Scale Invariance

Abstract

Scale-invariance is one of the concepts that appears more often in the context of complexity. The basic idea behind scale-invariance can be naively expressed as: ‘the system looks the same at every scale’. Or, ‘if we zoom in (or out) our view of the system, its features remain unchanged’. Although it is true that complex dynamics are often at work when a system exhibits scale-invariance, it is important to be aware that this is not always the case. For instance, the random walk [1] is a process that exhibits scale-invariance but with underlying dynamics that are far from complex in any sense of the word (see Chap. 5).

Appendices

Appendix 1: Numerical Generation of Fractional Noises

Various algorithms have been proposed in the literature to generate numerical time series that approximate either fGn, or fLm with arbitrary tail-index α, for a prescribed Hurst exponent H [36]. Here, we will discuss one that is based on rewriting fBm (Eq. 3.71) and fLm (Eq. 3.89) in the form,

$$\displaystyle \begin{aligned} y^{\mathrm{H},\alpha}(t) = y^{\mathrm{H},\alpha}(t_0) + \mbox{}_{t_0}D_t^{-(H+1/\alpha - 1)}\xi_\alpha. \end{aligned} $$

(3.119)

This equation introduces a new type of operators known as fractional operators [47]. Equation 3.119 will be formally introduced in Chap. 5 under the name of the fractional Langevin equation (see Sect. 5.3.2). It is straightforward to show that it reduces to fBm for α = 2, and yields all symmetric fLms if α < 2.

The operator that appears in the fractional Langevin equation is known as a Riemann-Liouville fractional operator. They are integro-differential operators defined as [47],

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \mbox{}_aD_t^p f(t) = \left\lbrace \begin{array}{lc} \displaystyle \frac{1}{\varGamma(k-p)}\frac{d^k}{dt^k}\left[\int_a^t dt' (t-t')^{k - p - 1}f(t')dt' \right], &\displaystyle p > 0 \\ &\displaystyle \\ \displaystyle \frac{1}{\varGamma(-p)} \left[\int_a^t dt' (t-t')^{-(p + 1)}f(t')dt' \right], &\displaystyle p < 0 \end{array} \right. \end{array} \end{aligned} $$

(3.120)

where k is the integer satisfying k − 1 ≤ p < k. They are called fractional derivatives if p > 0, and fractional integrals if p < 0. An introduction to the basic features of fractional operators⁵³ will be given in Appendix 2 of Chap. 5. For now, it suffices to say that fractional operators have many interesting properties. Among them, that for p = ±k, they coincide with the usual k-th order derivatives (+ ) or integrals (−) and, for p = 0, with the identity. They provide interpolations between the usual integrals and derivatives of integer order. Fractional operators were first introduced by Gottfried von Leibniz, as early as 1697, although the form given in Eq. 3.120 was not introduced until the middle of the eighteenth century by Bernhard Riemann and Joseph Liouville.

We focus now on the properties of fractional operators that will allow us to generate synthetic fBm and fLm data, following a method introduced by A.V. Chechkin and V. Yu. Gonchar [48]. The first property is that, when acted on the right by a normal derivative, they satisfy that (see Appendix 2 of Chap. 5, Eq. 5.135):

$$\displaystyle \begin{aligned} \frac{d^m}{dt^m}\cdot \mbox{}_aD_t^p f(t) = \mbox{}_aD_t^{p+m} f(t),\end{aligned} $$

(3.121)

for any positive integer m. Since fGn/fLn is essentially the derivative of fBm/fLm, fGn/fLn can be obtained by applying a normal derivative to Eq. 3.119 to get:

$$\displaystyle \begin{aligned} \varDelta y^{\mathrm{H},\alpha}:= \lim_{h\rightarrow 0}\varDelta_h y^{\mathrm{H,\alpha}}= \frac{dy^{\mathrm{H},\alpha}}{dt} = \mbox{}_{t_0}D_t^{-(H+1/\alpha - 2)}\xi_\alpha.\end{aligned} $$

(3.122)

The second property of interest has to do with the relation between the Fourier transforms (see Appendix 1 of Chap. 2 for an introduction to Fourier transforms) of a function and its fractional derivative/integral (see Appendix 2 of Chap. 5, Eq. 5.139)⁵⁴:

$$\displaystyle \begin{aligned} \mathrm{F}\left[ \mbox{}_{a}D_t^p f(t)\right] \simeq (-\i \omega)^p \hat f(\omega),\end{aligned} $$

(3.123)

where $\i = \sqrt {-1}$ and ω stands for the frequency. Applying the Fourier transform to Eq. 3.122, one thus obtains:

$$\displaystyle \begin{aligned} \varDelta \hat y^{\mathrm{H},\alpha}(\omega) =\frac{\hat \xi_\alpha (\omega) }{(-\i \omega)^{H+1/\alpha - 2}}. \end{aligned} $$

(3.124)

Thus, in order to generate a synthetic fGn/fLn series with arbitrary Hurst exponent H and tail-index α,⁵⁵ a possible procedure is simply to (see Problem 3.5):

• generate a random Gauss/Lévy noise sequence (see Appendix 2 of Chap. 2);

• carry out a discrete Fast Fourier transform (FFT) of the noise series;

• divide each Fourier harmonic by the corresponding factor (−ıω)^H+1/α−2;

• and Fourier invert the result to get the desired, approximated fGn/fLn series.

In addition, fBm/fLm synthetic series with arbitrary exponents H and α can easily be obtained by numerically integrating the fGn/fLn series for the same exponents that have been generated with this method. Or, by reusing the procedure just described but dividing instead the Gauss/Lévy noise series by (−ıω)^H+1/α−1.

Appendix 2: Detrended Fluctuation Analysis

Detrended fluctuation analysis (or DFA) [49, 50] is a popular method that can be applied to test for self-similarity in both stationary and non-stationary signals. In order to introduce the method, let’s consider the series,

$$\displaystyle \begin{aligned} y = \left\lbrace y_n, ~~ n = 1, 2, \cdots, N \right\rbrace, \end{aligned} $$

(3.125)

that might be stationary or not. The method considers first the associated integrated motion,

$$\displaystyle \begin{aligned} Y_n = \sum_{i = 1}^n y_i - n\bar y, \end{aligned} $$

(3.126)

where the overall average of the motion, $\bar y = \sum y_i /N$ is removed.

Next, for every possible scale l > 0, the integrated motion is divided in (possibly overlapping) windows of size l. Inside each of these windows, a local least-squared-fit to a polynomial is done to capture the local trend. That is, if a possible linear local trend is assumed,⁵⁶ the fit would be against a straight line, my + b. The total squared error, χ ², for each window k is then given by:

$$\displaystyle \begin{aligned} \chi^2_k = \frac{1}{l} \sum_{i\in W_k} \left(Y_i - im_k - b_k\right)^2,~~~ k = 1, N/l. \end{aligned} $$

(3.127)

where W _k stands for the k-th window. The fluctuation value at scale l, F(l), is given by the average of the squared-root of the error over all windows W _k of size l:

$$\displaystyle \begin{aligned} F^2(l) := \frac{l}{N}\sum_{k = 1}^{N/l} \chi^2_k.\end{aligned} $$

(3.128)

If the underlying process is self-similar once the trends have properly been removed,⁵⁷ it should happen that F(l) ∼ l ^a, for some exponent a. The interesting thing is that, if DFA is applied to fractional Gaussian noise, that is stationary, it is found that

$$\displaystyle \begin{aligned} F(l) \sim l^H,\end{aligned} $$

(3.129)

for all scales l >> δ, with H being the Hurst exponent that defines the associated fBm. But, more interestingly, DFA could also have been applied directly to the fractional Brownian motion signal itself, which is non-stationary. Then, one would find that,

$$\displaystyle \begin{aligned} F(l) \sim l^{H+1},\end{aligned} $$

(3.130)

so that the Hurst exponent can be obtained directly from fBm.

DFA can also be extended to probe for multifractality. The procedure is often referred to as M-DFA [51]. The twist here is to calculate, instead of the squared error per window, the quantity,

$$\displaystyle \begin{aligned} F_q(l) = \left[\frac{l}{N}\sum_{k=1}^{N/l} \left| \chi_k^2\right|{}^{q/2}\right]^{1/q}.\end{aligned} $$

(3.131)

The generalized Hurst exponent is then defined by assuming the scaling,

$$\displaystyle \begin{aligned} F_q(l) \propto l^{H(q)},\end{aligned} $$

(3.132)

that, for a monofractal⁵⁸ (i.e., fGn), reduces to H(q) = H, ∀q. The same techniques that were discussed in Sect. 3.3.7 to the estimation of the multifractal spectrum can be used here on the H(q) obtained from the application of MDFA.

Appendix 3: Multifractal Analysis Via Wavelets

Wavelets are another powerful technique to test for multifractal behaviour in time series. We will not discuss the basics of wavelets at any length, since there are some wonderful review papers and books available that discuss them much better than what we could do here [44, 45, 52]. Instead, we provide just a brief introduction to the topic, sufficiently long to clarify their relevance to the investigation of multifractal features (see also Problem 3.10).

Wavelets were introduced in the mid 1980s as an extension of Fourier analysis that permitted the examination of local properties in real time [53]. That is, while Fourier analysis expresses an arbitrary function as a linear combination of sines and cosines that are localized in frequency but not in time, the wavelet representation expresses the function as a linear combinations of rescaled versions of a prescribed basis function Φ, also called wavelet, that is localized both in frequency and in time.

The Φ-wavelet transform of a function, f(t), at a scale r and time t is defined by the integral [45]:

$$\displaystyle \begin{aligned} \hat f_\varPhi(r, t) =C^{-1/2} r^{-1/2}\int_{-\infty}^\infty \varPhi\left(\frac{t - t'}{r}\right)f(t')dt', \end{aligned} $$

(3.133)

where the constant C is defined by:

$$\displaystyle \begin{aligned} C := \int_{-\infty}^\infty \frac{|\hat\varPhi(\omega)|{}^2}{|\omega|} d\omega, \end{aligned} $$

(3.134)

where $\hat \varPhi (\omega )$ is the Fourier transform (in time) of the wavelet basis function. Clearly, C < ∞, if Φ is to provide a valid wavelet, which requires that $\hat \varPhi (0) = 0$ (or, in other words, that Φ has a zero mean). $\hat f_\varPhi (r,t)$ can be interpreted as the part of f(t) that contributes at time t to the scale r. A nice property of $\hat f_\varPhi$ is that, if:

$$\displaystyle \begin{aligned} |\,f(t + r) - f(t) | \sim r^\alpha,~~~r\rightarrow 0, \end{aligned} $$

(3.135)

then

$$\displaystyle \begin{aligned} \hat f_\varPhi(r, t) \sim r^\alpha. \end{aligned} $$

(3.136)

That is, the local Holder exponent α of f(t) at any given time can be recovered by the scaling of local wavelet spectrum with r at the same time.

Therefore, the wavelet analysis of a time process allows in principle the direct determination of the local Holder exponent as a function of time, which opens up many new avenues of characterizing multifractality. In addition, wavelet multifractal analysis has been reported to be more robust (i.e., less sensitive to noise, for instance) that some of the methods that we have discussed in this chapter. The use of wavelets has also its own complications. A significant one is how to choose the more proper choice of the basis function, Φ, in order to best characterize singularities in time processes. Another one lies in the fact that the rescaled wavelets that enter in the continuous formulation given in Eq. 3.133 do not form an orthonormal set. Therefore, they provide a redundant representation whose interpretation can sometimes be confusing. This problem can be partially resolved through the introduction of discrete, orthonormal wavelet representations [44].

Problems

3.1 Scale Invariance: Power-Laws

Derive Eq. 3.4 from the Gutenberg-Richter law (Eq. 3.3).

3.2 Fractals: Cantor Set

Prove that the self-similar dimension of the Cantor Set is $D_{\mathrm {ss}} = \log (2)/\log (3)$.

3.3 Fractals: The Mandelbrot Set

Consider the quadratic recurrence $z_{k+1} = z_k^2 + z_0$ in the complex plane. Mandelbrot set is composed of all the initial choices for z ₀ for which the orbit predicted by the recurrence relation does not tend to infinity. Build a code to generate the Mandelbrot set. Determine its BC fractal dimension.

3.4 Time Processes: Brownian Motion

Write a numerical code to generate Brownian motion trajectories in two dimensions. Use the box-counting procedure and show that the BC fractal dimension of 2D Brownian motion is equal to 2. That is, if given enough time, Brownian motion would fill the whole plane.

3.5 Time Processes: Fractional Brownian Motion

Prove, by exploiting the self-similarity of fBm, that the fractal dimension of the time traces of fBm is given by D = 2 − H.

3.6 Generation of Synthetic fGn/fLn Series

Write a code that implements the algorithm described in Appendix 1 in order to generate synthetic series of fGn/fLn with arbitrary tail-index α and Hurst exponent, H. Implement the possibility of generating fBm/fLm as well.

3.7 Running Sandpile: Scaling Behaviour for Various Overlapping Regimes

Use the sandpile code (see Problem 1.5) to generate time series for the total mass of the sandpile over the SOC state using L = 1000, N _f  = 30, Z _c  = 200, N _b  = 10, one for each the following values of p ₀ = 10⁻⁶, 10⁻⁵, 10⁻³ and 10⁻². Repeat the scale-invariance analysis discussed in Sect. 3.5 for each cases. What is the mesorange for each case? How does the monofractal behaviour change as a function of the figure-of-merit that controls avalanche overlapping, (p ₀ L)⁻¹?

3.8 Running Sandpile: Scaling Behaviour Below the Mesorange

Use the sandpile code (see Problem 1.5) and generate a time series for the total mass of the sandpile over the SOC state using L = 1000, N _f  = 30, Z _c  = 200, N _b  = 10 and p ₀ = 10⁻⁴. Then, repeat the scale-invariance analysis discussed in Sect. 3.5, but for the range of block sizes 1 < M < 30, 000. How are the results different from what was obtained within the mesorange?

3.9 Advanced Problem: Detrended Fluctuation Analysis

Write a code that implements DFA1 (see Appendix 2). Then, use the code to estimate the Hurst exponent of fGn series generated with Hurst exponents H = 0.25, 0.45, 0.65 and 0.85 (see Problem 3.6). Compare the performance of DFA with the moment methods discussed in this chapter, both for the fGn series and their integrated fBm processes.

3.10 Advanced Problem: Wavelet Analysis

Write a code to perform the local determination of the Holder exponent using wavelets (see Appendix 3). Then, use the code on a synthetic fBm generated with nominal exponent H = 0.76 and show that the process is indeed monofractal, and the instantaneous local Holder exponent is constant and given by H. Refer to [45] to decide on the best option for the wavelet basis function.