SpatioTemporal Estimation for Mixture Models and Gaussian Markov Random
Fields
Johan Lindström
Centre for Mathematical Sciences
Mathematical Statistics
Lund University
2008
ISBN 9789162875022
LUTFMS10332008

Abstract:

In this thesis computationally intensive methods are used to estimate models
and to make inference for large, spatiotemporal 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 spatiotemporal GMRF model is used to interpolate the precipitation
measurements. The model is created by extending a spatial GMRF to a
spatiotemporal model with a first order

autoregressive 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; spatiotemporal modelling; time series analysis; vegetation;
video segmentation;








