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Learning of state-space models with highly informative observations: A tempered sequential Monte Carlo solution
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Automatic control. (Syscon)
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Automatic control. (Syscon)ORCID iD: 0000-0001-5183-234X
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Automatic control. (Syscon)
2018 (English)In: Mechanical systems and signal processing, ISSN 0888-3270, E-ISSN 1096-1216, Vol. 104, p. 915-928Article in journal (Refereed) Published
Abstract [en]

Probabilistic (or Bayesian) modeling and learning offers interesting possibilities for systematic representation of uncertainty using probability theory. However, probabilistic learning often leads to computationally challenging problems. Some problems of this type that were previously intractable can now be solved on standard personal computers thanks to recent advances in Monte Carlo methods. In particular, for learning of unknown parameters in nonlinear state-space models, methods based on the particle filter (a Monte Carlo method) have proven very useful. A notoriously challenging problem, however, still occurs when the observations in the state-space model are highly informative, i.e. when there is very little or no measurement noise present, relative to the amount of process noise. The particle filter will then struggle in estimating one of the basic components for probabilistic learning, namely the likelihood p(datalparameters). To this end we suggest an algorithm which initially assumes that there is substantial amount of artificial measurement noise present. The variance of this noise is sequentially decreased in an adaptive fashion such that we, in the end, recover the original problem or possibly a very close approximation of it. The main component in our algorithm is a sequential Monte Carlo (SMC) sampler, which gives our proposed method a clear resemblance to the SMC2 method. Another natural link is also made to the ideas underlying the approximate Bayesian computation (ABC). We illustrate it with numerical examples, and in particular show promising results for a challenging Wiener-Hammerstein benchmark problem.

Place, publisher, year, edition, pages
ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD , 2018. Vol. 104, p. 915-928
Keywords [en]
Probabilistic modelling, Bayesian methods, Nonlinear system identification, Sequential Monte Carlo, Particle filter, Approximate Bayesian computations, Highly informative observations, Tempering, Wiener-Hammerstein model
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:uu:diva-350269DOI: 10.1016/j.ymssp.2017.09.016ISI: 000423652800057OAI: oai:DiVA.org:uu-350269DiVA, id: diva2:1206003
Funder
Swedish Research Council, 621-2013-5524, 2016-04278, 621-2016-06079Swedish Foundation for Strategic Research , RIT15-0012Available from: 2018-05-15 Created: 2018-05-15 Last updated: 2018-08-21Bibliographically approved
In thesis
1. 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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Svensson, AndreasSchön, Thomas B.Lindsten, Fredrik

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