Wrapped Cauchy distribution - Alchetron, the free social encyclopedia

CDF {displaystyle ,}

Parameters μ {displaystyle mu } Real γ > 0 {displaystyle gamma >0} Support − π ≤ θ < π {displaystyle -pi leq heta <pi } PDF 1 2 π sinh ⁡ ( γ ) cosh ⁡ ( γ ) − cos ⁡ ( θ − μ ) {displaystyle {rac {1}{2pi }},{rac {sinh(gamma )}{cosh(gamma )-cos( heta -mu )}}} Mean μ {displaystyle mu } (circular) Variance 1 − e − γ {displaystyle 1-e^{-gamma }} (circular)

In probability theory and directional statistics, a wrapped Cauchy distribution is a wrapped probability distribution that results from the "wrapping" of the Cauchy distribution around the unit circle. The Cauchy distribution is sometimes known as a Lorentzian distribution, and the wrapped Cauchy distribution may sometimes be referred to as a wrapped Lorentzian distribution.

Description

The probability density function of the wrapped Cauchy distribution is:

f W C ( θ ; μ , γ ) = ∑ n = − ∞ ∞ γ π ( γ 2 + ( θ − μ + 2 π n ) 2 )

where γ is the scale factor and μ is the peak position of the "unwrapped" distribution. Expressing the above pdf in terms of the characteristic function of the Cauchy distribution yields:

f W C ( θ ; μ , γ ) = 1 2 π ∑ n = − ∞ ∞ e i n ( θ − μ ) − | n | γ = 1 2 π sinh ⁡ γ cosh ⁡ γ − cos ⁡ ( θ − μ )

The PDF may also be expressed in terms of the circular variable z = e ^{i θ} and the complex parameter ζ =e ^{i(μ + i γ)}

f W C ( z ; ζ ) = 1 2 π 1 − | ζ | 2 | z − ζ | 2

where, as shown below, ζ = < z >.

In terms of the circular variable z = e i θ the circular moments of the wrapped Cauchy distribution are the characteristic function of the Cauchy distribution evaluated at integer arguments:

⟨ z n ⟩ = ∫ Γ e i n θ f W C ( θ ; μ , γ ) d θ = e i n μ − | n | γ .

where Γ is some interval of length 2 π . The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector:

⟨ z ⟩ = e i μ − γ

The mean angle is

⟨ θ ⟩ = A r g ⟨ z ⟩ = μ

and the length of the mean resultant is

R = | ⟨ z ⟩ | = e − γ

yielding a circular variance of 1-R.

Estimation of parameters

A series of N measurements z n = e i θ n drawn from a wrapped Cauchy distribution may be used to estimate certain parameters of the distribution. The average of the series z ¯ is defined as

z ¯ = 1 N ∑ n = 1 N z n

and its expectation value will be just the first moment:

⟨ z ¯ ⟩ = e i μ − γ

In other words, z ¯ is an unbiased estimator of the first moment. If we assume that the peak position μ lies in the interval [ − π , π ) , then Arg ( z ¯ ) will be a (biased) estimator of the peak position μ .

Viewing the z n as a set of vectors in the complex plane, the R ¯ 2 statistic is the length of the averaged vector:

R ¯ 2 = z ¯ z ∗ ¯ = ( 1 N ∑ n = 1 N cos ⁡ θ n ) 2 + ( 1 N ∑ n = 1 N sin ⁡ θ n ) 2

and its expectation value is

⟨ R ¯ 2 ⟩ = 1 N + N − 1 N e − 2 γ .

In other words, the statistic

R e 2 = N N − 1 ( R ¯ 2 − 1 N )

will be an unbiased estimator of e − 2 γ , and ln ⁡ ( 1 / R e 2 ) / 2 will be a (biased) estimator of γ .

Entropy

The information entropy of the wrapped Cauchy distribution is defined as:

H = − ∫ Γ f W C ( θ ; μ , γ ) ln ⁡ ( f W C ( θ ; μ , γ ) ) d θ

where Γ is any interval of length 2 π . The logarithm of the density of the wrapped Cauchy distribution may be written as a Fourier series in θ :

ln ⁡ ( f W C ( θ ; μ , γ ) ) = c 0 + 2 ∑ m = 1 ∞ c m cos ⁡ ( m θ )

where

c m = 1 2 π ∫ Γ ln ⁡ ( sinh ⁡ γ 2 π ( cosh ⁡ γ − cos ⁡ θ ) ) cos ⁡ ( m θ ) d θ

which yields:

c 0 = ln ⁡ ( 1 − e − 2 γ 2 π )

(c.f. Gradshteyn and Ryzhik 4.224.15) and

c m = e − m γ m f o r m > 0

(c.f. Gradshteyn and Ryzhik 4.397.6). The characteristic function representation for the wrapped Cauchy distribution in the left side of the integral is:

f W C ( θ ; μ , γ ) = 1 2 π ( 1 + 2 ∑ n = 1 ∞ ϕ n cos ⁡ ( n θ ) )

where ϕ n = e − | n | γ . Substituting these expressions into the entropy integral, exchanging the order of integration and summation, and using the orthogonality of the cosines, the entropy may be written:

H = − c 0 − 2 ∑ m = 1 ∞ ϕ m c m = − ln ⁡ ( 1 − e − 2 γ 2 π ) − 2 ∑ m = 1 ∞ e − 2 n γ n

The series is just the Taylor expansion for the logarithm of ( 1 − e − 2 γ ) so the entropy may be written in closed form as:

H = ln ⁡ ( 2 π ( 1 − e − 2 γ ) )

Circular Cauchy distribution

If X is Cauchy distributed with median μ and scale parameter γ, then the complex variable

Z = X − i X + i

has unit modulus and is distributed on the unit circle with density:

f C C ( θ , μ , γ ) = 1 2 π 1 − | ζ | 2 | e i θ − ζ | 2

where

ζ = ψ − i ψ + i

and ψ expresses the two parameters of the associated linear Cauchy distribution for x as a complex number:

ψ = μ + i γ

It can be seen that the circular Cauchy distribution has the same functional form as the wrapped Cauchy distribution in z and ζ (i.e. f_WC(z,ζ)). The circular Cauchy distribution is a reparameterized wrapped Cauchy distribution:

f C C ( θ , m , γ ) = f W C ( e i θ , m + i γ − i m + i γ + i )

The distribution f C C ( θ ; μ , γ ) is called the circular Cauchy distribution (also the complex Cauchy distribution) with parameters μ and γ. (See also McCullagh's parametrization of the Cauchy distributions and Poisson kernel for related concepts.)

The circular Cauchy distribution expressed in complex form has finite moments of all orders

E ⁡ [ Z n ] = ζ n , E ⁡ [ Z ¯ n ] = ζ ¯ n

for integer n ≥ 1. For |φ| < 1, the transformation

U ( z , ϕ ) = z − ϕ 1 − ϕ ¯ z

is holomorphic on the unit disk, and the transformed variable U(Z, φ) is distributed as complex Cauchy with parameter U(ζ, φ).

Given a sample z₁, ..., z_n of size n > 2, the maximum-likelihood equation

n − 1 U ( z , ζ ^ ) = n − 1 ∑ U ( z j , ζ ^ ) = 0

can be solved by a simple fixed-point iteration:

ζ ( r + 1 ) = U ( n − 1 U ( z , ζ ( r ) ) , − ζ ( r ) )

starting with ζ⁽⁰⁾ = 0. The sequence of likelihood values is non-decreasing, and the solution is unique for samples containing at least three distinct values.

The maximum-likelihood estimate for the median ( μ ^ ) and scale parameter ( γ ^ ) of a real Cauchy sample is obtained by the inverse transformation:

μ ^ ± i γ ^ = i 1 + ζ ^ 1 − ζ ^ .

For n ≤ 4, closed-form expressions are known for ζ ^ . The density of the maximum-likelihood estimator at t in the unit disk is necessarily of the form:

1 4 π p n ( χ ( t , ζ ) ) ( 1 − | t | 2 ) 2 ,

where

χ ( t , ζ ) = | t − ζ | 2 4 ( 1 − | t | 2 ) ( 1 − | ζ | 2 ) .

Formulae for p₃ and p₄ are available.

References

Wrapped Cauchy distribution Wikipedia

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Contents

Description

Estimation of parameters

Entropy

Circular Cauchy distribution

References