Statistical Outliers or Extremes, Dragon-Kings or Dragon-Fools?
Daniel Schertzer
LEESU, Université Paris Est, Ecole des Ponts ParisTech, Marne-la-Vallée, France

With floods and droughts of all magnitudes, hydrology may be seen as full of monsters as the Court of Miracles described by Victor Hugo. Indeed, these extremes have been statistically considered for a while as outliers with respect to rather mild distributions of more normal events. However, a characteristic magnitude separating the normal and abnormal events was understood as being at odd with the generic scaling behaviour of nonlinear systems, contrary to fat tailed distributions. More precisely, it can be shown [1] how the apparent monsters could be mere manifestations of a singular measure mishandled as a regular measure. Monstrous fluctuations are the rule, they are not outliers and are more frequent than usually thought. The most striking result is that the (theoretical) statistical moments can easily be infinite causing their empirical estimates to be erratic and to diverge with sample size. This can be physically understood as resulting from the fact that the upscaling cannot smooth out intense small scale events.

Nevertheless, it has also been argued, based on a few examples, that one should consider 'genuine' outliers of fat tailed distributions so monstrous that they can be called 'dragon-kings' [2]. We critically analyse these arguments, e.g. finite sample size and statistical estimates of the largest events, multifractal phase transition vs. more classical phase transition. We emphasize the fact that dragon-kings are not needed in order that the largest events are predictable. This is rather reminiscent of the Feast of Fools picturesquely described by Victor Hugo.

References:

[1] D. Schertzer, I. Tchiguirinskaia, S. Lovjeoy et P. Hubert (2010): No monsters, no miracles: in nonlinear sciences hydrology is not an outlier! Hydrological Sciences Journal, 55: 6.

[2] D. Sornette (2010): Dragon-Kings, Black Swans and the Prediction of Crises. International Journal of Terraspace Science and Engineering (in press).

Keywords: Extemes; Scaling; Singlular measure; Fat tailed distributions

Biography: Professor at Ecole des Ponts ParisTech and Head of the Chair “Hydrology for Resilient Cities” (Ecole des Ponts, supported by VEOLIA), has had responsibilities at union level of the Union European Geosciences Union and American Geophysical Union, at the bureau of the International Association for Hydrological Sciences. He is executive editor of the journal Nonlinear Processes in Geophysics.

He introduced the multifractal formalism and related techniques in hydrology, after having contributed to their development in turbulence. He also contributed to the development of the Fractional Fokker-Planck equation. His research has covered many domains of geophysics and environment, although with a particular emphasis on atmospheric dynamics, precipitations, extremes and remote-sensing techniques. One may note a model of stratified turbulence of dimension 23/9 and the genericity of power-laws for geophysical extremes.