1,323 research outputs found
Modeling and interpolation of the ambient magnetic field by Gaussian processes
Anomalies in the ambient magnetic field can be used as features in indoor
positioning and navigation. By using Maxwell's equations, we derive and present
a Bayesian non-parametric probabilistic modeling approach for interpolation and
extrapolation of the magnetic field. We model the magnetic field components
jointly by imposing a Gaussian process (GP) prior on the latent scalar
potential of the magnetic field. By rewriting the GP model in terms of a
Hilbert space representation, we circumvent the computational pitfalls
associated with GP modeling and provide a computationally efficient and
physically justified modeling tool for the ambient magnetic field. The model
allows for sequential updating of the estimate and time-dependent changes in
the magnetic field. The model is shown to work well in practice in different
applications: we demonstrate mapping of the magnetic field both with an
inexpensive Raspberry Pi powered robot and on foot using a standard smartphone.Comment: 17 pages, 12 figures, to appear in IEEE Transactions on Robotic
Stochastic partial differential equation based modelling of large space-time data sets
Increasingly larger data sets of processes in space and time ask for
statistical models and methods that can cope with such data. We show that the
solution of a stochastic advection-diffusion partial differential equation
provides a flexible model class for spatio-temporal processes which is
computationally feasible also for large data sets. The Gaussian process defined
through the stochastic partial differential equation has in general a
nonseparable covariance structure. Furthermore, its parameters can be
physically interpreted as explicitly modeling phenomena such as transport and
diffusion that occur in many natural processes in diverse fields ranging from
environmental sciences to ecology. In order to obtain computationally efficient
statistical algorithms we use spectral methods to solve the stochastic partial
differential equation. This has the advantage that approximation errors do not
accumulate over time, and that in the spectral space the computational cost
grows linearly with the dimension, the total computational costs of Bayesian or
frequentist inference being dominated by the fast Fourier transform. The
proposed model is applied to postprocessing of precipitation forecasts from a
numerical weather prediction model for northern Switzerland. In contrast to the
raw forecasts from the numerical model, the postprocessed forecasts are
calibrated and quantify prediction uncertainty. Moreover, they outperform the
raw forecasts, in the sense that they have a lower mean absolute error
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
The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensions
Inference and simulation in the context of high-dimensional dynamical systems
remain computationally challenging problems. Some form of dimensionality
reduction is required to make the problem tractable in general. In this paper,
we propose a novel approximate Gaussian filtering and smoothing method which
propagates low-rank approximations of the covariance matrices. This is
accomplished by projecting the Lyapunov equations associated with the
prediction step to a manifold of low-rank matrices, which are then solved by a
recently developed, numerically stable, dynamical low-rank integrator.
Meanwhile, the update steps are made tractable by noting that the covariance
update only transforms the column space of the covariance matrix, which is
low-rank by construction. The algorithm differentiates itself from existing
ensemble-based approaches in that the low-rank approximations of the covariance
matrices are deterministic, rather than stochastic. Crucially, this enables the
method to reproduce the exact Kalman filter as the low-rank dimension
approaches the true dimensionality of the problem. Our method reduces
computational complexity from cubic (for the Kalman filter) to \emph{quadratic}
in the state-space size in the worst-case, and can achieve \emph{linear}
complexity if the state-space model satisfies certain criteria. Through a set
of experiments in classical data-assimilation and spatio-temporal regression,
we show that the proposed method consistently outperforms the ensemble-based
methods in terms of error in the mean and covariance with respect to the exact
Kalman filter. This comes at no additional cost in terms of asymptotic
computational complexity.Comment: 12 pages main text (including references) + 9 pages appendix, 6
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