Particle Filtering Under General Regime Switching
Yousef El-Laham, Liu Yang, Petar M. Djuric, Monica F. Bugallo
PParticle Filtering Under General Regime Switching
Yousef El-Laham, Liu Yang, Petar M. Djuri´c, Mónica F. Bugallo
Department of Electrical & Computer EngineeringStony Brook University, Stony Brook (USA){yousef.ellaham, liu.yang.2, petar.djuric, monica.bugallo}@stonybrook.edu
Abstract —In this paper, we consider a new framework forparticle filtering under model uncertainty that operates beyondthe scope of Markovian switching systems. Specifically, wedevelop a novel particle filtering algorithm that applies to generalregime switching systems, where the model index is augmented asan unknown time-varying parameter in the system. The proposedapproach does not require the use of multiple filters and canmaintain a diverse set of particles for each considered modelthrough appropriate choice of the particle filtering proposaldistribution. The flexibility of the proposed approach allows forlong-term dependencies between the models, which enables itsuse to a wider variety of real-world applications. We validatethe method on a synthetic data experiment and show thatit outperforms state-of-the-art multiple model particle filteringapproaches that require the use of multiple filters.
I. I
NTRODUCTION
In the past three decades, particle filtering (PF) [1], [2]has emerged as one of the most powerful statistical toolsfor online state estimation in dynamical systems. PF methodsapproximate the posterior distribution of an unknown time-varying parameter vector in a state-space model (SSM) usinga set of weighted samples. The samples, which are also calledparticles, are drawn from a probability distribution called the proposal distribution and are weighted properly accordingto the principle of importance sampling [3]. Unlike Kalmanfiltering and its extensions [4], [5], PF can deal with SSMsthat exhibit both nonlinearities and non-Gaussianities. Thisflexibility has allowed PF methods to thrive in many appli-cations in fields as diverse as signal processing, economics,neuroscience, epidemiology, and ecology [6]–[10].Model uncertainty introduces an additional layer of com-plexity to stochastic filtering that is generally difficult to dealwith. In this situation, one must determine the model thatbest represents the system of interest from a set of candidatemodels, while also jointly estimating the unknown time-varying parameters of the chosen model. The issue of modeluncertainty is further complicated if the model can switch fromone time instant to the next. In signal processing, the well-known problem of tracking a maneuvering target falls withinthis class of model selection problems [11]. The trajectory of amaneuvering target is represented via a Markovian switchingsystem (i.e., jump Markov systems) [12], where the model
This work was carried out thanks to the support of the National ScienceFoundation (NSF) under Awards CCF-1617986 and CCF-1618999. The au-thors thank the Research Computing and Cyberinfrastructure and the Institutefor Advanced Computational Science at Stony Brook University for accessto the high-performance SeaWulf computing system sponsored by NSF ( dynamics change according to the state of a discrete-time,discrete-state Markov chain. More generally, however, systemswhose models (or regimes) can change from one time instantto the next are referred to as regime-switching systems .There have been mainly two classes of solutions proposedin the PF literature to deal with the challenge of modeluncertainty. In the first class of solutions, a model indexthat references the candidate models is augmented as anunknown state in the system and is jointly estimated withthe unknown time-varying parameters using a single particlefilter [12], [13]. While this solution is simple to implementand straightforwardly tackles the joint estimation problem,the disadvantage is that the number of samples assigned toeach candidate model cannot be controlled, which can leadto numerical issues and a lack of diversity in the consideredmodels. The second class of solutions employs the use of abank of particle filters that operate in parallel. Each filter isconditioned on one of the candidate models and state estimatesare obtained by weighting the results of each of the filters andthen fusing them. In [14], the filters are weighted accordingto the posterior probability of their respective models, whilein [15], they are weighted according to the predictive powersof their respective models. Hybrid solutions which combinethis class of approaches with interacting multiple models(IMM) filter have also been proposed to deal with Markovianswitching systems [16]. Unfortunately, because a separate filteris required for each model, this class of approaches can becomputationally intensive if the number of candidate modelsis large. We remark that, to the best of our knowledge, non-heuristic implementations of both methods have not beenformulated for more general systems, which may exhibit long-term dependencies in the regime switching dynamics beyondthose of the Markovian switching systems.In this work, we propose a novel PF algorithm for generalregime switching systems. Similar to the aforementioned so-lutions, the proposed PF method augments the model indexas an unknown in the system that is jointly estimated withthe time-varying parameters. Unlike previous approaches inthe literature, a diverse set of candidate models can alwaysbe considered in the proposed algorithm through appropriatechoice of the model index proposal distribution. Furthermore,since our method is not restricted to Markovian switchingsystems, it can handle more complicated processes that maydescribe the regime switching dynamics, such as the Pólya urnprocess [17]. Simulation results for synthetic data experimentsvalidate the performance of the proposed approach. a r X i v : . [ s t a t . C O ] S e p ig. 1: The considered regime switching SSM formulation fora time horizon T = 3 . The probability of a model depends onthe complete history of models at previous time instants.II. P ROBLEM F ORMULATION
Let x t ∈ R d x denote a latent state vector, y t ∈ R d y denotea measurement vector, and M t ∈ { , . . . , K } denote a modelindex from a set of K candidate models, where t denotes timeindex. We consider the generic stochastic filtering problemunder model uncertainty over a fixed time horizon T . Thegenerative process is assumed to have the following form: M t ∼ p ( M t |M t − ) , (1) x t ∼ p ( x t | x t − , M t ) , (2) y t ∼ p ( y t | x t , M t ) , (3)for t = 1 , . . . , T , where the initial model M is distributedaccording to M ∼ p ( M ) and the initial latent state x is distributed according to x ∼ p ( x |M ) . A graphicalrepresentation of the system is shown in Fig. 1 for T = 3 .Our goal is to jointly infer the unknown states x T (cid:44) [ x , . . . , x T ] ∈ R d x × ( T +1) and the sequence of unknownmodels M T (cid:44) ( M , . . . , M T ) based on the given obser-vations y T (cid:44) [ y , . . . , y T ] ∈ R d y × T under the Bayesianparadigm. In other words, we would like to approximate thejoint posterior distribution p ( x T , M T | y T ) in a recursivemanner using a Bayesian filtering solution.III. R EGIME S WITCHING P ARTICLE F ILTERING
In this section, we derive a generalized regime switchingPF (RSPF) algorithm. We begin by first establishing therecursiveness of the joint distribution and then we derive theimportance weights of the particles in the novel PF algorithm.We discuss different strategies for sampling models whichallow for model diversity. Finally, we elaborate on how onecan use the proposed algorithm to obtain the maximum aposteriori (MAP) estimate of the model at each time instant.We summarize the proposed approach in Algorithm 1.
A. Deriving the Joint Distribution
At time t , the distribution of interest is p ( x t , M t | y t ) .This joint distribution can be decomposed as p ( x t , M t | y t ) = p ( x t | y t , M t ) p ( M t | y t ) , (4) Algorithm 1
Regime Switching Particle Filtering (RSPF) Initialization:
Draw N samples from the prior of theinitial model to determine the model indexes M ( n )0 ∼ p ( M ) , n = 1 , . . . , N, and draw N samples from the prior of the initial stateconditioned on the sampled model indexes x ( n )0 ∼ p ( x |M ( n )0 ) , n = 1 , . . . , N. Set the weights as ˜ w ( n )0 = N for n = 1 , . . . , N . for t = 1 , . . . , T do Sampling models:
Draw N samples from the modelindex proposal distribution M ( n ) t ∼ q ( M t |M ( n )0: t − ) , n = 1 , . . . , N. Sampling states:
Draw N samples of the states condi-tioned on the drawn models x ( n ) t ∼ q ( x t | x ( n ) t − , M ( n ) t , y t ) , n = 1 , . . . , N. Weighting:
Compute the weights { ˜ w ( n ) t } Nn =1 accordingto (15) and normalize them as w ( n ) t = ˜ w ( n ) t (cid:80) Nj =1 ˜ w ( j ) t , n = 1 , . . . , N. Model selection:
Determine the model index estimate ˆ M t by solving the following maximization problem ˆ M t = arg max k ∈{ ,...,K } p ( M t = k | y t ) where p ( M t = k | y t ) ≈ (cid:80) Nn =1 w ( n ) t ( M ( n ) t = k ) for all k . State estimation:
Obtain the state estimate as ˆ x t = N (cid:88) n =1 w ( n ) t x ( n ) t . Resampling:
If necessary, resample the model indexesand the states using multinomial resampling and set theweights as ˜ w ( n ) t = N for n = 1 , . . . , N . end for where p ( x t | y t , M t ) is the posterior of the state tra-jectory x t conditioned on the model sequence M t and p ( M t | y t ) is the marginal posterior distribution of themodel sequence. Following previous analysis in the Bayesianfiltering literature, we can readily deduce the conditionalposterior of the state trajectory p ( x t | y t , M t ) as p ( x t | y t , M t ) = p ( y t | x t , M t ) p ( x t | x t − , M t ) p ( x t − | y t − , M t − ) p ( y t | y t − , M t ) . (5)Remark that we have made the appropriate assumption that p ( x t − | y t − , M t ) = p ( x t − | y t − , M t − ) , since theconditional posterior at time instant t − does not depend onthe model at time t . Applying Bayes’ theorem to the marginalosterior of the model sequence p ( M t | y t ) , we also have p ( M t | y t ) = p ( y t | y t − , M t ) p ( M t | y t − ) p ( y t | y t − ) . (6)We decompose p ( M t | y t − ) as p ( M t | y t − ) = p ( M t |M t − , y t − ) p ( M t − | y t − ) , (7)where one can show that p ( M t |M t − , y t − ) = p ( y t − |M t ) p ( M t |M t − ) p ( y t − |M t − ) (8) = p ( M t |M t − ) p ( y t − |M t − ) (cid:90) p ( x t − , y t − |M t ) d x t − (9) = p ( M t |M t − ) p ( y t − |M t − ) p ( y t − |M t − ) (10) = p ( M t |M t − ) , (11)since the joint distribution of x t − and y t − is condi-tionally independent from model M t given the sequence ofmodels up to time instant t − . We can now establish therecursive solution to the joint posterior p ( x t , M t | y t ) as p ( x t , M t | y t ) ∝ p ( y t | x t , M t ) p ( x t | x t − , M t ) × p ( M t |M t − ) p ( x t − , M t − | y t − ) . (12) B. Deriving the Particle Filtering Weights
Suppose we draw a set of N samples { ( x ( n )0: t , M ( n )0: t ) } Nn =1 ,where each sample ( x ( n )0: t , M ( n )0: t ) is drawn from a proposaldistribution q ( x t , M t | y t ) for n = 1 , . . . , N . Then, theimportance weight of each sample is determined according to ˜ w ( n ) t = p ( x ( n )0: t , M ( n )0: t | y t ) q ( x ( n )0: t , M ( n )0: t | y t ) , n = 1 , . . . , N. (13)Suppose that the proposal distribution can be factored as q ( x t , M t | y t ) = q ( x t − , M t − | y t − ) × q ( x t | x t − , M t , y t ) q ( M t |M t − ) . (14)Combining with results from (12), the importance weight ofeach sample ( x ( n )0: t , M ( n )0: t ) can thus be determined as ˜ w ( n ) t ∝ ˜ w ( n ) t − p ( y t | x ( n ) t , M ( n ) t ) p ( x ( n ) t | x ( n ) t − , M ( n ) t ) p ( M ( n ) t |M ( n )0: t − ) q ( x ( n ) t | x ( n ) t − , M ( n ) t , y t ) q ( M ( n ) t |M ( n )0: t − ) . (15)For sampling the states, the bootstrap implementation of thismethod would assume that the proposal distribution of thestates is identical to the state transition distribution, i.e., q ( x t | x t − , M t , y t ) = p ( x t | x t − , M t ) , and that the particlesare resampled after each time instant. For the bootstrap im-plementation, the importance weights are given by ˜ w ( n ) t ∝ p ( y t | x ( n ) t , M ( n ) t ) p ( M ( n ) t |M ( n )0: t − ) q ( M ( n ) t |M ( n )0: t − ) , (16)for n = 1 , . . . , N . The obtained solution is analogous tothe weighting function in bootstrap PF, except now, we musttaken into account that models can change according to p ( M t |M t − ) . We remark that p ( M t |M t − ) determineshow the model M t is determined from the history of models M t − and depends on the nature of system being considered. C. Discussion on Sampling Model Indexes
There are a variety of choices for the model index pro-posal distribution q ( M t |M t − ) . The most obvious choiceis the bootstrap approach, where we use the model transi-tion function as the proposal, i.e, we set q ( M t |M t − ) = p ( M t |M t − ) . Then, the importance weights simply becomethe joint likelihood of the sampled model indexes and states.Alternatively, one can use a discrete uniform proposal distribu-tion to sample the model indexes, i.e., q ( M t = k |M t − ) = K for all k . Then, each model has an equal chance to besampled at each step of the algorithm, and thus avoidingthe possibility of a model diversity issue. Finally, one can deterministically sample an equal number of particles for eachmodel. The weights in this case would be the same as if wehad sampled from the discrete uniform distribution. D. Online Maximum A Posteriori Model Selection
In order to select the most promising model at each timeinstant from the set of candidate models, we need to solve thefollowing optimization problem: ˆ M t = arg max k ∈{ ,...,K } p ( M t = k | y t ) , (17)where p ( M t = k | y t ) denotes the posterior probability ofthe k th model. This posterior probability of each model canbe estimated directly using the set of particles and weights p ( M t = k | y t ) ≈ (cid:80) Nn =1 ˜ w ( n ) t N (cid:88) n =1 ˜ w ( n ) t ( M ( n ) t = k ) , (18)for k = 1 , . . . , K , where ( · ) denotes the indicator function.Given the estimated posterior probabilities, one can obtain anapproximate solution to the optimization problem in (17).IV. E XAMPLES OF M ODEL S EQUENCE D YNAMICS
Here, we give examples of different regime switching dy-namics that can easily be treated using our proposed approach.
A. Independent Regime Dynamics
The simplest case is when the models are generated inde-pendently from one another, i.e., the joint distribution of themodels can be factored as: p ( M T ) = T (cid:89) t =0 p ( M t ) , (19)where the model independence assumption implies that p ( M t |M t − ) = p ( M t ) . The assumption that the modelsare independent may be unrealistic for most applications andrequires to specify the prior distribution of each model. B. Markovian Switching Dynamics
We also consider Markovian switching systems, where themodel at each time instant only depends on the model at theprevious time instant. The joint distribution of the modelsunder this assumption is given by p ( M T ) = p ( M ) T (cid:89) t =1 p ( M t |M t − ) , (20)here we have that p ( M t |M t − ) = p ( M t |M t − ) . Here,the model transition distribution p ( M t |M t − ) is representedby a transition probability matrix PP = p , . . . p ,K ... . . . ... p K, . . . p K,K , (21)where each element p i,j (cid:44) p ( M t = j |M t − = i ) is definedto be the probability of transitioning from model i to model j and each row of the matrix P satisfies (cid:80) Kj =1 p i,j = 1 . C. Pólya Urn Dynamics
Under a more general formulation, the model at a giventime instant t depends on the complete sequence of models M t − . Here, since there are no independence assumptions,the joint distribution of the models is given by p ( M T ) = p ( M ) T (cid:89) t =1 p ( M t |M t − ) . (22)If the number of models is finite and a priori known, onepossibility is to consider a Pólya urn process for the regimedynamics. For the Pólya urn process, the probability of transi-tioning to a particular model at time instant t depends on howmany times that model was chosen in previous time instants.Let α k,t = ( M t = k ) be variable indicating if model k wasvisited at time t for t = 1 , . . . , T and let β k ∈ N be anypositive integer for k = 1 , . . . , K . Then, the probability oftransitioning to model k at time t is given by p ( M t = k |M t − ) = β k + (cid:80) t − τ =0 α k,τ (cid:80) Kj =1 ( β j + (cid:80) t − τ =0 α j,τ ) . (23)V. S IMULATIONS
To validate the performance of the proposed RSPF, wegenerated synthetic measurement sequences of time length T = 50 based on eight candidate models, with each modelbeing of the form M k : (cid:40) x t = a k x t − + c k + u t y t = b k (cid:112) | x t | + d k + v t , (24)where the parameter settings are [ a , ..., a ] =[ − . , − . , − . , − . , . , . , . , . , [ c , ..., c ] =[0 , − , , − , , , − , , [ b , ..., b ] = [ a , ..., a ] , and [ d , ..., d ] = [ c , ..., c ] . The process noise u t andobservation noise v t are assumed to be i.i.d. zero-meanGaussian with equal variances, i.e., u t ∼ N (0 , σ u ) and v t ∼ N (0 , σ v ) with σ u = σ v = 0 . . The initial state x wasgenerated uniformly from -0.5 to 0.5. We tested the methodon two scenarios corresponding to regime switching based onMarkovian dynamics and Pólya urn dynamics, respectively. We first ran our novel algorithm with N = 2000 particlesper iteration when the model sequence dynamics is Markovian.The transition probability matrix in simulation was P = .
80 0 . (cid:15) · · · (cid:15)(cid:15) .
80 0 . · · · (cid:15) ... ... ... ... (cid:15) · · · .
80 0 . . (cid:15) · · · (cid:15) . , (25)where we set (cid:15) = so that each row of P summed to 1.Three different model index proposal distributions wereused (deterministic, uniform, and bootstrap). For comparison,we also ran the multiple model particle filtering (MMPF)algorithm presented in [14], [18], where we drew 250 samplesper model. Note that this algorithm considers a forgettingfactor parameter γ ∈ [0 , that determines how much theobservation history influences the model probabilities. Thecloser γ is to 1, the more the observation history influencesthe model probabilities. We tested four different settings ofthis method, each corresponding to a different forgetting factor γ ∈ { , . , . , } . The results are averaged over MonteCarlo runs and are summarized in Tables I and II. We can seethat the novel method, regardless of the choice of the modelindex proposal distribution, provides a smaller mean squarederror (MSE) and more accurate model selection results. Forreference, we also plot the average cumulative sum of theMSE in the state estimation in Fig. 2.Next, we conducted the proposed method on the Pólya urnprocess. The initial counts of the eight models were a randompermutation of the integers from 1 to 8. The parameter settingsfor RSPF and MMPF were the same as above. Table III andIV show the results, which are averaged over 500 MonteCarlo simulations. Again, we can see that the novel methodoutperforms MMPF with the settings γ = 0 . , . and by far,and slightly outperforms the MMPF with a forgetting factor of0 in terms of model selection accuracy, and in terms of state Average Best WorstNovel (Deterministic) 0.2443 0.0566 4.8527Novel (Uniform) 0.2446 0.0573 4.7388Novel (Bootstrap) 0.2462 0.0546 4.9030MMPF ( γ = 0 ) 0.5986 0.0792 9.0508MMPF ( γ = 0 . ) 9.9912 0.2635 279.7545MMPF ( γ = 0 . ) 51.4122 1.5156 900.7603MMPF ( γ = 1 ) 63.5191 1.5136 1002.7609 TABLE I: State estimation MSE (Markovian dynamics).
Average Best WorstNovel (Deterministic) 0.9407 1 0.5000Novel (Uniform) 0.9402 1 0.5000Novel (Bootstrap) 0.9419 1 0.5000MMPF ( γ = 0 ) 0.8437 1 0.5200MMPF ( γ = 0 . ) 0.5089 0.9200 0.1200MMPF ( γ = 0 . ) 0.2348 0.9200 0MMPF ( γ = 1 ) 0.2180 0.9200 0 TABLE II: Model selection accuracy (Markovian dynamics). verage Best WorstNovel (Deterministic) 0.4112 0.0663 2.3021Novel (Uniform) 0.4111 0.0672 2.2512Novel (Bootstrap) 0.4116 0.0644 2.3921MMPF ( γ = 0 ) 0.4995 0.0734 2.6993MMPF ( γ = 0 . ) 4.4573 0.3854 28.0283MMPF ( γ = 0 . ) 8.6751 1.9418 32.9856MMPF ( γ = 1 ) 11.1040 2.0316 43.6506 TABLE III: State estimation MSE (Pólya urn dynamics).
Average Best WorstNovel (Deterministic) 0.9003 1 0.6800Novel (Uniform) 0.8996 1 0.6600Novel (Bootstrap) 0.8996 1 0.6800MMPF ( γ = 0 ) 0.8526 1 0.6200MMPF ( γ = 0 . ) 0.2867 0.5600 0.0600MMPF ( γ = 0 . ) 0.1690 0.4200 0MMPF ( γ = 1 ) 0.1571 0.4800 0 TABLE IV: Model selection accuracy (Pólya urn dynamics).estimation MSE. The average cumulative sum of the MSEfor the state estimation is shown in Fig. 3.VI. C
ONCLUSIONS
In this paper, we introduced a novel particle filtering al-gorithm for regime switching systems. The proposed methodallows for the treatment of stochastic filtering problems undermodel uncertainty, where the model can change from one timeinstant to the next. Moreover, our algorithm does not have anyrestrictions on the regime switching dynamics and can workfor systems that are not Markovian switching systems. Wevalidated our method on two synthetic data experiments, wherein the first experiment we considered a Markovian switchingsystem and in the second experiment we considered a systemwhere regimes changed according to a Pólya urn process.R
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