Colloquium Mathematics - Prof. K. Podgórski

Title: Functional approach to excursions of general random processes and fields

Abstract:

The problem of a probabilistic description of random excursion sets is at the very center of the extreme value theory. We propose an advanced functional approach for several level distribution problems which is based on the Slepian model style representations of the behavior of random processes and fields at the crossing levels. The objective of the research is twofold, first, to provide mathematical foundations to approaches that have been earlier heuristically developed for applications in engineering and environmental sciences, and, second, to develop computational tools for a practitioner to benefit from conceptual and mathematical efficiency of the theory. From the methodological perspective, several specific topics are investigated including an approximation of the persistency exponent for stationary processes, calculus of the level crossing distributions for random fields, distributions for evolving geometric characteristics of random fields, Slepian-style models for a non-Gaussian field.

On the computational side, to utilize a functional form of the approach, machine-learning algorithms are employed for the choice of orthonormal functional bases using efficient orthogonalization of splines. We develop a computational open-source package in R to provide efficient computational tools for general analysis of functional data, including the ones that come from the functional level crossing data.

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