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Directional statistics - Entropy of wrapped normal (Jacobi theta) distribution

  1. Jul 2, 2011 #1

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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.
     
    Last edited: Jul 2, 2011
  2. jcsd
  3. Jul 3, 2011 #2
    You'd be lucky if a closed form expression exists. Perhaps try numerical integration or an asymptotic expansion?
     
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