scholarly journals Uniform convergence over time of a nested particle filtering scheme for recursive parameter estimation in state-space Markov models

2017 ◽  
Vol 49 (4) ◽  
pp. 1170-1200 ◽  
Author(s):  
Dan Crisan ◽  
Joaquín Míguez
Keyword(s):  
Monte Carlo ◽  
Uniform Convergence ◽  
State Space ◽  
Particle Filtering ◽  
Sequential Monte Carlo ◽  
Markov Models ◽  
Empirical Distribution ◽  
Model Parameters ◽  
Posterior Probability Distribution ◽  
Over Time

Abstract We analyse the performance of a recursive Monte Carlo method for the Bayesian estimation of the static parameters of a discrete-time state-space Markov model. The algorithm employs two layers of particle filters to approximate the posterior probability distribution of the model parameters. In particular, the first layer yields an empirical distribution of samples on the parameter space, while the filters in the second layer are auxiliary devices to approximate the (analytically intractable) likelihood of the parameters. This approach relates the novel algorithm to the recent sequential Monte Carlo square method, which provides a nonrecursive solution to the same problem. In this paper we investigate the approximation of integrals of real bounded functions with respect to the posterior distribution of the system parameters. Under assumptions related to the compactness of the parameter support and the stability and continuity of the sequence of posterior distributions for the state-space model, we prove that the Lp norms of the approximation errors vanish asymptotically (as the number of Monte Carlo samples generated by the algorithm increases) and uniformly over time. We also prove that, under the same assumptions, the proposed scheme can asymptotically identify the parameter values for a class of models. We conclude the paper with a numerical example that illustrates the uniform convergence results by exploring the accuracy and stability of the proposed algorithm operating with long sequences of observations.

scholarly journals Sequential Monte Carlo smoothing for general state space hidden Markov models

2011 ◽  
Vol 21 (6) ◽  
pp. 2109-2145 ◽  
Author(s):  
Randal Douc ◽  
Aurélien Garivier ◽  
Eric Moulines ◽  
Jimmy Olsson
Keyword(s):  
Monte Carlo ◽  
State Space ◽  
Hidden Markov Models ◽  
Sequential Monte Carlo ◽  
Markov Models ◽  
Hidden Markov ◽  
General State ◽  
General State Space

scholarly journals Characterization of tumor heterogeneity by latent haplotypes: a sequential Monte Carlo approach

PeerJ ◽  
2018 ◽  
Vol 6 ◽  
pp. e4838 ◽  
Author(s):  
Oyetunji E. Ogundijo ◽  
Xiaodong Wang
Keyword(s):  
Monte Carlo ◽  
State Space ◽  
Tumor Heterogeneity ◽  
Sequential Monte Carlo ◽  
State Space Model ◽  
Sequential Algorithm ◽  
Model Parameters ◽  
Single Nucleotide Variants ◽  
Time Step ◽  
Allocation Model

Tumor samples obtained from a single cancer patient spatially or temporally often consist of varying cell populations, each harboring distinct mutations that uniquely characterize its genome. Thus, in any given samples of a tumor having more than two haplotypes, defined as a scaffold of single nucleotide variants (SNVs) on the same homologous genome, is evidence of heterogeneity because humans are diploid and we would therefore only observe up to two haplotypes if all cells in a tumor sample were genetically homogeneous. We characterize tumor heterogeneity by latent haplotypes and present state-space formulation of the feature allocation model for estimating the haplotypes and their proportions in the tumor samples. We develop an efficient sequential Monte Carlo (SMC) algorithm that estimates the states and the parameters of our proposed state-space model, which are equivalently the haplotypes and their proportions in the tumor samples. The sequential algorithm produces more accurate estimates of the model parameters when compared with existing methods. Also, because our algorithm processes the variant allele frequency (VAF) of a locus as the observation at a single time-step, VAF from newly sequenced candidate SNVs from next-generation sequencing (NGS) can be analyzed to improve existing estimates without re-analyzing the previous datasets, a feature that existing solutions do not possess.

scholarly journals Bayesian Parameter Estimation for Nonlinear Dynamics Using Sensitivity Analysis

2019 ◽  
Author(s):  
Yi Chou ◽  
Sriram Sankaranarayanan
Keyword(s):  
Monte Carlo ◽  
Parameter Space ◽  
Sequential Monte Carlo ◽  
Measurement Data ◽  
Reachability Analysis ◽  
Model Parameters ◽  
Reference Trajectory ◽  
Nonlinear Odes ◽  
Ode Models ◽  
Inference Techniques

We investigate approximate Bayesian inference techniques for nonlinear systems described by ordinary differential equation (ODE) models. In particular, the approximations will be based on set-valued reachability analysis approaches, yielding approximate models for the posterior distribution. Nonlinear ODEs are widely used to mathematically describe physical and biological models. However, these models are often described by parameters that are not directly measurable and have an impact on the system behaviors. Often, noisy measurement data combined with physical/biological intuition serve as the means for finding appropriate values of these parameters. Our approach operates under a Bayesian framework, given prior distribution over the parameter space and noisy observations under a known sampling distribution. We explore subsets of the space of model parameters, computing bounds on the likelihood for each subset. This is performed using nonlinear set-valued reachability analysis that is made faster by means of linearization around a reference trajectory. The tiling of the parameter space can be adaptively refined to make bounds on the likelihood tighter. We evaluate our approach on a variety of nonlinear benchmarks and compare our results with Markov Chain Monte Carlo and Sequential Monte Carlo approaches.

Quantifying Monte Carlo Uncertainty in the Ensemble Kalman Filter

SPE Journal ◽  
2010 ◽  
Vol 16 (01) ◽  
pp. 172-182 ◽  
Author(s):  
Kristian Thulin ◽  
Geir Nævdal ◽  
Hans Julius Skaug ◽  
Sigurd Ivar Aanonsen
Keyword(s):  
Monte Carlo ◽  
Kalman Filter ◽  
Posterior Distribution ◽  
Ensemble Kalman Filter ◽  
Sequential Monte Carlo ◽  
Optimal Combination ◽  
Cumulative Distribution ◽  
Low Rank ◽  
Batch Size ◽  
Posterior Probability Distribution

Summary The ensemble Kalman filter (EnKF) is currently considered one of the most promising methods for conditioning reservoir-simulation models to production data. The EnKF is a sequential Monte Carlo method based on a low-rank approximation of the system covariance matrix. The posterior probability distribution of model variables may be estimated from the updated ensemble, but, because of the low-rank covariance approximation, the updated ensemble members become correlated samples from the posterior distribution. We suggest using multiple EnKF runs, each with a smaller ensemble size, to obtain truly independent samples from the posterior distribution. This allows a pointwise confidence interval to be constructed for the posterior cumulative distribution function (CDF). We investigate the methodology for finding an optimal combination of ensemble batch size n and number of EnKF runs m while keeping the total number of ensemble members n×m constant. The optimal combination of n and m is found through minimizing the integrated mean-square error (MSE) for the CDFs. We illustrate the approach on two models, first a small linear model and then a synthetic 2D model inspired by petroleum applications. In the latter case, we choose to define an EnKF run with 10,000 ensemble members as having zero Monte Carlo error. The proposed methodology should be applicable also to larger, more-realistic models.

scholarly journals On the Behaviour of the Backward Interpretation of Feynman-Kac Formulae Under Verifiable Conditions

2015 ◽  
Vol 52 (02) ◽  
pp. 339-359 ◽  
Author(s):  
Ajay Jasra
Keyword(s):  
Monte Carlo ◽  
Hidden Markov Models ◽  
Sequential Monte Carlo ◽  
Markov Models ◽  
Hidden Markov ◽  
Asymptotic Variance ◽  
Time Behaviour ◽  
Additive Functionals ◽  
Monte Carlo Estimate ◽  
Carlo Estimate

We consider the time behaviour associated to the sequential Monte Carlo estimate of the backward interpretation of Feynman-Kac formulae. This is particularly of interest in the context of performing smoothing for hidden Markov models. We prove a central limit theorem under weaker assumptions than adopted in the literature. We then show that the associated asymptotic variance expression for additive functionals grows at most linearly in time under hypotheses that are weaker than those currently existing in the literature. The assumptions are verified for some hidden Markov models.

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