arXiv subjects

Joaquin Miguez

Publications and source records attributed to Joaquin Miguez.

At least 19 recordsLinked to original sources

arXiv2026

On a class of constrained particle filters for continuous-discrete state space models

Particle filters (PFs) are recursive Monte Carlo algorithms for Bayesian tracking and prediction in state space models. This paper addresses continuous-discrete filtering problems, where the hidden state evolves as an It\^o stochastic differential equation (SDE) and observations arrive at discrete times. We propose a novel class of constrained PFs that enforce compact support on the state at each observation instant, thereby limiting exploration to plausible regions of the state space. Unlike earlier approaches that truncate the likelihood, the proposed method constrains the dynamics directly, yielding improved numerical stability. Under standard regularity assumptions, we prove convergence of the constrained filter, derive uniform-in-time error estimates, and extend the analysis to account for discretisation errors arising from numerical SDE solvers. A numerical study on a stochastic Lorenz-96 system demonstrates the practical application of the methodology when the constraint is implemented via barrier functions.

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arXiv2025

On a class of constrained Bayesian filters and their numerical implementation in high-dimensional state-space Markov models

Bayesian filtering is a key tool in many problems that involve the online processing of data, including data assimilation, optimal control, nonlinear tracking and others. Unfortunately, the implementation of filters for nonlinear, possibly high-dimensional, dynamical systems is far from straightforward, as computational methods have to meet a delicate trade-off involving stability, accuracy and computational cost. In this paper we investigate the design, and theoretical features, of constrained Bayesian filters for state space models. The constraint on the filter is given by a sequence of compact subsets of the state space that determines the sources and targets of the Markov transition kernels in the dynamical model. Subject to such constraints, we provide sufficient conditions for filter stability and approximation error rates with respect to the original (unconstrained) Bayesian filter. Then, we look specifically into the implementation of constrained filters in a continuous-discrete setting where the state of the system is a continuous-time stochastic It\^o process but data are collected sequentially over a time grid. We propose an implementation of the constraint that relies on a data-driven modification of the drift of the It\^o process using barrier functions, and discuss the relation of this scheme with methods based on the Doob $h$-transform. Finally, we illustrate the theoretical results and the performance of the proposed methods in computer experiments for a partially-observed stochastic Lorenz 96 model.

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arXiv2025

Data-driven informative priors for Bayesian inference with quasi-periodic data

Bayesian computational strategies for inference can be inefficient in approximating the posterior distribution in models that exhibit some form of periodicity. This is because the probability mass of the marginal posterior distribution of the parameter representing the period is usually highly concentrated in a very small region of the parameter space. Therefore, it is necessary to provide as much information as possible to the inference method through the parameter prior distribution. We intend to show that it is possible to construct a prior distribution from the data by fitting a Gaussian process (GP) with a periodic kernel. More specifically, we want to show that it is possible to approximate the marginal posterior distribution of the hyperparameter corresponding to the period in the kernel. Subsequently, this distribution can be used as a prior distribution for the inference method. We use an adaptive importance sampling method to approximate the posterior distribution of the hyperparameters of the GP. Then, we use the marginal posterior distribution of the hyperparameter related to the periodicity in order to construct a prior distribution for the period of the parametric model. This workflow is empirical Bayes, implemented as a modular (cut) transfer of a GP posterior for the period to the parametric model. We applied the proposed methodology to both synthetic and real data. We approximated the posterior distribution of the period of the GP kernel and then passed it forward as a posterior-as-prior with no feedback. Finally, we analyzed its impact on the marginal posterior distribution.

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arXiv2025

Nudging state-space models for Bayesian filtering under misspecified dynamics

Nudging is a popular algorithmic strategy in numerical filtering to deal with the problem of inference in high-dimensional dynamical systems. We demonstrate in this paper that general nudging techniques can also tackle another crucial statistical problem in filtering, namely the misspecification of the transition kernel. Specifically, we rely on the formulation of nudging as a general operation increasing the likelihood and prove analytically that, when applied carefully, nudging techniques implicitly define state-space models that have higher marginal likelihoods for a given (fixed) sequence of observations. This provides a theoretical justification of nudging techniques as data-informed algorithmic modifications of state-space models to obtain robust models under misspecified dynamics. To demonstrate the use of nudging, we provide numerical experiments on linear Gaussian state-space models and a stochastic Lorenz 63 model with misspecified dynamics and show that nudging offers a robust filtering strategy for these cases.

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arXiv2025

Sequential Filtering Techniques for Simultaneous Tracking and Parameter Estimation

The number of resident space objects is rising at an alarming rate. Mega-constellations and breakup events are proliferating in most orbital regimes, and safe navigation is becoming increasingly problematic. It is important to be able to track RSOs accurately and at an affordable computational cost. Orbital dynamics are highly nonlinear, and current operational methods assume Gaussian representations of the objects' states and employ linearizations which cease to hold true in observation-free propagation. Monte Carlo-based filters can provide a means to approximate the a posteriori probability distribution of the states more accurately by providing support in the portion of the state space which overlaps the most with the processed observations. Moreover, dynamical models are not able to capture the full extent of realistic forces experienced in the near-Earth space environment, and hence fully deterministic propagation methods may fail to achieve the desired accuracy. By modeling orbital dynamics as a stochastic system and solving it using stochastic numerical integrators, we are able to simultaneously estimate the scale of the process noise incurred by the assumed uncertainty in the system, and robustly track the state of the spacecraft. In order to find an adequate balance between accuracy and computational cost, we propose three algorithms which are capable of tracking a space object and estimating the magnitude of the system's uncertainty. The proposed filters are successfully applied to a LEO scenario, demonstrating the ability to accurately track a spacecraft state and estimate the scale of the uncertainty online, in various simulation setups.

astro-ph.IM↗

arXiv2025

On the contraction properties of Sinkhorn semigroups

We develop a novel stability theory for Sinkhorn semigroups based on Lyapunov techniques and quantitative contraction coefficients, and establish exponential convergence of Sinkhorn iterations on weighted Banach spaces. This operator-theoretic framework yields explicit exponential decay rates of Sinkhorn iterates toward Schr\"odinger bridges with respect to a broad class of $\phi$-divergences and Kantorovich-type distances, including relative entropy, squared Hellinger integrals, $\alpha$-divergences, weighted total variation norms, and Wasserstein distances. To the best of our knowledge, these results provide the first systematic contraction inequalities of this kind for entropic transport and the Sinkhorn algorithm. We further introduce Lyapunov contraction principles under minimal regularity assumptions, leading to quantitative exponential stability estimates for a large family of Sinkhorn semigroups. The framework applies to models with polynomially growing potentials and heavy-tailed marginals on general normed spaces, as well as to more structured boundary state-space models, including semicircle transitions and Beta, Weibull, and exponential marginals, together with semi-compact settings. Finally, our approach extends naturally to statistical finite mixtures of such models, including kernel-based density estimators arising in modern generative modeling.

math.PR↗

arXiv2024

A master-slave coupling scheme for synchronization and parameter estimation in the generalized Kuramoto-Sivashinsky equation

The problem of estimating the constant parameters of the Kuramoto-Sivashinsky (KS) equation from observed data has received attention from researchers in physics, applied mathematics and statistics. This is motivated by the various physical applications of the equation and, also, because it often serves as a test model for the study of space-time pattern formation. Remarkably, most existing inference techniques rely on statistical tools, which are computationally very costly yet do not exploit the dynamical features of the system. In this paper we introduce a simple, online parameter estimation method that relies on the synchronization properties of the KS equation. In particular, we describe a master-slave setup where the slave model is driven by observations from the master system. The slave dynamics are data-driven and designed to continuously adapt the model parameters until identical synchronization with the master system is achieved. We provide a simple analysis that supports the proposed approach and also present and discuss the results of an extensive set of computer simulations. Our numerical study shows that the proposed method is computationally fast and also robust to initialization errors, observational noise and variations in the spatial resolution of the numerical scheme used to integrate the KS equation.

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arXiv2024

Nested smoothing algorithms for inference and tracking of heterogeneous multi-scale state-space systems

Multi-scale problems, where variables of interest evolve in different time-scales and live in different state-spaces, can be found in many fields of science. Here, we introduce a new recursive methodology for Bayesian inference that aims at estimating the static parameters and tracking the dynamic variables of these kind of systems. Although the proposed approach works in rather general multi-scale systems, for clarity we analyze the case of a heterogeneous multi-scale model with 3 time-scales (static parameters, slow dynamic state variables and fast dynamic state variables). The proposed scheme, based on nested filtering methodology of Pérez-Vieites et al. (2018), combines three intertwined layers of filtering techniques that approximate recursively the joint posterior probability distribution of the parameters and both sets of dynamic state variables given a sequence of partial and noisy observations. We explore the use of sequential Monte Carlo schemes in the first and second layers while we use an unscented Kalman filter to obtain a Gaussian approximation of the posterior probability distribution of the fast variables in the third layer. Some numerical results are presented for a stochastic two-scale Lorenz 96 model with unknown parameters.

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arXiv2024

Sequential discretisation schemes for a class of stochastic differential equations and their application to Bayesian filtering

We introduce a predictor-corrector discretisation scheme for the numerical integration of a class of stochastic differential equations and prove that it converges with weak order 1.0. The key feature of the new scheme is that it builds up sequentially (and recursively) in the dimension of the state space of the solution, hence making it suitable for approximations of high-dimensional state space models. We show, using the stochastic Lorenz 96 system as a test model, that the proposed method can operate with larger time steps than the standard Euler-Maruyama scheme and, therefore, generate valid approximations with a smaller computational cost. We also introduce the theoretical analysis of the error incurred by the new predictor-corrector scheme when used as a building block for discrete-time Bayesian filters for continuous-time systems. Finally, we assess the performance of several ensemble Kalman filters that incorporate the proposed sequential predictor-corrector Euler scheme and the standard Euler-Maruyama method. The numerical experiments show that the filters employing the new sequential scheme can operate with larger time steps, smaller Monte Carlo ensembles and noisier systems.

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arXiv2021

Polynomial propagation of moments in stochastic differential equations

We address the problem of approximating the moments of the solution, $\boldsymbol{X}(t)$, of an Itô stochastic differential equation (SDE) with drift and a diffusion terms over a time-grid $t_0, t_1, \ldots, t_n$. In particular, we assume an explicit numerical scheme for the generation of sample paths $\hat{\boldsymbol{X}}(t_0), \ldots, \hat{\boldsymbol{X}}(t_n), \ldots$ and then obtain recursive equations that yield any desired non-central moment of $\hat{\boldsymbol{X}}(t_n)$ as a function of the initial condition $\boldsymbol{X}_0$. The core of the methodology is the decomposition of the numerical solution into a "central part" and an "effective noise" term. The central term is computed deterministically from the ordinary differential equation (ODE) that results from eliminating the diffusion term in the SDE, while the effective noise accounts for the stochastic deviation from the numerical solution of the ODE. For simplicity, we describe algorithms based on an Euler-Maruyama integrator, but other explicit numerical schemes can be exploited in the same way. We also apply the moment approximations to construct estimates of the 1-dimensional marginal probability density functions of $\hat{\boldsymbol{X}}(t_n)$ based on a Gram-Charlier expansion. Both for the approximation of moments and 1-dimensional densities, we describe how to handle the cases in which the initial condition is fixed (i.e., $\boldsymbol{X}_0 = \boldsymbol{x}_0$ for some known $\boldsymbol{x_0}$) or random. In the latter case, we resort to polynomial chaos expansion (PCE) schemes to approximate the target moments. The methodology has been inspired by the PCE and differential algebra (DA) methods used for uncertainty propagation in astrodynamics problems. Hence, we illustrate its application for the quantification of uncertainty in a 2-dimensional Keplerian orbit perturbed by a Wiener noise process.

math.NA↗

arXiv2019

Stable approximation schemes for optimal filters

A stable filter has the property that it asymptotically `forgets' initial perturbations. As a result of this property, it is possible to construct approximations of such filters whose errors remain small in time, in other words approximations that are uniformly convergent in the time variable. As uniform approximations are ideal from a practical perspective, finding criteria for filter stability has been the subject of many papers. In this paper we seek to construct approximate filters that stay close to a given (possibly) unstable filter. Such filters are obtained through a general truncation scheme and, under certain constraints, are stable. The construction enables us to give a characterisation of the topological properties of the set of optimal filters. In particular, we introduce a natural topology on this set, under which the subset of stable filters is dense.

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arXiv2018

The Incremental Proximal Method: A Probabilistic Perspective

In this work, we highlight a connection between the incremental proximal method and stochastic filters. We begin by showing that the proximal operators coincide, and hence can be realized with, Bayes updates. We give the explicit form of the updates for the linear regression problem and show that there is a one-to-one correspondence between the proximal operator of the least-squares regression and the Bayes update when the prior and the likelihood are Gaussian. We then carry out this observation to a general sequential setting: We consider the incremental proximal method, which is an algorithm for large-scale optimization, and show that, for a linear-quadratic cost function, it can naturally be realized by the Kalman filter. We then discuss the implications of this idea for nonlinear optimization problems where proximal operators are in general not realizable. In such settings, we argue that the extended Kalman filter can provide a systematic way for the derivation of practical procedures.

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arXiv2017

Analysis of a nonlinear importance sampling scheme for Bayesian parameter estimation in state-space models

The Bayesian estimation of the unknown parameters of state-space (dynamical) systems has received considerable attention over the past decade, with a handful of powerful algorithms being introduced. In this paper we tackle the theoretical analysis of the recently proposed {\it nonlinear} population Monte Carlo (NPMC). This is an iterative importance sampling scheme whose key features, compared to conventional importance samplers, are (i) the approximate computation of the importance weights (IWs) assigned to the Monte Carlo samples and (ii) the nonlinear transformation of these IWs in order to prevent the degeneracy problem that flaws the performance of conventional importance samplers. The contribution of the present paper is a rigorous proof of convergence of the nonlinear IS (NIS) scheme as the number of Monte Carlo samples, $M$, increases. Our analysis reveals that the NIS approximation errors converge to 0 almost surely and with the optimal Monte Carlo rate of $M^{-\frac{1}{2}}$. Moreover, we prove that this is achieved even when the mean estimation error of the IWs remains constant, a property that has been termed {\it exact approximation} in the Markov chain Monte Carlo literature. We illustrate these theoretical results by means of a computer simulation example involving the estimation of the parameters of a state-space model typically used for target tracking.

stat.CO↗

arXiv2016

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

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 this algorithm to the recent sequential Monte Carlo square (SMC$^2$) method, which provides a {\em non-recursive} solution to the same problem. In this paper, we investigate the approximation, via the proposed scheme, 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 $L_p$ 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.

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arXiv2015

An iterative importance sampler for Bayesian parameter estimation in stochastic models of multicellular clocks

We investigate a stochastic version of the synthetic multicellular clock model proposed by Garcia-Ojalvo, Elowitz and Strogatz. By introducing dynamical noise in the model and assuming that the partial observations of the system can be contaminated by additive noise, we enable a principled mechanism to represent experimental uncertainties in the synthesis of the multicellular system and pave the way for the design of probabilistic methods for the estimation of any unknowns in the model. Within this setup, we investigate the use of an iterative importance sampling scheme, termed nonlinear population Monte Carlo (NPMC), for the Bayesian estimation of the model parameters. The algorithm yields a stochastic approximation of the posterior probability distribution of the unknown parameters given the available data (partial and possibly noisy observations). We prove a new theoretical result for this algorithm, which indicates that the approximations converge almost surely to the actual distributions, even when the weights in the importance sampling scheme cannot be computed exactly. We also provide a detailed numerical assessment of the stochastic multicellular model and the accuracy of the proposed NPMC algorithm, including a comparison with the popular particle Metropolis-Hastings algorithm of Andrieu {\em et al.}, 2010, applied to the same model and an approximate Bayesian computation sequential Monte Carlo method introduced by Mariño {\em et al.}, 2013.

stat.CO↗

arXiv2015

A proof of uniform convergence over time for a distributed particle filter

Distributed signal processing algorithms have become a hot topic during the past years. One class of algorithms that have received special attention are particles filters (PFs). However, most distributed PFs involve various heuristic or simplifying approximations and, as a consequence, classical convergence theorems for standard PFs do not hold for their distributed counterparts. In this paper, we analyze a distributed PF based on the non-proportional weight-allocation scheme of Bolic {\em et al} (2005) and prove rigorously that, under certain stability assumptions, its asymptotic convergence is guaranteed uniformly over time, in such a way that approximation errors can be kept bounded with a fixed computational budget. To illustrate the theoretical findings, we carry out computer simulations for a target tracking problem. The numerical results show that the distributed PF has a negligible performance loss (compared to a centralized filter) for this problem and enable us to empirically validate the key assumptions of the analysis.

stat.CO↗

arXiv2015

A simple scheme for the parallelization of particle filters and its application to the tracking of complex stochastic systems

We investigate the use of possibly the simplest scheme for the parallelisation of the standard particle filter, that consists in splitting the computational budget into $M$ fully independent particle filters with $N$ particles each, and then obtaining the desired estimators by averaging over the $M$ independent outcomes of the filters. This approach minimises the parallelisation overhead yet displays highly desirable theoretical properties. Under very mild assumptions, we analyse the mean square error (MSE) of the estimators of 1-dimensional statistics of the optimal filtering distribution and show explicitly the effect of parallelisation scheme on the convergence rate. Specifically, we study the decomposition of the MSE into variance and bias components, to show that the former decays as $\frac{1}{MN}$, i.e., linearly with the total number of particles, while the latter converges towards $0$ as $\frac{1}{N^2}$. Parallelisation, therefore, has the obvious advantage of dividing the running times while preserving the (asymptotic) performance of the particle filter. Following this lead, we propose a time-error index to compare schemes with different degrees of parallelisation. Finally, we provide two numerical examples. The first one deals with the tracking of a Lorenz 63 chaotic system with dynamical noise and partial (noisy) observations, while the second example involves a dynamical network of modified FitzHugh-Nagumo (FH-N) stochastic nodes. The latter is a large dimensional system ($\approx3,000$ state variables in our computer experiments) designed to numerically reproduce typical electrical phenomena observed in the atria of the human heart. In both examples, we show how the proposed parallelisation scheme attains the same approximation accuracy as a centralised particle filter with only a small fraction of the running time, using a standard multicore computer.

stat.CO↗

arXiv2015

A nonlinear population Monte Carlo scheme for the Bayesian estimation of parameters of $α$-stable distributions

The class of $α$-stable distributions enjoys multiple practical applications in signal processing, finance, biology and other areas because it allows to describe interesting and complex data patterns, such as asymmetry or heavy tails, in contrast with the simpler and widely used Gaussian distribution. The density associated with a general $α$-stable distribution cannot be obtained in closed form, which hinders the process of estimating its parameters. A nonlinear population Monte Carlo (NPMC) scheme is applied in order to approximate the posterior probability distribution of the parameters of an $α$-stable random variable given a set of random realizations of the latter. The approximate posterior distribution is computed by way of an iterative algorithm and it consists of a collection of samples in the parameter space with associated nonlinearly-transformed importance weights. A numerical comparison of the main existing methods to estimate the $α$-stable parameters is provided, including the traditional frequentist techniques as well as a Markov chain Monte Carlo (MCMC) and a likelihood-free Bayesian approach. It is shown by means of computer simulations that the NPMC method outperforms the existing techniques in terms of parameter estimation error and failure rate for the whole range of values of $α$, including the smaller values for which most existing methods fail to work properly. Furthermore, it is shown that accurate parameter estimates can often be computed based on a low number of observations. Additionally, numerical results based on a set of real fish displacement data are provided.

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