A Kernel-Based Approach for Modelling Gaussian Processes with Functional Information

Gaussian processes are among the most useful tools in modeling continuous processes in machine learning and statistics. If the value of a process is known at a finite collection of points, one may use Gaussian processes to construct a surface which interpolates these values to be used for prediction and uncertainty quantification in other locations. However, it is not always the case that the available information is in the form of a finite collection of points. For example, boundary value problems contain information on the boundary of a domain, which is an uncountable collection of points that cannot be incorporated into typical Gaussian process techniques. In this paper we construct a Gaussian process model which utilizes reproducing kernel Hilbert spaces to unify the typical finite case with the case of having uncountable information by exploiting the equivalence of conditional expectation and orthogonal projections. We discuss this construction in statistical models, including numerical considerations and a proof of concept.

Bayesian Regularization for Functional Graphical Models

Graphical models, used to express conditional dependence between random variables observed at various nodes, are used extensively in many fields such as genetics, neuroscience, and social network analysis. While most current statistical methods for estimating graphical models focus on scalar data, there is interest in estimating analogous dependence structures when the data observed at each node are functional, such as signals or images. In this paper, we propose a fully Bayesian regularization scheme for estimating functional graphical models. We first consider a direct Bayesian analog of the functional graphical lasso proposed by Qiao et al. (2019). We then propose a regularization strategy via the graphical horseshoe. We compare these approaches via simulation study and apply our proposed functional graphical horseshoe to two motivating applications, electroencephalography data for comparing brain activation between an alcoholic group and controls, as well as changes in structural connectivity in the presence of traumatic brain injury (TBI). Our results yield insight into how the brain attempts to compensate for disconnected networks after injury.

Coupling material and mechanical design processes via computer model calibration

Computer model calibration typically operates by choosing parameter values in a computer model so that the model output faithfully predicts reality. By using performance targets in place of observed data, we show that calibration techniques can be repurposed to wed engineering and material design, two processes that are traditionally carried out separately. This allows materials to be designed with specific engineering targets in mind while quantifying the associated sources of uncertainty. We demonstrate our proposed approach by "calibrating" material design settings to performance targets for a wind turbine blade.