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Scalable Bayesian spatial analysis with Gaussian Markov random fields
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.ORCID iD: 0000-0002-5115-5657
2020 (English)Doctoral thesis, comprehensive summary (Other academic)Alternative title
Skalbar Bayesiansk spatial analys med Gaussiska Markov-fält (Swedish)
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 [en]
Spatial statistics, Bayesian statistics, Gaussian Markov random fields, fMRI, Machine learning
Keywords [sv]
Spatial statistik, Bayesiansk statistik, Gaussiska Markov-fält, fMRI, Maskininlärning
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:liu:diva-165872DOI: 10.3384/diss.diva-165872ISBN: 9789179298180 (print)OAI: oai:DiVA.org:liu-165872DiVA, id: diva2:1433819
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
List of papers
1. Fast Bayesian whole-brain fMRI analysis with spatial 3D priors
Open this publication in new window or tab >>Fast Bayesian whole-brain fMRI analysis with spatial 3D priors
2017 (English)In: NeuroImage, ISSN 1053-8119, E-ISSN 1095-9572, Vol. 146, p. 211-225Article in journal (Refereed) Published
Abstract [en]

Spatial whole-brain Bayesian modeling of task-related functional magnetic resonance imaging (fMRI) is a great computational challenge. Most of the currently proposed methods therefore do inference in subregions of the brain separately or do approximate inference without comparison to the true posterior distribution. A popular such method, which is now the standard method for Bayesian single subject analysis in the SPM software, is introduced in Penny et al. (2005b). The method processes the data slice-by-slice and uses an approximate variational Bayes (VB) estimation algorithm that enforces posterior independence between activity coefficients in different voxels. We introduce a fast and practical Markov chain Monte Carlo (MCMC) scheme for exact inference in the same model, both slice-wise and for the whole brain using a 3D prior on activity coefficients. The algorithm exploits sparsity and uses modern techniques for efficient sampling from high-dimensional Gaussian distributions, leading to speed-ups without which MCMC would not be a practical option. Using MCMC, we are for the first time able to evaluate the approximate VB posterior against the exact MCMC posterior, and show that VB can lead to spurious activation. In addition, we develop an improved VB method that drops the assumption of independent voxels a posteriori. This algorithm is shown to be much faster than both MCMC and the original VB for large datasets, with negligible error compared to the MCMC posterior.

Place, publisher, year, edition, pages
Elsevier, 2017
Keywords
fMRI, Spatial priors, Variational Bayes, Markov chain Monte Carlo, Gaussian Markov random fields, General linear model
National Category
Medical Engineering
Identifiers
urn:nbn:se:liu:diva-132945 (URN)10.1016/j.neuroimage.2016.11.040 (DOI)000394560700019 ()27876654 (PubMedID)2-s2.0-84999622353 (Scopus ID)
Note

Funding agencies: Swedish Research Council (Vetenskapsradet) [20135229]; Knut and Alice Wallenberg Foundation [KAW 20012.0067]

Available from: 2016-12-05 Created: 2016-12-05 Last updated: 2020-06-29Bibliographically approved
2. Efficient Covariance Approximations for Large Sparse Precision Matrices
Open this publication in new window or tab >>Efficient Covariance Approximations for Large Sparse Precision Matrices
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
Keywords
Gaussian Markov random fields; Selected inversion; Sparse precision matrix; Spatial analysis; Stochastic approximation
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-153715 (URN)10.1080/10618600.2018.1473782 (DOI)000453029500018 ()
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
3. Deep Gaussian Markov Random Fields
Open this publication in new window or tab >>Deep Gaussian Markov Random Fields
2020 (English)In: Proceedings of the 37th International Conference on Machine Learning / [ed] Hal Daumé III, Aarti Singh, PMLR , 2020, Vol. 119, p. 8916-8926Conference paper, Published paper (Refereed)
Abstract [en]

Gaussian Markov random fields (GMRFs) are probabilistic graphical models widely used in spatial statistics and related fields to model dependencies over spatial structures. We establish a formal connection between GMRFs and convolutional neural networks (CNNs). Common GMRFs are special cases of a generative model where the inverse mapping from data to latent variables is given by a 1-layer linear CNN. This connection allows us to generalize GMRFs to multi-layer CNN architectures, effectively increasing the order of the corresponding GMRF in a way which has favorable computational scaling. We describe how well-established tools, such as autodiff and variational inference, can be used for simple and efficient inference and learning of the deep GMRF. We demonstrate the flexibility of the proposed model and show that it outperforms the state-of-the-art on a dataset of satellite temperatures, in terms of prediction and predictive uncertainty.

Place, publisher, year, edition, pages
PMLR, 2020
Series
Proceedings of Machine Learning Research, ISSN 2640-3498 ; 119
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-171582 (URN)
Conference
International Conference on Machine Learning, 13-18 July 2020, Virtual
Available from: 2020-11-24 Created: 2020-11-24 Last updated: 2020-11-24
4. Real-Time Robotic Search using Structural Spatial Point Processes
Open this publication in new window or tab >>Real-Time Robotic Search using Structural Spatial Point Processes
Show others...
2020 (English)In: 35TH UNCERTAINTY IN ARTIFICIAL INTELLIGENCE CONFERENCE (UAI 2019), Association For Uncertainty in Artificial Intelligence (AUAI) , 2020, Vol. 115, p. 995-1005Conference paper, Published paper (Refereed)
Abstract [en]

Aerial robots hold great potential for aiding Search and Rescue (SAR) efforts over large areas, such as during natural disasters. Traditional approaches typically search an area exhaustively, thereby ignoring that the density of victims varies based on predictable factors, such as the terrain, population density and the type of disaster. We present a probabilistic model to automate SAR planning, with explicit minimization of the expected time to discovery. The proposed model is a spatial point process with three interacting spatial fields for i) the point patterns of persons in the area, ii) the probability of detecting persons and iii) the probability of injury. This structure allows inclusion of informative priors from e.g. geographic or cell phone traffic data, while falling back to latent Gaussian processes when priors are missing or inaccurate. To solve this problem in real-time, we propose a combination of fast approximate inference using Integrated Nested Laplace Approximation (INLA), and a novel Monte Carlo tree search tailored to the problem. Experiments using data simulated from real world Geographic Information System (GIS) maps show that the framework outperforms competing approaches, finding many more injured in the crucial first hours.

Place, publisher, year, edition, pages
Association For Uncertainty in Artificial Intelligence (AUAI), 2020
Series
Proceedings of Machine Learning Research (PMLR), E-ISSN 2640-3498 ; 115
National Category
Computer and Information Sciences
Identifiers
urn:nbn:se:liu:diva-159698 (URN)000722423500092 ()2-s2.0-85084016675 (Scopus ID)
Conference
Proceedings of the 35th Conference on Uncertainty in Artificial Intelligence (UAI 2019), Tel Aviv, Israel, July 22-25, 2019
Note

Funding: Wallenberg AI, Autonomous Systems and Software Program (WASP); WASP Autonomous Research Arenas - Knut and Alice Wallenberg Foundation; Swedish Foundation for Strategic Research (SSF)Swedish Foundation for Strategic Research; ELLIIT Excellence Center at Link opingLund for Information Technology

Available from: 2019-08-19 Created: 2019-08-19 Last updated: 2023-04-05Bibliographically approved

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Sidén, Per

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