Models and Methods for Random Fields in Spatial Statistics with Computational Efficiency from Markov Properties
Author
Summary, in English
A large part is devoted to a number of extensions to the newly proposed Stochastic Partial Differential Equation (SPDE) approach for representing Gaussian fields using Gaussian Markov Random Fields (GMRFs). The method is based on that Gaussian Matérn field can be viewed as solutions to a certain SPDE, and is useful for large spatial problems where traditional methods are too computationally intensive to use. A variation of the method using wavelet basis functions is proposed and using a simulation-based study, the wavelet approximations are compared with two of the most popular methods for efficient approximations of Gaussian fields. A new class of spatial models, including the Gaussian Matérn fields and a wide family of fields with oscillating covariance functions, is also constructed using nested SPDEs. The SPDE method is extended to this model class and it is shown that all desirable properties are preserved, such as computational efficiency, applicability to data on general smooth manifolds, and simple non-stationary extensions. Finally, the SPDE method is extended to a larger class of non-Gaussian random fields with Matérn covariance functions, including certain Laplace Moving Average (LMA) models. In particular it is shown how the SPDE formulation can be used to obtain an efficient simulation method and an accurate parameter estimation technique for a LMA model.
A method for estimating spatially dependent temporal trends is also developed. The method is based on using a space-varying regression model, accounting for spatial dependency in the data, and it is used to analyze temporal trends in vegetation data from the African Sahel in order to find regions that have experienced significant changes in the vegetation cover over the studied time period. The problem of estimating such regions is investigated further in the final part of the thesis where a method for estimating excursion sets, and the related problem of finding uncertainty regions for contour curves, for latent Gaussian fields is proposed. The method is based on using a parametric family for the excursion sets in combination with Integrated Nested Laplace Approximations (INLA) and an importance sampling-based algorithm for estimating joint probabilities.
Department/s
Publishing year
2012
Language
English
Publication/Series
Doctoral Theses in Mathematical Sciences
Volume
2012:2
Full text
- Available as PDF - 10 MB
- Download statistics
Document type
Dissertation
Publisher
Faculty of Engineering, Centre for Mathematical Sciences, Mathematical Statistics, Lund University
Topic
- Probability Theory and Statistics
Keywords
- random fields
- Gaussian Markov random fields
- Matérn covariances
- stochastic partial differential equations
- Computational efficiency
Status
Published
Supervisor
ISBN/ISSN/Other
- ISSN: 1404-0034
- ISBN: 978-91-7473-336-5
Defence date
8 June 2012
Defence time
13:15
Defence place
Room MH:A, Centre for Mathematical Sciences, Sölvegatan 18, Lund University Faculty of Engineering
Opponent
- Michael Stein (Professor)