Parameter Estimation and Seasonal Modification of the Fractional Poisson Process with Application to Vorticity Extremes over the North Atlantic

TL;DR

This paper introduces a new parameter estimation method for the fractional Poisson process, incorporating seasonality, and applies it to analyze vorticity extremes in the North Atlantic, enhancing modeling accuracy in meteorology.

Contribution

It proposes a novel estimation technique for the FPP and a seasonal modification method, validated through simulations and applied to meteorological data.

Findings

The new estimation method outperforms existing estimators in simulations.

Seasonal modeling improves the fit of the FPP to real-world vorticity data.

Application reveals insights into the timing and frequency of extreme vorticity events.

Abstract

The fractional Poisson process (FPP) generalizes the standard Poisson process by replacing exponentially distributed return times with Mittag-Leffler distributed ones with an extra tail parameter, allowing for greater flexibility. The FPP has been applied in various fields, such as modeling occurrences of extratropical cyclones in meteorology and solar flares in physics. We propose a new estimation method for the parameters of the FPP, based on minimizing the distance between the empirical and the theoretical distribution at selected quantiles. We conduct an extensive simulation study to evaluate the advantages and limitations of the new estimation method and to compare it with several competing estimators, some of which have not yet been examined in the Mittag-Leffler setting. To enhance the applicability of the FPP in real-world scenarios, particularly in meteorology, we propose a method for incorporating seasonality into the FPP through distance-based weighting. We then analyze the return times of relative vorticity extremes in the North Atlantic-European region using our seasonal modeling approach.