The Stochastic Processes and Applied 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 is an exciting research topic that combines methods from statistics and applied mathematics in order to construct more flexible statistical models and better computational methods for statistical inference. Some current areas of focus are:

Random fields on metric graphs

In many statistical applications, there is a need to model data on networks such as street networks in a city, or river networks. Such spatial domains are examples of metric graphs, and for applications on these, there is a need to create flexible random field models defined directly on the metric graph. In a recent paper, we defined the class of Gaussian Whittle-Matérn fields on metric graphs. These random fields are defined through a fractional stochastic differential equation on the metric graph, and we have shown that they are as far as we know the only known class of Gaussian random fields that are well defined on general metric graphs. As we have shown in another recent paper, these can be extended to generalized Gaussian Whittle-Matérn field, where the parameters can be spatially varying, and we have derived theoretical guarantees for convergence rates of finite element approximations of such models. We are currently working on a number of extensions, such as extensions to spatio-temporal models and point process data, and we are implementing these models in a new R package to be released soon.

Non-Gaussian Spatial models and robustification of latent Gaussian models

In many statistical applications, there is a need for non-Gaussian spatial models. Such models can be constructed as solutions to partial differential equations driven by non-Gaussian noise. The investigation of models of this type have been conducted by the group in a series of papers, including a RSS discussion paper and a JRSS-B paper. Currently, we are working on the theory of these models, various extensions, and on user friendly software that implements the models. 

Fractional-order stochastic partial differential equations

One area that we have been working on extensively is efficient numerical methods for fractional-order stochastic partial differential equations. This has been done in a series of papers in both numerical analysis journals and statistics journals. In a recent paper, we proposed an approach that is compatible in the R-INLA and inlabru R packages, and thus facilitates using the method in a wide range of Bayesian hierarchical models. This approach has been implemented in the rSPDE package, and we are currently working on various extensions to spatio-temporal and non-Gaussian models. 

Statistical theory for random fields

Besides the research on flexible models and numerical methods for spatial and spatio-temporal data, we are also working on the statistical theory for random fields in general. Two recent examples of this work is the AOS paper on kriging prediction and the Bernoulli paper on Gaussian random fields with fractional-order covariance operators. 

Applications

In parallell with the theoretical research, the group works on applications in a wide range of areas, ranging 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. 

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