# acg: Angular Central Gaussian Distribution In Riemann: Learning with Data on Riemannian Manifolds

## Description

For a hypersphere \mathcal{S}^{p-1} in \mathbf{R}^p, Angular Central Gaussian (ACG) distribution ACG_p (A) is defined via a density

f(x\vert A) = |A|^{-1/2} (x^\top A^{-1} x)^{-p/2}

with respect to the uniform measure on \mathcal{S}^{p-1} and A is a symmetric positive-definite matrix. Since f(x\vert A) = f(-x\vert A), it can also be used as an axial distribution on real projective space, which is unit sphere modulo \lbrace{+1,-1\rbrace}. One constraint we follow is that f(x\vert A) = f(x\vert cA) for c > 0 in that we use a normalized version for numerical stability by restricting tr(A)=p.

## Usage

 1 2 3 4 5 dacg(datalist, A) racg(n, A) mle.acg(datalist, ...) 

## Arguments

 datalist a list of length-p unit-norm vectors. A a (p\times p) symmetric positive-definite matrix. n the number of samples to be generated. ... extra parameters for computations, including maxitermaximum number of iterations to be run (default:50). epstolerance level for stopping criterion (default: 1e-5).

## Value

dacg gives a vector of evaluated densities given samples. racg generates unit-norm vectors in \mathbf{R}^p wrapped in a list. mle.acg estimates the SPD matrix A.

## References

\insertRef

tyler_statistical_1987Riemann

\insertRef

mardia_directional_1999Riemann

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 # ------------------------------------------------------------------- # Example with Angular Central Gaussian Distribution # # Given a fixed A, generate samples and estimate A via ML. # ------------------------------------------------------------------- ## GENERATE AND MLE in R^5 # Generate data Atrue = diag(5) # true SPD matrix sam1 = racg(50, Atrue) # random samples sam2 = racg(100, Atrue) # MLE Amle1 = mle.acg(sam1) Amle2 = mle.acg(sam2) # Visualize opar <- par(no.readonly=TRUE) par(mfrow=c(1,3), pty="s") image(Atrue[,5:1], axes=FALSE, main="true SPD") image(Amle1[,5:1], axes=FALSE, main="MLE with n=50") image(Amle2[,5:1], axes=FALSE, main="MLE with n=100") par(opar) 

Riemann documentation built on June 20, 2021, 5:07 p.m.