402 research outputs found
Iterated Posterior Linearization Smoother
This note considers the problem of Bayesian smoothing in nonlinear state-space models with additive noise using Gaussian approximations. Sigma-point approximations to the general Gaussian Rauch-Tung-Striebel smoother are widely used methods to tackle this problem. These algorithms perform statistical linear regression (SLR) of the nonlinear functions considering only the previous measurements. We argue that SLR should be done taking all measurements into account. We propose the iterated posterior linearization smoother (IPLS), which is an iterated algorithm that performs SLR of the nonlinear functions with respect to the current posterior approximation. The algorithm is demonstrated to outperform conventional Gaussian nonlinear smoothers in two numerical examples
Iterated Filters for Nonlinear Transition Models
A new class of iterated linearization-based nonlinear filters, dubbed
dynamically iterated filters, is presented. Contrary to regular iterated
filters such as the iterated extended Kalman filter (IEKF), iterated unscented
Kalman filter (IUKF) and iterated posterior linearization filter (IPLF),
dynamically iterated filters also take nonlinearities in the transition model
into account. The general filtering algorithm is shown to essentially be a
(locally over one time step) iterated Rauch-Tung-Striebel smoother. Three
distinct versions of the dynamically iterated filters are especially
investigated: analogues to the IEKF, IUKF and IPLF. The developed algorithms
are evaluated on 25 different noise configurations of a tracking problem with a
nonlinear transition model and linear measurement model, a scenario where
conventional iterated filters are not useful. Even in this "simple" scenario,
the dynamically iterated filters are shown to have superior root mean-squared
error performance as compared with their respective baselines, the EKF and UKF.
Particularly, even though the EKF diverges in 22 out of 25 configurations, the
dynamically iterated EKF remains stable in 20 out of 25 scenarios, only
diverging under high noise.Comment: 8 pages. Accepted to IEEE International Conference on Information
Fusion 2023 (FUSION 2023). Copyright 2023 IEE
Levenberg-Marquardt and Line-Search Extended Kalman Smoothers
The aim of this article is to present Levenberg–Marquardt and line-search extensions of the classical iterated extended Kalman smoother (IEKS) which has previously been shown to be equivalent to the Gauss–Newton method. The algo- rithms are derived by rewriting the algorithm’s steps in forms that can be efficiently implemented using modified EKS iter- ations. The resulting algorithms are experimentally shown to have superior convergence properties over the classical IEKS
A Probabilistic State Space Model for Joint Inference from Differential Equations and Data
Mechanistic models with differential equations are a key component of
scientific applications of machine learning. Inference in such models is
usually computationally demanding, because it involves repeatedly solving the
differential equation. The main problem here is that the numerical solver is
hard to combine with standard inference techniques. Recent work in
probabilistic numerics has developed a new class of solvers for ordinary
differential equations (ODEs) that phrase the solution process directly in
terms of Bayesian filtering. We here show that this allows such methods to be
combined very directly, with conceptual and numerical ease, with latent force
models in the ODE itself. It then becomes possible to perform approximate
Bayesian inference on the latent force as well as the ODE solution in a single,
linear complexity pass of an extended Kalman filter / smoother - that is, at
the cost of computing a single ODE solution. We demonstrate the expressiveness
and performance of the algorithm by training, among others, a non-parametric
SIRD model on data from the COVID-19 outbreak.Comment: 12 pages (+ 5 pages appendix), 7 figures. In: Advances in Neural
Information Processing Systems (NeurIPS 2021
Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation
In this paper, we present the optimization formulation of the Kalman
filtering and smoothing problems, and use this perspective to develop a variety
of extensions and applications. We first formulate classic Kalman smoothing as
a least squares problem, highlight special structure, and show that the classic
filtering and smoothing algorithms are equivalent to a particular algorithm for
solving this problem. Once this equivalence is established, we present
extensions of Kalman smoothing to systems with nonlinear process and
measurement models, systems with linear and nonlinear inequality constraints,
systems with outliers in the measurements or sudden changes in the state, and
systems where the sparsity of the state sequence must be accounted for. All
extensions preserve the computational efficiency of the classic algorithms, and
most of the extensions are illustrated with numerical examples, which are part
of an open source Kalman smoothing Matlab/Octave package.Comment: 46 pages, 11 figure
Maximum likelihood estimation of time series models: the Kalman filter and beyond
The purpose of this chapter is to provide a comprehensive treatment of likelihood inference for state space models. These are a class of time series models relating an observable time series to quantities called states, which are characterized by a simple temporal dependence structure, typically a first order Markov process. The states have sometimes substantial interpretation. Key estimation problems in economics concern latent variables, such as the output gap, potential output, the non-accelerating-inflation rate of unemployment, or NAIRU, core inflation, and so forth. Time-varying volatility, which is quintessential to finance, is an important feature also in macroeconomics. In the multivariate framework relevant features can be common to different series, meaning that the driving forces of a particular feature and/or the transmission mechanism are the same. The objective of this chapter is reviewing this algorithm and discussing maximum likelihood inference, starting from the linear Gaussian case and discussing the extensions to a nonlinear and non Gaussian framework
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