Complete spectral scaling of time series: Towards a classification of 1/f noise

Abstract

The standard spectral scaling, S(f)∼1/fβ, has been traditionally used as a correlation measure characterizing the dynamical behavior of time series. The ubiquity of 1/f and 1/f2 spectra in many processes certainly suggests the existence of universal mechanisms, but also gives rise to the suspicion that some important features are not included in this scaling. In this paper we argue that a complete spectral scaling, including as a main variable the size of the series, S(f,T)∼Tη/fβ, which is usually considered irrelevant, gives an insight into this problem. Using synthetically generated series we show that, in general, the scaling exponent β is too generic, while the exponent associated with the size, η, gives a more specific information. Hence, we propose the use of both exponents in a scheme to classify series into different universality classes. In this way many of the processes appearing in the literature could be better identified, and much of the ambiguity that surrounds the standard spectral scaling could be clarified.