In this talk I will discuss recent joint work with Mike McCourt (SigOpt, San Francisco) that has led to progress on the numerically stable computation of certain quantities of interest when working with positive definite kernels to solve scattered data interpolation (or kriging) problems.
In particular, I will draw upon insights from both numerical analysis and modeling with Gaussian processes which will allow us to connect quantities such as, e.g., (deterministic) error estimates in terms of the power function with the kriging variance. This provides new kernel parametrization criteria as well as new ways to compute known criteria such as MLE. Some numerical examples will illustrate the effectiveness of this approach.[-]

The term ‘Public Access Defibrillation' (PAD) is referred to programs based on the placement of Automated External Defibrillators (AED) in key locations along cities' territory together with the development of a training plan for users (first responders). PAD programs are considered necessary since time for intervention in cases of sudden cardiac arrest outside of a medical environment (out-of-hospital cardiocirculatory arrest, OHCA) is strongly limited: survival potential decreases from a 67% baseline by 7 to 10% for each minute of delay in first defibrillation. However, it is widely recognized that current PAD performance is largely below its full potential. We provide a Bayesian spatio-temporal statistical model for predidicting OHCAs. Then we construct a risk map for Ticino, adjusted for demographic covariates, that explains and forecasts the spatial distribution of OHCAs, their temporal dynamics, and how the spatial distribution changes over time. The objective is twofold: to efficiently estimate, in each area of interest, the occurrence intensity of the OHCA event and to suggest a new optimized distribution of AEDs that accounts for population exposure to the geographic risk of OHCA occurrence and that includes both displacement of current devices and installation of new ones.[-]

We study random fields taking values in a separable Hilbert space H. First, we focus on their local structure and establish a counterpart to Falconer's characterization of tangent fields. That is, we show (under general conditions) that the tangent fields to a H-valued process are self-similar and almost all of them have stationary increments. We go a bit further and study higher-order tangent fields. This leads naturally to the study of self-similar intrinsic random functions (IRF) taking values in a Hilbert space. To this end, we begin by extending Matheron's theory of scalar-valued IRFs and provide the spectral representation of H-valued IRFs. We then use this theory to characterize large classes of operator self-similar H-valued IRF processes, which in the Gaussian case can be viewed as the H-valued counterparts to fractional Brownian fields. These general results may find applications to the study of long-range dependence for random fields taking values in a Hilbert space as well as to modeling function-valued spatial data.[-]