# Complex Systems, Emergence, and Multiscale Analysis: A Tutorial and Brief Survey
## 2 Basics of Complex Systems and Emergence
### Basics of Random Fractal Theory
#### 2.4.2 Overview of Random Fractal Theory  
Gaussian white noise is the most extensively studied noise in engineering. In complex systems, however, the temporal or spatial fluctuations often cannot be modeled by Gaussian white noise. Rather, they are characterized by a power law decaying spectrum in the frequency domain, denoted as \(1/f^{\alpha}\) noise [38]. Its dimensionality cannot be reduced by popular methods such as principle component analysis [124]. Interesting examples of such processes include genetics [125; 126; 127; 128; 129], human cognition [130] and coordination [131], posture [132], vision [133; 134], physiological signals [135; 136; 137; 138; 139; 140; 141; 142; 143], neuronal firing [144; 145], urban traffic [146], tree rings [147], global terrorism [148], human response to natural and social phenomena [149], foreign exchange rate [76], and the distribution of prime numbers [150]. Basic Definitions and Equations  
Denote a covariance stationary stochastic process as \(X=\{X_{t}:t=0,1,2,\dots\}\). Its mean is \(\mu\), variance is \(\sigma^{2}\), and autocorrelation function \(r(k),k\geq 0\) has the following form  
\[r(k)\sim k^{2H-2},\;\;as\;\;k\rightarrow\infty, \tag{71}\]  
where \(H\) is a parameter called the Hurst exponent. It is in the unit interval, \(0<H<1\). The exponent \(\alpha\) for the spectra of the process, \(1/f^{\alpha}\), is related to \(H\) by a simple equation,  
\[\alpha=2H-1. \tag{72}\]  
When \(0<H<1/2\), the process is said to have anti-persistent correlations; when \(H=1/2\), the process is memoryless or only has short memory, when \(1/2<H<1\), the process is said to have persistent long-range correlations. In this case, it is easy to prove \(\sum_{k}r(k)=\infty\). This is why the process is said to have long-range correlation [38]. Let us now smooth the process \(X\) to obtain a time series \(X^{(m)}=\{X^{(m)}_{t}:t=1,2,3,\dots\}\), \(m=1,2,3,\dots\), where  
\[X^{(m)}_{t}=(X_{tm-m+1}+\dots+X_{tm})/m,\;\;t\geq 1\;. \tag{73}\]  
The smoothing is carried out in a non-overlapping fashion; therefore, the length of \(\{X^{(m)}_{t}\}\) is the largest integer that is not larger than \(N/m\), where \(N\) is the length of \(\{X_{t}\}\). Is there a relation between the variance of \(X^{(m)}_{t}\), which is denoted by \(V_{m}=var(X^{(m)})\), and that of the original process, which is denoted by \(\sigma^{2}\)? It is given by  
\[var(X^{(m)})=\sigma^{2}m^{2H-2} \tag{74}\]  
Equation (74) is often called the variance-time relation. It is fundamental and can help us understand the \"little smoothing\" phenomenon: when \(H=0.5\), \(var(X^{(m)})=10^{-2}\sigma^{2}\) when \(m=100\). When \(H=0.75\), for \(var(X^{(m)})\) to drop as much, \(m\) has to be 10,000. However, if \(H=0.25\), then \(var(X^{(m)})\approx 10^{-2}\sigma^{2}\) when \(m\approx 23\). Therefore, when \(H\) increases to 1, smoothing has little effect in reducing the variance of the process. As we have mentioned, the power spectral density (PSD) for \(X\) is  
\[S_{X}(f)\sim f^{-\alpha}=f^{-(2H-1)}. \tag{75}\]  
The integration of the \(X\) process, called the random walk process,  
\[y_{k}=\sum_{i=1}^{k}(X_{i}-\overline{X}), \tag{76}\]where \(\overline{X}\) is the mean of \(X\), has a PSD  
\[S_{Y}(f)\sim f^{-a-2}=f^{-(2H+1)}. \tag{77}\]  
It is easy to see that the following relation is equivalent to Equation (74)  
\[\left\langle|y(i+m)-y(i)|^{2}\right\rangle\sim m^{2H}, \tag{78}\]  
where the angle brackets denote averaging over \(i\). Equation (78) is often called fluctuation analysis (FA). The superiority of Equation (78) over Equation (74) is that it can be readily generalized to a multifractal formulation. The Fractional Brownian Motion (fBm) Process  
The fBm process is the prototypical random walk model for \(1/f^{\alpha}\) process [121]. It is a zero-mean Gaussian process, with stationary increments and variance  
\[E[(B_{H}(t))^{2}]=t^{2H} \tag{79}\]  
and covariance:  
\[E[B_{H}(s)B_{H}(t)]=\frac{1}{2}\{s^{2H}+t^{2H}-|s-t|^{2H}\} \tag{80}\]  
where \(H\) is the Hurst parameter.