Nicholas W. Watkins
University of Warwick
Mandelbrot's fractional renewal model for 1/f noise: weak ergodicity breaking in the 60s?
I will give an update on work that I presented at PKS at the beginning
of this year, and which I have been pursuing in the current ASG led by
Rainer Klages. The talk links two topics: one physical, the range of
possible models for generating "1/f" noise, and one historical, how
this range was anticipated by, and embodied in, some still very little
known work by Mandelbrot in 1963-67. Running through both threads is
the idea of ergodicity, in the sense widely used in both signal
processing and statistical mechanics, whereby a time average
of a single trajectory (or time series) can be treated as estimating
its ensemble average. Ergodicity has been fundamental to the
interpretation and widespread use of techniques such as power spectra
and autocorrelation, and so in the education of most physicist (and
many engineers).
In recent years, however, there has been a great increase of interest
in physical (and other) systems which break ergodicity, motivated
in part by new experimental time series from, for example, single
particle tracking, and blinking quantum dots. In particular,
paradigmatic models of the fractional renewal type, with long-tailed
waiting times between changes in discrete levels, have been shown to
be non-ergodic, having both an explicit dependence of spectra, and
correlation functions on the observed length of the time series and
strongly fluctuating estimates of quantities such as power
spectra. This is in contrast to the long range dependent, fractional
Gaussian noise model (fGn) studied by Mandelbrot, van Ness and Wallis
from 1965, which is ergodic, and "better" behaved, at the cost of
assuming what has sometimes been seen as an unphysical and fully
non-Markovian correlation structure.
It has thus been fascinating to me to discover that many of these new
findings were prefigured in Mandelbrot's own work, on a model for
"1/f" noise that was quite different to his much better known fGn. I
will summarise what I have so far been able to understand about his
fractional renewal models, inspired by the need to model telephone
errors; and his results, especially those in a remarkably far-sighted
paper from 1967. I speculate on how the full scope of the work
came to be largely overlooked by both the statistics and physics
literatures since the 60s, and how this has, in my view, affected the
history of complexity science, and even the way the subject is
currently taught. I will conclude by setting the fGn and fractional
renewal classes in their wider modern context, as part of what he
called the "panorama of grid bound variability".
talk