Gaussian Process Latent Force Models for Learning and Stochastic Control of Physical Systems

© 1963-2012 IEEE. This paper is concerned with learning and stochastic control in physical systems that contain unknown input signals. These unknown signals are modeled as Gaussian processes (GP) with certain parameterized covariance structures. The resulting latent force models can be seen as hybrid models that contain a first-principle physical model part and a nonparametric GP model part. We briefly review the statistical inference and learning methods for this kind of models, introduce stochastic control methodology for these models, and provide new theoretical observability and controllability results for them.The work of S. Sarkka was financially supported by the Academy of Finland. The work of M. A. Alvarez was supported in part by the EPSRC under Research Project EP/N014162/1

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

Spin dynamics at the singlet-triplet crossings in a double quantum dot

Peer reviewe

Sound-source position tracking from direction-of-arrival measurements: Application to distributed first-order spherical microphone arrays

Rendering 6-degrees-of-freedom (6DoF) spatial audio requires sound-source position tracking. Without further assumptions, directional receivers, such as a spherical microphone array (SMA), can estimate the direction of arrival (DoA), but not reliably estimate sound-source distance. By utilizing multiple, distributed SMAs, further methods are available that directly infer the position in 3-D space. Typically used DoA intersection by triangulation delivers problematically noisy estimates, therefore, statistical filters are better suited. In this study, we compare the performance of different DoA to position tracking strategies. DoA angles suffer from the well-known angle wrapping problem, which is especially problematic in Gaussian filters. However, these filters are attractive due to their low computational complexity. Using circular and spherical statistics, the non- linear extensions of the Kalman filter can be formulated to explicitly treat the discontinuity of DoA angles. Furthermore, we introduce a time adaptive regularization of the filter update by the instantaneous sound-field diffuseness estimate. An experiment with three first-order SMAs in a reverberant room shows an improved distance error compared to the mean DoA intersection baseline. The results highlight the importance of treating the angle wrapping and the stabilization when incorporating the sound-field diffuseness estimate.publishedVersionNon peer reviewe

Bayesian ODE Solvers: The Maximum A Posteriori Estimate

It has recently been established that the numerical solution of ordinary differential equations can be posed as a nonlinear Bayesian inference problem, which can be approximately solved via Gaussian filtering and smoothing, whenever a Gauss--Markov prior is used. In this paper the class of $\nu$ times differentiable linear time invariant Gauss--Markov priors is considered. A taxonomy of Gaussian estimators is established, with the maximum a posteriori estimate at the top of the hierarchy, which can be computed with the iterated extended Kalman smoother. The remaining three classes are termed explicit, semi-implicit, and implicit, which are in similarity with the classical notions corresponding to conditions on the vector field, under which the filter update produces a local maximum a posteriori estimate. The maximum a posteriori estimate corresponds to an optimal interpolant in the reproducing Hilbert space associated with the prior, which in the present case is equivalent to a Sobolev space of smoothness $\nu+1$. Consequently, using methods from scattered data approximation and nonlinear analysis in Sobolev spaces, it is shown that the maximum a posteriori estimate converges to the true solution at a polynomial rate in the fill-distance (maximum step size) subject to mild conditions on the vector field. The methodology developed provides a novel and more natural approach to study the convergence of these estimators than classical methods of convergence analysis. The methods and theoretical results are demonstrated in numerical examples