Advanced Computational Methods in Statistics: Lecture 5 Sequential Monte Carlo/Particle Filtering
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1 Advanced Computational Methods in Statistics: Lecture 5 Sequential Monte Carlo/Particle Filtering Axel Gandy Department of Mathematics Imperial College London London Taught Course Centre for PhD Students in the Mathematical Sciences Autumn 2014
2 Outline Introduction Setup Examples Finitely Many States Kalman Filter Particle Filtering Improving the Algorithm Further Topics Summary Axel Gandy Particle Filtering 2
3 Sequential Monte Carlo/Particle Filtering Particle filtering introduced in Gordon et al. (1993) Most of the material on particle filtering is based on Doucet et al. (2001) and on the tutorial Doucet & Johansen (2008). Examples: Tracking of Objects Robot Localisation Financial Applications Axel Gandy Particle Filtering 3
4 Setup - Hidden Markov Model x 0, x 1, x 2,... : unobserved Markov chain - hidden states y 1, y 2,... : observations; y 1 y 2 y 3 y 4 x 0 x 1 x 2 x 3 x 4... y 1, y 2,... are conditionally independent given x 0, x 1,... Model given by π(x0 ) - the initial distribution f (xt x t 1 ) for t 1 - the transition kernel of the Markov chain g(yt x t ) for t 1 - the distribution of the observations Notation: x0:t = (x 0,..., x t ) - hidden states up to time t y1:t = (y 1,..., y t ) - observations up to time t Interested in the posterior distribution p(x 0:t y 1:t ) or in p(x t y 1:t )
5 Some Remarks No explicit solution in the general case - only in special cases. Will focus on the time-homogeneous case, i.e. the transition densities f and the density of the observations g will be the same for each step. Extensions to inhomogeneous case straightforward. It is important that one is able to update quickly as new data becomes available, i.e. if y t+1 is observed want to be able to quickly compute p(x t+1 y 1:t+1 ) based on p(x t y 1:t ) and y t+1. Axel Gandy Particle Filtering 5
6 Example- Bearings Only Tracking Gordon et al. (1993); Ship moving in the two-dimensional plane Stationary observer sees only the angle to the ship. Hidden states: position x t,1, x t,2, speed x t,3, x t,4. Speed changes randomly x t,3 N(x t 1,3, σ 2 ), x t,4 N(x t 1,4, σ 2 ) Position changes accordingly x t,1 = x t 1,1 + x t 1,3, x t,2 = x t 1,2 + x t 1,4 Observations: y t N(tan 1 (x t,1 /x t,2 ), η 2 )
7 Example- Stochastic Volatility Returns: y t N(0, β 2 exp(x t )) (observable from price data) Volatility: x t N(αx t 1, σ 2 (1 α) 2 ), x 1 N(0, σ 2 (1 α) 2 ), σ = 1, β = 0.5, α = Volatility x t Returns y t t Axel Gandy Particle Filtering 7
8 Hidden Markov Chain with finitely many states If the hidden process x t takes only finitely many values then p(x t+1 y 1:t+1 ) can be computed recursively via p(x t+1 y 1:t+1 ) g(y t+1 x t+1 ) x t f (x t+1 x t )p(x t y 0:t ) May not be practicable if there are too many states! Axel Gandy Particle Filtering 8
9 Kalman Filter Kalman (1960) Linear Gaussian state space model: x 0 N(ˆx 0, P 0 ) x t = Ax t 1 + w t yt = Hx t + v t wt N(0, Q) iid, v t N(0, R) iid A, H deterministic matrices; ˆx0, P 0, A, H, Q, R known Explicit computation and updating of the posterior possible: Posterior: x t y 1,..., y t N(ˆx t, P t ) ˆx 0, P 0 P t Time Update ˆx t = Aˆx t 1 = AP t 1 A + Q Measurement Update K t = P t H (HP t H + R) 1 ˆx t = ˆx t + K t (y t Hˆx t ) P t = (I K t H)P t
10 Kalman Filter - Simple Example x 0 N(0, 1) x t = 0.9x t 1 + w t y t = 2x t + v t w t N(0, 1) iid, v t N(0, 1) iid y t t true x t mean x t y 1,...,y t 95% credible interval
11 Kalman Filter - Remarks I Updating very easy - only involves linear algebra. Very widely used A, H, Q and R can change with time A linear control input can be incorporated, i.e. the hidden state can evolve according to where u t can be controlled. x t = Ax t 1 + Bu t 1 + w t 1 Normal prior/updating/observations can be replaced through appropriate conjugate distributions. Continuous Time Version: Kalman-Bucy filter See Øksendal (2003) for a nice introduction. Axel Gandy Particle Filtering 11
12 Kalman Filter - Remarks II Extended Kalman Filter extension to nonlinear dynamics: xt = f (x t 1, u t 1, w t 1 ) y t = g(x t, v t ). where f and g are nonlinear functions. The extended Kalman filter linearises the nonlinear dynamics around the current mean and covariance. To do so it uses the Jacobian matrices, i.e. the matrices of partial derivatives of f and g with respect to its components. The extended Kalman filter does no longer compute precise posterior distributions. Axel Gandy Particle Filtering 12
13 Outline Introduction Particle Filtering Introduction Bootstrap Filter Example-Tracking Example-Stochastic Volatility Example-Football Theoretical Results Improving the Algorithm Further Topics Summary Axel Gandy Particle Filtering 13
14 What to do if there were no observations... y 1 y 2 y 3 y 4 x 0 x 1 x 2 x 3 x 4... Without observations y i the following simple approach would work: Sample N particles following the initial distribution x (1) 0,..., x(n) 0 π(x 0 ) For every step propagate each particle according to the transition kernel of the Markov chain: x (j) i+1 f ( x(j) i ), j = 1,..., N After each step there are N particles that approximate the distribution of x i. Note: very easy to update to the next step.
15 Importance Sampling Cannot sample from x 0:t y 1:t directly. Main idea: Change the density we are sampling from. Interested in E(φ(x 0:t ) y 1:t ) = φ(x 0:t )p(x 0:t y 1:t )dx 0:t For any density h, E(φ(x 0:t ) y 1:t ) = φ(x 0:t ) p(x 0:t y 1:t ) h(x 0:t )dx 0:t, h(x 0:t ) Thus an unbiased estimator of E(φ(x 0:t ) y 1:t ) is Î = 1 N N φ(x i 0:t)wt i, i=1 where wt i = p(xi 0:t y 1:t ) h(x i 0:t ) and x 1 0:t,..., xn 0:t h iid. How to evaluate p(x i 0:t y 1:t)? How to choose h? Can importance sampling be done recursively? Axel Gandy Particle Filtering 15
16 Sequential Importance Sampling I Recursive definition and sampling of the importance sampling distribution: h(x 0:t ) = h(x t x 0:t 1 )h(x 0:t 1 ) Can the weights be computed recursively? By Bayes Theorem: w t = p(x 0:t y 1:t ) h(x 0:t ) = p(y 1:t x 0:t )p(x 0:t ) h(x 0:t )p(y 1:t ) Hence, w t = g(y t x t )p(y 1:t 1 x 0:t 1 )f (x t x t 1 )p(x 0:t 1 ) h(x t x 0:t 1 )h(x 0:t 1 )p(y 1:t ) Thus, g(y w t = w t x t )f (x t x t 1 ) p(y 1:t 1 ) t 1 h(x t x 0:t 1 ) p(y 1:t ) Axel Gandy Particle Filtering 16
17 Sequential Importance Sampling I Recursive definition and sampling of the importance sampling distribution: h(x 0:t ) = h(x t x 0:t 1 )h(x 0:t 1 ) Can the weights be computed recursively? By Bayes Theorem: w t = p(x 0:t y 1:t ) h(x 0:t ) = p(y 1:t x 0:t )p(x 0:t ) h(x 0:t )p(y 1:t ) Hence, w t = g(y t x t )p(y 1:t 1 x 0:t 1 )f (x t x t 1 )p(x 0:t 1 ) h(x t x 0:t 1 )h(x 0:t 1 )p(y 1:t ) Thus, g(y w t = w t x t )f (x t x t 1 ) p(y 1:t 1 ) t 1 h(x t x 0:t 1 ) p(y 1:t ) Axel Gandy Particle Filtering 16
18 Sequential Importance Sampling I Recursive definition and sampling of the importance sampling distribution: h(x 0:t ) = h(x t x 0:t 1 )h(x 0:t 1 ) Can the weights be computed recursively? By Bayes Theorem: w t = p(x 0:t y 1:t ) h(x 0:t ) = p(y 1:t x 0:t )p(x 0:t ) h(x 0:t )p(y 1:t ) Hence, w t = g(y t x t )p(y 1:t 1 x 0:t 1 )f (x t x t 1 )p(x 0:t 1 ) h(x t x 0:t 1 )h(x 0:t 1 )p(y 1:t ) Thus, g(y w t = w t x t )f (x t x t 1 ) p(y 1:t 1 ) t 1 h(x t x 0:t 1 ) p(y 1:t ) Axel Gandy Particle Filtering 16
19 Sequential Importance Sampling II Can work with normalised weights: w i t = the recursion w t i w t 1 i g(y t x i t)f (x i t x i t 1 ) h(x i t x i 0:t 1 ) w t i ; then one gets j w j t If one uses uses the prior distribution h(x 0 ) = π(x 0 ) and h(x t x 0:t 1 ) = f (x t x t 1 ) as importance sampling distribution then the recursion is simply w i t w i t 1g(y t x i t) Axel Gandy Particle Filtering 17
20 Failure of Sequential Importance Sampling Weights degenerate as t increases. Example: x 0 N(0, 1), x t+1 N(x t, 1), y t N(x t, 1). N = 100 particles Plot of the empirical cdfs of the normalised weights w 1 t,..., wt N t= 1 t= 10 t= 20 t= 40 Fn(x) Fn(x) Fn(x) Fn(x) e 52 1e 41 1e 30 1e 19 1e 08 1e 52 1e 41 1e 30 1e 19 1e 08 1e 52 1e 41 1e 30 1e 19 1e 08 x x x Note the log-scale on the x-axis. Most weights get very small. 1e 52 1e 41 1e 30 1e 19 1e 08 x Axel Gandy Particle Filtering 18
21 Resampling Goal: Eliminate particles with very low weights. Suppose Q = N w t i δ x i t i=1 is the current approximation to the distribution of x t. Then one can obtain a new approximation as follows: Sample N iid particles x i t from Q The new approximation is Q = N i=1 1 N δ x i t Axel Gandy Particle Filtering 19
22 The Bootstrap Filter 1. Sample x (i) 0 π(x 0 ) and set t = 1 2. Importance Sampling Step For i = 1,..., N: Sample x (i) t f (x t x (i) t 1 ) and set x(i) 0:t = (x(i) 0:t 1, x(i) Evaluate the importance weights w (i) t 3. Selection Step: t ) = g(y t x (i) t ). (i) Resample with replacement N particles (x 0:t ; i = 1,..., N) from the set { x (1) } according to the normalised importance weights 0:t,..., x(1) 0:t. w (i) t N j=1 w (j) t t:=t+1; go to step 2. Axel Gandy Particle Filtering 20
23 Illustration of the Bootstrap Filter N=10 particles g(y 1 x) g(y 2 x) g(y 3 x)
24 Example- Bearings Only Tracking N = particles true position estimated position observer Axel Gandy Particle Filtering 22
25 Example- Stochastic Volatility Bootstrap Particle Filter with N = 1000 xtrue true volatility mean posterior volatility 95% credible interval t Axel Gandy Particle Filtering 23
26 Example: Football Data: Premier League 2007/08 x t,j strength of the jth team at time t, j = 1,..., 20 y t result of the games on date t Note: not time-homogeneous (different teams playing one-another - different time intervals between games). Model: Initial distribution of the strength: xt,j N(0, 1) Evolution of strength: xt,j N((1 /β) 1/2 x t,j, /β) will use β = 50 Result of games conditional on strength: Match between team H of strength x H (at home) against team A of strength x A. Goals scored by the home team Poisson(λ H exp(x H x A )) Goals scored by the away team Poisson(λ A exp(x A x H )) λ H and λ A constants chosen based on the average number of goals scored at home/away. Axel Gandy Particle Filtering 24
27 Mean Team Strength Arsenal Aston Villa Birmingham Blackburn Bolton Chelsea Derby Everton Fulham Liverpool t Man City Man United Middlesbrough Newcastle Portsmouth Reading Sunderland Tottenham West Ham Wigan t (based on N = particle)
28 League Table at the end of 2007/08 1 Man Utd 87 2 Chelsea 85 3 Arsenal 83 4 Liverpool 76 5 Everton 65 6 Aston Villa 60 7 Blackburn 58 8 Portsmouth 57 9 Manchester City West Ham Utd Tottenham Newcastle Middlesbrough Wigan Athletic Sunderland Bolton Fulham Reading Birmingham Derby County 11
29 Influence of the Number N of Particles N=100 Arsenal Aston Villa Birmingham Blackburn Bolton Chelsea Derby Everton Fulham Liverpool N=1000 Arsenal Aston Villa Birmingham Blackburn Bolton Chelsea Derby Everton Fulham Liverpool N=10000 Arsenal Aston Villa Birmingham Blackburn Bolton Chelsea Derby Everton Fulham Liverpool N= Arsenal Aston Villa Birmingham Blackburn Bolton Chelsea Derby Everton Fulham Liverpool
30 Theoretical Results Convergence results are as N Laws of Large Numbers Central limit theorems see e.g. Chopin (2004) The central limit theorems yield an asymptotic variance. This asymptotic variance can be used for theoretical comparisons of algorithms. Axel Gandy Particle Filtering 28
31 Outline Introduction Particle Filtering Improving the Algorithm General Proposal Distribution Improving Resampling Further Topics Summary Axel Gandy Particle Filtering 29
32 General Proposal Distribution Algorithm with a general proposal distribution h: 1. Sample x (i) 0 π(x 0 ) and set t = 1 2. Importance Sampling Step For i = 1,..., N: Sample x (i) t h t (x t x (i) t 1 ) and set x(i) 0:t = (x(i) Evaluate the importance weights w (i) t 3. Selection Step: = g(y t x(i) t 0:t 1, x(i) t ) )f ( x (i) t x (i) t 1 ). h t( x (i) t x (i) t 1 ) Resample with replacement N particles (x (i) 0:t ; i = 1,..., N) from the set { x (1) } according to the normalised importance weights 0:t,..., x(1) 0:t. w (i) t N j=1 w (j) t t:=t+1; go to step 2. Optimal proposal distribution depends on quantity to be estimated. generic choice: choose proposal to minimise the variance of the normalised weights. Axel Gandy Particle Filtering 30
33 Improving Resampling Resampling was introduced to remove particles with low weights Downside: adds variance Axel Gandy Particle Filtering 31
34 Other Types of Resampling Goal: Reduce additional variance in the resampling step Standardised weights W 1,..., W N ; N i - number of offspring of the ith element. Need E N i = W i N for all i. Want to minimise the resulting variance of the weights. Multinomial Resampling - resampling with replacement Systematic Resampling Sample U 1 U(0, 1/N). Let U i = U 1 + i 1 N, i = 2,..., N N i = {j : i 1 k=1 W k U j i k=1 W k Residual Resampling Idea: Guarantee at least Ñi = W i N offspring of the ith element; N 1,..., N n : Multinomial sample of N Ñ i items with weights W i W i Ñ i /N. Set N i = Ñ i + N i.
35 Adaptive Resampling Resampling was introduced to remove particles with low weights. However, resampling introduces additional randomness to the algorithm. Idea: Only resample when weights are too uneven. Can be assessed by computing the variance of the weights and comparing it to a threshold. Equivalently, one can compute the effective sample size (ESS): ( n ) 1 ESS = (wt i ) 2. i=1 (w 1 t,..., w n t are the normalised weights) Intuitively the effective sample size describes how many samples from the target distribution would be roughly equivalent to importance sampling with the weights w i t. Thus one could decide to resample only if ESS < k where k can be chosen e.g. as k = N/2.
36 Outline Introduction Particle Filtering Improving the Algorithm Further Topics Path Degeneracy Smoothing Parameter Estimates Summary Axel Gandy Particle Filtering 34
37 Path Degeneracy Let s N. #{x i 0:s : i = 1,..., N} 1 (as # of steps t infty) Example x 0 N(0, 1), x t+1 N(x t, 1), y t N(x t, 1). N = 100 particles. t=20 t=40 t= Axel Gandy Particle Filtering 35
38 Particle Smoothing Estimate the distribution of the state x t given all the observations y 1,..., y τ up to some late point τ > t. Intuitively, a better estimation should be possible than with filtering (where only information up to τ = t is available). Trajectory estimates tend to be smoother than those obtained by filtering. More sophisticated algorithms are needed. Axel Gandy Particle Filtering 36
39 Filtering and Parameter Estimates Recall that the model is given by π(x0 ) - the initial distribution f (xt x t 1 ) for t 1 - the transition kernel of the Markov chain g(yt x t ) for t 1 - the distribution of the observations In practical applications these distributions will not be known explicitly - they will depend on unknown parameters themselves. Two different starting points Bayesian point of view: parameters have some prior distribution Frequentist point of view: parameters are unknown constants. Examples: σ Stoch. Volatility: x t N(αx t 1, 2 σ ), x (1 α) 2 1 N(0, 2 ), (1 α) 2 y t N(0, β 2 exp(x t )) Unknown parameters: σ, β, α Football: Unknown Parameters: β, λ H, λ A. How to estimate these unknown parameters? Axel Gandy Particle Filtering 37
40 Maximum Likelihood Approach Let θ Θ contain all unknown parameters in the model. Would need marginal density p θ (y 1:t ). Can be estiamted by running the particle filter for each θ of interest and by muliplying the unnormalised weights, see the slides Sequential Importance Sampling I/II earlier in the lecture for some intuition. Axel Gandy Particle Filtering 38
41 Artificial(?) Random Walk Dynamics Allow the parameters to change with time - give them some dynamic. More precisely: Suppose we have a parameter vector θ Allow it to depend on time (θt ), assign a dynamic to it, i.e. a prior distribution and some transition probability from θ t 1 to θ t incorporate θ t in the state vector x t May be reasonable in the Stoch. Volatility and Football Example Axel Gandy Particle Filtering 39
42 Bayesian Parameters Prior on θ Want: Posterior p(θ y 1,..., y t ). Naive Approach: Incorporate θ into xt ; transition for these components is just the identity Resampling will lead to θ degenerating - after a moderate number of steps only few (or even one) θ will be left New approaches: Particle filter within an MCMC algorithm, Andrieu et al. (2010) - computationally very expensive. SMC 2 by Chopin, Jacob, Papaspiliopoulos, arxiv: Axel Gandy Particle Filtering 40
43 Outline Introduction Particle Filtering Improving the Algorithm Further Topics Summary Concluding Remarks Axel Gandy Particle Filtering 41
44 Concluding Remarks Active research area A collection of references/resources regarding SMC http: // Collection of application-oriented articles: Doucet et al. (2001) Brief introduction to SMC: (Robert & Casella, 2004, Chapter 14) R-package implementing several methods: pomp Axel Gandy Particle Filtering 42
45 References Part I Appendix Axel Gandy Particle Filtering 43
46 References References I Andrieu, C., Doucet, A. & Holenstein, R. (2010). Particle markov chain monte carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 72, Chopin, N. (2004). Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist. 32, Doucet, A., Freitas, N. D. & Gordon, N. (eds.) (2001). Sequential Monte Carlo Methods in Practice. Springer. Doucet, A. & Johansen, A. M. (2008). A tutorial on particle filtering and smoothing: Fifteen years later. Available at Gordon, N., Salmond, D. & Smith, A. (1993). Novel approach to nonlinear/non-gaussian bayesian state estimation. Radar and Signal Processing, IEE Proceedings F 140, Kalman, R. (1960). A new approach to linear filtering and prediction problems. Journal of Basic Engineering 82, Axel Gandy Particle Filtering 44
47 References References II Øksendal, B. (2003). Stochastic Differential Equations: An Introduction With Applications. Springer. Robert, C. & Casella, G. (2004). Monte Carlo Statistical Methods. Second ed., Springer. Axel Gandy Particle Filtering 45
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