinery to solve (Kitagawa Also, see multiple iterative filtering (Ionides Likelihood Methods: Use sample based likelihoods T [y 1:T θ] = L(θ y 1:T = [y t x t,θ][x t y 1:t 1,θ]dx t t=1 T t=1 n 1 (i p(y t x t t 1 n i=1

27 Summary 1. State space model is sta's'cal framework for considering dynamic models and measurements 2. Importance sampling provides theore'cal basis for par'cle filters 3. Sequen'al importance resampling (SIR yields a straighforward and workable algorithm for the filtering problem 4. For the realis'c problems in data assimila'on, par'cle filters must be modified either via approxima'ons, or clever use of proposal distribu'ons.

28 Some Key References Jazwinski AH Stochas'c Processes and Filtering Theory. Academic Press: New York, 376pp. Gordon NJ, Salmond DJ, Smith AFM Novel approach to nonlinear/non- Gaussian Bayesian state es'ma'on. IEE Proceedings- F 140(2: Kitagawa G Monte Carlo filter and smoother for non- Gaussian nonlinear state space models. Journal of Computa'onal and Graphical Sta's'cs 5: Kitagawa G A self- organizing state space model. Journal of the American Sta's'cal Associa'on. 93(443: Arulampalam MS, Maskell S, Gordon N, Clapp T, A tutorial on par'cle filters for online nonlinear non- Gaussian Bayesian tracking. IEEE Transac'ons on Signal Processing 50(2: Kunsch HR Recursive Monte Carlo filters: algorithms and theore'cal analysis. The Annals of Sta's'cs. 33(5: Godsill SJ, Doucet A, West M Monte Carlo Smoothing for Nonlinear Time Series. Journal of the American Sta's'cal Associa'on. 99(465: Ionides EL, Breto C, King AA Inference for nonlinear dynamical systems. Proceedings of the Na'onal Academy of Sciences 103:

Par$cle Filters Part I: Theory. Peter Jan van Leeuwen Data- Assimila$on Research Centre DARC University of Reading

Par$cle Filters Part I: Theory Peter Jan van Leeuwen Data- Assimila$on Research Centre DARC University of Reading Reading July 2013 Why Data Assimila$on Predic$on Model improvement: - Parameter es$ma$on