The singularity spectrum is a function used in
Multifractal analysis
A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed ...
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 Hölder:
* ''Hölder, Hoelder'' as surname
* Hölder condition
* Hölder's inequality
* Hölder mean
* Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modul ...
. Intuitively, the singularity spectrum gives a value for how "fractal" a set of points are in a function.
More formally, the singularity spectrum
of a function,
, is defined as:
:
Where
is the function describing the
Hölder exponent Hölder:
* ''Hölder, Hoelder'' as surname
* Hölder condition
* Hölder's inequality
* Hölder mean
* Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modul ...
,
of
at the point
.
is the
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
of a point set.
See also
*
Multifractal analysis
A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed ...
*
Holder exponent
Holder may refer to:
Law
* Holder (law), a person that has in their custody a promissory note, bill of exchange or cheque
* ''Holder v Holder'', an English trusts law case
* ''Holder v. Humanitarian Law Project'' (2010), a U.S. Supreme Court decis ...
*
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
*
Fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
*
Fractional Brownian motion In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gauss ...
References
* .
Fractals
{{fractal-stub