Multiplicative cascade


In mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process.


Model I :
Model II :
'Model III :
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. Each new cell is then assigned a probability randomly from the set without replacement, where. 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. 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 cell is then populated depending on whether the assigned number is lesser than or greater or equal to 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 dimensions can be theoretically predicted. It can be shown that as,
where N is the level of the grid refinement and,