Singularity spectrum explained

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)=DF\{x,\alpha(x)=\alpha\}

Where

\alpha(x)

is the function describing the Hölder exponent,

\alpha(x)

of

f(x)

at the point

x

.

DF\{\}

is the Hausdorff dimension of a point set.

See also

References