Outliers, extreme events, and multiscaling

Extreme events have an important role which is sometimes catastrophic in a variety of natural phenomena, including climate, earthquakes, and turbulence, as well as in manmade environments such as financial markets. Statistical analysis and predictions in such systems are complicated by the fact that on the one hand extreme events may appear as ``outliers'' whose statistical properties do not seem to conform with the bulk of the data, and on the other hand they dominate the tails of the probability distributions and the scaling of high moments, leading to ``abnormal'' or ``multiscaling.'' We employ a shell model of turbulence to show that it is very useful to examine in detail the dynamics of onset and demise of extreme events. Doing so may reveal dynamical scaling properties of the extreme events that are characteristic to them, and not shared by the bulk of the fluctuations. As the extreme events dominate the tails of the distribution functions, knowledge of their dynamical scaling properties can be turned into a prediction of the functional form of the tails. We show that from the analysis of relatively short-time horizons (in which the extreme events appear as outliers) we can predict the tails of the probability distribution functions, in agreement with data collected in very much longer time horizons. The conclusion is that events that may appear unpredictable on relatively short time horizons are actually a consistent part of a multiscaling statistics on longer time horizons.