About Research OverviewThe Stochastic Processes and Mathematical Statistics group, led by Prof. David Bolin, develops methodology for statistical models involving stochastic processes and random fields. A main focus is the development of statistical methods based on stochastic partial differential equations. This exciting line of research combines methods from statistics and applied mathematics in order to construct flexible and physically interpretable statistical models, and efficient computational methods for statistical inference. Current areas of focus include the following:Random fields on metric graphsMany statistical applications require modeling data on networks such as city streets for air quality risk assessment, river network for water quality management, and road networks for traffic safety improvement. Such networks are spatial domains that can be represented as metric graphs. Applications on metric graphs require creating flexible and valid random field models defined directly on the graph. This defines the line of research carried out for this area of focus, to overcome challenges imposed by the graph structure, inter alia and simultaneously: developing flexible classes of Gaussian random fields formulated as solutions to fractional-order stochastic differential equations on the metric graphs,developing the general mathematical and statistical theory for Gaussian processes on metric graphs, developing the R package MetricGraph implementing developed theory while containing user-friendly tools for statistical data on metric graphs applications.Non-Gaussian Spatial models and robustification of latent Gaussian modelsAlthough Gaussian models are fundamental in statistics and machine learning, there is often a need for non-Gaussian random fields. Such models can be constructed as solutions to partial differential equations driven by non-Gaussian noise. The group is developing such models both from theoretical and applied point of views. We support this line of research by continuously developing the R package ngme2, which implements methods for using the developed models in statistical applications. Fractional-order stochastic partial differential equationsOne area that we have been working on extensively is the development of efficient numerical methods for fractional-order stochastic partial differential equations. These numerical methods are necessary for both metric graph models and non-Gaussian models, and for flexible Gaussian random fields on manifolds and Euclidean domains. This research is being published in both numerical analysis journals and statistics journals. In a recent publication, we proposed an approach that is compatible with the R-INLA and inlabru R packages, and consequently facilitates using the method in a wide range of Bayesian hierarchical models. This approach has been implemented in the R package rSPDE which is continuously being extended with new models and methods.Statistical theory for random fieldsBesides the research on flexible models and efficient numerical methods for spatial and spatio-temporal data, we are also working on the statistical theory for random fields in general. Recent developments of this line of resaerch are (i) necessary and sufficient conditions for asymptotic optimality of kriging prediction under model misspecification (ii) general theory for equivalence of measures of Gaussian random fields with fractional-order covariance operators, and (iii) theory for spatial self-confounding in spatial regression models. ApplicationsIn parallel with the theoretical research, the group works applications covering a broad spectrum, from brain imaging to environmental sciences. These applications are in many cases deciding the direction of the theoretical developments, and the students in the group are typically developing methods with a specific application in mind.
