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 $D(\alpha )$ of a function, $f(x)$, is defined as:

$D(\alpha )=D_{F}\{x,\alpha (x)=\alpha \}$

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

See also

• Fractal
• Fractional Brownian motion
• Hausdorff dimension

References

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