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Efficient Covariance Approximations for Large Sparse Precision Matrices
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.
Univ Edinburgh, Scotland.
Chalmers, Sweden; Univ Gothenburg, Sweden.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.
2018 (English)In: Journal of Computational And Graphical Statistics, ISSN 1061-8600, E-ISSN 1537-2715, Vol. 27, no 4, p. 898-909Article in journal (Refereed) Published
Abstract [en]

The use of sparse precision (inverse covariance) matrices has become popular because they allow for efficient algorithms for joint inference in high-dimensional models. Many applications require the computation of certain elements of the covariance matrix, such as the marginal variances, which may be nontrivial to obtain when the dimension is large. This article introduces a fast Rao-Blackwellized Monte Carlo sampling-based method for efficiently approximating selected elements of the covariance matrix. The variance and confidence bounds of the approximations can be precisely estimated without additional computational costs. Furthermore, a method that iterates over subdomains is introduced, and is shown to additionally reduce the approximation errors to practically negligible levels in an application on functional magnetic resonance imaging data. Both methods have low memory requirements, which is typically the bottleneck for competing direct methods.

Place, publisher, year, edition, pages
AMER STATISTICAL ASSOC , 2018. Vol. 27, no 4, p. 898-909
Keywords [en]
Gaussian Markov random fields; Selected inversion; Sparse precision matrix; Spatial analysis; Stochastic approximation
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:liu:diva-153715DOI: 10.1080/10618600.2018.1473782ISI: 000453029500018OAI: oai:DiVA.org:liu-153715DiVA, id: diva2:1276178
Note

Funding Agencies|Swedish Research Council (Vetenskapsradet) [2013-5229, 2016-04187]; European Unions Horizon 2020 Programme for Research and Innovation [640171]

Available from: 2019-01-07 Created: 2019-01-07 Last updated: 2020-06-29
In thesis
1. Scalable Bayesian spatial analysis with Gaussian Markov random fields
Open this publication in new window or tab >>Scalable Bayesian spatial analysis with Gaussian Markov random fields
2020 (English)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[sv]
Skalbar Bayesiansk spatial analys med Gaussiska Markov-fält
Abstract [en]

Accurate statistical analysis of spatial data is important in many applications. Failing to properly account for spatial autocorrelation may often lead to false conclusions. At the same time, the ever-increasing sizes of spatial datasets pose a great computational challenge, as many standard methods for spatial analysis are limited to a few thousand data points.

In this thesis, we explore how Gaussian Markov random fields (GMRFs) can be used for scalable analysis of spatial data. GMRFs are closely connected to the commonly used Gaussian processes, but have sparsity properties that make them computationally cheap both in time and memory. The Bayesian framework enables a GMRF to be used as a spatial prior, comprising the assumption of smooth variation over space, and gives a principled way to estimate the parameters and propagate uncertainty.

We develop new algorithms that enable applying GMRF priors in 3D to the brain activity inherent in functional magnetic resonance imaging (fMRI) data, with millions of observations. We show that our methods are both faster and more accurate than previous work. A method for approximating selected elements of the inverse precision matrix (i.e. the covariance matrix) is also proposed, which is important for evaluating the posterior uncertainty. In addition, we establish a link between GMRFs and deep convolutional neural networks, which have been successfully used in countless machine learning tasks for images, resulting in a deep GMRF model. Finally, we show how GMRFs can be used in real-time robotic search and rescue operations, for modeling the spatial distribution of injured persons.

Abstract [sv]

Tillförlitlig statistisk analys av spatiala data är viktigt inom många tillämpningar. Om inte korrekt hänsyn tas till spatial autokorrelation kan det ofta leda till felaktiga slutsatser. Samtidigt ökar ständigt storleken på de spatiala datamaterialen vilket utgör en stor beräkningsmässig utmaning, eftersom många standardmetoder för spatial analys är begränsade till några tusental datapunkter.

I denna avhandling utforskar vi hur Gaussiska Markov-fält (eng: Gaussian Markov random fields, GMRF) kan användas för mer skalbara analyser av spatiala data. GMRF-modeller är nära besläktade med de ofta använda Gaussiska processerna, men har gleshetsegenskaper som gör dem beräkningsmässigt effektiva både vad gäller tids- och minnesåtgång. Det Bayesianska synsättet gör det möjligt att använda GMRF som en spatial prior som innefattar antagandet om långsam spatial variation och ger ett principiellt tillvägagångssätt för att skatta parametrar och propagera osäkerhet.

Vi utvecklar nya algoritmer som gör det möjligt att använda GMRF-priors i 3D för den hjärnaktivitet som indirekt kan observeras i hjärnbilder framtagna med tekniken fMRI, som innehåller milliontals datapunkter. Vi visar att våra metoder är både snabbare och mer korrekta än tidigare forskning. En metod för att approximera utvalda element i den inversa precisionsmatrisen (dvs. kovariansmatrisen) framförs också, vilket är viktigt för att kunna evaluera osäkerheten i posteriorn. Vidare gör vi en koppling mellan GMRF och djupa neurala faltningsnätverk, som har använts framgångsrikt för mängder av bildrelaterade problem inom maskininlärning, vilket mynnar ut i en djup GMRF-modell. Slutligen visar vi hur GMRF kan användas i realtid av autonoma drönare för räddningsinsatser i katastrofområden för att modellera den spatiala fördelningen av skadade personer.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2020. p. 53
Series
Linköping Studies in Arts and Sciences, ISSN 0282-9800 ; 790Linköping Studies in Statistics, ISSN 1651-1700 ; 15
Keywords
Spatial statistics, Bayesian statistics, Gaussian Markov random fields, fMRI, Machine learning, Spatial statistik, Bayesiansk statistik, Gaussiska Markov-fält, fMRI, Maskininlärning
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-165872 (URN)10.3384/diss.diva-165872 (DOI)9789179298180 (ISBN)
Public defence
2020-09-18, Ada Lovelace, B Building, Campus Valla, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2020-08-17 Created: 2020-06-01 Last updated: 2020-11-24Bibliographically approved

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