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

The standard spectral scaling, $S\left(f\right)\sim 1/f^{\beta }$, has been traditionally used as a correlation measure characterizing the dynamical behavior of time series. The ubiquity of $1/f$ and $1/f^2$ 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, $S(f,T)\sim T^{\eta }/f^{\beta }$, which is usually considered irrelevant, gives an insight into this problem. Using synthetically generated series we show that, in general, the scaling exponent $\beta$ is too generic, while the exponent associated with the size, $\eta$, 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.

Figure 1

Classification of time series by their complete scaling spectra showing the main classes: stationary noises (SN) and stationary fractal curves (SF), self-affine curves (SA), and pure $1/f$ processes (1F). Points $P1–P8$ correspond to examples shown in the figures: (+, black) additive and (x, red) multiplicative processes, (o, blue) uncorrelated and (@, brown) correlated pulses.

Figure 2

Spectra (upper) and local widths (lower) of SA $\left(P1,\alpha ={\alpha }_{\mathrm{loc}}={\alpha }_s=0.1\right)$ and SN $\left(P2,\alpha ={\alpha }_{\mathrm{loc}}=0,{\alpha }_s=-0.1\right)$ series generated from a fractional-Brownian method as explained in the text. The initial condition is a Gaussian with ${\sigma }^2=2.14$, the number of samples is 100, and the sizes of series are $T=2^9,2^{11},2^{13}$.

Figure 3

Plot of individual series (with $T=2^{11})$ showing cases with single scaling: In the SA class, $P1$ $\left(\alpha ={\alpha }_s=0.1\right)$, stationary noise, $P2$ $\left(\alpha ={\alpha }_s=-0.1\right)$, and stationary $1f$ process, $P6$ $\left(\alpha ={\alpha }_s=0\right)$. Fractality of $P1$ and (less evident) $P6$ can be observed by eye. $P2$ and $P6$ are displaced upwards by 5 and 10 units respectively.

Figure 4

Spectra (upper) and local widths (lower) with their respective collapses (insets) of a multiplicative process $\left(P3,\alpha =1,{\alpha }_{\mathrm{loc}}={\alpha }_s=0.5\right)$ generated as the $q=2$ power of a random walk (RW). The initial condition is Gaussian with $\sigma =1$, the number of samples is 1000, and the sizes of series are $T=2^9,2^{11},2^{13}$.

Figure 5

Spectra and collapses (insets) of two multiplicative processes generated as the $q$ power of a RW with ${\sigma }_0=1$ as initial condition. Upper with $q=0.1$ $\left(P4,\alpha =0.05,{\alpha }_s=0.3\right)$ in the limit of the SF class. Lower with $q=-1/2$ $\left(P5,\alpha =-1/4,{\alpha }_s=0\right)$ in the 1F class. As in the previous figure $T=2^9,2^{11},2^{13}$ with 1000 samples.

Figure 6

Spectra and collapse (inset) of series of uncorrelated pulses. Upper, a typical example of stationary 1F noise $\left(P6,\alpha ={\alpha }_s=0\right)$ generated from exponential pulses with a uniform distribution of widths and a Poissonian renewal time. Lower, an example of the SF class $\left(P7,\alpha =0,{\alpha }_s=1/4\right)$ generated from exponential pulses with the same width than their renewal time, which is power law distributed. The same sizes and number of samples as in the previous figures.

Figure 7

Plot of individual series (with $T=2^{11})$ showing samples nearly of the SF class $\left(\alpha \sim 0,{\alpha }_s\ge 0\right)$ with distinct spectral exponents: ${\alpha }_s=0.25$ $\left(P7\right)$, ${\alpha }_s=0.3$ $\left(P4\right)$, ${\alpha }_s=0$ $\left(P6\right)$. $P4$ and $P7$ have closed exponents but their shape is very different. $P4$ and $P6$ are displaced upwards by five and ten units respectively. $P4$ is amplified by a factor of 5.

Figure 8

Spectra of series of correlated pulses as examples of the 1F class $\left(P8,\alpha =-1/3,{\alpha }_s=0\right)$. Upper, with a positive Wiener process with ${\sigma }_0=1$ for the interevent times and distinct shapes of pulses: flat (black), exponential (red), and Lorentzian (blue). Lower, for series of exponential pulses with distinct sizes, from top to bottom $T=2^9,2^{11},2^{13}$. The inset shows the collapse of spectra with $\alpha =-1/3$. In all cases each spectrum is an average of 1000 samples.

Figure 9

Plot of individual series showing samples in the 1F class $\left({\alpha }_s=0\right)$ with distinct global exponent: $\alpha =-0.25$ $\left(P5\right)$, $\alpha =-0.33$ $\left(P8\right)$, and $\alpha =0$ $\left(P6\right)$. Despite the distinct nature of the generating process $P5$ and $P8$ look similar. $P6$ and $P8$ are displaced upwards by 10 and 20 units respectively. $P8$ is amplified by a factor of 5.