Multiplicative cascade

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 [0,1]. 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

To produce the plots above, the probability density field is filled 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 that as N \rightarrow \infty,

: D_q=\frac{\log_2\left( f^q_1+f^q_2+f^q_3+f^q_4\right)}{1-q},

where N is the level of the grid refinement and,

: f_i=\frac{p_i}{\sum_i p_i}.

References

References

1. (September 1987). "Diffusion-limited aggregation on multifractal lattices: A model for fluid-fluid displacement in porous media". Physical Review A.
2. [https://arxiv.org/abs/0803.3212 Cristano G. Sabiu, Luis Teodoro, Martin Hendry, arXiv:0803.3212v1 ''Resolving the universe with multifractals'']
3. Martinez et al. ApJ 357 50M "Clustering Paradigms and Multifractal Measures" [http://adsabs.harvard.edu/abs/1990ApJ...357...50M]