Stochastic Differential Equation Methods for Spatio-Temporal Gaussian Process Regression

Research output: ThesisDoctoral ThesisCollection of Articles


Gaussian processes (GPs) are widely used tools for non-parametric 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 state-space 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 one-dimensional (temporal) covariance functions and linear time-invariant stochastic differential equations. This class of models covers a wide range of both stationary and non-stationary GP models for encoding, for example, continuity, smoothness, or periodicity. The framework also extends to spatio-temporal models, where the GP is represented as an evolution type stochastic partial differential equation and inference conducted by infinite-dimensional 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 spatio-temporal Gaussian process models are useful in a multitude of data-intensive applications. Examples in this work include brain image analysis, weather modelling, financial forecasting, and tracking applications.
Translated title of the contributionStokastisia differentiaaliyhtälömenetelmiä spatiotemporaaliseen regressioon gaussisilla prosesseilla
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
  • Lampinen, Jouko, Supervising Professor
  • Särkkä, Simo, Thesis Advisor
Print ISBNs978-952-60-6710-0
Electronic ISBNs978-952-60-6711-7
Publication statusPublished - 2016
MoE publication typeG5 Doctoral dissertation (article)


  • stochastic differential equation
  • Gaussian process
  • state-space model
  • spatio-temporal data


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