Apparent universality of $1/f$ spectra as an artifact of finite-size effects

Abstract

Power spectral density scaling with frequency $f$ as $1/f^\beta$ and $\beta \approx 1$ is widely found in natural and socio-economic systems. Consequently, it has been suggested that such self-similar spectra reflect the universal dynamics of complex phenomena. Here, we show that for a superposition of uncorrelated pulses with a power-law distribution of duration times the estimated scaling exponents $\bar{\beta}$ depend on the system size. We derive a parametrized, closed-form expression for the power spectral density, and demonstrate that for $\beta \in [0,2]$ the estimated scaling exponents have a bias towards $\bar{\beta}=1$. For $\beta=0$ and $\beta=2$ the explicit logarithmic corrections to frequency scaling are derived. The bias is particularly strong when the scale invariance spans less than four decades in frequency. Since this is the case for the majority of empirical data, the boundedness of systems well described by the superposition of uncorrelated pulses may contribute to overemphasizing the universality of $1/f$.