Gaussian random fields on metric graphs

Seminar in Mathematics

Abstract: There is a growing interest in modeling data on networks and graphs. In statistics, typical applications are modeling traffic accidents or street crimes on road networks and modeling temperature or pollutants in river networks. In these cases, the statistical model is defined on both the edges and vertices of the graph, where the edges represent the roads or river segments. Importantly, distance between locations should then be measured on the network, and not by using the Euclidean distance. A graph coupled with a metric like this is referred to as a metric graph.

Gaussian random fields are essential for modeling spatially dependent data; however, the study of Gaussian random fields on metric graphs is challenging because it is difficult to construct valid covariance functions which are isotropic in the metric on the graph. In this talk, I will give an overview of our ongoing work on developing a complete mathematical and statistical theory for Gaussian random fields on metric graphs, complemented by an easy-to-use statistical software.

I will specifically introduce a new flexible class of Gaussian random fields, formulated via fractional-order stochastic differential equations on metric graphs, and discuss their statistical properties and numerical approximation via finite element methods. I will also discuss Markov random fields on metric graphs and their connection to graphical models, and show that a subset of the proposed models have Markov properties which facilitate exact inference. Finally, I will introduce the MetricGraph R package and use it to illustrate the proposed models in a traffic data application.