It has long been known that for many processes, the likelihood of encountering an extreme event, or sequence of large events, is in fact much larger than would be expected from the statistics we are familiar with. Examples range from fluctuations in financial markets, to the height of the river Nile, to explosive solar flares releasing energy from the Sun. These processes are connected by the mathematics used to describe them: some are examples of long-range dependent time series, meaning that the value of the process in the far past has a tangible effect on its value in the future, allowing long correlated “runs” of values to occur; others exemplify heavy-tailed time series, where values which would be rare in a Gaussian distribution are seen much more often.

These two mechanisms were named by Mandelbrot as the ‘"Joseph"’ and "‘Noah"’ effects. The "‘Joseph’" effect is named after the Biblical character’'s prediction of seven years of good harvests and seven years of bad, i.e. increased likelihood of sequential values being similar. The "‘Noah"’ effect captures the occurrence of dramatic extreme events, which while rare can dominate the action of the process (for example, the extreme rainfall that caused Noah’'s flood). Each of these effects is described by a parameter in the heavy-tailed ARFIMA model; these parameters must be accurately determined to effectively model the process, in particular its extreme behaviour.

Our algorithm uses a Bayesian approach to estimate both parameters simultaneously. This approach is both more general and computationally “"better value"” than past algorithms, which estimate the two parameters separately. We first test the accuracy of the algorithm using synthetic data, randomly generated from known ARFIMA models, then apply it to solar X-ray data from NASA’s Geostationary Operational Environmental System (GOES) satellites. Our results suggest that a heavy-tailed ARFIMA model is appropriate for the solar flare time series.