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Can any one point me to a research article or post the actual algorithm to generate random numbers from the multivariate normal distribution in spherical coordinates? I need both $N_p(0,I_p)$ , the standard multivariate normal case in spherical coordinates and $N_p(0,\Sigma)$ where $\Sigma \neq I_p$ |
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This book has this reference to this book here. Not exactly cheap, but I think it's the one you are looking for. ;) |
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Given how easy it is to generate the normal variates, I would do just that, and then convert to the spherical coordinates directly. If you need $N_p(0,I_p)$, where $p$ is the dimension, then in spherical coordinates, $r\sim \chi^2_p = \Gamma(p/2,1/2)$, and the angles are all independent of each other and the length, with uniform distribution on their respective ranges $[0,\pi)$ for the first one (the one that goes with the cosine only), $[0,2\pi)$ for the remaining ones. If $p$ is even, $p=2m$, you can generate $r_p = \sum_{k=1}^{m} (-\frac12) \ln U_k$, $U_k \sim \mbox{i.i.d. } U[0,1]$. If $p$ is odd, $p=2m+1$, then you need to add another square of a normal to this (see above how to generate them), $r_p = r_{p-1} + z^2$, $z\sim N(0,1)$. |
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