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