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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 {\displaystyle D} of a function, f {\displaystyle f} , is defined as:
Where α {\displaystyle \alpha } is the function describing the Hölder exponent, α {\displaystyle \alpha } of f {\displaystyle f} at the point x {\displaystyle x}. D F { ⋅ } {\displaystyle D_{F}\{\cdot \}} is the Hausdorff dimension of a point set.