21,852 research outputs found
Optimal Transport-based Nonlinear Filtering in High-dimensional Settings
This paper addresses the problem of nonlinear filtering, i.e., computing the
conditional distribution of the state of a stochastic dynamical system given a
history of noisy partial observations. The primary focus is on scenarios
involving degenerate likelihoods or high-dimensional states, where traditional
sequential importance resampling (SIR) particle filters face the weight
degeneracy issue. Our proposed method builds on an optimal transport
interpretation of nonlinear filtering, leading to a simulation-based and
likelihood-free algorithm that estimates the Brenier optimal transport map from
the current distribution of the state to the distribution at the next time
step. Our formulation allows us to harness the approximation power of neural
networks to model complex and multi-modal distributions and employ stochastic
optimization algorithms to enhance scalability. Extensive numerical experiments
are presented that compare our method to the SIR particle filter and the
ensemble Kalman filter, demonstrating the superior performance of our method in
terms of sample efficiency, high-dimensional scalability, and the ability to
capture complex and multi-modal distributions.Comment: 24 pages, 15 figure
Particle Learning and Smoothing
Particle learning (PL) provides state filtering, sequential parameter
learning and smoothing in a general class of state space models. Our approach
extends existing particle methods by incorporating the estimation of static
parameters via a fully-adapted filter that utilizes conditional sufficient
statistics for parameters and/or states as particles. State smoothing in the
presence of parameter uncertainty is also solved as a by-product of PL. In a
number of examples, we show that PL outperforms existing particle filtering
alternatives and proves to be a competitor to MCMC.Comment: Published in at http://dx.doi.org/10.1214/10-STS325 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Continuous-Discrete Path Integral Filtering
A summary of the relationship between the Langevin equation,
Fokker-Planck-Kolmogorov forward equation (FPKfe) and the Feynman path integral
descriptions of stochastic processes relevant for the solution of the
continuous-discrete filtering problem is provided in this paper. The practical
utility of the path integral formula is demonstrated via some nontrivial
examples. Specifically, it is shown that the simplest approximation of the path
integral formula for the fundamental solution of the FPKfe can be applied to
solve nonlinear continuous-discrete filtering problems quite accurately. The
Dirac-Feynman path integral filtering algorithm is quite simple, and is
suitable for real-time implementation.Comment: 35 pages, 18 figures, JHEP3 clas
Filtering and Smoothing with Score-Driven Models
We propose a methodology for filtering, smoothing and assessing parameter and
filtering uncertainty in misspecified score-driven models. Our technique is
based on a general representation of the well-known Kalman filter and smoother
recursions for linear Gaussian models in terms of the score of the conditional
log-likelihood. We prove that, when data are generated by a nonlinear
non-Gaussian state-space model, the proposed methodology results from a
first-order expansion of the true observation density around the optimal
filter. The error made by such approximation is assessed analytically. As shown
in extensive Monte Carlo analyses, our methodology performs very similarly to
exact simulation-based methods, while remaining computationally extremely
simple. We illustrate empirically the advantages in employing score-driven
models as misspecified filters rather than purely predictive processes.Comment: 33 pages, 5 figures, 6 table
Ensemble Kalman methods for high-dimensional hierarchical dynamic space-time models
We propose a new class of filtering and smoothing methods for inference in
high-dimensional, nonlinear, non-Gaussian, spatio-temporal state-space models.
The main idea is to combine the ensemble Kalman filter and smoother, developed
in the geophysics literature, with state-space algorithms from the statistics
literature. Our algorithms address a variety of estimation scenarios, including
on-line and off-line state and parameter estimation. We take a Bayesian
perspective, for which the goal is to generate samples from the joint posterior
distribution of states and parameters. The key benefit of our approach is the
use of ensemble Kalman methods for dimension reduction, which allows inference
for high-dimensional state vectors. We compare our methods to existing ones,
including ensemble Kalman filters, particle filters, and particle MCMC. Using a
real data example of cloud motion and data simulated under a number of
nonlinear and non-Gaussian scenarios, we show that our approaches outperform
these existing methods
Nonlinear state space smoothing using the conditional particle filter
To estimate the smoothing distribution in a nonlinear state space model, we
apply the conditional particle filter with ancestor sampling. This gives an
iterative algorithm in a Markov chain Monte Carlo fashion, with asymptotic
convergence results. The computational complexity is analyzed, and our proposed
algorithm is successfully applied to the challenging problem of sensor fusion
between ultra-wideband and accelerometer/gyroscope measurements for indoor
positioning. It appears to be a competitive alternative to existing nonlinear
smoothing algorithms, in particular the forward filtering-backward simulation
smoother.Comment: Accepted for the 17th IFAC Symposium on System Identification
(SYSID), Beijing, China, October 201
Universal Nonlinear Filtering Using Feynman Path Integrals II: The Continuous-Continuous Model with Additive Noise
In this paper, the Feynman path integral formulation of the
continuous-continuous filtering problem, a fundamental problem of applied
science, is investigated for the case when the noise in the signal and
measurement model is additive. It is shown that it leads to an independent and
self-contained analysis and solution of the problem. A consequence of this
analysis is Feynman path integral formula for the conditional probability
density that manifests the underlying physics of the problem. A corollary of
the path integral formula is the Yau algorithm that has been shown to be
superior to all other known algorithms. The Feynman path integral formulation
is shown to lead to practical and implementable algorithms. In particular, the
solution of the Yau PDE is reduced to one of function computation and
integration.Comment: Interdisciplinary, 41 pages, 5 figures, JHEP3 class; added more
discussion and reference
A Probabilistic Perspective on Gaussian Filtering and Smoothing
We present a general probabilistic perspective on Gaussian filtering and smoothing. This allows us to show that common approaches to Gaussian filtering/smoothing can be distinguished solely by their methods of computing/approximating the means and covariances of joint probabilities. This implies that novel filters and smoothers can be derived straightforwardly by providing methods for computing these moments. Based on this insight, we derive the cubature Kalman smoother and propose a novel robust filtering and smoothing algorithm based on Gibbs sampling
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