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 ( α ) of a function, f ( x ) , is defined as:

Where α ( x ) is the function describing the Holder exponent, α ( x ) of f ( x ) at the point x . D F { ⋅ } is the Hausdorff dimension of a point set.