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 ${\displaystyle D(\alpha )}$ of a function, ${\displaystyle f(x)}$, is defined as:

${\displaystyle D(\alpha )=D_{F}\{x,\alpha (x)=\alpha \}}$

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