# Public defence in Bioelectronics and Instrumentation, M.Sc.(Tech) Juha Sarmavuori

New state estimation and numerical integration methods are developed based on Hilbert-space theory.

Public defence from the Aalto University School of Electrical Engineering, Department of Electrical Engineering and Automation

The title of the thesis: Hilbert Space Projection Methods for Numerical Integration and State Estimation

Doctoral student: Juha Sarmavuori
Opponent: Prof. Jindřich Duník, University of West Bohemia, Czeck Republic
Custos: Prof. Simo Särkkä, Aalto University School of Electrical Engineering, Department of Electrical Engineering and Automation

Numerical integration is necessary for nonlinear state-space estimation. State-space models have many applications in real-world problems like communications, medicine or positioning. New methods are developed for Gaussian approximation and less restrictive arbitrary order moment method.

Gaussian approximation means that the probability density of the state-space model is approximated as Gaussian. For this approximation, both filtering and smoothing are considered using Fourier–Hermite series. In ﬁltering, the state is estimated at a given time instant based on measurements up to the given time instant. In smoothing, measurements after the given time instant are used as well. A new method is developed for computing terms of the Fourier–Hermite series by using partial differentials of a Weierstrass transform of a nonlinear function.

We also developed a new general numerical integration method, for which non-Gaussian state-space estimation is one application. This new method is based on approximating multiplication operator in Hilbert space with projections. It is a generalisation of Gaussian quadrature and has many similar properties, which are analysed using the theory of linear operators in Hilbert space. An example application of the new numerical integration method is the generalisation of the product rule for numerical integration. Ordinarily, the product rule is restricted to independent variables. This new generalisation to dependent variables is used for non-linear filtering using arbitrary order of moments.

Key words: Fourier–Hermite series, Numerical integration as a multiplication operator, stochastic ﬁltering and smoothing, Hilbert space

Thesis available for public display 10 days prior to the defence at: https://aaltodoc.aalto.fi/doc_public/eonly/riiputus/

Yhteystiedot:

 Sähköposti [email protected] Puhelinnumero +358 50 486 5075

Doctoral theses in the School of Electrical Engineering: https://aaltodoc.aalto.fi/handle/123456789/53

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