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Models and Methods for Random Fields in Spatial Statistics with Computational Efficiency from Markov Properties

Bolin, David LU (2012) In Doctoral theses in mathematical sciences 2012:2.
Abstract (Swedish)
Popular Abstract in Swedish

Beräkningsmässigt skalar många av de klassiska metoderna inom spatiell statistik kubiskt med antalet observationer, vilket är opraktiskt om datamängderna är stora. I de traditionella tillämpningarna av statistik på spatiella datamänger begränsades ofta antalet observationer på grund av höga mätkostnader och metodernas beräkningskomplexitet inte sällan ett problem. Numera är ofta mätningar relativt billiga att åstakomma med hjälp av automatiserade mätplattformar och således kan datamängderna vara så stora att de klassiska metoderna inte går att använda på grund av begränsade beräkningsmöjligheter. Samtidigt gör större datamängder det möjligt att använda mer komplicerade modeller som bättre... (More)
Popular Abstract in Swedish

Beräkningsmässigt skalar många av de klassiska metoderna inom spatiell statistik kubiskt med antalet observationer, vilket är opraktiskt om datamängderna är stora. I de traditionella tillämpningarna av statistik på spatiella datamänger begränsades ofta antalet observationer på grund av höga mätkostnader och metodernas beräkningskomplexitet inte sällan ett problem. Numera är ofta mätningar relativt billiga att åstakomma med hjälp av automatiserade mätplattformar och således kan datamängderna vara så stora att de klassiska metoderna inte går att använda på grund av begränsade beräkningsmöjligheter. Samtidigt gör större datamängder det möjligt att använda mer komplicerade modeller som bättre förklarar den spatiella variationen i observationerna. Det finns således en önskan att kunna konstruera flexibla statistiska modeller som samtidigt är beräkningseffektiva även för stora datamängder, och det är konstruktionen av sådana modeller som är det huvudsakliga målet med detta arbete.



En av de mest populära modellerna inom spatiell statistik är den så kallade Gaussiska Matérn-modellen, där det underliggande fältet antas vara normalfördelat med en speciell typ av beroendestruktur. I arbetet studeras en ny metod för att åstakomma beräkningseffektiva representationer av den Gaussiska Matérn-modellen, och ett antal utvidgningar av modellen görs genom att tillåta mer generella ickestationtionära beroendestrukturer och fördelningstyper. En av dessa modeller används för att studera globala ozonmätningar.



En medod för att skatta spatiellt beroende tidstrender i miljödata utvecklas också och denna används för att studera tidsuteckligen av vegetation i Sahelregionen i norra Afrika. Av speciellt intresse är att hitta områden där vegetationen signifikant har förändrats över den studerade tidsperioden. Problemet att hitta sådana områden undersöks också i mer detalj och en metod för att skatta dessa så kallade exkursionsmängder för latenta Gaussiska fält föreslås. Metoden används för Saheldatan men även för luftföroreningsmätningar från en region i norra Italien och områden där luftföroreningarna överskrider satta grändvärden skattas. (Less)
Abstract
The focus of this work is on the development of new random field models and methods suitable for the analysis of large environmental data sets.



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... (More)
The focus of this work is on the development of new random field models and methods suitable for the analysis of large environmental data sets.



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. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Professor Stein, Michael, Department of Statistics, The University of Chicago, USA
organization
publishing date
type
Thesis
publication status
published
subject
keywords
random fields, Gaussian Markov random fields, Matérn covariances, stochastic partial differential equations, Computational efficiency
in
Doctoral theses in mathematical sciences
volume
2012:2
pages
223 pages
publisher
Faculty of Engineering, Centre for Mathematical Sciences, Mathematical Statistics, Lund University
defense location
Room MH:A, Centre for Mathematical Sciences, Sölvegatan 18, Lund University Faculty of Engineering
defense date
2012-06-08 13:15
ISSN
1404-0034
ISBN
978-91-7473-336-5
project
BECC
MERGE
language
English
LU publication?
yes
id
58ab2c60-acf8-495c-bb8d-6ab696d786de (old id 2539400)
date added to LUP
2012-05-15 13:13:56
date last changed
2016-09-19 08:44:50
@phdthesis{58ab2c60-acf8-495c-bb8d-6ab696d786de,
  abstract     = {The focus of this work is on the development of new random field models and methods suitable for the analysis of large environmental data sets. <br/><br>
<br/><br>
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. <br/><br>
<br/><br>
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.},
  author       = {Bolin, David},
  isbn         = {978-91-7473-336-5},
  issn         = {1404-0034},
  keyword      = {random fields,Gaussian Markov random fields,Matérn covariances,stochastic partial differential equations,Computational efficiency},
  language     = {eng},
  pages        = {223},
  publisher    = {Faculty of Engineering, Centre for Mathematical Sciences, Mathematical Statistics, Lund University},
  school       = {Lund University},
  series       = {Doctoral theses in mathematical sciences},
  title        = {Models and Methods for Random Fields in Spatial Statistics with Computational Efficiency from Markov Properties},
  volume       = {2012:2},
  year         = {2012},
}