Singularity spectrum

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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.

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