Timeline for Simulation under Wishart-like constraint in $\mathbb{R}^{k\times p}$
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:44 | history | edited | CommunityBot |
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| Sep 26, 2016 at 23:34 | comment | added | whuber♦ | See stats.stackexchange.com/questions/2746 concerning generating random orthogonal matrices. Many of the techniques described there for generating random correlation matrices can be adapted. | |
| Sep 23, 2016 at 16:40 | comment | added | whuber♦ | That approach in (2) should work--it ought to be the equivalent of my quick-and-dirty method. There's a tiny chance the columns will appear to be collinear (up to floating point error). That's why I recommend using one or a few extra rows: the collinearity becomes that much less likely. The proof of uniformity is inductive (on the number of columns). The base case reduces to the fact that the standard Normal distribution in $k$ dimensions is spherically symmetric. The induction step relies on the independence of the columns and the retention of symmetry upon projection to a subspace. | |
| Sep 23, 2016 at 16:34 | comment | added | Xi'an | @whuber: (1) Thank you for pointing this out: I am interested in one parameterisation with a manageable prior, so picking a uniform prior on $\mathcal{O}(k)$ is good enough. (2) I was thinking of simulating a matrix of $k^2$ iid standard normal and of brute-force orthonormalising the columns. I may be missing the uniform distribution though. | |
| Sep 23, 2016 at 15:27 | comment | added | whuber♦ | (+1) (1) Note that this does not produce vectors that are "uniformly" distributed unless all the eigenvalues of $H$ are equal. (That's why I left my reply in a comment). (2) A quick and dirty way to construct orthogonal matrices is to generate an $m\times n$ array of iid standard Normal variates with $m \ge n$ (preferably a little larger to make sure the array is not rank deficient) and diagonalize its covariance matrix. The diagonalization yields an orthogonal transformation of the basis represented as an orthogonal $n\times n$ matrix. | |
| Sep 23, 2016 at 14:53 | history | answered | Xi'an | CC BY-SA 3.0 |