Multiplicative cascade

In mathematics, a multiplicative cascade^[1][2] is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process.

Definition

The plots above are examples of multiplicative cascade multifractals.

To create these distributions there are a few steps to take. Firstly, we must create a lattice of cells which will be our underlying probability density field.

Secondly, an iterative process is followed to create multiple levels of the lattice: at each iteration the cells are split into four equal parts (cells). Each new cell is then assigned a probability randomly from the set [math]\displaystyle{ \lbrace p_1,p_2,p_3,p_4 \rbrace }[/math] without replacement, where [math]\displaystyle{ p_i \in [0,1] }[/math]. This process is continued to the Nth level. For example, in constructing such a model down to level 8 we produce a 4⁸ array of cells.

Thirdly, the cells are filled as follows: We take the probability of a cell being occupied as the product of the cell's own p_i and those of all its parents (up to level 1). A Monte Carlo rejection scheme is used repeatedly until the desired cell population is obtained, as follows: x and y cell coordinates are chosen randomly, and a random number between 0 and 1 is assigned; the (x, y) cell is then populated depending on whether the assigned number is lesser than (outcome: not populated) or greater or equal to (outcome: populated) the cell's occupation probability.

Examples

Three multiplicative cascades.
Generators (left to right): [math]\displaystyle{ \lbrace p_1,p_2,p_3,p_4 \rbrace = \lbrace 1,1,1,0 \rbrace }[/math], [math]\displaystyle{ \lbrace p_1,p_2,p_3,p_4 \rbrace = \lbrace 1,0.75,0.75,0.5 \rbrace }[/math], [math]\displaystyle{ \lbrace p_1,p_2,p_3,p_4 \rbrace = \lbrace 1,0.5,0.5,0.25 \rbrace }[/math]

To produce the plots above we filled the probability density field with 5,000 points in a space of 256 × 256.

An example of the probability density field:

The fractals are generally not scale-invariant and therefore cannot be considered standard fractals. They can however be considered multifractals. The Rényi (generalized) dimensions can be theoretically predicted. It can be shown ^[3] that as [math]\displaystyle{ N \rightarrow \infty }[/math],

[math]\displaystyle{ D_q=\frac{\log_2\left( f^q_1+f^q_2+f^q_3+f^q_4\right)}{1-q}, }[/math]

where N is the level of the grid refinement and,

[math]\displaystyle{ f_i=\frac{p_i}{\sum_i p_i}. }[/math]

See also

Wikimedia Commons has media related to fractals.

• Fractal dimension
• Hausdorff dimension
• Scale invariance

References