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If p(x) is a probability distribution on the real number line, the "wrapped" distribution around the unit circle is:
[tex]p_w(\theta)=\sum_{n=-\infty}^\infty p(\theta+2\pi n)[/tex]
which is periodic with period 2π. The (differential) entropy is:
[tex]H=-\int_\Gamma p_w(\theta)\ln[p_w(\theta)]\,d\theta[/tex]
where [itex]\Gamma[/itex] is any interval of length 2π. I am having trouble finding the entropy for the wrapped normal distribution. For the wrapped normal distribution with zero mean,
[tex]p_w(\theta)=\frac{\vartheta _3\left(\frac{\theta }{2},e^{-\frac{\sigma ^2}{2}}\right)}{2 \pi }[/tex]
which is in Mathematica notation, [itex]\vartheta _3(\cdot)[/itex] is the Jacobi theta function:
[tex]\vartheta _3(u,q)=1+2\sum_{n=1}^\infty q^{n^2}\cos(2nu)[/tex]
Is anyone familiar enough with Jacobi theta functions to give some guidance in calculating the entropy? Mathematica is not giving much help.
[tex]p_w(\theta)=\sum_{n=-\infty}^\infty p(\theta+2\pi n)[/tex]
which is periodic with period 2π. The (differential) entropy is:
[tex]H=-\int_\Gamma p_w(\theta)\ln[p_w(\theta)]\,d\theta[/tex]
where [itex]\Gamma[/itex] is any interval of length 2π. I am having trouble finding the entropy for the wrapped normal distribution. For the wrapped normal distribution with zero mean,
[tex]p_w(\theta)=\frac{\vartheta _3\left(\frac{\theta }{2},e^{-\frac{\sigma ^2}{2}}\right)}{2 \pi }[/tex]
which is in Mathematica notation, [itex]\vartheta _3(\cdot)[/itex] is the Jacobi theta function:
[tex]\vartheta _3(u,q)=1+2\sum_{n=1}^\infty q^{n^2}\cos(2nu)[/tex]
Is anyone familiar enough with Jacobi theta functions to give some guidance in calculating the entropy? Mathematica is not giving much help.
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