Mandelbrot’s other model of 1/f noise: “Conditional stationarity” and weak ergodicity breaking as seen from the mid sixties
More than 100 years ago, Thomson (Kelvin) and Tait’s classic “Treatise on Natural Philosophy” cautioned its readers against “considering the formula and not the fact as physical reality”. My own research into the modelling of time series from complex natural systems, including space plasmas, atmospheric temperature and animal foraging datasets [1,2,3]-and into the history of such models [4,5]-has exposed me to many instances of the problem Thomson & Tait identified. I have certainly fallen prey to it myself.
This talk focuses on one facet of the problem-the “1/f” spectral shape seen in many areas of physics, recognised by Schottky in the early 20th century. I will briefly recap the relatively well known story  of part of Mandelbrot’s quest to explain 1/f noise-from the amplitude domain, and heavy tails in cotton price fluctuations in 1963; via the Fourier spectral domain, and a stationary long range dependent (LRD) fractional Gaussian noise (fGn) model for the water levels in the Nile in 1968; and then on to multifractal models for turbulence and finance from 1972 on. I will also discuss a much less well known part of his journey, the family of models he dubbed “conditionally stationary” [1963-67]. These also gave a 1/f spectrum-but in contrast to fGn-he showed it depended explicitly on the length of the time series. These models achieved a 1/f effect by having long tailed waiting times, and I will talk about their differences from fGn, and similarities to the continuous time random walks, renewal reward process, and alternating fractional renewal process, now studied in different contexts. They form the “missing link” between heavy tails in amplitude in 1963, and the long range dependent kernel of ’65-68.
I will recount how in the 1990s, late in his life, Mandelbrot made a special effort in his Selecta volumes to revisit the differences between his models. He urged us to use our eyes as well as formalism, making him an unexpected (to some) ally of Thomson and Tait. The current interest in weak ergodicity breaking has meant that concepts similar to Mandelbrot’s conditionally stationary models are now being considered (e.g. Niemann et al, PRL, 2013), so I will end by trying to place them in a current context to indicate where we can still learn from them, and will speculate on how their relative neglect may have influenced the history of complexity science [4,5].
 Watkins, Bunched Black Swans, Geophys. Res. Lett., 2013
 Watkins and Freeman, Natural Complexity, Science, 2008
 Edwards et al, Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer, Nature, 2007
 Graves et al, A Brief History of Long Memory, arXiv:1406.6018v2[stat.OT]
 Watkins, 25 Years of Self-organized Criticality: Concepts and Controversies, Space Science Reviews, 2015