Extreme Value Theory and the Solar Cycle

TL;DR

This paper applies extreme value theory to analyze the statistical properties of solar cycle extremes, estimating upper limits and return periods for sunspot numbers based on 250 years of data.

Contribution

It introduces a methodology for fitting generalized Pareto distributions to sunspot data and assesses the sensitivity of the results to threshold choices.

Findings

The sunspot number distribution is bounded with an upper limit of 324.

The estimated return value for a solar cycle maximum is approximately 188.

Large solar events like cycle 19 occur roughly once every 700 years.

Abstract

We investigate the statistical properties of the extreme events of the solar cycle as measured by the sunspot number. The recent advances in the methodology of the theory of extreme values is applied to the maximal extremes of the time series of sunspots. We focus on the extreme events that exceed a carefully chosen threshold and a generalized Pareto distribution is fitted to the tail of the empirical cumulative distribution. A maximum likelihood method is used to estimate the parameters of the generalized Pareto distribution and confidence levels are also given to the parameters. Due to the lack of an automatic procedure for selecting the threshold, we analyze the sensitivity of the fitted generalized Pareto distribution to the exact value of the threshold. According to the available data, that only spans the previous ~250 years, the cumulative distribution of the time series is bounded, yielding an upper limit of 324 for the sunspot number. We also estimate that the return value for each solar cycle is ~188, while the return value for a century increases to ~228. Finally, the results also indicate that the most probable return time for a large event like the maximum at solar cycle 19 happens once every ~700 years and that the probability of finding such a large event with a frequency smaller than ~50 years is very small. In spite of the essentially extrapolative character of these results, their statistical significance is very large.