Research Projects

In many statistical applications, there is a need to model data on networks such as street networks in a city, or river networks. These spatial domains are examples of metric graphs and thus, this project focuses on the study of Gaussian random field models on metric graphs for such applications. One approach to define such stochastic processes or random fields on metric graphs is to describe them in terms of a covariance function. However, this approach incurs certain limitations on the graphs and the parameters of the model which makes it not suitable for real-life applications. Therefore, to overcome these limitations, we shall consider the alternative approach of defining the Gaussian random fields as a solution to the stochastic partial differential equations (SPDEs) on metric graphs, particularly Whittle-Matérn fields. The rationale behind considering the Whittle-Matérn models is that when such models are considered on the Euclidean domain, it has Gaussian random fields with the Matérn covariance function as stationary solutions and over the years it has been proven a very useful model in spatial statistics and machine learning. In the existing literature, some methods have been proposed for the underlying approach; however, it does not permit the inference on the smoothness parameter of the model, which, in general, is very essential for the applications. We shall develop fast and computationally efficient methods for the the Whittle-Matérn fields on metric graphs, that also permit inference on the smoothness parameter. For those models, we shall perform the likelihood-based inference such as log-likelihood evaluation, sampling, and predictions based on the point observations from the process on the graph. Moreover, for the applications of spatio-temporal data on network-like domains, one needs to describe the spatio-temporal models, i.e., stochastic evolution equations directly on the metric graphs which also have not been explored yet. We shall analyze the appropriate space-time models for such applications, for instance, the parabolic stochastic evolution equations on metric graphs. For such space-time models, our focus would be to develop the theoretical concepts for these problems such as the existence of solutions, sample path regularity and the expression for the covariance function, which would serve as a mathematical framework for analyzing spatio-temporal data on these structures. These theoretical advancements will form the basis of a computational framework for the approximation of corresponding random fields in spatio-temporal settings. The outcomes of this project will establish a comprehensive mathematical and computational framework for analyzing spatial and spatio-temporal data on network-like domains and it would pave the way for efficient and impactful applications in the area of stochastic PDEs, spatial statistics, and related fields.