# Complex Systems, Emergence, and Multiscale Analysis: A Tutorial and Brief Survey
## 2 Basics of Complex Systems and Emergence
### The Random Cascade Model  
In the study of multifractals, it is important to have a constructive model. This is provided by the random cascade model. It is among the most powerful models to understand the intermittency phenomenon of turbulence [158; 159; 160; 161; 162]. Here, we will use the notations developed for modeling Internet traffic and geophysical data [163; 164; 38; 165] to present the model.  
Consider a unit mass unevenly distributed on a unit interval. Let us divide the unit interval into two parts: call them the left and the right segments. By doing so, we have also partitioned the mass into two fractions, \(r\) and \(1-r\), which are on the left and right segments correspondingly. In general, the multiplier \(r\) is a random variable, having a probability density function (PDF) \(P(r)\), \(0\leq r\leq 1\). Always with this rule we can further partition each new subinterval and the weight attached to it into two parts, ad infinitum. Figure 21 shows the procedure schematically. To facilitate mathematical analysis, the multiplier \(r\) has been rewritten as \(r_{ij}\), where \(i\) indicates the stage number and \(j\) indicates the positions of a weight on that stage (we only use odd numbers, leaving even numbers for \(1-r_{ij}\)). For many types of data analysis, it is important to explicitly introduce the notion of scale. This is provided by the interval length, which is \(2^{-i}\) at stage \(i\). Assuming bilateral symmetry, then we have to require that \(P(r)\) is symmetric about \(r=1/2\). Let \(P(r)\) have successive moments\(\mu_{1},\mu_{2},\cdots\). Hence, \(r_{ij}\) and \(1-r_{ij}\) both have marginal distribution \(P(r)\). The weights at the stage \(N\), \(\{w_{n},n=1,...,2^{N}\}\), can be expressed as  
\[w_{n}=u_{1}u_{2}\cdots u_{N}, \tag{87}\]  
where \(u_{l},l=1,\cdots,N\), are either \(r_{ij}\) or \(1-r_{ij}\). Thus, \(\{u_{i},i\geq 1\}\) are IID random variables all having PDF \(P(r)\).  
The cascade model has many interesting properties. We list a few here:  
* The weights at stage \(N\) are log-normally distributed. To see this, one can take logarithm on both sides of Equation (87), then the multiplication becomes summation, and one can use the central limit theorem.
* We can readily derive that \[\tau(q)=-\ln(2\mu_{q})/\ln 2.\] (88)
* We can also derive that \[H(q)\sim-\frac{1}{q}\ln\mu_{q}/\ln 2,\] (89) and \[\tau(q)=qH(q)-1.\] (90)  
We now illustrate Equations (88) and (89) using an example, the random binomial model, whose \(P(r)\) is  
\[P(r)=[\delta(r-p)+\delta(r-(1-p))]/2 \tag{91}\]  
where \(\delta\) denotes the Dirac function. Therefore, \(P(r=p)=P(r=1-p)=1/2\). Here, the \(q\)th moment \(\mu_{q}=[p^{q}+(1-p)^{q}]/2\). We thus find  
\[\tau(q)=-\ln[p^{q}+(1-p)^{q}]/\ln 2 \tag{92}\]  
and  
\[H(q)=\frac{1}{q}\{1-\ln[p^{q}+(1-p)^{q}]/\ln 2\} \tag{93}\]  
Clearly, \(H(q)\) is a non-decreasing function of \(q\). Without loss of generality, we may take \(p\leq 1/2\). When \(q\to-\infty\), \(H\) converges to its tight upper bound of \(-\ln p/\ln 2>1\). When \(q\to\infty\), \(H\) converges to its tight lower bound of \(-\ln(1-p)/\ln 2<1\).  
Figure 21: Schematic showing how a multiplicative multifractal is constructed.  
In the cascade model, many different functional shapes for \(P(r)\) can be used [163; 164], and the model can simulate a random function with very high accuracy. Two examples are shown in Figure 22 for sea-clutter amplitude data [38; 166]. The model can also be readily generalized to the high-dimensional case. The case for 2D is shown in Figure 23.