1

For spherical coordinates with angles $\Theta$ (polar) following truncated normal distribution and $\Phi$ (azimuth) following circular uniform distribution, is there any closed form distribution for its cartesian coordinates ($X,Y,Z$)?

Let's say $\Theta$ ~ truncated normal distribution around $\pi/2$ with pdf $f_{\Theta}(\theta)$ and $\Phi$ ~ circular uniform distribution with pdf $f_{\Phi}(\phi)$. And, let $C_\Theta = \cos(\Theta), S_\Theta = \sin(\Theta), C_\Phi = \cos(\Phi)$, and $S_\Phi = \sin(\Phi)$. Then,

\begin{align} f_{C_\Theta}(c) &= \frac{f_\Theta(\arccos(c))}{\sqrt{1-c^2}},\quad \mathrm{for} ~-1 \leq c \leq 1,\\ f_{S_\Theta}(s) &= \frac{2f_\Theta(\arcsin(s))}{\sqrt{1-s^2}},\quad \mathrm{for} ~0 \leq s \leq 1, \\ f_{C_\Phi}(c) &= \frac{1}{\pi\sqrt{1-c^2}},\quad \mathrm{for} ~-1 \leq c \leq 1, \mathrm{and} \\ f_{S_\Phi}(s) &= \frac{1}{\pi\sqrt{1-s^2}},\quad \mathrm{for} ~-1 \leq s \leq 1. \end{align}

Since $X = \sin(\Theta)\cos(\Phi), Y = \sin(\Theta)\sin(\Phi)$, and $Z = \cos(\Theta)$, then we have

\begin{align} f_X(x) &= \int\limits_{-\infty}^{\infty} f_{C_\Theta}(\tau) f_{S_\Phi}(x/\tau)/|\tau| \mathrm{d}\tau,\\ f_Y(y) &= \int\limits_{-\infty}^{\infty} f_{S_\Theta}(\tau) f_{S_\Phi}(y/\tau)/|\tau| \mathrm{d}\tau, \quad \mathrm{and} \\ f_Z(z) &= f_{C_\Phi}(z). \end{align}

Is there any closed form for $f_X(x), f_Y(y)$, and $f_Z(z)$? Could you please point me to some references related to this problem because I just noticed that there is a study about circular statistics, and I am quite new to this topic?

kjetil b halvorsen
  • 63,378
  • 26
  • 142
  • 467
DEVA
  • 137
  • 8
  • A truncated Normal distribution for spherical or polar coordinates is unnatural. You won't find any analytic formulas. Have you considered using a [von Mises distribution](https://en.wikipedia.org/wiki/Von_Mises%E2%80%93Fisher_distribution) instead? – whuber Dec 05 '16 at 20:09
  • Thank you for your suggestion @whuber. I'm considering it right now. However, since I think this is pretty common problem (transforming from spherical to cartesian coordinates given a priori distribution), there should be works that have been done before, for example [link](https://arxiv.org/pdf/1306.0290.pdf). Have you found similar works? – DEVA Dec 05 '16 at 21:07
  • I doubt anyone has done this because--as I wrote--positing a truncated Normal distribution for a spherical coordinate is unnatural. The Cartesian version will have a PDF that is not differentiable along a ray, so we can expect significant mathematical difficulties in expressing it. – whuber Dec 05 '16 at 21:11
  • I used the truncated Normal distribution because I refer to a paper that has done some experiments and approximates the data with the distribution. Adding keyword 'von Mises distribution' doesn't bring me to references I'm looking for. Thank you very much for your comments @whuber. At least I won't think that this problem's as common as I thought it was. – DEVA Dec 05 '16 at 21:19

0 Answers0