Iterative and Geometric Methods for State Estimation in Non-linear Models
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School of Electrical Engineering |
Doctoral thesis (article-based)
| Defence date: 2020-01-31
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Authors
Date
2020
Major/Subject
Mcode
Degree programme
Language
en
Pages
90 + app. 75
Series
Aalto University publication series DOCTORAL DISSERTATIONS, 12/2020
Abstract
Many problems in science and engineering involve estimating a dynamic signal from indirect measurements subject to noise, where points can either evolve in continuous time or in discrete time. These problems are often formalised as inference in probabilistic state-space models, which are also frequently assumed to be Markovian. For inferring the value of the signal at a particular point in time, methods of inference can be divided into different classes, namely prediction, filtering (tracking), and smoothing. In prediction, only past measurements of the signal are used to infer its present value, whereas in filtering both past and present measurements are used, and in smoothing past, present, and future measurements are used. Prediction is useful in situations where decisions need to be made contingent on a future value of the signal before future measurements are made. On the other hand, filtering is useful when the signal needs to be inferred as the measurements arrive, that is on-line. Lastly, smoothing is the preferred choice when none of the aforementioned constraints are present as it allows the use of the entire sequence of measurements to infer the signal. In this thesis, the filtering and smoothing problems and their applications are examined. In particular, iterative Gaussian filters and smoothers are developed for both inferring continuous and discrete time signals. Furthermore, it is shown that methods for inference in state-space models can be applied to the field of probabilistic numerics. More specifically, estimating the solutions to ordinary differential equations can be formulated as inference in a probabilistic state-space model, hence the solutions can be inferred using either Gaussian filtering methods or sequential Monte Carlo. Another theme of this thesis is the exploitation of geometry - in a broad sense. Firstly, the geometry of probability densities, namely information geometry, is exploited to approximately infer the signal in filtering. Geometry is also exploited in terms of the geometry of the state-space, the space where the signal takes its values. That is, for tracking a time-varying unit vector, a continuous-time dynamic model is posed that respects the geometry of the unit sphere. Subsequently, a filtering algorithm is developed based on the von Mises-Fisher distribution for inference in this model. The method is demonstrated to have applications in tracking the local gravity and magnetic field vectors using a smartphone. Lastly, the geometry of L_2 spaces is used to approximate a stochastic differential equation with an ordinary differential equation with random coefficients. On this basis filtering and smoothing algorithms are developed.Description
Supervising professor
Särkkä, Simo, Prof., Aalto University, Department of Electrical Engineering and Automation, FinlandThesis advisor
Särkkä, Simo, Prof., Aalto University, Department of Electrical Engineering and Automation, FinlandKeywords
information geometry, iterative Gaussian filters, smoothing, models
Other note
Parts
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[Publication 1]: Filip Tronarp, Simo Särkkä. Iterative statistical linear regression for Gaussian smoothing in continuous-time non-linear stochastic dynamic systems. Signal Processing, 2019, Volume 159, pages 1–12.
Full text in Acris/Aaltodoc: http://urn.fi/URN:NBN:fi:aalto-201902251828DOI: 10.1016/j.sigpro.2019.01.013 View at publisher
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[Publication 2]: Filip Tronarp, Ángel García–Fernández, Simo Särkkä. Iterative Filtering and Smoothing in Nonlinear and Non-Gaussian Systems Using Conditional Moments. IEEE Signal Processing Letters, 2018, 25, 3, 408–412.
Full text in Acris/Aaltodoc: http://urn.fi/URN:NBN:fi:aalto-201803231782DOI: 10.1109/LSP.2018.2794767 View at publisher
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[Publication 3]: Filip Tronarp, Hans Kersting, Simo Särkkä, Philipp Hennig. Probabilistic Solutions to Ordinary Differential Equations as Non-Linear Bayesian Filtering: A New Perspective. Statistics and Computing, 2019, Volume 29, Issue 6, pages 1297–1315. Full text in Acris/Aalto: http://urn.fi/URN:NBN:fi:aalto-202001021360.
DOI: 10.1007/s11222-019-09900-1 View at publisher
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[Publication 4]: Filip Tronarp, Simo Särkkä. Updates in Bayesian Filtering by Continuous Projections on a Manifold of Densities. In Proceedings of the International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brighton, UK, pages 5032–5036, May 2019.
Full text in Acris/Aaltodoc: http://urn.fi/URN:NBN:fi:aalto-201908154668DOI: 10.1109/ICASSP.2019.8682279 View at publisher
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[Publication 5]: Filip Tronarp, Roland Hostettler, Simo Särkkä. Continuous-Discrete von Mises-Fisher Filtering on S2 for Reference Vector Tracking. In Proceedings of the 21st International Conference on Information Fusion (FUSION), Cambridge, UK, pages 1345–1352, July 2018.
Full text in Acris/Aaltodoc: http://urn.fi/URN:NBN:fi:aalto-201812106213DOI: 10.23919/ICIF.2018.8455299 View at publisher
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[Publication 6]: Filip Tronarp, Simo Särkkä. Non-Linear Continuous-Discrete Smoothing By Basis Function Expansions Of Brownian Motion. In Proceedings of the 21st International Conference on Information Fusion (FUSION), Cambridge, UK, pages 1114–1121, July 2018.
Full text in Acris/Aaltodoc: http://urn.fi/URN:NBN:fi:aalto-201902251807DOI: 10.23919/ICIF.2018.8455493 View at publisher
- [Errata file]: Errata of P3, P4, P6