{{Short description|Fractal distribution of random points}} In mathematics, a '''multiplicative cascade'''<ref>{{cite journal|last1=Meakin|first1=Paul|title=Diffusion-limited aggregation on multifractal lattices: A model for fluid-fluid displacement in porous media|journal=Physical Review A|date=September 1987|volume=36|issue=6|pages=2833–2837|doi=10.1103/PhysRevA.36.2833|pmid=9899187|url=http://journals.aps.org/pra/abstract/10.1103/PhysRevA.36.2833|url-access=subscription}}</ref><ref>[https://arxiv.org/abs/0803.3212 Cristano G. Sabiu, Luis Teodoro, Martin Hendry, arXiv:0803.3212v1 ''Resolving the universe with multifractals'']</ref> 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>\lbrace p_1,p_2,p_3,p_4 \rbrace</math> without replacement, where <math>p_i \in [0,1]</math>. This process is continued to the ''N''th level. For example, in constructing such a model down to level 8 we produce a 4<sup>8</sup> 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''<sub>''i''</sub> 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==
thumb|Three multiplicative cascades.<br/>Generators (left to right): <math>\lbrace p_1,p_2,p_3,p_4 \rbrace = \lbrace 1,1,1,0 \rbrace</math>, <math>\lbrace p_1,p_2,p_3,p_4 \rbrace = \lbrace 1,0.75,0.75,0.5 \rbrace</math>, <math>\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, the probability density field is filled with 5,000 points in a space of 256 × 256.
An example of the probability density field:<br /> Image:Multifractal density field.jpg
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 <ref>Martinez et al. ApJ 357 50M "Clustering Paradigms and Multifractal Measures" [http://adsabs.harvard.edu/abs/1990ApJ...357...50M]</ref> that as <math>N \rightarrow \infty</math>,
: <math>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>f_i=\frac{p_i}{\sum_i p_i}.</math>
== See also == {{Commons|Fractal|fractals}} * Fractal dimension * Hausdorff dimension * Scale invariance
==References== {{reflist}}
Category:Fractals