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A flexible state–space model for learning nonlinear dynamical systems
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Systems and Control. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Automatic control.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Systems and Control. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Automatic control.ORCID iD: 0000-0001-5183-234X
2017 (English)In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 80, p. 189-199Article in journal (Refereed) Published
Abstract [en]

We consider a nonlinear state-space model with the state transition and observation functions expressed as basis function expansions. The coefficients in the basis function expansions are learned from data. Using a connection to Gaussian processes we also develop priors on the coefficients, for tuning the model flexibility and to prevent overfitting to data, akin to a Gaussian process state-space model. The priors can alternatively be seen as a regularization, and helps the model in generalizing the data without sacrificing the richness offered by the basis function expansion. To learn the coefficients and other unknown parameters efficiently, we tailor an algorithm using state-of-the-art sequential Monte Carlo methods, which comes with theoretical guarantees on the learning. Our approach indicates promising results when evaluated on a classical benchmark as well as real data.

Place, publisher, year, edition, pages
2017. Vol. 80, p. 189-199
Keywords [en]
System identification, Nonlinear models, Regularization, Probabilistic models, Bayesian learning, Gaussian processes, Monte Carlo methods
National Category
Control Engineering
Identifiers
URN: urn:nbn:se:uu:diva-311584DOI: 10.1016/j.automatica.2017.02.030ISI: 000401391800023OAI: oai:DiVA.org:uu-311584DiVA, id: diva2:1060732
Funder
Swedish Research Council, 621-2013-5524Swedish Foundation for Strategic Research
Note

The material in this paper was partially presented at the 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), December 13-16, 2015, Cancun, Mexico and at the 19th International Conference on Artificial Intelligence and Statistics (AISTATS), May 9-11, 2016, Cadiz, Spain.

Available from: 2017-03-28 Created: 2016-12-29 Last updated: 2018-08-21Bibliographically approved
In thesis
1. Learning probabilistic models of dynamical phenomena using particle filters
Open this publication in new window or tab >>Learning probabilistic models of dynamical phenomena using particle filters
2016 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Dynamical behavior can be seen in many real-life phenomena, typically as a dependence over time. This thesis studies and develops methods and probabilistic models for statistical learning of such dynamical phenomena.

A probabilistic model is a mathematical model expressed using probability theory. Statistical learning amounts to constructing such models, as well as adjusting them to data recorded from real-life phenomena. The resulting models can be used for, e.g., drawing conclusions about the phenomena under study and making predictions.

The methods in this thesis are primarily based on the particle filter and its generalizations, sequential Monte Carlo (SMC) and particle Markov chain Monte Carlo (PMCMC). The model classes considered are nonlinear state-space models and Gaussian processes.

The following contributions are included. Starting with a Gaussian-process state-space model, a general, flexible and computationally feasible nonlinear state-space model is derived in Paper I. In Paper II, a benchmark is performed between the two alternative state-of-the-art methods SMCs and PMCMC. Paper III considers PMCMC for solving the state-space smoothing problem, in particular for an indoor positioning application. In Paper IV, SMC is used for marginalizing the hyperparameters in the Gaussian-process state-space model, and Paper V is concerned with learning of jump Markov linear state-space models. In addition, the thesis also contains an introductory overview covering statistical inference, state-space models, Gaussian processes and some advanced Monte Carlo methods, as well as two appendices summarizing some useful technical results.

Place, publisher, year, edition, pages
Uppsala University, 2016
Series
Information technology licentiate theses: Licentiate theses from the Department of Information Technology, ISSN 1404-5117 ; 2016-011
National Category
Control Engineering
Research subject
Electrical Engineering with specialization in Automatic Control
Identifiers
urn:nbn:se:uu:diva-311585 (URN)
Supervisors
Available from: 2016-11-18 Created: 2016-12-29 Last updated: 2016-12-29Bibliographically approved
2. Machine learning with state-space models, Gaussian processes and Monte Carlo methods
Open this publication in new window or tab >>Machine learning with state-space models, Gaussian processes and Monte Carlo methods
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Numbers are present everywhere, and when they are collected and recorded we refer to them as data. Machine learning is the science of learning mathematical models from data. Such models, once learned from data, can be used to draw conclusions, understand behavior, predict future evolution, and make decisions. This thesis is mainly concerned with two particular statistical models for this purpose: the state-space model and the Gaussian process model, as well as a combination thereof. To learn these models from data, Monte Carlo methods are used, and in particular sequential Monte Carlo (SMC) or particle filters.

The thesis starts with an introductory background on state-space models, Gaussian processes and Monte Carlo methods. The main contribution lies in seven scientific papers. Several contributions are made on the topic of learning nonlinear state-space models with the use of SMC. An existing SMC method is tailored for learning in state-space models with little or no measurement noise. The SMC-based method particle Gibbs with ancestor sampling (PGAS) is used for learning an approximation of the Gaussian process state-space model. PGAS is also combined with stochastic approximation expectation maximization (EM). This  method, which we refer to as particle stochastic approximation EM, is a general method for learning parameters in nonlinear state-space models. It is later applied to the particular problem of maximum likelihood estimation in jump Markov linear models. An alternative and non-standard approach for how to use SMC to estimate parameters in nonlinear state-space models is also presented.

There are also two contributions not related to learning state-space models. One is how SMC can be used also for learning hyperparameters in Gaussian process regression models. The second is a method for assessing consistency between model and data. By using the model to simulate new data, and compare how similar that data is to the observed one, a general criterion is obtained which follows directly from the model specification. All methods are implemented and illustrated, and several are also applied to various real-world examples.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2018. p. 74
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1709
Keywords
Machine learning, State-space models, Gaussian processes
National Category
Signal Processing Probability Theory and Statistics
Research subject
Electrical Engineering with specialization in Automatic Control
Identifiers
urn:nbn:se:uu:diva-357611 (URN)978-91-513-0417-5 (ISBN)
Public defence
2018-10-12, ITC 2446, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2018-09-18 Created: 2018-08-21 Last updated: 2018-10-02

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