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