Stochastic Differential Equation Methods for SpatioTemporal Gaussian Process Regression
Research output: Thesis › Doctoral Thesis › Collection of Articles
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Abstract
Gaussian processes (GPs) are widely used tools for nonparametric probabilistic modelling in machine learning, spatial statistics, and signal processing. Their strength lies in flexible model specification, where prior beliefs of the model functions are encoded by the GP model. This way they can also be interpreted as specifying a probability distribution over the space of functions. In signal processing GPs are typically represented as statespace models, whereas the kernel (covariance function) representation is favoured in machine learning. Under the kernel formalism, the naïve solution to a GP regression problem scales cubically in the number of data points, which makes the approach computationally infeasible for large data sets. This work explores the link between the two representations, which enables the use of efficient sequential Kalman filtering based methods for solving the inference problem. These methods have linear time complexity with respect to the number of data points. The interest is in presenting an explicit connection between a large class of covariance functions and state space models. This is done for onedimensional (temporal) covariance functions and linear timeinvariant stochastic differential equations. This class of models covers a wide range of both stationary and nonstationary GP models for encoding, for example, continuity, smoothness, or periodicity. The framework also extends to spatiotemporal models, where the GP is represented as an evolution type stochastic partial differential equation and inference conducted by infinitedimensional Kalman filtering methods. Both separable and non separable models are considered, and implementation techniques for numerical solutions are discussed. The link between stochastic differential equations and standard covariance functions widens the applicability of Gaussian processes in combination with mechanistic physical differential equation models. Temporal and spatiotemporal Gaussian process models are useful in a multitude of dataintensive applications. Examples in this work include brain image analysis, weather modelling, financial forecasting, and tracking applications.
Details
Translated title of the contribution  Stokastisia differentiaaliyhtälömenetelmiä spatiotemporaaliseen regressioon gaussisilla prosesseilla 

Original language  English 
Qualification  Doctor's degree 
Awarding Institution  
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Publisher 

Print ISBNs  9789526067100 
Electronic ISBNs  9789526067117 
Publication status  Published  2016 
MoE publication type  G5 Doctoral dissertation (article) 
 stochastic differential equation, Gaussian process, statespace model, spatiotemporal data
Research areas
ID: 18880563