Singularity spectrum

The singularity spectrum is a function used in multifractal analysis to describe the fractal dimension of a subset of points of a function belonging to a group of points that have the same Hölder exponent. Intuitively, the singularity spectrum gives a value for how "fractal" a set of points are in a function.

More formally, the singularity spectrum $$D(\alpha)$$ of a function, $$f(x)$$, is defined as:


 * $$D(\alpha) = D_F\{x, \alpha(x) = \alpha\}$$

Where $$\alpha(x)$$ is the function describing the Hölder exponent, $$\alpha(x)$$ of $$f(x)$$ at the point $$x$$. $$D_F\{\cdot\}$$ is the Hausdorff dimension of a point set.