Particle Filters in High Dimensions

Transcription

1 Particle Filters in High Dimensions Dan Crisan Imperial College London Workshop - Simulation and probability: recent trends The Henri Lebesgue Center for Mathematics 5-8 June 2018 Rennes Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

2 . Part 1: Theoretical Considerations Stochastic Filtering Particle filters/ Sequential Monte Carlo methods Convergence Result Final remarks DC, Particle Filters. A Theoretical Perspective, Sequential Monte Carlo Methods in Practice, DC, A Doucet, A survey of convergence results on particle filtering methods for practitioners, IEEE Transactions on signal processing, A Doucet, AM Johansen, A tutorial on particle filtering and smoothing: Fifteen years later, The Oxford handbook of nonlinear filtering, P. Del Moral. Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer, A. Bain, DC, Fundamentals of Stochastic Filtering, Springer, DC, B Rozovskii, The Oxford handbook of nonlinear filtering, Oxford University Press, Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

3 . What is stochastic filtering? Stochastic Filtering: The process of using partial observations and a stochastic model to make inferences about an evolving dynamical system. X the signal process - hidden component Y the observation process - the data Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

4 . What is stochastic filtering? The filtering problem: Find the conditional distribution of the signal X t given Y t = σ(y s, s [0, t]), i.e., π t (A) = P(X t A Y t ), t 0, A B(R d ). Discrete framework: {X t, Y t } t 0 Markov process The signal process {X t } t 0 Markov chain, X 0 π 0 (dx 0 ) P (X t dx t X t 1 = x t 1 ) = K t (x t 1, dx t ) = f t (x t x t 1 )dt, Example : X t = b (X t 1 ) + σ (X t 1 ) B t, The observationprocess B t N (0, 1) i.i.d. P ( Y t dy t X [0,t] = x [0,t], Y [0,t 1] = y [0,t 1] ) = P (Yt dy t X t = x t ) = g t (y t x t )dy Example : Y t = h (X t ) + V t, where X [0,t] (X 0,..., X t ), x [0,t] (x 0,..., x t ). V t N (0, 1) i.i.d. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

5 . What is stochastic filtering? Notation: posterior measure: the conditional distribution of the signal X t given Y t π t (A) = P(X t A Y t ), t 0, A B(R d ). predictive measure: the conditional distribution of the signal X t given Y t 1 p t (A) = P(X t A Y t 1 ), t 0, A B(R d ). If μ is a measure and f is a function, then μ (f ) f (x)μ (dx). If f is a function and k is a kernel, then kf (x) f (y)k (x, dy). If μ is a measure and k is a kernel, then μk (A) μ (dx) k (x, A). Bayes recursion. Prediction Updating p t = π t 1 K t π t = g t p t In other words, dπt dp t = C 1 t g t, where C t R d g t (y t, x t ) p t (dx t ). Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

6 . Particle filters 1. Class of approximations: Particle filters/sequential Monte Carlo Methods: SMC (a j (t), vj 1 (t),..., vj d (t)) N j=1 }{{}}{{} weight position π t πt N = N j=1 a j (t) δ vj (t) 2. The law of evolution of the approximation: π N t 1 SMC selection {}}{ pt N {}}{ πt N mutation 3. The measure of the approximating error: sup E [ πt n (ϕ) π t (ϕ) ], ˆπ t ˆπ t n. ϕ B(R d ) Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

7 . The classical/standard/bootstrap/garden-variety particle filter π n = {π n (t), t 0} the occupation measure of a system of weighted particles π n (0) = n i=1 1 n δ xi n π n (t) = n i=1 ā n i (t)δ V n i (t). DC, Particle Filters. A Theoretical Perspective, Sequential Monte Carlo Methods in Practice, P. Del Moral. Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer, Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

8 1. Initialisation [t = 0]. The Filtering Problem For i = 1,..., N, sample x (i) 0 from π 0, Framework: discrete/continuous time π N 0 = 1 N N i=1 δ (i) x Iteration [t 1 to t]. Let x (i) t 1, i = 1,..., n be the positions of the particles at time t 1. π N t 1 = 1 N N i=1 δ (i) x. t 1 Step 1. For i = 1,..., n, sample ˉx (i) t from f t 1 (x t x (i) t 1 )dx t. p N t = 1 N N i=1 (i). δˉx t Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

9 Framework: discrete/continuous time Compute the (normalized) weight ā (i) t = g t (ˉx (i) t )/( n j=1 g t(ˉx (j) t )). ˉπ N t = N i=1 ā (i) (i) t = g δˉx t pt N. t Step 2. Replace each particle by ξ (i) t offsprings such that n i=1 ξ(i) t = n. [Sample with replacement n-times from ˉx (i) t, ] Denote the positions of the particles by x (i) t, i = 1,..., n. Further details in: π N t = 1 N N i=1 δ (i) x. t Bain, A., DC, Fundamentals of Stochastic Filtering, Series: Stochastic Modelling and Applied Probability, Vol. 60, Springer Verlag, Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

10 Framework: discrete/continuous time Theorem π n converges to π. Moreover sup sup t [0,T ] { ϕ 1} E Y [ π N t (ϕ) π t (ϕ) ] c T N. and N(π N π) converges to a measure valued process ū = {ū t, t 0}. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

11 Notation: The Filtering Problem Framework: discrete/continuous time Error(π, T, N) = sup t [0,T ] sup { ϕ 1} EY [ π N t (ϕ) π t (ϕ) ] Error(p, T, N) = sup t [0,T ] sup { ϕ 1} EY [ p N t (ϕ) p t (ϕ) ] Theorem sup sup t [0,T ] { ϕ 1} For all T > 0, there exists c T such that E Y [ π N t (ϕ) π t (ϕ) ] c T N. Error(π, T, N) c T N, Error(p, T, N) c T N if and only if Error(π, 0, N) c 0 N and, for all T > 0, there exists c T such that sup sup t [0,T ] { ϕ 1} sup sup t [0,T ] { ϕ 1} E Y [ p N t (ϕ) π N t 1K t (ϕ) ] c T N E Y [ π N t (ϕ) ˉπ N t (ϕ) ] c T N. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

12 Framework: discrete/continuous time Proof. Immediate from the following two inequalities pt N ϕ πt 1K N t ϕ pt N ϕ p t ϕ + π t 1 (K t ϕ) πt 1(K N t ϕ), πt N ϕ ˉπ t N ϕ πt N ϕ π t ϕ + π t ϕ ˉπ t N ϕ where we used the fact that p t = π t 1 K t. Induction. The case t = 0 is assumed. The induction step is obtained as follows: Since p t = π t 1 K t by the triangle inequality Also p N t ϕ p t ϕ p N t ϕ π N t 1K t ϕ + π N t 1K t ϕ π t 1 K t ϕ. ˉπ N t ϕ π t ϕ= pn t (ϕg t ) p N t g t p t(ϕg t ) p t g t = pn t (ϕg t ) p N t g t p t g t (p N t g t p t g t )+ ( p N t (ϕg t ) p t(ϕg t ) p t g t p t g t and as pt N (ϕg t ) ϕ pt N g t, ˉπ t N ϕ π t ϕ ϕ p N 1 p p t g t g t p t g t + N t p t g t (ϕg t ) p t (ϕg t ). t Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55 ),

13 Framework: discrete/continuous time Remarks: Particle filters are recursive algorithms: The approximation for π t and Y t+1 are the only information used in order to obtain the approximation for π t+1. In other words, the information gained from Y 1,..., Y t is embedded in the current approximation. The generic SMC method involves sampling from the prior distribution of the signal and then using a weighted bootstrap technique (or equivalent) with weights defined by the likelihood of the most recent observation data. Step 2 can be done by means of sampling with replacement (SIR algorithm), stratified sampling, Bernoulli sampling, Carpenter-Clifford-Fearnhead-Whitley genetic algorithm, Crisan-Lyons TBBA algorithm. All these methods satisfy the convergence requirement. If d is small to moderate, then the standard particle filter can perform very well in the time parameter n. Under certain conditions, the Monte Carlo error of the estimate of the filter can be uniform with respect to the time parameter. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

14 Remarks: The Filtering Problem Framework: discrete/continuous time The function x k g(x k, y k ) can convey a lot of information about the hidden state, especially so in high dimensions. If this is the case, using the prior transition kernel f (x k 1, x k ) as proposal will be ineffective. It is then known that the standard particle filter will typically perform poorly in this context, often requiring that N = O(κ d ). Wall clock time per time step (seconds) Algorithm PF STPF Dimension Figure: Computational cost per time step to achieve a predetermined RMSE versus model dimension, for standard particle filter (PF) and STPF. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

15 Application to high-dimensional problems Why is the high-dimensional filtering problem hard? A running example Using particle filers to solve high-dimensional filtering problems Final remarks Research partially supported by EPSRC grant EP/N023781/1. Numerical work done by Wei Pan (Imperial College London). A. Beskos, DC, A. Jasra, Ajay; K. Kamatani, Y. Zhou, Y A stable particle filter for a class of high-dimensional state-space models. Adv. in Appl. Probab. 49 (2017). A. Beskos, DC, A. Jasra, On the stability of sequential Monte Carlo methods in high dimensions, Ann. Appl. Probab. 24 (2014). C.J. Cotter, DC, D.D. Holm, W. Pan, I. Shevchenko, Numerically Modelling Stochastic Lie Transport in Fluid Dynamics, C.J. Cotter, DC, D.D. Holm, W. Pan, I. Shevchenko, Sequential Monte Carlo for Stochastic Advection by Lie Transport (SALT): A case study for the damped and forced incompressible 2D stochastic Euler equation, in preparation. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

16 Why is the high-dimensional problem hard? Consider Π 0 = N (0, 1) (mean 0 and variance matrix 1). Π 1 = N (1, 1) (mean 1 and variance matrix 1). Π d = N (d, 1) (mean d and variance matrix 1). d(π 0, Π 1 ) TV = 2 P [ X 1/2 ], X N(0, 1). d(π 0, Π d ) TV = 2 P [ X d/2 ], X N(0, 1). as d increases, the two measures get further and further apart, becoming singular w.r.t. each other. as d increases, it becomes increasingly harder to use standard importance sampling, to construct a sample from Π 3 by using a proposal from Π 1, weighting it using dπ d dπ 0 and (possibly) resample from it. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

17 Why is the high-dimensional problem hard? Consider Π 0 = N ((0,..., 0), I d ) (mean (0,..., 0) and covariance matrix I d ). Π d = N ((1,..., 1), I d ) (mean (1,..., 1) and covariance matrix I d ). d(π 0, Π d ) TV = 2 P [ X d/2 ], X N(0, 1). as d increases, the two measures get further and further apart, becoming singular w.r.t. each other exponentially fast. it becomes increasingly harder to use standard importance sampling, to construct a sample from Π d by using a proposal from Π 0. Moving from Π 0 to Π d is equivalent to moving from a standard normal distribution N (0, 1) to a normal distribution N (d, 1) (the total variation distance between N (0, 1) and N (d, 1) is the same as that between Π 1 and Π 2 ). Add-on techniques: Tempering * Optimal transport prior posterior Sequential DA in space * Jittering * Model Reduction (High Low Res)* Nudging Hybrid models Hamiltonian Monte Carlo Informed priors Localization Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

18 The Filtering Problem What is DA? State estimation in Numerical Weather Prediction Data Assimilation at the UK Met Office set of methodologies that combines past knowledge of a system in the form of a numerical model with new information about that system in the form of observations of that system. designed to improve forecasting, reduce model uncertainties and adjust model parameters. termen used mainly in the computational geoscience community major component of Numerical Weather Prediction Variational DA: combines the model and the data through the optimisation of a given criterion (minimisation of a so-called cost-function). Ensemble based DA: uses a set of model trajectories/possible scenarios that are intermittently updated according to data and are used to infer the past, current or future position of a system. Hurricane Irma forecast: a. ECMWF, b. USA Global Forecast Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

19 A stochastic transport model Consider a two dimensional incompressible fluid flow u defined on 2D-torus Ω = [0, L x ] [0, L y ] modelled by the two-dimensional Euler equations with forcing and dampening. Let q = ẑ curl u denote the vorticity of u, where ẑ denotes the z-axis. For a scalar field g : Ω R, we write g = ( y g, x g) T. Let ψ : Ω [0, ) R denote the stream function. t q + (u ) q = Q rq u = ψ Δψ = q. Q is the forcing term given by Q = 0.1 sin (8πx) r is a positive constant - the large scale dissipation time scale. we consider slip flow boundary condition ψ Ω = 0. evolution of Lagrangian fluid parcels dx t dt = u(x t, t). Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

20 A stochastic transport model Domain is [0, 1] 2 PDE System SPDE System t ω + u ω = Q rω dq + ū qdt + i ξ i q dw i t = (Q rq) dt u = ψ ū = ψ Δψ = ω Δ ψ = q Q = 0.1 sin (8πx), r = Boundary Condition ψ Ω = 0 and ψ = 0. Ω PDE SPDE Grid Resolution 512x512 64x64 Numerical Δt Spin-up 40 ett ett: eddy turnover time L/u L 2.5 time units. Numerical scheme: a mixed continuous and discontinuous Galerkin finite element scheme + an optimal third order strong stability preserving Runge-Kutta, [Bernsen et al 2006, Gottlieb 2005]. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

21 Initial configuration for the vorticity A stochastic transport model ω spin = sin(8πx) sin(8πy) cos(6πx) cos(6πy) cos(10πx) cos(4πy) sin(2πy) sin(2πx) from which we spin up the system until an energy equilibrium state seems to have been reached. This equilibrium state, denoted by ω initial, is then chosen as the initial condition. (1) Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

22 A stochastic transport model Plot of the numerical PDE solution at the initial time t initial and its coarse-grained version done via spatial averaging and projection of the fine grid stream-function to the coarse grid. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

23 A stochastic transport model Plot of the numerical PDE solution at the final time t = t initial large eddy turnover times (ett). The coarse-graining is done via spatial averaging and projection of the fine grid streamfunction to the coarse grid. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

24 A stochastic transport model Observations: u is observed on a subgrid of the signal grid (9 9 points) { u SPDE t (x) + αz x, z x N (0, 1) Experiment 1 Y t (x) = u PDE t (x) + αz x, z x N (0, 1) Experiment 2 α is calibrated to the standard deviation of the true solution over a coarse grid cell. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

25 Initial condition Initial Condition A good choice of the initial condition is esential for the successful implementation of the filter. In practice it is a reflection of the level of uncertainty of the estimate of initial position of the dynamical system. We use the initial condition is to obtain an ensemble which contain particles that are reasonably close to the truth. Choice for the running example deformation - physically consistent with the system, casimirs preserved. We take a nominal value ω t0 and deform it using the following modified Euler equation: tω + β i u(τ i ) ω = 0 (2) where β i N (0, ɛ), i = 1,..., N p are centered Gaussian weights with an apriori variance parameter ɛ, and τ i U (t initial, t 0 ), i = 1,..., N p are uniform random numbers. Thus each u (τ i ) corresponds to a PDE solution in the time period [t initial, t 0 ). Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

26 Initial condition Alternative choices q + ζ where ζ is gaussian random field, doable but not physical, only works for q because it s the least smooth of the three fields of interest. The other fields are spatially smooth. also this breaks the SPDE well-posedness theorem (function space regularity). Figure (ux,uy) Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

27 Initial condition directly perturb ψ, by ψ + ˉψ where ˉψ = (I κδ) 1 ζ invert elliptic operator with boundary condition ˉψ = 0. Figure (ux, uy) Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

28 Add-on techniques Model Reduction (High Low Res)* Model reduction is a valuable methodology that can lead to substantial computational savings: For the current numerical example we perform and state space order reduction from to = However if applied erroneously order reduction it can introduce large errors. Recall that the recursion formula for the conditional distribution of the signal where dπt dp t p t = π t 1 K t π t = g t p t, = C 1 t g t, where C t R d g t (y t, x t ) p t (dx t ). Remark. The conditional distribution of the signal is a continuous function of (π 0, g 1,..., g t, K 1,..., K t ). In other words if lim ε 0 (πε 0, g1, ε..., gt ε, K1 ε,..., Kt ε ) = (π 0, g 1,..., g t, K 1,..., K t ) in a suitably chosen topology and p ε t Δ = π ε t 1K ε t π ε t Δ = g ε t p ε t, (3) then lim ε 0 π ε t = π t and lim ε 0 p ε t = p t (again, in a suitably chosen topology). Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

29 Add-on techniques NB. Note that πt ε is no long the solution of a filtering problem, but simply the solution of the iteration (3) Order reduction can be theoretically justified through the continuity of the conditional distribution of the signal on (π 0, g 1,..., g t, K 1,..., K t ). This is the case when the order reduction is performed through a coarsening of the grid used for the numerical algorithm that approximates the dynamical system. Example: We use a Stochastic PDE defined on a coarser grid: t q + (u ) q + (ξ k ) q dbt k = Q rq k=1 u = ψ Δψ = q. ξ k are divergence free given vector fields ξ k are computed from the true solution by using an empirical orthogonal functions (EOFs) procedure Bt k are scalar independent Brownian motions dx t = u(x t, t) dt + i ξ i (x t ) dw i (t). Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

30 Methodology to calibrate the noise The reason for this stochastic parametrization is grounded in solid physical considerations, see D.D. Holm, Variational principles for stochastic fluids, Proc.Roy. Soc. A, dx t = u f t (x t )dt dx t = ut c (x t )dt + ξ i (x t ) dw i (t) For each m = 0, 1,..., M 1 i 1 Solve dxij f (t)/dt = uf t (x ij f f (t)) with initial condition xij (mδt ) = x ij. 2 Compute ut c by low-pass filtering ut f along the trajectory. 3 Compute xij c c (t) by solving dxij (t)/dt = uc t (x ij f (t)) with the same initial condition. 4 Compute the difference Δxij m = xij f c ((m + 1)ΔT ) xij ((m + 1)ΔT ), which measures the error between the fine and coarse trajectory. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

31 Methodology to calibrate the noise Having obtained Δxij m, we would like to extract the basis for the noise. This amounts to a Gaussian model of the form Δxij m = Δx ij + δt N ξij k ΔW m, k where ΔWm k are i.i.d. Normal random variables with mean zero, variance 1. We estimate ξ by minimising 2 Δxij m E Δx N ij ξij k ΔW m k, δt ijm where the choice of N can be informed by using Empirical Orthogonal Function (EOFs). k=1 k=1 Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

32 Methodology to calibrate the noise Number of EoFs decide on a case by case basis too many will slow down the algorithm On the left: Number of EOFs 90% variance vs 50% (no change). On the right: Normalised spectrum of the Δx covariance operator, showing number of eofs required to capture 50%, 70% and 90% of total variance Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

33 Methodology to calibrate the noise Model reduction UQ pictures sytems 512x512 vs 128x128 vs 64x64 (a) ux (b) uy Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

34 Methodology to calibrate the noise (a) psi (b) q Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

35 Methodology to calibrate the noise The aim of the calibration is to capture the statistical properties of the fast fluctuations, rather than the trajectory of the flow. Validation of stochastic parameterisation in terms of uncertainty quantification for the SPDE. Performance of DA algorithm relies on the correct modelling of the unresolved scales. Evaluating the uncertainty arising from the choice of EOFs is be part of the particle filter implementation. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

36 Methodology to calibrate the noise Ensemble Distance from the truth Velocity Field d ({ˆq } ) i, i = 1,..., N p, ω, t := mini {1,...,Np} ω(t) ˆq i (t) L 2 (D) ω(t) L 2 (D) Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

37 Methodology to calibrate the noise Implementation issues Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

38 Implementation issues Number of Particles decide on a case by case basis too few will not give a reasonable solution too many will slow down the algorithm Picture Number of particles 225 (good) vs 500 (no change), 225 (good) vs 25 (less good) 25 seems ok but we want as many as computationally feasible to tune the algorithm (a) psi 225 vs 500 (b) psi 225 vs 25 Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

39 SIR fails Histogram of weights Classical Particle Filter fails! Figure: example: loglikelihoods histogram, period 1 ett, 100 particles Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

40 Framework: The Filtering Problem Tempering For i = 1 to d {X t } t 0 Markov chain P (X t dx t X t 1 = x t 1 ) = f t (x t x t 1 )dx t, {X t, Y t } t 0 P (Y t dy t X t = x t ) = g t (y t x t )dy t reweight the particle using g 1 d t A tempering procedure and (possibly) resample from it move particles using an MCMC that leaves g k d t f t π [0,t 1] invariant Beskos, DC, Jasra, On the stability of SMC methods in high dimensions, Kantas, Beskos, Jasra, Sequential Monte Carlo for inverse problems, Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

41 Tempering Initialisation t=0: For i = 1,..., n, sample q i 0 from π 0. Iteration (t i 1, t i ]: Given the ensemble {X n (t i 1 )} n=1,...,n, 1 Evolve X n (t i 1 ) using the SPDE to obtain X n (t i ). 2 Given X := {X n (t i )} n=1,...,n, define normalised tempered weights ˉλ n,i (φ, X) := exp ( φλ n,i) m exp ( φλ m,i) where the dependence on X means the Λ n,i are computed using X. Define effective sample size Set φ = 1. ESS i (φ, X) := ˉλ i (φ, X) 1 l While ESS i (φ, X) < N threshold do: (a) Find 1 φ < φ < 1 such that ESS i (φ (1 φ), X) N threshold. Resample according to ˉλ n,i (φ (1 φ), X) and apply MCMC if required (i.e. when there are duplicated particles), to obtain a new set of particles X (φ ). Set φ = 1 φ and X = X (φ ). (b) If ESS i N threshold then STOP and go to the next filtering step with {( Xn (t i ), ˉλn,i )} n=1,...,n. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

42 Tempering MCMC Mutation Algorithm Given the ensemble {X n,k (t i )} n=1,...,n corresponding to the k th tempering step with temperatureφ k, and proposal step size ρ [0, 1], repeat the following steps. Propose ( X n (t i ) = G X n (t i 1 ), ρw (t i 1 : t i ; ω) + ) 1 ρ 2 Z (t i 1 : t i ; ω) where X n (t i ) = G (X n (t i 1 ), W (t i 1 : t i ; ω)), and W Z. Accept X n (t i ) with probability ( ˉλ φ k, X ) n (t i ) 1 ˉλ (φ k, X n (t i )) where λ (φ, x) = exp ( φλ(x)) is the unnormalised weight function. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

43 Resampling Intervals Resampling Intervals small resampling intervals lead to an unreasonable increase in the computational effort large resampling intervals make the algorithm fail the ESS can be used as criterion for choosing the resampling time adapted resampling time can be used ESS evolution in time/observation noise Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

44 Results DA Solution for DA periods: 1 ETT and 0.2 ETT (a) ux (b) uy (c) psi (d) q Figure: DA: obs spde, period 0.2 ett Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

45 Results (a) ux (b) uy (c) psi (d) q Figure: DA: obs pde, period 0.2 ett (a) ux (b) uy (c) psi (d) q Figure: DA: obs pde, period 1 ett Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

46 Results Number of tempering steps/average MCMC steps Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

47 . Sequential DA Space-Time Particle Filter Assume that there exists an increasing sequence of sets {A k,j } τ k,d j=1, with A k,1 A k,2 A k,τk,d = {1 : d}, for some integer 0 < τ k,d d, such that we can factorize: τ k,d g(x k, y k )f (x k 1, x k ) = α k,j (y k, x k 1, x k (A k,j )), for appropriate functions α k,j ( ), where x k (A) = {x k (j) : j A} R A. Example: j=1 X n (j) = j 1 β d j+i+1 (X n (i)) + i=1 d ˉβ i j+1 (X n 1 (i)) + ɛ j n i=j Y n (j) = X n (j) + ξ j n where ɛ n (j) i.i.d. N (0, σ 2 x) and ξ n (j) i.i.d. N (0, σ 2 y), j {1,..., d}. Beskos, CD, Jasra, Kamatani, Zhou, A Stable Particle Filter in High-Dimensions, Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

48 . Sequential DA Within a sequential Monte Carlo context, one can think of augmenting the sequence of distributions of increasing dimension X 1:k Y 1:k, 1 k n, moving from R d(k 1) to R dk, with intermediate laws on R d(k 1)+ A k,j, for j = 1,..., τ k,d. This holds when: one can obtain a factorization for the prior term f (x k 1, x k ) by marginalising over subsets of co-ordinates. the likelihood component g(x k, y k ) can be factorized when the model assumes a local dependence structure for the observations. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

49 For j = 1 to τ d 1. Sequential DA Move particle according to q k+1,j (x k+1 (A k+1,j ) x k, x k+1 (A k+1,j 1 )). α k+1,j (y k+1,x k,x k+1 (A k+1,j 1 )) q k+1,j (x k+1 (A k+1,j ) x k,x k+1 (A k+1,j 1 )) weight the particle using and (possibly) resample from it. Remarks. Since particle filters work well with regards to the time parameter (they are sequential), we exploit the model structure in to build up a particle filter in space and time. We break the k-th time-step of the particle filter into τ k,d space-steps and run a system of N independent particle filters for these steps. It is necessary that the factorisation is such that allows for a gradual introduction of the full likelihood term g(x k, y k ) along the τ k,d steps. For instance, trivial choices like α k,j = f (x k 1, x k )dx k (j + 1 : d)/ f (x k 1, x k )dx k (j : d), 1 j d 1, and α k,d = ( f (x k 1, x k )/ ) f (x k 1, x k )dx k (d) g(x k, y k ) are ineffective, as they only introduce the complete likelihood term in the last step. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

50 . Sequential DA Numerical Test Let X n R d be such that we have X 0 = 0 d (the d-dimensional vector of zeros) and j 1 d X n (j) = β d j+i+1 X n (i) + β i j+1 X n 1 (i) + ɛ n i=1 where ɛ n i.i.d. N (0, σ 2 x) and β 1:d are known static parameters. For the observations, we set Y n = X n + ξ n where ξ n (j) i.i.d. N (0, σ 2 y ), j {1,..., d}. Comparison between the SIR algorithm and the STPF. Parameters: σ 2 x = σ 2 y = 1, n = 1000, d-dimensional observations, d {16, 128, 1024}. Both filters use the model transitions as the proposal and the likelihood function as the potential and adaptive resampling. STPF: N = 100 and M d = d SIR algorithm NM d particles. i=j Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

51 . Sequential DA The averages of estimators for the posterior mean of the first co-ordinate X n (1) given all data up to time n are illustrated below. The SIR collapses when the dimension becomes moderate or large. No meaningful estimates when d = 1024 (the estimates completely lose track of the observations and the analytical mean). The STPF performs reasonably well in all three cases. Observation Kalman Filter Estimator Standard Particle Filter Space Time Particle Filter Mean of estimator for X(1) d = 16 d = 128 d = Time Figure: Mean of estimators of X n(1) for across 100 runs. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

52 . Sequential DA The ESS (calculated over global filter for STPF) scaled by the number of particles for each time step of the two algorithms. The standard filter struggles significantly even in the case d = 16 and collapses when d = 128. The performance of the STPF is deteriorating (but not collapsing) when the dimension increases, due to the path degeneracy effect. However, even for d = 1024, it still retains an acceptable level of ESS. Standard Particle Filter d = 16 Space Time Particle Filter ESS (scaled by the number of particles) d = 128 d = Time Figure: Effective Sample Size plots from a single run. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

53 . Sequential DA The variance per time step for the estimators of the posterior mean of the first co-ordinate X n (1) (given the data up to time n) across 100 runs: Standard Particle Filter d = 16 Space Time Particle Filter Variance of estimator for X(1) d = 128 d = Time Figure: Logarithmic scale variance. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

54 . Jittering Jittering Procedure employed to reduce sample degeneracy. The particles are moved using a suitable chosen kernel The moves are controlled so that the size of the (stochastic) perturbation remains of the same order as the particle filter error (1/ N) D. C., Joaquin Miguez, Nested particle filters for online parameter estimation in discrete-time state-space Markov models, D. C., Joaquin Miguez, Uniform convergence over time of a nested particle filtering scheme for recursive parameter estimation in state space Markov models, Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55

55 . Final Remarks Particle filters/sequential Monte Carlo methods are theoretically justified algorithms for approximating the state of dynamical systems partially (and noisily) observed. Standard particle filters do NOT work in high dimensions. Properly calibrated and modified, particle filters can be used to solve high dimensional problems (see also the work of Peter Jan van Leeuwen, Roland Potthast, Hans Kunsch). Important parameters: initial condition, number of particle, number of observations, correction times, observation error, etc. One can use methodology to assess the reliability of ensemble forecasting system. Dan Crisan (Imperial College London) Particle Filters in High Dimensions 7-8 June / 55