In mathematics, a multiplicative cascade[1] [2] is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process.
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
\lbracep1,p2,p3,p4\rbrace
pi\in[0,1]
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.
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
N → infty
| D | ||||||||||||||||
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,
where N is the level of the grid refinement and,
| f | ||||
|
.