Emergence of bi-fractal time series from noise via allometric filters

We show how fractal time series can naturally emerge from an input signal, which might be also random noise, when processed by some allometric mechanisms. In particular, we introduce and study the properties of allometric low and high-pass filters. We also argue that in nature pure monofractal signals could be unphysical, while bi-scaling signals might be extremely common, a fact that might be due to allometric geometrical constraint. Finally, we suggest how allometric physical mechanisms could give an approximative interpretation of the fractional calculus. Copyright c © EPLA, 2007 Introduction. – Innumerable physical, geophysical, biophysical and econophysical phenomena have been classified as complex and their patterns described by appropriate scaling laws and scaling exponents [1–5]. Herein we argue that a large set of these physical observations easily emerge because complex systems might often process much simpler stimuli, and the complexity of the response depends on both the properties of the input and the complexity of the processing mechanism. Indeed, chaotic complex behavior emerge even from randomly polarized pulses processed by simple filters under time reversal [6]. However, time reversal is not physical. We show that allometric filters and relaxation processes, which should be quite common in nature, might indeed let fractal patterns naturally emerge from simpler signals, such as random fluctuations. These filters produce bi-scaling patterns that suggest candidate explanations for the generating mechanisms. Furthermore, such allometric filters might help disentangle the complexity of the forcing input signal from the complexity of the processing mechanism producing the final output. Hurst [7] first observed long-range fractal correlation in a Nile River minima series. Later, Mandelbrot [1] introduced fractal signals as those characterized by a power spectrum fulfilling the following scaling property