Grupo de Tratamiento de Señal y Aprendizajehttp://hdl.handle.net/10016/90412017-11-24T22:08:23Z2017-11-24T22:08:23ZA population Monte Carlo scheme with transformed weights and its application to stochastic kinetic modelsKoblents Lapteva, EugeniaMíguez Arenas, Joaquínhttp://hdl.handle.net/10016/259012017-11-23T16:21:08Z2015-03-01T00:00:00ZA population Monte Carlo scheme with transformed weights and its application to stochastic kinetic models
Koblents Lapteva, Eugenia; Míguez Arenas, Joaquín
This paper addresses the Monte Carlo approximation of posterior probability distributions. In particular, we consider the population Monte Carlo (PMC) technique, which is based on an iterative importance sampling (IS) approach. An important drawback of this methodology is the degeneracy of the importance weights (IWs) when the dimension of either the observations or the variables of interest is high. To alleviate this difficulty, we propose a new method that performs a nonlinear transformation of the IWs. This operation reduces the weight variation, hence it avoids degeneracy and increases the efficiency of the IS scheme, specially when drawing from proposal functions which are poorly adapted to the true posterior. For the sake of illustration, we have applied the proposed algorithm to the estimation of the parameters of a Gaussian mixture model. This is a simple problem that enables us to discuss the main features of the proposed technique. As a practical application, we have also considered the challenging problem of estimating the rate parameters of a stochastic kinetic model (SKM). SKMs are multivariate systems that model molecular interactions in biological and chemical problems. We introduce a particularization of the proposed algorithm to SKMs and present numerical results.
2015-03-01T00:00:00ZAdapting the number of particles in sequential Monte Carlo methods through an online scheme for convergence assessmentElvira Arregui, VíctorMíguez Arenas, JoaquínDjuric, Petar M.http://hdl.handle.net/10016/258982017-11-21T01:15:08Z2016-12-08T00:00:00ZAdapting the number of particles in sequential Monte Carlo methods through an online scheme for convergence assessment
Elvira Arregui, Víctor; Míguez Arenas, Joaquín; Djuric, Petar M.
Particle filters are broadly used to approximate posterior distributions of hidden states in state-space models by means of sets of weighted particles. While the convergence of the filter is guaranteed when the number of particles tends to infinity, the quality of the approximation is usually unknown but strongly dependent on the number of particles. In this paper, we propose a novel method for assessing the convergence of particle filters in an online manner, as well as a simple scheme for the online adaptation of the number of particles based on the convergence assessment. The method is based on a sequential comparison between the actual observations and their predictive probability distributions approximated by the filter. We provide a rigorous theoretical analysis of the proposed methodology and, as an example of its practical use, we present simulations of a simple algorithm for the dynamic and online adaptation of the number of particles during the operation of a particle filter on a stochastic version of the Lorenz 63 system.
2016-12-08T00:00:00ZParticle-kernel estimation of the filter density in state-space modelsCrisan, DanMíguez Arenas, Joaquínhttp://hdl.handle.net/10016/258962017-11-21T01:14:59Z2014-10-01T00:00:00ZParticle-kernel estimation of the filter density in state-space models
Crisan, Dan; Míguez Arenas, Joaquín
Sequential Monte Carlo (SMC) methods, also known as particle filters, are simulation-based recursive algorithms for the approximation of the a posteriori probability measures generated by state-space dynamical models. At any given time t, a SMC method produces a set of samples over the state space of the system of interest (often termed "particles") that is used to build a discrete and random approximation of the posterior probability distribution of the state variables, conditional on a sequence of available observations. One potential application of the methodology is the estimation of the densities associated to the sequence of a posteriori distributions. While practitioners have rather freely applied such density approximations in the past, the issue has received less attention from a theoretical perspective. In this paper, we address the problem of constructing kernel-based estimates of the posterior probability density function and its derivatives, and obtain asymptotic convergence results for the estimation errors. In particular, we find convergence rates for the approximation errors that hold uniformly on the state space and guarantee that the error vanishes almost surely as the number of particles in the filter grows. Based on this uniform convergence result, we first show how to build continuous measures that converge almost surely (with known rate) toward the posterior measure and then address a few applications. The latter include maximum a posteriori estimation of the system state using the approximate derivatives of the posterior density and the approximation of functionals of it, for example, Shannon's entropy.
2014-10-01T00:00:00ZNested particle filters for online parameter estimation in discrete-time state-space Markov modelsCrisan, DanMíguez Arenas, Joaquínhttp://hdl.handle.net/10016/258792017-11-18T01:15:40Z2017-05-10T00:00:00ZNested particle filters for online parameter estimation in discrete-time state-space Markov models
Crisan, Dan; Míguez Arenas, Joaquín
We address the problem of approximating the posterior probability distribution of the fixed parameters of a state-space dynamical system using a sequential Monte Carlo method. The proposed approach relies on a nested structure that employs two layers of particle filters to approximate the posterior probability measure of the static parameters and the dynamic state variables of the system of interest, in a vein similar to the recent "sequential Monte Carlo square" (SMC2) algorithm. However, unlike the SMC2 scheme, the proposed technique operates in a purely recursive manner. In particular, the computational complexity of the recursive steps of the method introduced herein is constant over time. We analyse the approximation of integrals of real bounded functions with respect to the posterior distribution of the system parameters computed via the proposed scheme. As a result, we prove, under regularity assumptions, that the approximation errors vanish asymptotically in Lp (p≥1) with convergence rate proportional to 1N√+1M√, where N is the number of Monte Carlo samples in the parameter space and N×M is the number of samples in the state space. This result also holds for the approximation of the joint posterior distribution of the parameters and the state variables. We discuss the relationship between the SMC2 algorithm and the new recursive method and present a simple example in order to illustrate some of the theoretical findings with computer simulations.
Documento depositado en el repositorio arXiv.org. Versión: arXiv:1308.1883v5 [stat.CO]
2017-05-10T00:00:00Z