Singularity spectrum

The singularity spectrum is a function used in <a href="/facts/Multifractal_analysis/Fly3MCTb">multifractal analysis</a> to describe the fractal dimension of a subset of points of a function belonging to a group of points that have the same <a href="/facts/H%25C3%25B6lder_exponent/CI0yp6kD">Hölder exponent</a>. Intuitively, the singularity spectrum gives a value for how "fractal" a set of points are in a function.
More formally, the singularity spectrum 
 
 
 
 D
 (
 α
 )
 
 
 {\displaystyle D(\alpha )}
 
 of a function, 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
, is defined as:

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

Where 
 
 
 
 α
 (
 x
 )
 
 
 {\displaystyle \alpha (x)}
 
 is the function describing the Hölder exponent, 
 
 
 
 α
 (
 x
 )
 
 
 {\displaystyle \alpha (x)}
 
 of 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 at the point 
 
 
 
 x
 
 
 {\displaystyle x}
 
. 
 
 
 
 
 D
 
 F
 
 
 {
 ⋅
 }
 
 
 {\displaystyle D_{F}\{\cdot \}}
 
 is the <a href="/facts/Hausdorff_dimension/rtX9ryrp">Hausdorff dimension</a> of a point set.

Singularity spectrum open-in-new

Singularity spectrum