Multifractals

Visual interpretations of f(α) curves

Addresses 1 and 2 appear to have about the same number of points, more than those in address 3, in turn more than those in address 4.

p₁ = p₂ > p₃ > p₄

Because r₁ = r₂ = r₃ = r₄, the minimum probability corresponds to the maximum α, and the maximum probability corresponds to the minimum α.

Then α_min = log(p₁)/log(1/2) and α_max = log(p₄)/log(1/2)

The maximum probability occurs when the transformations T₁ and T₂ are applied. This generates the line between corners 1 and 2, so this line fills in most densely, which we observe. Then f(α_min) is the dimension of the line between corners 1 and 2, that is, f(α_min) = 1. We see this on the left endpoint of the f(α) curve.

The minimum probability occurs when just the transformation T₄ is applied. This generates the point at corner 4, so the vicinity of this point fills in least densely, which we observe. Then f(α_max) is the dimension of the point at corner 4, that is, f(α_max) = 0. We see this on the right endpoint of the f(α) curve.

The four transformations T₁, T₂, T₃, and T₄ generate the filled-in unit square, a 2-dimensional shape. consequently max(f(α)) = 2. We see this as the highest point of the f(α) curve.

Return to visual interpretations of f(α) curves.


Addresses 1 and 2 appear to have about the same number of points, more than those in address 3, in turn more than those in address 4.
p₁ = p₂ > p₃ > p₄
Because r₁ = r₂ = r₃ = r₄, the minimum probability corresponds to the maximum α, and the maximum probability corresponds to the minimum α.
Then α_min = log(p₁)/log(1/2) and α_max = log(p₄)/log(1/2)
The maximum probability occurs when the transformations T₁ and T₂ are applied. This generates the line between corners 1 and 2, so this line fills in most densely, which we observe. Then f(α_min) is the dimension of the line between corners 1 and 2, that is, f(α_min) = 1. We see this on the left endpoint of the f(α) curve.
The minimum probability occurs when just the transformation T₄ is applied. This generates the point at corner 4, so the vicinity of this point fills in least densely, which we observe. Then f(α_max) is the dimension of the point at corner 4, that is, f(α_max) = 0. We see this on the right endpoint of the f(α) curve.
The four transformations T₁, T₂, T₃, and T₄ generate the filled-in unit square, a 2-dimensional shape. consequently max(f(α)) = 2. We see this as the highest point of the f(α) curve.