Multiplicative Cascade

In mathematics, a multiplicative cascade 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 $\lbrace p_1,p_2,p_3,p_4 \rbrace$ without replacement, where p_i \in ,1/math>. This process is continued to the ''N''th 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

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 Martinez et al. ApJ 357 50M "Clustering Paradigms and Multifractal Measures

/ref> that as $N \rightarrow \infty$,

: $D_q=\frac,$

where N is the level of the grid refinement and,

: $f_i=\frac.$

See also

* Fractal dimension
* Hausdorff dimension
* Scale invariance

References

{{reflist

Fractals