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Wrapped distribution
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<h2 id="theory">Theory</h2> <p>In most situations, a process involving <a href="/facts/Circular_statistics/ucnUv52T">circular statistics</a> produces angles ( ϕ {\displaystyle \phi } ) which lie in the interval ( − ∞ , ∞ ) {\displaystyle (-\infty ,\infty )} , and are described by an "unwrapped" probability density function p ( ϕ ) {\displaystyle p(\phi )} . However, a measurement will yield an angle θ {\displaystyle \theta } which lies in some interval of length 2 π {\displaystyle 2\pi } (for example, 0 to 2 π {\displaystyle 2\pi } ). In other words, a measurement cannot tell whether the true angle ϕ {\displaystyle \phi } or a wrapped angle θ = ϕ + 2 π a {\displaystyle \theta =\phi +2\pi a} , where a {\displaystyle a} is some unknown integer, has been measured. </p><p>If we wish to calculate the <a href="/facts/Expected_value/1XV0JKL8">expected value</a> of some function of the measured angle it will be: </p> ⟨ f ( θ ) ⟩ = ∫ − ∞ ∞ p ( ϕ ) f ( ϕ + 2 π a ) d ϕ {\displaystyle \langle f(\theta )\rangle =\int _{-\infty }^{\infty }p(\phi )f(\phi +2\pi a)d\phi } . <p>We can express the integral as a sum of integrals over periods of 2 π {\displaystyle 2\pi } : </p> ⟨ f ( θ ) ⟩ = ∑ k = − ∞ ∞ ∫ 2 π k 2 π ( k + 1 ) p ( ϕ ) f ( ϕ + 2 π a ) d ϕ {\displaystyle \langle f(\theta )\rangle =\sum _{k=-\infty }^{\infty }\int _{2\pi k}^{2\pi (k+1)}p(\phi )f(\phi +2\pi a)d\phi } . <p>Changing the variable of integration to θ ′ = ϕ − 2 π k {\displaystyle \theta '=\phi -2\pi k} and exchanging the order of integration and summation, we have </p> ⟨ f ( θ ) ⟩ = ∫ 0 2 π p w ( θ ′ ) f ( θ ′ + 2 π a ′ ) d θ ′ {\displaystyle \langle f(\theta )\rangle =\int _{0}^{2\pi }p_{w}(\theta ')f(\theta '+2\pi a')d\theta '} <p>where p w ( θ ′ ) {\displaystyle p_{w}(\theta ')} is the PDF of the wrapped distribution and a ′ {\displaystyle a'} is another unknown integer ( a ′ = a + k ) {\displaystyle (a'=a+k)} . The unknown integer a ′ {\displaystyle a'} introduces an ambiguity into the expected value of f ( θ ) {\displaystyle f(\theta )} , similar to the problem of calculating <a href="/facts/Angular_mean/eMWmRu8P">angular mean</a>. This can be resolved by introducing the parameter z = e i θ {\displaystyle z=e^{i\theta }} , since z {\displaystyle z} has an unambiguous relationship to the true angle ϕ {\displaystyle \phi } : </p> z = e i θ = e i ϕ {\displaystyle z=e^{i\theta }=e^{i\phi }} . <p>Calculating the expected value of a function of z {\displaystyle z} will yield unambiguous answers: </p> ⟨ f ( z ) ⟩ = ∫ 0 2 π p w ( θ ′ ) f ( e i θ ′ ) d θ ′ {\displaystyle \langle f(z)\rangle =\int _{0}^{2\pi }p_{w}(\theta ')f(e^{i\theta '})d\theta '} . <p>For this reason, the z {\displaystyle z} parameter is preferred over measured angles θ {\displaystyle \theta } in circular statistical analysis. This suggests that the wrapped distribution function may itself be expressed as a function of z {\displaystyle z} such that: </p> ⟨ f ( z ) ⟩ = ∮ p w z ( z ) f ( z ) d z {\displaystyle \langle f(z)\rangle =\oint p_{wz}(z)f(z)\,dz} <p>where p w ( z ) {\displaystyle p_{w}(z)} is <a href="/facts/Probability_density_function/zvfybna4">defined</a> such that p w ( θ ) | d θ | = p w z ( z ) | d z | {\displaystyle p_{w}(\theta )\,|d\theta |=p_{wz}(z)\,|dz|} . This concept can be extended to the multivariate context by an extension of the simple sum to a number of F {\displaystyle F} sums that cover all dimensions in the feature space: </p> p w ( θ → ) = ∑ k 1 , . . . , k F = − ∞ ∞ p ( θ → + 2 π k 1 e 1 + ⋯ + 2 π k F e F ) {\displaystyle p_{w}({\vec {\theta }})=\sum _{k_{1},...,k_{F}=-\infty }^{\infty }{p({\vec {\theta }}+2\pi k_{1}\mathbf {e} _{1}+\dots +2\pi k_{F}\mathbf {e} _{F})}} <p>where e k = ( 0 , … , 0 , 1 , 0 , … , 0 ) T {\displaystyle \mathbf {e} _{k}=(0,\dots ,0,1,0,\dots ,0)^{\mathsf {T}}} is the k {\displaystyle k} th Euclidean basis vector. </p> <h2 id="expression-in-terms-of-characteristic-functions">Expression in terms of characteristic functions</h2> <p>A fundamental wrapped distribution is the <a href="/facts/Dirac_comb/0tx3xslS">Dirac comb</a>, which is a wrapped <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a>: </p> Δ 2 π ( θ ) = ∑ k = − ∞ ∞ δ ( θ + 2 π k ) {\displaystyle \Delta _{2\pi }(\theta )=\sum _{k=-\infty }^{\infty }{\delta (\theta +2\pi k)}} . <p>Using the delta function, a general wrapped distribution can be written </p> p w ( θ ) = ∑ k = − ∞ ∞ ∫ − ∞ ∞ p ( θ ′ ) δ ( θ − θ ′ + 2 π k ) d θ ′ {\displaystyle p_{w}(\theta )=\sum _{k=-\infty }^{\infty }\int _{-\infty }^{\infty }p(\theta ')\delta (\theta -\theta '+2\pi k)\,d\theta '} . <p>Exchanging the order of summation and integration, any wrapped distribution can be written as the convolution of the unwrapped distribution and a Dirac comb: </p> p w ( θ ) = ∫ − ∞ ∞ p ( θ ′ ) Δ 2 π ( θ − θ ′ ) d θ ′ {\displaystyle p_{w}(\theta )=\int _{-\infty }^{\infty }p(\theta ')\Delta _{2\pi }(\theta -\theta ')\,d\theta '} . <p>The Dirac comb may also be expressed as a sum of exponentials, so we may write: </p> p w ( θ ) = 1 2 π ∫ − ∞ ∞ p ( θ ′ ) ∑ n = − ∞ ∞ e i n ( θ − θ ′ ) d θ ′ {\displaystyle p_{w}(\theta )={\frac {1}{2\pi }}\,\int _{-\infty }^{\infty }p(\theta ')\sum _{n=-\infty }^{\infty }e^{in(\theta -\theta ')}\,d\theta '} . <p>Again exchanging the order of summation and integration: </p> p w ( θ ) = 1 2 π ∑ n = − ∞ ∞ ∫ − ∞ ∞ p ( θ ′ ) e i n ( θ − θ ′ ) d θ ′ {\displaystyle p_{w}(\theta )={\frac {1}{2\pi }}\,\sum _{n=-\infty }^{\infty }\int _{-\infty }^{\infty }p(\theta ')e^{in(\theta -\theta ')}\,d\theta '} . <p>Using the definition of ϕ ( s ) {\displaystyle \phi (s)} , the <a href="/facts/Characteristic_function_(probability_theory)/rl3uAGKM">characteristic function</a> of p ( θ ) {\displaystyle p(\theta )} yields a <a href="/facts/Laurent_series/sSMevFU8">Laurent series</a> about zero for the wrapped distribution in terms of the characteristic function of the unwrapped distribution: </p> p w ( θ ) = 1 2 π ∑ n = − ∞ ∞ ϕ ( n ) e − i n θ {\displaystyle p_{w}(\theta )={\frac {1}{2\pi }}\,\sum _{n=-\infty }^{\infty }\phi (n)\,e^{-in\theta }} <p>or </p> p w z ( z ) = 1 2 π ∑ n = − ∞ ∞ ϕ ( n ) z − n {\displaystyle p_{wz}(z)={\frac {1}{2\pi }}\,\sum _{n=-\infty }^{\infty }\phi (n)\,z^{-n}} <p>Analogous to linear distributions, ϕ ( m ) {\displaystyle \phi (m)} is referred to as the characteristic function of the wrapped distribution (or more accurately, the characteristic <a href="/facts/Sequence/gV2S4uqp">sequence</a>).<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> This is an instance of the <a href="/facts/Poisson_summation_formula/CDM4uRPs">Poisson summation formula</a>, and it can be seen that the coefficients of the <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> for the wrapped distribution are simply the coefficients of the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> of the unwrapped distribution at integer values. </p> <h2 id="moments">Moments</h2> <p>The moments of the wrapped distribution p w ( z ) {\displaystyle p_{w}(z)} are defined as: </p> ⟨ z m ⟩ = ∮ p w z ( z ) z m d z {\displaystyle \langle z^{m}\rangle =\oint p_{wz}(z)z^{m}\,dz} . <p>Expressing p w ( z ) {\displaystyle p_{w}(z)} in terms of the characteristic function and exchanging the order of integration and summation yields: </p> ⟨ z m ⟩ = 1 2 π ∑ n = − ∞ ∞ ϕ ( n ) ∮ z m − n d z {\displaystyle \langle z^{m}\rangle ={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }\phi (n)\oint z^{m-n}\,dz} . <p>From the <a href="/facts/Residue_theorem/M5ITFIyx">residue theorem</a> we have </p> ∮ z m − n d z = 2 π δ m − n {\displaystyle \oint z^{m-n}\,dz=2\pi \delta _{m-n}} <p>where δ k {\displaystyle \delta _{k}} is the <a href="/facts/Kronecker_delta/gcGAy0bm">Kronecker delta</a> function. It follows that the moments are simply equal to the characteristic function of the unwrapped distribution for integer arguments: </p> ⟨ z m ⟩ = ϕ ( m ) {\displaystyle \langle z^{m}\rangle =\phi (m)} . <h2 id="generation-of-random-variates">Generation of random variates</h2> <p>If X {\displaystyle X} is a random variate drawn from a linear probability distribution P {\displaystyle P} , then Z = e i X {\displaystyle Z=e^{iX}} is a circular variate distributed according to the wrapped P {\displaystyle P} distribution, and θ = arg ( Z ) {\displaystyle \theta =\arg(Z)} is the angular variate distributed according to the wrapped P {\displaystyle P} distribution, with − π < θ ≤ π {\displaystyle -\pi <\theta \leq \pi } . </p> <h2 id="entropy">Entropy</h2> <p>The <a href="/facts/Entropy_(information_theory)/NLg4NLvt">information entropy</a> of a circular distribution with probability density p w ( θ ) {\displaystyle p_{w}(\theta )} is defined as: </p> H = − ∫ Γ p w ( θ ) ln ( p w ( θ ) ) d θ {\displaystyle H=-\int _{\Gamma }p_{w}(\theta )\,\ln(p_{w}(\theta ))\,d\theta } <p>where Γ {\displaystyle \Gamma } is any interval of length 2 π {\displaystyle 2\pi } .<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> If both the probability density and its logarithm can be expressed as a <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> (or more generally, any <a href="/facts/Integral_transform/AB2yjHJJ">integral transform</a> on the circle), the <a href="/facts/Orthogonal_basis/H4DgNCgR">orthogonal basis</a> of the series can be used to obtain a <a href="/facts/Closed_form_expression/y0xCXlSk">closed form expression</a> for the entropy. </p><p>The moments of the distribution ϕ ( n ) {\displaystyle \phi (n)} are the Fourier coefficients for the Fourier series expansion of the probability density: </p> p w ( θ ) = 1 2 π ∑ n = − ∞ ∞ ϕ n e − i n θ {\displaystyle p_{w}(\theta )={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }\phi _{n}e^{-in\theta }} . <p>If the logarithm of the probability density can also be expressed as a Fourier series: </p> ln ( p w ( θ ) ) = ∑ m = − ∞ ∞ c m e i m θ {\displaystyle \ln(p_{w}(\theta ))=\sum _{m=-\infty }^{\infty }c_{m}e^{im\theta }} <p>where </p> c m = 1 2 π ∫ Γ ln ( p w ( θ ) ) e − i m θ d θ {\displaystyle c_{m}={\frac {1}{2\pi }}\int _{\Gamma }\ln(p_{w}(\theta ))e^{-im\theta }\,d\theta } . <p>Then, exchanging the order of integration and summation, the entropy may be written as: </p> H = − 1 2 π ∑ m = − ∞ ∞ ∑ n = − ∞ ∞ c m ϕ n ∫ Γ e i ( m − n ) θ d θ {\displaystyle H=-{\frac {1}{2\pi }}\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }c_{m}\phi _{n}\int _{\Gamma }e^{i(m-n)\theta }\,d\theta } . <p>Using the orthogonality of the Fourier basis, the integral may be reduced to: </p> H = − ∑ n = − ∞ ∞ c n ϕ n {\displaystyle H=-\sum _{n=-\infty }^{\infty }c_{n}\phi _{n}} . <p>For the particular case when the probability density is symmetric about the mean, c − m = c m {\displaystyle c_{-m}=c_{m}} and the logarithm may be written: </p> ln ( p w ( θ ) ) = c 0 + 2 ∑ m = 1 ∞ c m cos ( m θ ) {\displaystyle \ln(p_{w}(\theta ))=c_{0}+2\sum _{m=1}^{\infty }c_{m}\cos(m\theta )} <p>and </p> c m = 1 2 π ∫ Γ ln ( p w ( θ ) ) cos ( m θ ) d θ {\displaystyle c_{m}={\frac {1}{2\pi }}\int _{\Gamma }\ln(p_{w}(\theta ))\cos(m\theta )\,d\theta } <p>and, since normalization requires that ϕ 0 = 1 {\displaystyle \phi _{0}=1} , the entropy may be written: </p> H = − c 0 − 2 ∑ n = 1 ∞ c n ϕ n {\displaystyle H=-c_{0}-2\sum _{n=1}^{\infty }c_{n}\phi _{n}} . <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Wrapped_normal_distribution/N16Nx35y">Wrapped normal distribution</a></li> <li><a href="/facts/Wrapped_Cauchy_distribution/bhJ7Qgdq">Wrapped Cauchy distribution</a></li> <li><a href="/facts/Wrapped_exponential_distribution/Aw7NKsCA">Wrapped exponential distribution</a></li></ul> <ul><li>Borradaile, Graham (2003). <a href="https://books.google.com/books?id=R3GpDglVOSEC"><i>Statistics of Earth Science Data</i></a>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-43603-4.</li> <li>Fisher, N. I. (1996). <a href="https://books.google.com/books?id=IIpeevaNH88C"><i>Statistical Analysis of Circular Data</i></a>. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-56890-6.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.codeproject.com/Articles/190833/Circular-Values-Math-and-Statistics-with-Cplusplus">Circular Values Math and Statistics with C++11</a>, A C++11 infrastructure for circular values (angles, time-of-day, etc.) mathematics and statistics</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Mardia, Kantilal; Jupp, Peter E. (1999). Directional Statistics. Wiley. ISBN 978-0-471-95333-3. <a href="978-0-471-95333-3" target="_blank">978-0-471-95333-3</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Mardia, K. (1972). Statistics of Directional Data. New York: Academic press. ISBN 978-1-4832-1866-3. <a href="978-1-4832-1866-3" target="_blank">978-1-4832-1866-3</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Mardia, Kantilal; Jupp, Peter E. (1999). Directional Statistics. Wiley. ISBN 978-0-471-95333-3. <a href="978-0-471-95333-3" target="_blank">978-0-471-95333-3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>