Abstract
The aim of this thesis is to develop Hilbert space methods for approximation of integrals appearing in filtering and smoothing of nonlinear state-space models. State-space models have many applications in real-world problems and have been studied extensively for almost a century. In filtering, the state is estimated at a given time instant based on measurements up to the time instant. In smoothing, measurements after the given time instant are used as well. The used state-space models are stochastic and hence need to be estimated in probabilistic terms, which requires solving probability integrals. We consider two kinds of state-space models: discrete-time and continuous-discrete-time ones. In the latter case, the dynamics model is continuous time the measurements are obtained in discrete time instants. In linear state-space models with additive Gaussian noise, closed-form solutions are known for both filtering and smoothing problems. In a nonlinear case, we can use Gaussian approximations, which means that we approximate the probability distributions with Gaussian distributions. We study how to use Fourier–Hermite series for smoothing and filtering with Gaussian approximations. For computing terms of the Fourier–Hermite series, we develop a new method that uses partial differentials of a Weierstrass transform of a nonlinear function. Even with the simplifying Gaussian approximation, in general, we cannot solve the resulting Gaussian integrals in closed form, but we need numerical approximations instead. We develop a new numerical integration method based on an approximation of a multiplication operator with a finite matrix, and it is not only applicable to Gaussian integrals but can be used for more general numerical integration. This new numerical integration method generalises Gaussian quadrature and has many similar properties, which are analysed using the theory of linear operators in Hilbert space. Specifically, we prove convergence for a large class of functions. In the case of independent variables, it is possible to compute multidimensional integrals by product rule of unidimensional numerical integrals. With the new numerical integration method, we can generalise the product rule for non-independent variables. We apply this generalised product rule to filtering with arbitrary order moments.
Translated title of the contribution | Hilbertin avaruuden projektiomenetelmät numeerisessa integroinnissa ja tilaestimoinnissa |
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Original language | English |
Qualification | Doctor's degree |
Awarding Institution |
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Supervisors/Advisors |
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Print ISBNs | 978-952-64-1984-8 |
Electronic ISBNs | 978-952-64-1985-5 |
Publication status | Published - 2024 |
MoE publication type | G5 Doctoral dissertation (article) |
Keywords
- Fourier–Hermite series
- numerical integration as a multiplication operator
- stochastic filtering and smoothing
- Hilbert space