Compound Extremes and Bunched Black (or Grouped Grey) Swans

Abstract:

Observed “wild” natural fluctuations may differ substantially in their character. Some events may be genuinely
unforeseen (and unforeseeable), as with Taleb’s “black swans”. These may occur singly, or may have their impact
further magnified by being “bunched” in time. Some of the others may, however, be the rare extreme events from a
light-tailed underlying distribution. Studying their occurrence may then be tractable with the methods of extreme
value theory [e.g. Coles, 2001], suitably adapted to allow correlation if that is observed to be present.

Yet others may belong to a third broad class, described in today’s presentation [ reviewed in Watkins, GRL
Frontiers, 2013, doi: 10.1002/grl.50103]. Such “bursty” time series may show comparatively frequent high
amplitude events, and/or long range correlations between successive values. The frequent large values due to the
first of these effects, modelled in economics by Mandelbrot in 1963 using heavy- tailed probability distributions,
can give rise to an “IPCC type I” burst composed of successive wild events. Conversely, long range dependence,
even in a light-tailed Gaussian model like Mandelbrot and van Ness’ fractional Brownian motion, can integrate
“mild” events into an extreme “IPCC type III” burst.

I will show how a standard statistical time series model, linear fractional stable motion (LFSM), which de-
scends from the two special cases advocated by Mandelbrot, allows these two effects to be varied independently,
and will present results from a preliminary study of such bursts in LFSM. The consequences for burst scaling when
low frequency effects due to dissipation (FARIMA models), and multiplicative cascades (such as multifractals)
are included will also be discussed, and the physical assumptions and constraints associated with making a given
choice of model.