Spatio-Temporal Estimation for Mixture Models and Gaussian Markov Random Fields

Johan Lindström

Centre for Mathematical Sciences
Mathematical Statistics
Lund University
2008

ISBN 978-91-628-7502-2
LUTFMS-1033-2008

Abstract:

In this thesis computationally intensive methods are used to estimate models and to make inference for large, spatio-temporal data sets. The thesis is divided into two parts: the first two papers are concerned with video analysis, while the last three papers model and investigate environmental data from the Sahel area in northern Africa.

In the first part of the thesis, mixture models are used to distinguish between moving (foreground) and stationary (background) pixels in video sequences. A recursive estimator for mixtures of

Gaussians is derived using an expectation maximisation (EM) algorithm. It is shown that the recursive estimator can be interpreted in a Bayesian framework. With some additional steps, the estimator is used to construct an algorithm that segments video frames into foreground and background pixels.

Additionally, an extension to existing segmentation algorithms that detects and adjusts for rapid changes in illumination is presented. This extension is shown to work for two segmentation

algorithms that model the pixel values using Gaussian mixtures.

In the second part of the thesis, environmental data sets, consisting of precipitation measurements and satellite derived vegetation indices, are examined. First, calibration issues for the vegetation index data are investigated. Thereafter, a Gaussian Markov random field (GMRF) model for estimation of spatially dependent trends is constructed. The parameters in the GMRF model are estimated using an EM algorithm, and the performance of the model is evaluated using simulated data. The model is used to analyse temporal trends in the vegetation data.

Finally, a spatio-temporal GMRF model is used to interpolate the precipitation measurements. The model is created by extending a spatial GMRF to a spatio-temporal model with a first order

auto-regressive dependence in time. The spatial part of the model consists of a GMRF that approximates a field with isotropic Matérn covariance. To obtain a model that is defined where the precipitation measurements were taken the spatial GMRF is constructed on a set of irregularly spaced points on the globe. The model is estimated using a Markov chain Monte Carlo approach and the formulation as a Markov field allows for efficient computations, even though the field has more than 30000 nodes.

Key words:

adaptive Gaussian mixtures; African Sahel; Bayesian recursive estimation;  change point detection; expectation maximisation; Gaussian Markov random fields; Markov chain Monte Carlo;

precipitation; spatio-temporal modelling; time series analysis; vegetation; video segmentation;