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 Holder 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:

$D(\alpha) = D_F\{x, \alpha(x) = \alpha\}$

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

See also

• Multifractal analysis
• Holder exponent
• Hausdorff dimension
• Fractal
• Fractional Brownian motion

References

• van den Berg, J. C. (2004), Wavelets in Physics, Cambridge, ISBN 978-0-521-53353-9.