An Introduction to Sequential Monte Carlo for Filtering and Smoothing
|
|
- Sabrina Haynes
- 6 years ago
- Views:
Transcription
1 An Introduction to Sequential Monte Carlo for Filtering and Smoothing Olivier Cappé LTCI, TELECOM ParisTech & CNRS cappe/ Acknowlegdment: Eric Moulines (TELECOM ParisTech) & Tobias Rydén (Lund)
2 Bayesian Dynamic Models Bayesian Dynamic Models Hidden Markov Models and State-Space Models Extensions 2 The Filtering and Smoothing Recursions 3 Sequential Importance Sampling 4 Sequential Importance Sampling with Resampling 5 The Auxiliary Particle Filter 6 Smoothing 7 Mixture Kalman Filter
3 Bayesian Dynamic Models Hidden Markov Models and State-Space Models Hidden Markov Model (HMM) The Hidden State Process {X k } k 0 is a Markov chain with initial probability density function (pdf) t 0 (x) and transition density function t(x, x ) such that * k p(x 0:k ) = t 0 (x 0 ) t(x l, x l+ ). The Observed Process {Y k } k 0 is such that the conditional joint density of y 0:k given x 0:k has the conditional independence (product) form p(y 0:k x 0:k ) = l=0 k l(x l, y l ). l=0 * x 0:k denotes the collection x 0,..., x k.
4 Bayesian Dynamic Models Hidden Markov Models and State-Space Models Graphical Representation of the Dependence Structure The HMM can be represented pictorially by a Bayesian network which depicts the conditional independence relations: X k X k+ Y k Y k+
5 Bayesian Dynamic Models Hidden Markov Models and State-Space Models State-Space Form Here the model is described in a functional form: X k+ = a(x k, U k ), Y k = b(x k, V k ), where {U k } k 0 and {V k } k 0 are mutually independent i.i.d. sequences of random variables (also independent of X 0 ). Remark The term state-space model often refers to the case where a and b are linear functions of their arguments (and {U k }, {V k }, X 0 are jointly Gaussian). Likewise, the term HMM is sometimes used (not in this talk!) more restrictively for the case where X is a finite set.
6 Bayesian Dynamic Models Hidden Markov Models and State-Space Models HMM Examples source coding ion channel modelling ergodic tracking, vision quantitative finance finite state space continuous state space speech recognition handwritting recognition left-right trajectory models
7 Bayesian Dynamic Models Extensions Beyond HMMs For sequential Monte Carlo methods, the key point is the structure of the conditional p(x 0:k y 0:k ): the methods described in this talk directly apply in cases where the conditional may be factored as k p(x 0:k y 0:k ) = p(x 0 y 0 ) p(x l+ x l, y 0:l+ ) l=0 Example: Switching Autoregressive Model If the observation equation is replaced by Y k = b(x k )Y k + V k then p(x 0:k y 0:k ) = p(x 0 y 0 ) k l=0 p(x l+ x l, y l+, y l )
8 Bayesian Dynamic Models Extensions Hierarchical HMMs C k C k+ Z k Z k+ Y k Y k+ Bayesian network for a hierarchical HMM: The state sequence {X k } is composed of two chains {C k } and {Z k } that evolve in parallel and the observation Y k depends on both component of the chain.
9 Bayesian Dynamic Models Extensions In hierarchical HMMs, sequential Monte Carlo is applicable to the marginalized posterior p(c 0:k Y 0:k ) (despite the fact that it doesn t factorizes as previously requested due to the marginalization of Z 0:k ) when p(z k c 0:k, y 0:k ) is computable Conditionally Gaussian Linear State-Space Model Dynamic equation Observation equation Z k+ = A(C k+ )Z k + R(C k+ )U k Y k = B(C k )Z k + S(C k )V k where {C k } is a finite-valued Markov Chain.
10 The Filtering and Smoothing Recursions Bayesian Dynamic Models 2 The Filtering and Smoothing Recursions Basic Recursions Computational Filtering and Smoothing Approaches 3 Sequential Importance Sampling 4 Sequential Importance Sampling with Resampling 5 The Auxiliary Particle Filter 6 Smoothing 7 Mixture Kalman Filter
11 The Filtering and Smoothing Recursions Basic Recursions Tasks of interest for HMMs State Inference How to make probabilistic statements on the state sequence given the model and the observations? Filtering π k k (x k ) = p(x k Y 0:k ) Prediction π k+ k (x k+ ) = p(x k+ Y 0:k ) Smoothing π 0:k k (x 0:k ) = p(x 0:k Y 0:k ) (fixed-interval: π l k for l = 0,..., k; fixed-lag: π k k+ for k = 0,... ) Parameter Inference How to tune the model parameters based on the observations?
12 The Filtering and Smoothing Recursions Basic Recursions Recursive Structure of the Joint Smoothing Density By Bayes rule π 0:k+ k+ (x 0:k+ ) = (L k+ (Y 0:k+ )) t 0 (x 0 ) k l=0 k+ t(x l, x l+ ) l(x l, Y l ) ( ) Lk+ (Y 0:k+ ) = π L k (Y 0:k ) 0:k k(x 0:k) t(x k, x k+)l(x k+, Y k+), where the normalization constants L k, i.e., the likelihood of the observations, is usually not computable. l=0
13 The Filtering and Smoothing Recursions Basic Recursions The Joint Smoothing Recursion π 0:k+ k+ (x 0:k+ ) = ( ) Lk+ L k π 0:k k (x 0:k ) t(x k, x k+ )l(x k+, Y k+ ) The marginal recursion may be decomposed in two steps: Prediction π k+ k (x k+ ) = π k k (x k )t(x k, x k+ )dx k Filtering π k+ k+ (x k+ ) = ( Lk+ L k ) π k+ k(x k+)l(x k+, Y k+)
14 The Filtering and Smoothing Recursions Computational Filtering and Smoothing Approaches Exact Implementation of the Filtering and Smoothing Recursions When X is finite (Baum et al., 970) The associated computational cost is X 2 per time index (for the filtering part). In linear Gaussian state-space models (Kalman & Bucy, 96) The filtering and prediction recursion is implemented by the Kalman filter (L k+ /L k is interpreted as the likelihood of the (k + )-th innovation). Such finite dimensional filters exist only in very specific models (see, e.g., Runggaldier & Spizzichino, 200)
15 The Filtering and Smoothing Recursions Computational Filtering and Smoothing Approaches Approximate Implementations of the Filtering and Smoothing Recursions EKF (Extended Kalman Filter) Linearization-based approach (for non-linear Gaussian state space models) UKF (Unscented Kalman Filter, Julier & Uhlmann, 997) Point-based approach Variational Methods (e.g., Valpola & Karhunen, 2002) Based on parametric density approximation arguments. Exact Suboptimal Filters In particular, Kalman filter viewed as minimum mean square error linear filtering.
16 The Filtering and Smoothing Recursions Computational Filtering and Smoothing Approaches Sequential Monte Carlo (SMC) Sequential Monte Carlo (sometimes called particle filtering) is a method which uses pseudo-random simulations to produce successive populations of weighted particles Xk :n and associated weights Wk :n such that n Wk i f(xi k ) i= for all functions f of interest. f(x)π k k (x)dx, The SMC process is sequential in the sense that given Xk :n, Wk :n and the observations Y 0:k+, Xk+ :n and W :n k+ are conditionally independent of previous populations of particles. SMC is based on importance sampling and resampling.
17 Sequential Importance Sampling Bayesian Dynamic Models 2 The Filtering and Smoothing Recursions 3 Sequential Importance Sampling Self-Normalized Importance Sampling Sequential Importance Sampling (SIS) Weight Degeneracy 4 Sequential Importance Sampling with Resampling 5 The Auxiliary Particle Filter 6 Smoothing 7 Mixture Kalman Filter
18 Sequential Importance Sampling Self-Normalized Importance Sampling Self-Normalized Importance Sampling, or IS (Hammersley & Handscomb, 964) IS is a weighted form of Monte Carlo approximation, in which expectations under a target pdf π π(f) = E π [f(x)] are estimated as where X i iid q, ω i = π q (Xi ). ˆπ n q (f) = n ω i n f(x i ), j= }{{ ωj } W i i= This form of IS (sometimes also called Bayesian IS) does not necessitate that π be properly normalized.
19 Sequential Importance Sampling Self-Normalized Importance Sampling Performance of IS Assuming that E π [ π q (X)( + f 2 (X))] <, ˆπ q n (f) is consistent and asymptotically normal, with asymptotic variance given by [ ] π υ q (f) = E π q (X) (f(x) π(f))2. The asymptotic variance can be estimated from the IS sample by ˆυ n q (f) = n n (W i ) 2 {f(x i ) ˆπ q n (f)} 2, i= where W i = ω i / n j= ωj are the normalized weights.
20 Sequential Importance Sampling Self-Normalized Importance Sampling Elements of proof Consistency If π = π/c, where c = π(x)dx, is a pdf, E q [ω i f(x i )] = E q [ π q (X)f(X) ] = c E π [f(x)]. CLT n(ˆπ n q (f) π(f)) = n n i= ωi (f(x i ) π(f)) n n j= ωj and Var[ω i (f(x i ) π(f))] = c 2 E π [ π q (X)(f(X) π(f))2 ]. Empirical estimate ˆυ n q (f) = n i= W i n π q (Xi ) n j= π q (Xj ) {f(xi ) ˆπ q n (f)} 2
21 Sequential Importance Sampling Self-Normalized Importance Sampling Empirical diagnostic tools for IS Effective sample size ESS n q = [ n i= ( W i ) ] 2 ESS n q n 2 n/ ESS n q is an estimate of E π [ π q (X)] which is the maximal IS asymptotic variance for functions f such that f(x) π(f) (note: these functions have maximal Monte Carlo variance of under π ). 3 n/ ESS n q is an estimator of the χ 2 divergence (π(x) q(x)) 2 dx q(x) (n/ ESS n q is also the square of the empirical coefficient of variation associated with the normalized weights).
22 Sequential Importance Sampling Self-Normalized Importance Sampling Another Empirical diagnostic tools for IS (Shannon) Entropy of the Normalized Weights N ENT n q = W i log(w i ) i= exp(ent n q ) n 2 exp(ent n q )/n is an estimate of exp[ K(π q)], where K(π q) is the Kullback-Leibler divergence between π and q.
23 Sequential Importance Sampling Self-Normalized Importance Sampling Optimal IS Instrumental Density When considering large classes of functions f, IS generally appears to perform worse than Monte Carlo under π (e.g., for functions such that f(x) π(f), the maximal Monte Carlo asymptotic variance is whereas, the maximal IS asymptotic variance is E π [ π q (X)] which is strictly larger than by Jensen s inequality, unless q = π). However, for a fixed function f, the optimal IS density is not π but f(x) π(f) π(x) f(x ) π(f) π(x )dx as Cauchy-Schwarz inequality implies that E π [ π q (X) (f(x) π(f))2 ] E π [ q π (X) ] } {{ } = E 2 π [ f(x) π(f) ].
24 Sequential Importance Sampling Self-Normalized Importance Sampling Example (Bayesian Posterior) Consider a simple model in which X 0 N(, 2 2 ), Y 0 X 0 N(X0, 2 σ 2 ). And apply IS to compute expectations under the posterior for Y 0 = 2, using the prior as instrumental pdf.
25 Sequential Importance Sampling Self-Normalized Importance Sampling Instrumental Target.4 Instrumental Target σ = 2 ESS n q /n 0.66 υ q (x x).69 υ π (x x).25 σ = 0.5 ESS n q /n 0.25 υ q (x x) 5.35 υ π (x x).25
26 Sequential Importance Sampling Sequential Importance Sampling (SIS) Back to the Filtering and Smoothing Problem How to estimate expectations under the posterior π 0:k k (x 0:k ) = p(x 0:k Y 0:k ) in the model k p(x 0:k ) = t 0 (x 0 ) t(x l, x l+ ), p(y 0:k x 0:k ) = using a sequential algorithm? l=0 k l(x l, y l ), l=0
27 Sequential Importance Sampling Sequential Importance Sampling (SIS) Sequential Smoothing through IS, or SIS (Handschin & Mayne, ) Propose n independent particle trajectories {X i 0:k+ } i n under a Markovian scheme such that p(x 0:k+ ) = ρ 0:k+ (x 0:k+ ) = q 0 (x 0 ) 2 Compute importance weights sequentially: k q l (x l, x l+ ). l= ω i k+ = π 0:k+ k+(x i 0:k+ ) ρ 0:k+ (X i 0:k+ ) = ω i k t(xi k, Xi k+ )l(xi k+, Y k+) q k (X i k, Xi k+ ). Then, n i= ω i k+ n j= ωj k+ f(x i 0:k+ ) is an estimate of E [f(x 0:k+ ) Y 0:k+ ].
28 Sequential Importance Sampling Sequential Importance Sampling (SIS) FILT. INSTR. FILT. + One step of the SIS algorithm with just seven particles.
29 Sequential Importance Sampling Sequential Importance Sampling (SIS) Choice of the Instrumental Kernel The so-called optimal choice of q k, consists in setting q k (x, x ) = q opt k (x, x ) = t(x, x )l(x, Y k+ ) t(x, x )l(x, Y k+ )dx for which the normalized weights are predictable, i.e. Wk+ i depend on Xk i but not Xi k+ (hence, for this instrumental kernel the conditional variance cancels). This is however usually not feasible and common choices include the prior q k = t (and then ωk i l(xi k, Y k)) 2 approximations (sometimes heuristic) to q opt k matching, use of EKF or UKF,... ), (moment 3 tuning parameters of q k so as to maximize the effective sample size or entropy criterions
30 Sequential Importance Sampling Weight Degeneracy Weight Degeneracy Empirically, the SIS approach always fail when the time-horizon k is more than a few tens; the IS weights ωk :n usually become very unbalanced with a few weights dominating all the other To understand why it is the case, consider the (silly) model where { t(x, x ) = t(x ) = t 0 (x ), (Independent states) l(x, y) = l(y), (Non-informative observations) and the instrumental kernel is such that q l (x, x ) = q 0 (x ) = q(x )
31 Sequential Importance Sampling Weight Degeneracy Weight Degeneracy (Contd.) For a function of interest f that only depends on the last coordinate x k of the trajectory x 0:k, the asymptotic variance of the SIS approximation to π k k (f) = E πk k [f(x)] is given by υ k (f) = ( k = l=0 ) 2 t (f(xk q (x l) ) π ) 2 k k k(f) l=0 q(x l ) dx 0... dx k ( t ) k t q (x)t(x)dx q (x) ( f(x) π k k (f) ) 2 t(x)dx. }{{} > In practise, this situation can usually be detected by monitoring the effective sample size or entropy criterions, which become abnormally small.
32 Sequential Importance Sampling Weight Degeneracy Application to the Stochastic Volatility Model This is a non-linearly observed state-space model used to represent log-returns in quantitative finance: where X k+ = φx k + σu k φ <, Y k = β exp(x k /2)V k, {U k } k 0 and {V k } k 0 are independent standard Gaussian white noise processes. X 0 N(0, σ 2 /( φ 2 ). We consider trajectories of length 50 simulated under the model with φ = 0.98, σ = 0.7 and β = 0.64 and apply SIS with q k = q and n =, 000 particles.
33 Sequential Importance Sampling Weight Degeneracy Typical Evolution of ESS for a Single Trajectory 000 ESS (500 indep. replications of the particles) time index
34 Sequential Importance Sampling Weight Degeneracy Evolution of ESS When Averaging Over Trajectories ESS (500 indep. simulations) time index
35 Sequential Importance Sampling Weight Degeneracy Typical Evolution of the Weight Distribution Importance Weights (base 0 logarithm) Histograms of the base 0 logarithm of the normalized importance weights after (from top to bottom), 0 and 00 iterations for the stochastic volatility model (improved instrumental kernel based on moment matching).
36 Sequential Importance Sampling Weight Degeneracy Filtering Results 0.08 Density Time Index State Waterfall representation of the sequence of estimated filtering pdfs (weighted kernel smooth) together with the actual state for,000 particles (improved instrumental kernel based on moment matching).
37 Sequential Importance Sampling with Resampling Bayesian Dynamic Models 2 The Filtering and Smoothing Recursions 3 Sequential Importance Sampling 4 Sequential Importance Sampling with Resampling Sampling Importance Resampling Sequential Importance Sampling with Resampling (SISR) Case Study 5 The Auxiliary Particle Filter 6 Smoothing 7 Mixture Kalman Filter
38 Sequential Importance Sampling with Resampling Sampling Importance Resampling In IS, it is indeed possible to reset the weights to a constant value at the price of a, usually moderate, increase in variance. Sampling Importance Resampling (Rubin, 987) Replace {X :n, W :n } by { X :Ñ, W :Ñ} such that the discrepancy between the resampled weights { W :Ñ} is reduced and Ñ W i= k iδ is a good approximation to n Xi i= W i δ X i. In general the resampling is random and subject to the constraints Ñ = n, W [ i = /Ñ, Ñ E i= { X ] i = X j } X :n, W :n = ÑW j ( j n). The last condition is often referred to as unbiasedness or proper weighting.
39 Sequential Importance Sampling with Resampling Sampling Importance Resampling Multinomial Resampling Draw, conditionally independently given {X :n, W :n }, n discrete random variables (J,..., J n ) taking their values in the set {,..., n} with probabilities (W,..., W n ). 2 Set, for i =,..., n, Xi = X J i and W i = /n. Let C i = n j= { X j = X i } (i =,..., n) denote the number of times each particle is duplicated in the resampling process. The counts (C,..., C n ) follow a multinomial distribution with parameters n, (W,..., W n ), conditionally to {X :n, W :n }. INSTRUMENTAL Resampled particles TARGET
40 Sequential Importance Sampling with Resampling Sampling Importance Resampling Some Results on SIR D Xi π are n (some extensions of this result) n 2 n i= f( X i ) is an asymptotically normal estimator of π(f) (assuming E π [ π q (X)( + f 2 (X)) + f 2 (X)] < ) with asymptotic variance given by [ ] π [ υ q (f) = E π q (X) (f(x) π(f))2 + E π (f(x) π(f)) 2] }{{}}{{} υ Var q(f) π[f(x)] If n is sufficiently large, the cost of resampling is very moderate in situation that are challenging for IS, i.e., when υ q (f) Var π [f(x)].
41 Sequential Importance Sampling with Resampling Sampling Importance Resampling Elements of Proof For a bounded function f, [ E f( X ] [ ( i ) = E E f( X i ) X :n, W :n)] n = E W j f(x j ) j= and n j= W j f(x j ) a.s. π(f) ( n j= W j f(x j ) f ). For the CLT, an ingredient of the proof is that [ ] [ n n ] n Var f( n X i ) = n Var W i f(x i ) i= i= }{{} υ q(f) [ n ( n + E W i f 2 (X i ) W i f(x i ) i= i= ) 2 } {{ } a.s. Var π[f(x)] ]
42 Sequential Importance Sampling with Resampling Sampling Importance Resampling There Exists Resampling Schemes with Reduced Variance Residual Resampling For i =,..., n set C i = nw i + C i, where C,..., C n are distributed, conditionally to W :n, according to the multinomial distribution Mult(n R, W,..., W n ) with R = n i= nw i and W i = nw i nw i, i =,..., n. n R This scheme is obviously unbiased and induces an additional variance which is always lower than Var π [f] (the same is true for the bootstrap filter based on residual resampling, Douc & Moulines, 2005).
43 Sequential Importance Sampling with Resampling Sequential Importance Sampling with Resampling (SISR) The Simplest Functional Algorithm (Gordon et al., 993) Regular resampling is added to avoid weight degeneracy and to guarantee the long-term (k ) stability of the particle filter. The Bootstrap filter X :n k Given, propose new positions Xi k+ independently under the prior dynamic t( X k i, ), for i =,..., n; 2 Compute the weights ωk+ i = l(xi k+, Y k+), for i =,..., n and normalize them (Wk+ i = ωi k+ / n 3 Resample to obtain X :n k+ indices J i k+ such that P ( J i k+ = j W :n k+ setting X i k+ = XJ i k+ k+ j= ωj k+ );, e.g., by drawing independent ) = W j k+ and (Multinomial Resampling). To avoid unnecessary resampling, 3 can be used only when the ESS statistics of the weights Wk+ :n falls below a threshold.
44 Sequential Importance Sampling with Resampling Sequential Importance Sampling with Resampling (SISR) FILT. FILT. FILT. + FILT. + FILT. +2 FILT. +2 SIS (left) and SISR (right).
45 Sequential Importance Sampling with Resampling Case Study AR() Model Observed in Pulsated Noise We consider a simple Gaussian linear state-space model to allow for comparison with analytical computations: X k+ = φx k + σu k, Y k = X k + η k V k, where φ = 0.99, σ = 0.2, and η k is varied (periodically) between 0. and 3. The observation sequence is simulated from the model but we also consider the influence of an out-of-model outlying observation at time 460.
46 Sequential Importance Sampling with Resampling Case Study 0 5 state obs. filt. smooth time 6 state obs. filt. smooth. 0 8 state obs. filt. smooth time time
47 Sequential Importance Sampling with Resampling Case Study Sequential Importance Sampling (prior kernel, n = 00) 00 ESS filt. mean filt. std time
48 Sequential Importance Sampling with Resampling Case Study Bootstrap Filter (prior kernel, n = 00, resamp. ESS < 0) ESS filt. mean filt. std time
49 Sequential Importance Sampling with Resampling Case Study Bootstrap Filter (prior kernel, n = 00, resamp. always) filt. mean filt. std time
50 Sequential Importance Sampling with Resampling Case Study Bootstrap Filter (prior kernel, n = 00, resamp. ESS < 0) ESS filt. mean filt. std time
51 Sequential Importance Sampling with Resampling Case Study Bootstrap Filter (n = 5, 000, resamp. ESS < 500) ESS filt. mean filt. std time
52 Sequential Importance Sampling with Resampling Case Study Conclusions With resampling, SISR achieves long-term stability. The increase in variance due to resampling is very moderate, especially when resampling is applied only when needed. The method is still sensitive to outliers, model misspecification, etc., which may necessitate the use of more elaborate instrumental kernels.
53 The Auxiliary Particle Filter Bayesian Dynamic Models 2 The Filtering and Smoothing Recursions 3 Sequential Importance Sampling 4 Sequential Importance Sampling with Resampling 5 The Auxiliary Particle Filter A New Interpretation of SISR The Auxiliary Trick 6 Smoothing 7 Mixture Kalman Filter
54 The Auxiliary Particle Filter A New Interpretation of SISR Alternatives to SISR The resampling step in the SISR algorithm can be seen as a method to sample approximately under the distribution obtained when plugging the current particle approximation into the filtering update. This alternative way of thinking about resampling suggests several sequential Monte Carlo variants.
55 The Auxiliary Particle Filter A New Interpretation of SISR The Filtering Recursion Revisited Recall that (with more general notations) π k+ k+ (f) = f(x )π k k (dx)t(x, dx )l(x, Y k+ ) πk k (dx)t(x, dx )l(x, Y k+ ). Now, consider what happens when considering the empirical filtering distribution ˆπ n k k = n Wk i δ Xk i i= as an approximation to π k k and plugging it into the previous relation.
56 The Auxiliary Particle Filter A New Interpretation of SISR Filtering Target One obtains a mixture target pdf π n k+ k+ (x) = = n i= W k it(xi k, x)l(x, Y k+) W i k t(xk i, x)l(x, Y k+)dx n i= n i= W i k γ k(x i k ) n j= W j k γ k(x j k ) qopt k (Xi k, x), where γ k (x) = t(x, dx )l(x, Y k+ ), q opt k (x, x ) = t(x, x )l(x, Y k+ ). γ k (x)
57 The Auxiliary Particle Filter A New Interpretation of SISR Filtering Step 0.2 Approximation Prediction likelihood 0.3 Prediction likelihood 0.2 predictive distribution 0.2 predictive distribution Filtering Filtering
58 The Auxiliary Particle Filter A New Interpretation of SISR The previous reinterpretation of SISR shows that resampling and trajectory update can be integrated into a single global IS step that targets π n k+ k+. But, how to chose the global instrumental kernel q glob k ((Xk :n, W k :n ), x)? as πk+ k+ n is an n-mixture density, evaluation of the IS weights may necessitate of the order of n 2 computations. Remark The global instrumental kernel that corresponds to the bootstrap filter (with resampling) is q glob k ((X :n k, W :n k ), x) = n Wk i t(xi k, x). i=
59 The Auxiliary Particle Filter The Auxiliary Trick The Auxiliary Trick Assume that the global instrumental kernel is a mixture of the form q glob k ((X :n k, W :n k ), x) = n τk i q k(xk i, x). To avoid the n 2 update, one can use data augmentation, introducing the mixture component as an auxiliary variable: i= q aux k ((X :n k, W :n k ), (i, x)) = τ i k q k(x i k, x) defines a pdf on {,..., n} X, whose marginal is q k ((Xk :n, W k :n ), x). Likewise, π n,aux k+ k+ (i, x) = c W i k t(xi k, x)l(x, Y k+), where c = n i= W i k t(x i k, x)l(x, Y k+)dx, is the auxiliary target with marginal π n k+ k+ (x).
60 The Auxiliary Particle Filter The Auxiliary Trick Auxiliary Sampling Algorithm (Pitt & Shephard, 999) Auxiliary Particle Filter Repeat n times independently, for i =,..., n, Sample an index J i in {,..., n} with probabilities (τ k,..., τ n k ). 2 Sample a position X i k+ from q k(x J i k, x). 3 Compute the (unnormalized) auxiliary IS weight ωk+ i = W J i k t(xj i k, Xi k+ )l(xi k+, Y k+) τ J i k q k(x J i k, Xi k+ ).
61 The Auxiliary Particle Filter The Auxiliary Trick The Auxiliary View Provides New Degrees of Freedom In the original auxiliary particle filter, Pitt & Shephard used qk i = t and proposed an heuristic rule for setting the weights τk i based on Xk i and Y k+. The auxiliary IS weight may also be rewritten in normalized form as ωk+ i = W J i k γ k(x J i k )qopt k (XJ i k, Xi k+ ) τ J i k t(xj i k, Xi k+ ). showing that the optimal choice of τk i, in the sense of minimizing the conditional (to Xk :n, W k :n ) variance of n i= W k+ i f(xi k+ ) for a function f, is an instance of the Neyman allocation problem. And thus the optimal choice is τ i k W i k γ k(x i k ) υ i (f) where υ i (f) is the asymptotic variance of IS for f, target q opt k (Xi k, x), and, instrumental pdf t(xi k, x) (Olsson et al., 2007).
62 Smoothing Bayesian Dynamic Models 2 The Filtering and Smoothing Recursions 3 Sequential Importance Sampling 4 Sequential Importance Sampling with Resampling 5 The Auxiliary Particle Filter 6 Smoothing Using the Ancestry Tree Backward Reweighting Smoothing for Sum Functionals 7 Mixture Kalman Filter
63 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
64 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
65 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
66 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
67 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
68 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
69 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
70 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
71 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
72 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
73 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
74 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
75 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
76 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
77 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
78 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
79 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
80 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
81 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
82 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
83 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
84 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
85 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
86 Smoothing Using the Ancestry Tree state time index state time index Predictive densities and evolution of the particle ancestry tree
87 Smoothing Backward Reweighting Backward Smoothing Recursion There are several options for computing the marginal smoothing pdfs π l k such that π l k (x) = p(x l Y 0:k ) for l = 0,..., k. The backward smoothing recursion is based on the observation that the conditional time-reversed state sequence has a non-homogeneous Markovian structure (for conditionally Gaussian linear state-space models, this is known as RTS, or Rauch-Tung-Striebel, 965, smoothing).
88 Smoothing Backward Reweighting Backward Smoothing Recursion Define the backward smoothing kernels: b l (x l+, x l ) = π l l(x l )t(x l, x l+ ) πl l (x)t(x, x l+ )dx, for l = k, k 2,..., 0. Then π l k (x l ) = π l+ k (x l+ )b l (x l+, x l )dx l+. As p(x l, x l+ Y 0:k ) = p(x l x l+, Y 0:k ) p(x }{{} l+ Y 0:k ) }{{} p(x l x l+, Y 0:l ) π l+ k (x l+ ) }{{} b l (x l+,x l )
89 Smoothing Backward Reweighting Particle Reweighting Scheme The natural approximation to the backward smoothing recursion consists in computing smoothing weights Wl k i backwards according to the recursion Wk k i = W k i, for i =,..., n. For l = k, k 2,..., 0, W i l k = for i =,..., n. n W j l+ k j= W i l t(xi l, Xj l+ ) n i = W i l t(x i l, Xj l+ ), The approximation to E[f(X l ) Y 0:k ] is given by n i= W i l k f(xi l ).
90 Smoothing Backward Reweighting Back to Our Case Study AR() Model Observed in Pulsated Noise 0 8 state obs. filt. smooth time
91 Smoothing Backward Reweighting Using the Ancestry Tree (n = 00) 4 smooth. mean smooth. std time
92 Smoothing Backward Reweighting With Backward Reweighting (n = 00) 4 smooth. mean smooth. std time
93 Smoothing Backward Reweighting Backward Particle Reweighting Appears to be efficient and stable in the long term (although this hasn t been proved yet). Yet, it is not sequential (in particular, one needs to store all particle positions and weights); it has a potential numerical complexity proportional to the number n of particles squared (but not further likelihood evaluation is needed).
94 Smoothing Smoothing for Sum Functionals Sum Functionals and Parameter Estimation For parameter estimation, one requires smoothing of particular sum functionals of the hidden states. In the example of the stochastic volatility model, one needs to evaluate E [s i (X 0:n ) Y 0:n ], for 0 i 4 with n s 0 (x 0:n ) = x 2 0, s (x 0:n ) = x 2 k, s 2(x 0:n ) = s 3 (x 0:n ) = k=0 n x k x k, s 4 (x 0:n ) = k= n k=0 n x 2 k, k= Y 2 k exp( x k). We consider the simple case of s 0 (X 0 ) using N i= W n i { X i 0:n (0) } 2 as the sequential Monte Carlo estimate of E (s 0 (X 0 ) Y 0:n ) (for sum functionals, the same applies for each term in the sum).
95 Smoothing Smoothing for Sum Functionals Smoothing for s 0 (X 0 ) 0 2 Particles 0 3 Particles 0 4 Particles time time time Box and whisker plots of particle estimates of R x 2 π 0 k (dx) for k =, 5, 20, 30 and 500, and particle population sizes n = 0 2, 0 3 and 0 4.
96 Smoothing Smoothing for Sum Functionals Smoothing for s 0 (X 0 ), Contd. 2.5 State Value Time Index Particle trajectories at time n = 70 for the stochastic volatility model with n = 00 particles and systematic resampling. Using k < n is beneficial! Properly setting k corresponds to a bias-variance tradeoff: k bias decreases, k variance decreases. Hence, the general principle for sequential smoothing of sum functionals: use fixed-lag smoothing (Olsson et al., 2008).
97 Mixture Kalman Filter Bayesian Dynamic Models 2 The Filtering and Smoothing Recursions 3 Sequential Importance Sampling 4 Sequential Importance Sampling with Resampling 5 The Auxiliary Particle Filter 6 Smoothing 7 Mixture Kalman Filter The Mixture Kalman Filter (MKF) Algorithm An Application to Change Point Detection
98 Mixture Kalman Filter The Mixture Kalman Filter (MKF) Algorithm Filtering in Conditionally Gaussian Linear State-Space Models There are several cases of interest where the particles are more complicated than just elements of the state-space X. Conditionally Gaussian Linear State-Space Model Dynamic equation Observation equation Z k = A(C k )Z k + R(C k )U k Y k = B(C k )Z k + S(C k )V k where {C k } is itself a finite-valued Markov Chain. Many applications: Non-Gaussian noises modelled as mixtures, switching models, applications in digital communications, etc.
99 Mixture Kalman Filter The Mixture Kalman Filter (MKF) Algorithm The Exact Filtering Recursions Given C 0:k, p(z k Y 0:k, C 0:k ) is a Gaussian distribution with mean vector Ẑk k(y 0:k, C 0:k ) and covariance matrix Σ k k (Y 0:k, C 0:k ), which can be determined recursively using the Kalman filter. Hence p(x k Y 0:k ) is the mixture of C k+ Gaussian densities p(x k Y 0:k ) L k (Y 0:k c 0:k )p(c 0:k ) c 0:k C ( k+ ) N x k ; Ẑk k(y 0:k, c 0:k ), Σ k k (Y 0:k, c 0:k ). This is obviously of no use when k is not very small.
100 Mixture Kalman Filter The Mixture Kalman Filter (MKF) Algorithm Mixture Kalman Filtering Using (Z k, C k ) as state variable, one an use the SMC approaches described so far with (Z C)-valued particles. One can also use a related algorithm, where each particle represents a trajectory C 0:k summarized by the Kalman statistics Ẑk k(y 0:k, C 0:k ) and Σ k k (Y 0:k, C 0:k ). The resulting algorithm mixes systematic exploration of the trajectory continuations (in C), Kalman update to compute the likelihood factor and resampling.
101 Mixture Kalman Filter The Mixture Kalman Filter (MKF) Algorithm MKF: computation of the weights For i =,..., n and j =,..., r, compute Ẑ k+ k (C i 0:k, j) = A(j)Ẑk k(c i 0:k ), Σ k+ k (C i 0:k, j) = A(j)Σ k k(c i 0:k )At (j) + R(j)R t (j), Ŷ k+ k (C i 0:k, j) = B(j)Ẑk+ k(c i 0:k, j), Γ k+ (C i 0:k, j) = B(j)Σ k+ k(c i 0:k, j)bt (j) + S(j)S t (j), ω i,j k+ = W i k N(Y k+ ; Ŷk+ k(c i 0:k, j), Γ k+(c i 0:k, j)) Q C(C i k, j).
102 Mixture Kalman Filter The Mixture Kalman Filter (MKF) Algorithm MKF: Importance Sampling Step For i =,..., n, draw Jk+ i with probabilities proportional to ω i, k+,..., ωi,r k+, conditionally independently of the particle history, and set C0:k+ i = (Ci 0:k, J k+ i / ), r n Wk+ i = j= ω i,j k+ r i= j= ω i,j k+, K k+ (C i 0:k+ ) = Σ k+ k(c i 0:k, J i k+ )Bt (J i k+ )Γ k+ (Ci 0:k+, J i k+ ), Ẑ k+ k+ (C i 0:k+ ) = Ẑk+ k(c i 0:k, J i k+ ) + K k+ (C i 0:k+ ){Y k+ Ŷk+ k(c i 0:k, J i k+ )}, Σ k+ k+ (C i 0:k+ ) = {I K k+(c i 0:k+ )B(J k+)}σ k+ k (C i 0:k, J i k+ ).
103 Mixture Kalman Filter An Application to Change Point Detection Application to a Change Point Detection Task.5 x 05 x Nuclear response Nuclear response Time Time Left: well-log data waveform with a median smoothing estimate of the state. Right: median smoothing residual.
104 Mixture Kalman Filter An Application to Change Point Detection Change Point Modelling To model this situation we put C = {0, }, where C k = 0 means that there is no change point at time index k whereas C k = means that a change point has occurred. The state space model is Z k+ = A(C k+ )Z k + R(C k+ )U k, Y k = Z k + V k, where A(0) = I, R(0) = 0 and A() = 0 and R() = R. The simplest model consists in taking for {C k } k 0 an i.i.d. sequence of Bernoulli random variables with probability of success p. The time between two change points (period of time during which the state variable is constant) is then distributed as a geometric random variable with mean /p.
105 Mixture Kalman Filter An Application to Change Point Detection Outliers Modelling 6 x 04 4 Quantiles of Input Sample Standard Normal Quantiles Quantile-quantile regression of empirical quantiles of the well-log data residuals with respect to quantiles of the standard normal distribution.
106 Mixture Kalman Filter An Application to Change Point Detection Outliers Modelling, Contd. The normal distribution does not fit the measurement noise well in the tails. We model the measurement noise as a mixture of two Gaussian distributions: Z k+ = A(C k+, )Z k + R(C k+, )U k, U k N(0, ), Y k = µ(c k,2 ) + B(C k,2 )Z k + S(C k,2 )V k, V k N(0, ), where C k, {0, } and C k,2 {0, } are indicators of the presence of a change point and of an outlier, respectively. {C k,2 } is modelled as a two-state Markov chain which represents the clustering behavior of outliers.
107 Mixture Kalman Filter An Application to Change Point Detection 5 x 04 Data Posterior Probability of Jump Posterior Probability of Outlier Time On-line analysis of the well-log data, using 00 particles with detection delay d = 0. Top: data; middle: posterior probability of a jump; bottom: posterior probability of an outlier.
108 Mixture Kalman Filter An Application to Change Point Detection 5 x 04 Data Posterior Probability of Jump Posterior Probability of Outlier Time On-line analysis of the well-log data, using 00 particles with detection delay d = 5 (same display as above).
109 Conclusions There are many more important issues in SMC such as Analysis of Performance Convergence of more elaborate algorithms, less restrictive conditions for long-term stability, analysis of smoothing algorithms Resampling Variants Conditional variance reduction schemes, triggered resampling, varying number of particles... Choice of the Instrumental Moves Lookahead moves, combination with deterministic approximation, adaptive moves
110 Conclusions Some General References on SMC Doucet, A., De Freitas, N. and Gordon, N. (eds.) (200) Sequential Monte Carlo Methods in Practice. Springer. Ristic, B., Arulampalam, M. and Gordon, A. (2004) Beyond Kalman Filters: Particle Filters for Target Tracking. Artech House. Cappé, 0., Moulines, E. and Rydén, T. (2005) Inference in Hidden Markov Models. Springer. Doucet, A., Godsill, S. and Andrieu, C. (2000) On sequential Monte-Carlo sampling methods for Bayesian filtering. Stat. Comput., 0, Arulampalam, M., Maskell, S., Gordon, N. and Clapp, T. (2002) A tutorial on particle filters for on line non-linear/non-gaussian Bayesian tracking. IEEE Trans. Signal Process., 50, Cappé, O., Godsill, S. J. and Moulines, E. (2007) An overview of existing methods and recent advances in sequential Monte Carlo, IEEE Proc., 95,
cappe/
Particle Methods for Hidden Markov Models - EPFL, 7 Dec 2004 Particle Methods for Hidden Markov Models Olivier Cappé CNRS Lab. Trait. Commun. Inform. & ENST département Trait. Signal Image 46 rue Barrault,
More informationComputer Intensive Methods in Mathematical Statistics
Computer Intensive Methods in Mathematical Statistics Department of mathematics johawes@kth.se Lecture 7 Sequential Monte Carlo methods III 7 April 2017 Computer Intensive Methods (1) Plan of today s lecture
More informationSequential Monte Carlo Methods for Bayesian Computation
Sequential Monte Carlo Methods for Bayesian Computation A. Doucet Kyoto Sept. 2012 A. Doucet (MLSS Sept. 2012) Sept. 2012 1 / 136 Motivating Example 1: Generic Bayesian Model Let X be a vector parameter
More informationAuxiliary Particle Methods
Auxiliary Particle Methods Perspectives & Applications Adam M. Johansen 1 adam.johansen@bristol.ac.uk Oxford University Man Institute 29th May 2008 1 Collaborators include: Arnaud Doucet, Nick Whiteley
More informationSensor Fusion: Particle Filter
Sensor Fusion: Particle Filter By: Gordana Stojceska stojcesk@in.tum.de Outline Motivation Applications Fundamentals Tracking People Advantages and disadvantages Summary June 05 JASS '05, St.Petersburg,
More informationComputer Intensive Methods in Mathematical Statistics
Computer Intensive Methods in Mathematical Statistics Department of mathematics johawes@kth.se Lecture 16 Advanced topics in computational statistics 18 May 2017 Computer Intensive Methods (1) Plan of
More informationLecture 2: From Linear Regression to Kalman Filter and Beyond
Lecture 2: From Linear Regression to Kalman Filter and Beyond January 18, 2017 Contents 1 Batch and Recursive Estimation 2 Towards Bayesian Filtering 3 Kalman Filter and Bayesian Filtering and Smoothing
More informationL09. PARTICLE FILTERING. NA568 Mobile Robotics: Methods & Algorithms
L09. PARTICLE FILTERING NA568 Mobile Robotics: Methods & Algorithms Particle Filters Different approach to state estimation Instead of parametric description of state (and uncertainty), use a set of state
More informationA Note on Auxiliary Particle Filters
A Note on Auxiliary Particle Filters Adam M. Johansen a,, Arnaud Doucet b a Department of Mathematics, University of Bristol, UK b Departments of Statistics & Computer Science, University of British Columbia,
More informationLecture 2: From Linear Regression to Kalman Filter and Beyond
Lecture 2: From Linear Regression to Kalman Filter and Beyond Department of Biomedical Engineering and Computational Science Aalto University January 26, 2012 Contents 1 Batch and Recursive Estimation
More informationAn Brief Overview of Particle Filtering
1 An Brief Overview of Particle Filtering Adam M. Johansen a.m.johansen@warwick.ac.uk www2.warwick.ac.uk/fac/sci/statistics/staff/academic/johansen/talks/ May 11th, 2010 Warwick University Centre for Systems
More informationThe Kalman Filter ImPr Talk
The Kalman Filter ImPr Talk Ged Ridgway Centre for Medical Image Computing November, 2006 Outline What is the Kalman Filter? State Space Models Kalman Filter Overview Bayesian Updating of Estimates Kalman
More informationLecture 7: Optimal Smoothing
Department of Biomedical Engineering and Computational Science Aalto University March 17, 2011 Contents 1 What is Optimal Smoothing? 2 Bayesian Optimal Smoothing Equations 3 Rauch-Tung-Striebel Smoother
More informationLecture 6: Bayesian Inference in SDE Models
Lecture 6: Bayesian Inference in SDE Models Bayesian Filtering and Smoothing Point of View Simo Särkkä Aalto University Simo Särkkä (Aalto) Lecture 6: Bayesian Inference in SDEs 1 / 45 Contents 1 SDEs
More informationState Estimation using Moving Horizon Estimation and Particle Filtering
State Estimation using Moving Horizon Estimation and Particle Filtering James B. Rawlings Department of Chemical and Biological Engineering UW Math Probability Seminar Spring 2009 Rawlings MHE & PF 1 /
More informationAn introduction to Sequential Monte Carlo
An introduction to Sequential Monte Carlo Thang Bui Jes Frellsen Department of Engineering University of Cambridge Research and Communication Club 6 February 2014 1 Sequential Monte Carlo (SMC) methods
More informationAN EFFICIENT TWO-STAGE SAMPLING METHOD IN PARTICLE FILTER. Qi Cheng and Pascal Bondon. CNRS UMR 8506, Université Paris XI, France.
AN EFFICIENT TWO-STAGE SAMPLING METHOD IN PARTICLE FILTER Qi Cheng and Pascal Bondon CNRS UMR 8506, Université Paris XI, France. August 27, 2011 Abstract We present a modified bootstrap filter to draw
More informationIntroduction. log p θ (y k y 1:k 1 ), k=1
ESAIM: PROCEEDINGS, September 2007, Vol.19, 115-120 Christophe Andrieu & Dan Crisan, Editors DOI: 10.1051/proc:071915 PARTICLE FILTER-BASED APPROXIMATE MAXIMUM LIKELIHOOD INFERENCE ASYMPTOTICS IN STATE-SPACE
More informationPARTICLE FILTERS WITH INDEPENDENT RESAMPLING
PARTICLE FILTERS WITH INDEPENDENT RESAMPLING Roland Lamberti 1, Yohan Petetin 1, François Septier, François Desbouvries 1 (1) Samovar, Telecom Sudparis, CNRS, Université Paris-Saclay, 9 rue Charles Fourier,
More informationRao-Blackwellized Particle Filter for Multiple Target Tracking
Rao-Blackwellized Particle Filter for Multiple Target Tracking Simo Särkkä, Aki Vehtari, Jouko Lampinen Helsinki University of Technology, Finland Abstract In this article we propose a new Rao-Blackwellized
More informationThe Particle Filter. PD Dr. Rudolph Triebel Computer Vision Group. Machine Learning for Computer Vision
The Particle Filter Non-parametric implementation of Bayes filter Represents the belief (posterior) random state samples. by a set of This representation is approximate. Can represent distributions that
More informationComputer Intensive Methods in Mathematical Statistics
Computer Intensive Methods in Mathematical Statistics Department of mathematics johawes@kth.se Lecture 5 Sequential Monte Carlo methods I 31 March 2017 Computer Intensive Methods (1) Plan of today s lecture
More information13: Variational inference II
10-708: Probabilistic Graphical Models, Spring 2015 13: Variational inference II Lecturer: Eric P. Xing Scribes: Ronghuo Zheng, Zhiting Hu, Yuntian Deng 1 Introduction We started to talk about variational
More informationTSRT14: Sensor Fusion Lecture 8
TSRT14: Sensor Fusion Lecture 8 Particle filter theory Marginalized particle filter Gustaf Hendeby gustaf.hendeby@liu.se TSRT14 Lecture 8 Gustaf Hendeby Spring 2018 1 / 25 Le 8: particle filter theory,
More informationAdaptive Population Monte Carlo
Adaptive Population Monte Carlo Olivier Cappé Centre Nat. de la Recherche Scientifique & Télécom Paris 46 rue Barrault, 75634 Paris cedex 13, France http://www.tsi.enst.fr/~cappe/ Recent Advances in Monte
More informationAdvanced Computational Methods in Statistics: Lecture 5 Sequential Monte Carlo/Particle Filtering
Advanced Computational Methods in Statistics: Lecture 5 Sequential Monte Carlo/Particle Filtering Axel Gandy Department of Mathematics Imperial College London http://www2.imperial.ac.uk/~agandy London
More informationEVALUATING SYMMETRIC INFORMATION GAP BETWEEN DYNAMICAL SYSTEMS USING PARTICLE FILTER
EVALUATING SYMMETRIC INFORMATION GAP BETWEEN DYNAMICAL SYSTEMS USING PARTICLE FILTER Zhen Zhen 1, Jun Young Lee 2, and Abdus Saboor 3 1 Mingde College, Guizhou University, China zhenz2000@21cn.com 2 Department
More informationECE276A: Sensing & Estimation in Robotics Lecture 10: Gaussian Mixture and Particle Filtering
ECE276A: Sensing & Estimation in Robotics Lecture 10: Gaussian Mixture and Particle Filtering Lecturer: Nikolay Atanasov: natanasov@ucsd.edu Teaching Assistants: Siwei Guo: s9guo@eng.ucsd.edu Anwesan Pal:
More informationInferring biological dynamics Iterated filtering (IF)
Inferring biological dynamics 101 3. Iterated filtering (IF) IF originated in 2006 [6]. For plug-and-play likelihood-based inference on POMP models, there are not many alternatives. Directly estimating
More informationSelf Adaptive Particle Filter
Self Adaptive Particle Filter Alvaro Soto Pontificia Universidad Catolica de Chile Department of Computer Science Vicuna Mackenna 4860 (143), Santiago 22, Chile asoto@ing.puc.cl Abstract The particle filter
More informationSequential Monte Carlo Methods (for DSGE Models)
Sequential Monte Carlo Methods (for DSGE Models) Frank Schorfheide University of Pennsylvania, PIER, CEPR, and NBER October 23, 2017 Some References These lectures use material from our joint work: Tempered
More informationParticle Filters. Outline
Particle Filters M. Sami Fadali Professor of EE University of Nevada Outline Monte Carlo integration. Particle filter. Importance sampling. Degeneracy Resampling Example. 1 2 Monte Carlo Integration Numerical
More informationThe Unscented Particle Filter
The Unscented Particle Filter Rudolph van der Merwe (OGI) Nando de Freitas (UC Bereley) Arnaud Doucet (Cambridge University) Eric Wan (OGI) Outline Optimal Estimation & Filtering Optimal Recursive Bayesian
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 13: SEQUENTIAL DATA
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 13: SEQUENTIAL DATA Contents in latter part Linear Dynamical Systems What is different from HMM? Kalman filter Its strength and limitation Particle Filter
More informationMonte Carlo Approximation of Monte Carlo Filters
Monte Carlo Approximation of Monte Carlo Filters Adam M. Johansen et al. Collaborators Include: Arnaud Doucet, Axel Finke, Anthony Lee, Nick Whiteley 7th January 2014 Context & Outline Filtering in State-Space
More informationRAO-BLACKWELLISED PARTICLE FILTERS: EXAMPLES OF APPLICATIONS
RAO-BLACKWELLISED PARTICLE FILTERS: EXAMPLES OF APPLICATIONS Frédéric Mustière e-mail: mustiere@site.uottawa.ca Miodrag Bolić e-mail: mbolic@site.uottawa.ca Martin Bouchard e-mail: bouchard@site.uottawa.ca
More informationRobert Collins CSE586, PSU Intro to Sampling Methods
Intro to Sampling Methods CSE586 Computer Vision II Penn State Univ Topics to be Covered Monte Carlo Integration Sampling and Expected Values Inverse Transform Sampling (CDF) Ancestral Sampling Rejection
More informationRAO-BLACKWELLIZED PARTICLE FILTER FOR MARKOV MODULATED NONLINEARDYNAMIC SYSTEMS
RAO-BLACKWELLIZED PARTICLE FILTER FOR MARKOV MODULATED NONLINEARDYNAMIC SYSTEMS Saiat Saha and Gustaf Hendeby Linöping University Post Print N.B.: When citing this wor, cite the original article. 2014
More informationHmms with variable dimension structures and extensions
Hmm days/enst/january 21, 2002 1 Hmms with variable dimension structures and extensions Christian P. Robert Université Paris Dauphine www.ceremade.dauphine.fr/ xian Hmm days/enst/january 21, 2002 2 1 Estimating
More informationMini-course 07: Kalman and Particle Filters Particle Filters Fundamentals, algorithms and applications
Mini-course 07: Kalman and Particle Filters Particle Filters Fundamentals, algorithms and applications Henrique Fonseca & Cesar Pacheco Wellington Betencurte 1 & Julio Dutra 2 Federal University of Espírito
More informationRobert Collins CSE586, PSU Intro to Sampling Methods
Intro to Sampling Methods CSE586 Computer Vision II Penn State Univ Topics to be Covered Monte Carlo Integration Sampling and Expected Values Inverse Transform Sampling (CDF) Ancestral Sampling Rejection
More informationBayesian Monte Carlo Filtering for Stochastic Volatility Models
Bayesian Monte Carlo Filtering for Stochastic Volatility Models Roberto Casarin CEREMADE University Paris IX (Dauphine) and Dept. of Economics University Ca Foscari, Venice Abstract Modelling of the financial
More informationIntroduction to Particle Filters for Data Assimilation
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,
More informationBayesian Methods for Machine Learning
Bayesian Methods for Machine Learning CS 584: Big Data Analytics Material adapted from Radford Neal s tutorial (http://ftp.cs.utoronto.ca/pub/radford/bayes-tut.pdf), Zoubin Ghahramni (http://hunch.net/~coms-4771/zoubin_ghahramani_bayesian_learning.pdf),
More informationLecture 6: Multiple Model Filtering, Particle Filtering and Other Approximations
Lecture 6: Multiple Model Filtering, Particle Filtering and Other Approximations Department of Biomedical Engineering and Computational Science Aalto University April 28, 2010 Contents 1 Multiple Model
More informationAn Introduction to Particle Filtering
An Introduction to Particle Filtering Author: Lisa Turner Supervisor: Dr. Christopher Sherlock 10th May 2013 Abstract This report introduces the ideas behind particle filters, looking at the Kalman filter
More informationSequential Bayesian Updating
BS2 Statistical Inference, Lectures 14 and 15, Hilary Term 2009 May 28, 2009 We consider data arriving sequentially X 1,..., X n,... and wish to update inference on an unknown parameter θ online. In a
More informationExercises Tutorial at ICASSP 2016 Learning Nonlinear Dynamical Models Using Particle Filters
Exercises Tutorial at ICASSP 216 Learning Nonlinear Dynamical Models Using Particle Filters Andreas Svensson, Johan Dahlin and Thomas B. Schön March 18, 216 Good luck! 1 [Bootstrap particle filter for
More informationSensitivity analysis in HMMs with application to likelihood maximization
Sensitivity analysis in HMMs with application to likelihood maximization Pierre-Arnaud Coquelin, Vekia, Lille, France pacoquelin@vekia.fr Romain Deguest Columbia University, New York City, NY 10027 rd2304@columbia.edu
More informationAdaptive Monte Carlo methods
Adaptive Monte Carlo methods Jean-Michel Marin Projet Select, INRIA Futurs, Université Paris-Sud joint with Randal Douc (École Polytechnique), Arnaud Guillin (Université de Marseille) and Christian Robert
More informationLinear Dynamical Systems
Linear Dynamical Systems Sargur N. srihari@cedar.buffalo.edu Machine Learning Course: http://www.cedar.buffalo.edu/~srihari/cse574/index.html Two Models Described by Same Graph Latent variables Observations
More informationSequential Monte Carlo and Particle Filtering. Frank Wood Gatsby, November 2007
Sequential Monte Carlo and Particle Filtering Frank Wood Gatsby, November 2007 Importance Sampling Recall: Let s say that we want to compute some expectation (integral) E p [f] = p(x)f(x)dx and we remember
More informationPattern Recognition and Machine Learning. Bishop Chapter 2: Probability Distributions
Pattern Recognition and Machine Learning Chapter 2: Probability Distributions Cécile Amblard Alex Kläser Jakob Verbeek October 11, 27 Probability Distributions: General Density Estimation: given a finite
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.utstat.utoronto.ca/~rsalakhu/ Sidney Smith Hall, Room 6002 Lecture 11 Project
More informationNonlinear Estimation Techniques for Impact Point Prediction of Ballistic Targets
Nonlinear Estimation Techniques for Impact Point Prediction of Ballistic Targets J. Clayton Kerce a, George C. Brown a, and David F. Hardiman b a Georgia Tech Research Institute, Georgia Institute of Technology,
More informationEvolution Strategies Based Particle Filters for Fault Detection
Evolution Strategies Based Particle Filters for Fault Detection Katsuji Uosaki, Member, IEEE, and Toshiharu Hatanaka, Member, IEEE Abstract Recent massive increase of the computational power has allowed
More informationParticle Filtering a brief introductory tutorial. Frank Wood Gatsby, August 2007
Particle Filtering a brief introductory tutorial Frank Wood Gatsby, August 2007 Problem: Target Tracking A ballistic projectile has been launched in our direction and may or may not land near enough to
More information9 Multi-Model State Estimation
Technion Israel Institute of Technology, Department of Electrical Engineering Estimation and Identification in Dynamical Systems (048825) Lecture Notes, Fall 2009, Prof. N. Shimkin 9 Multi-Model State
More informationConsistency of the maximum likelihood estimator for general hidden Markov models
Consistency of the maximum likelihood estimator for general hidden Markov models Jimmy Olsson Centre for Mathematical Sciences Lund University Nordstat 2012 Umeå, Sweden Collaborators Hidden Markov models
More informationIntroduction to Probability and Statistics (Continued)
Introduction to Probability and Statistics (Continued) Prof. icholas Zabaras Center for Informatics and Computational Science https://cics.nd.edu/ University of otre Dame otre Dame, Indiana, USA Email:
More informationKalman filtering and friends: Inference in time series models. Herke van Hoof slides mostly by Michael Rubinstein
Kalman filtering and friends: Inference in time series models Herke van Hoof slides mostly by Michael Rubinstein Problem overview Goal Estimate most probable state at time k using measurement up to time
More informationSEQUENTIAL MONTE CARLO METHODS FOR THE CALIBRATION OF STOCHASTIC RAINFALL-RUNOFF MODELS
XIX International Conference on Water Resources CMWR 2012 University of Illinois at Urbana-Champain June 17-22,2012 SEQUENTIAL MONTE CARLO METHODS FOR THE CALIBRATION OF STOCHASTIC RAINFALL-RUNOFF MODELS
More informationSEQUENTIAL MONTE CARLO METHODS WITH APPLICATIONS TO COMMUNICATION CHANNELS. A Thesis SIRISH BODDIKURAPATI
SEQUENTIAL MONTE CARLO METHODS WITH APPLICATIONS TO COMMUNICATION CHANNELS A Thesis by SIRISH BODDIKURAPATI Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of
More informationLinear Dynamical Systems (Kalman filter)
Linear Dynamical Systems (Kalman filter) (a) Overview of HMMs (b) From HMMs to Linear Dynamical Systems (LDS) 1 Markov Chains with Discrete Random Variables x 1 x 2 x 3 x T Let s assume we have discrete
More informationWhy do we care? Measurements. Handling uncertainty over time: predicting, estimating, recognizing, learning. Dealing with time
Handling uncertainty over time: predicting, estimating, recognizing, learning Chris Atkeson 2004 Why do we care? Speech recognition makes use of dependence of words and phonemes across time. Knowing where
More informationProbabilistic Graphical Models
Probabilistic Graphical Models Lecture 12 Dynamical Models CS/CNS/EE 155 Andreas Krause Homework 3 out tonight Start early!! Announcements Project milestones due today Please email to TAs 2 Parameter learning
More informationParticle filters, the optimal proposal and high-dimensional systems
Particle filters, the optimal proposal and high-dimensional systems Chris Snyder National Center for Atmospheric Research Boulder, Colorado 837, United States chriss@ucar.edu 1 Introduction Particle filters
More informationRECURSIVE OUTLIER-ROBUST FILTERING AND SMOOTHING FOR NONLINEAR SYSTEMS USING THE MULTIVARIATE STUDENT-T DISTRIBUTION
1 IEEE INTERNATIONAL WORKSHOP ON MACHINE LEARNING FOR SIGNAL PROCESSING, SEPT. 3 6, 1, SANTANDER, SPAIN RECURSIVE OUTLIER-ROBUST FILTERING AND SMOOTHING FOR NONLINEAR SYSTEMS USING THE MULTIVARIATE STUDENT-T
More information2D Image Processing (Extended) Kalman and particle filter
2D Image Processing (Extended) Kalman and particle filter Prof. Didier Stricker Dr. Gabriele Bleser Kaiserlautern University http://ags.cs.uni-kl.de/ DFKI Deutsches Forschungszentrum für Künstliche Intelligenz
More informationRecent Developments in Auxiliary Particle Filtering
Chapter 3 Recent Developments in Auxiliary Particle Filtering Nick Whiteley 1 and Adam M. Johansen 2 3.1 Background 3.1.1 State Space Models State space models (SSMs; sometimes termed hidden Markov models,
More informationParticle Filtering for Data-Driven Simulation and Optimization
Particle Filtering for Data-Driven Simulation and Optimization John R. Birge The University of Chicago Booth School of Business Includes joint work with Nicholas Polson. JRBirge INFORMS Phoenix, October
More informationThe Hierarchical Particle Filter
and Arnaud Doucet http://go.warwick.ac.uk/amjohansen/talks MCMSki V Lenzerheide 7th January 2016 Context & Outline Filtering in State-Space Models: SIR Particle Filters [GSS93] Block-Sampling Particle
More information1 What is a hidden Markov model?
1 What is a hidden Markov model? Consider a Markov chain {X k }, where k is a non-negative integer. Suppose {X k } embedded in signals corrupted by some noise. Indeed, {X k } is hidden due to noise and
More informationTowards inference for skewed alpha stable Levy processes
Towards inference for skewed alpha stable Levy processes Simon Godsill and Tatjana Lemke Signal Processing and Communications Lab. University of Cambridge www-sigproc.eng.cam.ac.uk/~sjg Overview Motivation
More informationRobert Collins CSE586, PSU Intro to Sampling Methods
Robert Collins Intro to Sampling Methods CSE586 Computer Vision II Penn State Univ Robert Collins A Brief Overview of Sampling Monte Carlo Integration Sampling and Expected Values Inverse Transform Sampling
More informationParticle Filters. Pieter Abbeel UC Berkeley EECS. Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics
Particle Filters Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics Motivation For continuous spaces: often no analytical formulas for Bayes filter updates
More informationHuman Pose Tracking I: Basics. David Fleet University of Toronto
Human Pose Tracking I: Basics David Fleet University of Toronto CIFAR Summer School, 2009 Looking at People Challenges: Complex pose / motion People have many degrees of freedom, comprising an articulated
More informationMarkov Chain Monte Carlo Methods for Stochastic Optimization
Markov Chain Monte Carlo Methods for Stochastic Optimization John R. Birge The University of Chicago Booth School of Business Joint work with Nicholas Polson, Chicago Booth. JRBirge U of Toronto, MIE,
More informationTarget Tracking and Classification using Collaborative Sensor Networks
Target Tracking and Classification using Collaborative Sensor Networks Xiaodong Wang Department of Electrical Engineering Columbia University p.1/3 Talk Outline Background on distributed wireless sensor
More informationParticle Filtering Approaches for Dynamic Stochastic Optimization
Particle Filtering Approaches for Dynamic Stochastic Optimization John R. Birge The University of Chicago Booth School of Business Joint work with Nicholas Polson, Chicago Booth. JRBirge I-Sim Workshop,
More informationWhy do we care? Examples. Bayes Rule. What room am I in? Handling uncertainty over time: predicting, estimating, recognizing, learning
Handling uncertainty over time: predicting, estimating, recognizing, learning Chris Atkeson 004 Why do we care? Speech recognition makes use of dependence of words and phonemes across time. Knowing where
More informationAn efficient stochastic approximation EM algorithm using conditional particle filters
An efficient stochastic approximation EM algorithm using conditional particle filters Fredrik Lindsten Linköping University Post Print N.B.: When citing this work, cite the original article. Original Publication:
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 218 Outlines Overview Introduction Linear Algebra Probability Linear Regression 1
More informationRobotics. Mobile Robotics. Marc Toussaint U Stuttgart
Robotics Mobile Robotics State estimation, Bayes filter, odometry, particle filter, Kalman filter, SLAM, joint Bayes filter, EKF SLAM, particle SLAM, graph-based SLAM Marc Toussaint U Stuttgart DARPA Grand
More information27 : Distributed Monte Carlo Markov Chain. 1 Recap of MCMC and Naive Parallel Gibbs Sampling
10-708: Probabilistic Graphical Models 10-708, Spring 2014 27 : Distributed Monte Carlo Markov Chain Lecturer: Eric P. Xing Scribes: Pengtao Xie, Khoa Luu In this scribe, we are going to review the Parallel
More informationSensor Tasking and Control
Sensor Tasking and Control Sensing Networking Leonidas Guibas Stanford University Computation CS428 Sensor systems are about sensing, after all... System State Continuous and Discrete Variables The quantities
More informationVariantes Algorithmiques et Justifications Théoriques
complément scientifique École Doctorale MATISSE IRISA et INRIA, salle Markov jeudi 26 janvier 2012 Variantes Algorithmiques et Justifications Théoriques François Le Gland INRIA Rennes et IRMAR http://www.irisa.fr/aspi/legland/ed-matisse/
More informationMarkov Chain Monte Carlo Methods for Stochastic
Markov Chain Monte Carlo Methods for Stochastic Optimization i John R. Birge The University of Chicago Booth School of Business Joint work with Nicholas Polson, Chicago Booth. JRBirge U Florida, Nov 2013
More informationBrief Introduction of Machine Learning Techniques for Content Analysis
1 Brief Introduction of Machine Learning Techniques for Content Analysis Wei-Ta Chu 2008/11/20 Outline 2 Overview Gaussian Mixture Model (GMM) Hidden Markov Model (HMM) Support Vector Machine (SVM) Overview
More informationKernel Sequential Monte Carlo
Kernel Sequential Monte Carlo Ingmar Schuster (Paris Dauphine) Heiko Strathmann (University College London) Brooks Paige (Oxford) Dino Sejdinovic (Oxford) * equal contribution April 25, 2016 1 / 37 Section
More informationSTA 414/2104: Machine Learning
STA 414/2104: Machine Learning Russ Salakhutdinov Department of Computer Science! Department of Statistics! rsalakhu@cs.toronto.edu! http://www.cs.toronto.edu/~rsalakhu/ Lecture 9 Sequential Data So far
More informationFast Sequential Monte Carlo PHD Smoothing
14th International Conference on Information Fusion Chicago, Illinois, USA, July 5-8, 11 Fast Sequential Monte Carlo PHD Smoothing Sharad Nagappa and Daniel E. Clark School of EPS Heriot Watt University
More informationPattern Recognition and Machine Learning
Christopher M. Bishop Pattern Recognition and Machine Learning ÖSpri inger Contents Preface Mathematical notation Contents vii xi xiii 1 Introduction 1 1.1 Example: Polynomial Curve Fitting 4 1.2 Probability
More informationApril 20th, Advanced Topics in Machine Learning California Institute of Technology. Markov Chain Monte Carlo for Machine Learning
for for Advanced Topics in California Institute of Technology April 20th, 2017 1 / 50 Table of Contents for 1 2 3 4 2 / 50 History of methods for Enrico Fermi used to calculate incredibly accurate predictions
More informationCPSC 540: Machine Learning
CPSC 540: Machine Learning MCMC and Non-Parametric Bayes Mark Schmidt University of British Columbia Winter 2016 Admin I went through project proposals: Some of you got a message on Piazza. No news is
More informationNON-LINEAR NOISE ADAPTIVE KALMAN FILTERING VIA VARIATIONAL BAYES
2013 IEEE INTERNATIONAL WORKSHOP ON MACHINE LEARNING FOR SIGNAL PROCESSING NON-LINEAR NOISE ADAPTIVE KALMAN FILTERING VIA VARIATIONAL BAYES Simo Särä Aalto University, 02150 Espoo, Finland Jouni Hartiainen
More informationChris Bishop s PRML Ch. 8: Graphical Models
Chris Bishop s PRML Ch. 8: Graphical Models January 24, 2008 Introduction Visualize the structure of a probabilistic model Design and motivate new models Insights into the model s properties, in particular
More informationNonlinear Filtering. With Polynomial Chaos. Raktim Bhattacharya. Aerospace Engineering, Texas A&M University uq.tamu.edu
Nonlinear Filtering With Polynomial Chaos Raktim Bhattacharya Aerospace Engineering, Texas A&M University uq.tamu.edu Nonlinear Filtering with PC Problem Setup. Dynamics: ẋ = f(x, ) Sensor Model: ỹ = h(x)
More informationExpectation Propagation Algorithm
Expectation Propagation Algorithm 1 Shuang Wang School of Electrical and Computer Engineering University of Oklahoma, Tulsa, OK, 74135 Email: {shuangwang}@ou.edu This note contains three parts. First,
More informationExpectation Propagation in Dynamical Systems
Expectation Propagation in Dynamical Systems Marc Peter Deisenroth Joint Work with Shakir Mohamed (UBC) August 10, 2012 Marc Deisenroth (TU Darmstadt) EP in Dynamical Systems 1 Motivation Figure : Complex
More information