[SciPy-user] How to generate random positive definite matrix
Mon Jun 11 02:01:20 CDT 2007
On 10/06/07, David Cournapeau <firstname.lastname@example.org> wrote:
> Hi there,
> I need to generate random positive definite matrix, mainly for
> testing purpose. Before, I was generating them using a random matrix A
> given by randn, and computing A'A. Unfortunately, if A is singular, so
> is A'A. Is there a better way to do than testing whether A is singular ?
> They do not need to follow a specific distribution (but I would like to
> avoid them to follow a really special "pattern").
Every symmetric positive definite matrix (you didn't say symmetric but
I assume you want it...) can be diagonalized using an orthogonal
matrix. That is, you can always factor it as O'DO, where D is diagonal
and has positive elements on the diagonal. So to get a random one, do
it backwards: pick a set of eigenvalues from some distribution, then
for each pair of axes, rotate the matrix by a random angle in the
plane they form (which you can do with a cos/sin/-sin/cos matrix) with
a randomly-decided reflection. This gives a distribution that's
invariant under orthonormal change of basis; your only nonuniformities
come in the choice of eigenvalues, and those you will probably want to
control in testing. Eigenvalues of wildly differing sizes tend to lead
to big numerical headaches, so you can experiment with the size of
headache you want to give your code. If you want them all roughly the
same size you could take normal variates and square them, but for
better control I'd go with exponentiating a normal distribution.
There's an alternative, probably faster, approach based on the
uniqueness of the Cholesky square root of a matrix - just choose your
eigenvalues, again, put their square roots on the diagonal, and
randomly generate the upper triangle. Then multiply A'A and you should
be able to get an arbitrary (symmetric) positive definite matrix.
Unless you are cleverer than I am about choosing the distributions of
the entries, it won't be nicely direction-independent.
Now, a sort-of interesting question is, is there a natural
distribution on the cone of positive definite matrices one could hope
to draw from? Apart from direction invariance, the only other
criterion I can think of to include is some sort of convexity
condition - all the matrices on the line segment connecting two
positive definite matrices are positive definite - so perhaps one
could want the probability densities for all those matrices to be at
least as high as for the endpoints? It's not clear to me that this is
a sufficient (or appropriate) constraint on the PDF.
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