Wrapped distribution

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a> and <a href="/facts/Directional_statistics/ucnUv52T">directional statistics</a>, a wrapped probability distribution is a continuous <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> that describes data points that lie on a unit <a href="/facts/N-sphere/bFu2S7E2">n-sphere</a>. In one dimension, a wrapped distribution consists of points on the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a>. If 
 
 
 
 ϕ
 
 
 {\displaystyle \phi }
 
 is a random variate in the interval 
 
 
 
 (
 −
 ∞
 ,
 ∞
 )
 
 
 {\displaystyle (-\infty ,\infty )}
 
 with <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> (PDF) 
 
 
 
 p
 (
 ϕ
 )
 
 
 {\displaystyle p(\phi )}
 
, then 
 
 
 
 z
 =
 
 e
 
 i
 ϕ
 
 
 
 
 {\displaystyle z=e^{i\phi }}
 
 is a circular variable distributed according to the wrapped distribution 
 
 
 
 
 p
 
 w
 z
 
 
 (
 θ
 )
 
 
 {\displaystyle p_{wz}(\theta )}
 
 and 
 
 
 
 θ
 =
 arg
 ⁡
 (
 z
 )
 
 
 {\displaystyle \theta =\arg(z)}
 
 is an angular variable in the interval 
 
 
 
 (
 −
 π
 ,
 π
 ]
 
 
 {\displaystyle (-\pi ,\pi ]}
 
 distributed according to the wrapped distribution 
 
 
 
 
 p
 
 w
 
 
 (
 θ
 )
 
 
 {\displaystyle p_{w}(\theta )}
 
.
Any probability density function 
 
 
 
 p
 (
 ϕ
 )
 
 
 {\displaystyle p(\phi )}
 
 on the line can be "wrapped" around the circumference of a circle of unit radius. That is, the PDF of the wrapped variable

θ
        =
        ϕ
        
        mod
        
        
        2
        π
      
    
    {\displaystyle \theta =\phi \mod 2\pi }
  
 in some interval of length 
  
    
      
        2
        π
      
    
    {\displaystyle 2\pi }

is

p
          
            w
          
        
        (
        θ
        )
        =
        
          ∑
          
            k
            =
            −
            ∞
          
          
            ∞
          
        
        
          p
          (
          θ
          +
          2
          π
          k
          )
        
      
    
    {\displaystyle p_{w}(\theta )=\sum _{k=-\infty }^{\infty }{p(\theta +2\pi k)}}

which is a <a href="/facts/Periodic_summation/x3jMCU4y">periodic sum</a> of period 
 
 
 
 2
 π
 
 
 {\displaystyle 2\pi }
 
. The preferred interval is generally 
 
 
 
 (
 −
 π
 <
 θ
 ≤
 π
 )
 
 
 {\displaystyle (-\pi <\theta \leq \pi )}
 
 for which 
 
 
 
 ln
 ⁡
 (
 
 e
 
 i
 θ
 
 
 )
 =
 arg
 ⁡
 (
 
 e
 
 i
 θ
 
 
 )
 =
 θ
 
 
 {\displaystyle \ln(e^{i\theta })=\arg(e^{i\theta })=\theta }
 
.

Wrapped distribution open-in-new

Wrapped distribution