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 [math]\displaystyle{ D(\alpha) }[/math] of a function, [math]\displaystyle{ f(x) }[/math], is defined as:
- [math]\displaystyle{ D(\alpha) = D_F\{x, \alpha(x) = \alpha\} }[/math]
Where [math]\displaystyle{ \alpha(x) }[/math] is the function describing the Hölder exponent, [math]\displaystyle{ \alpha(x) }[/math] of [math]\displaystyle{ f(x) }[/math] at the point [math]\displaystyle{ x }[/math]. [math]\displaystyle{ D_F\{\cdot\} }[/math] is the Hausdorff dimension of a point set.
See also
References
- van den Berg, J. C. (2004), Wavelets in Physics, Cambridge, ISBN 978-0-521-53353-9.
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