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 { p 1 , p 2 , p 3 , p 4 } {\displaystyle \lbrace p_{1},p_{2},p_{3},p_{4}\rbrace } without replacement, where p i ∈ [ 0 , 1 ] {\displaystyle 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 pi 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 → ∞ {\displaystyle N\rightarrow \infty } ,
D q = log 2 ( f 1 q + f 2 q + f 3 q + f 4 q ) 1 − q , {\displaystyle D_{q}={\frac {\log _{2}\left(f_{1}^{q}+f_{2}^{q}+f_{3}^{q}+f_{4}^{q}\right)}{1-q}},}
where N is the level of the grid refinement and,
f i = p i ∑ i p i . {\displaystyle f_{i}={\frac {p_{i}}{\sum _{i}p_{i}}}.}
See also
Fractal dimension Hausdorff dimension Scale invariance
References
