Stochastic Differential Equation Methods for Spatio-Temporal Gaussian Process Regression
Arno Solin · 2016
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 naive 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.; Gaussiset prosessit (GP) ovat ei-parametrisia tilastollisia tyokaluja, joita kaytetaan yleisesti koneoppimisessa, spatiaalisessa tilastotieteessa seka signaalinkasittelyssa. Niiden avulla mallifunktioita koskevat oletukset voidaan maaritella joustavasti. Gaussisen prosessin voi ajatella maarittelevan todennakoisyysjakauman funktioavaruuden yli. Signaalinkasittelyssa gaussiset prosessit maaritellaan yleensa tila-avaruusmallin avulla, kun taas koneoppimisessa suositaan ytimeen (kovarianssifunktio) perustuvaa esitystapaa. Jalkimmaisessa esitystavassa GP-regressio-ongelman ratkaisun laskennallinen vaativuus skaalautuu kuutiollisesti aineiston koon suhteen, mika tekee lahestymistavasta laskennallisesti raskaan suurilla tietoaineistoilla. Tassa tyossa tarkastellaan tila-avaruusmallien ja kovarianssifunktioiden yhteytta, jonka avulla voidaan kayttaa tilastollisessa…