Matrix Angular Central Gaussian Distribution

For Stiefel and Grassmann manifolds $St(r,p)$ and $Gr(r,p)$, the matrix variant of ACG distribution is known as Matrix Angular Central Gaussian (MACG) distribution $MACG_{p,r}(\Sigma)$ with density $$f(X\vert \Sigma) = |\Sigma|^{-r/2} |X^\top \Sigma^{-1} X|^{-p/2}$$ where $\Sigma$ is a $(p\times p)$ symmetric positive-definite matrix. Similar to vector-variate ACG case, we follow a convention that $tr(\Sigma)=p$.

Usage

dmacg(datalist, Sigma, log = FALSE)

rmacg(n, r, Sigma)

mle.macg(datalist, ...)

Arguments

datalist

a list of $(p\times r)$ orthonormal matrices.

Sigma

a $(p\times p)$ symmetric positive-definite matrix.

log

logical; return log densities when TRUE.

n

a positive integer number of samples.

r

the number of basis.

...

extra parameters for computations, including

maxiter: maximum number of iterations to be run (default:50).
eps: tolerance level for stopping criterion (default: 1e-5).

Value

dmacg gives a vector of evaluated densities given samples. rmacg generates $(p\times r)$ orthonormal matrices wrapped in a list. mle.macg estimates the SPD matrix $\Sigma$.

Details

The density is relative to the invariant uniform probability measure on the Stiefel manifold, or its push-forward on the Grassmann quotient, and equals one at $\Sigma=I$. Cholesky/log-determinant evaluation avoids determinant products that overflow at extreme parameter scales.

Sampling uses the polar factor of a Gaussian matrix. The fixed-point estimate is trace normalized and carries convergence, iteration, step, and negative mean-log-likelihood attributes. Failed convergence warns; degenerate updates error. Full observed span does not certify the subspace-dispersion conditions needed for an identifiable, unique interior estimate. When $r=p>1$, the density is uniform regardless of $\Sigma$, so estimation errors.

References

Chikuse Y (1990). “The matrix angular central Gaussian distribution.” Journal of Multivariate Analysis, 33(2), 265–274. ISSN 0047259X.

Mardia KV, Jupp PE (eds.) (1999). Directional Statistics, Wiley Series in Probability and Statistics. John Wiley and Sons, Inc., Hoboken, NJ, USA. ISBN 978-0-470-31697-9 978-0-471-95333-3.

Kent JT, Ganeiber AM, Mardia KV (2013). "A new method to simulate the Bingham and related distributions in directional data analysis with applications." arXiv:1310.8110.

Examples

# -------------------------------------------------------------------
#          Example with Matrix Angular Central Gaussian Distribution
#
# Given a fixed Sigma, generate samples and estimate Sigma via ML.
# -------------------------------------------------------------------
## GENERATE AND MLE in St(2,5)/Gr(2,5)
#  Generate data
Strue = diag(5)                  # true SPD matrix
sam1  = rmacg(n=50,  r=2, Strue) # random samples
sam2  = rmacg(n=100, r=2, Strue) # random samples

#  MLE
Smle1 = mle.macg(sam1)
Smle2 = mle.macg(sam2)

#  Visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
image(Strue[,5:1], axes=FALSE, main="true SPD")
image(Smle1[,5:1], axes=FALSE, main="MLE with n=50")
image(Smle2[,5:1], axes=FALSE, main="MLE with n=100")

par(opar)