Speaker: Ao Li, Ph.D. Candidate, Department of Statistics, Indiana University
Title: Topics in Spatial Statistics: Nonparametric Variogram estimation, Lévy's Brownian Motion and White Noise Space on the circle.
Spatial statistics plays an important role in various fields, where data are observed with location information, including environmental science, geography and epidemiology. Such data are usually modeled through spatial random process. %with dependence functions as the focus of the study. In this research, we investigate spatial random process in both Euclidean space and on the circle.
In Euclidean space, variogram is popularly used to study the spatial dependence. A valid variogram needs to be conditionally negative definite. In this research, we obtain novel inversion formulae between variogram and its associated spectral function. Then, a valid nonparametric variogram estimator is proposed based on these inversions through the constrained optimization. We demonstrate this approach through simulation studies.
On the circle, a proper probability space for the white noise is investigated. We first demonstrate that the commonly applied Brownian motion on the circle constructed by Lévy is problematic. We show that it is a regular Euclidean Brownian motion on the half-circle with its own mirror image on the half-circle, and is degenerated in the sense of Minlos's study. We then formally define the white noise space and its associated Brownian bridge on the circle.
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