SBmedian (version 0.1.2)

mpost.spd: Median Posterior for Subset Posterior Samples in SPD manifold

Description

SPD manifold is a collection of matrices that are symmetric and positive-definite and it is well known that using Euclidean geometry for data on the manifold is rather inaccurate. Here, we propose a function for dealing with SPD matrices specifically where valid examples include full-rank covariance and precision matrices. Note that \(N_M = \sum_{m=1}^M n_m\).

Usage

mpost.spd(
  splist,
  sigma = 0.1,
  maxiter = 121,
  abstol = 1e-06,
  show.progress = FALSE
)

Value

a named list containing:

med.atoms

a \((p\times p\times N_M)\) 3d array whose slices are atoms aggregated.

med.weights

a weight vector that sums to 1 corresponding to med.atoms.

weiszfeld.weights

a weight for \(M\) subset posteriors.

weiszfeld.history

updated parameter values. Each row is for iteration, while columns are weights corresponding to weiszfeld.weights.

Arguments

splist

a list of length \(M\) containing \((p\times p)\) matrix or 3d array of size \((p\times p\times n_m)\) whose slices are SPD matrices from subset posterior samples respectively.

sigma

bandwidth parameter for RBF kernel.

maxiter

maximum number of iterations for Weiszfeld algorithm.

abstol

stopping criterion for Weiszfeld algorithm.

show.progress

a logical; TRUE to show iteration mark, FALSE otherwise.

Examples

## Median Posteior from 5-dimension Wishart distribution
## Visualization will be performed for distribution of larget eigenvalue
## where RED is for estimated density and BLUE is density from all samples.

#  Step 1. let's build a list of atoms whose numbers differ
set.seed(8128)                   # for reproducible results
mydata = list()
mydata[[1]] = stats::rWishart(96, df=10, Sigma=diag(5))
mydata[[2]] = stats::rWishart(78, df=10, Sigma=diag(5))
mydata[[3]] = stats::rWishart(65, df=10, Sigma=diag(5))
mydata[[4]] = stats::rWishart(77, df=10, Sigma=diag(5))

#  Step 2. Let's run the algorithm
myrun = mpost.spd(mydata, show.progress=TRUE)

#  Step 3. Compute largest eigenvalues for the samples
eig4 = list()
for (i in 1:4){
  spdmats = mydata[[i]]        # SPD atoms
  spdsize = dim(spdmats)[3]    # number of atoms
  eigvals = rep(0,spdsize)     # compute largest eigenvalues
  for (j in 1:spdsize){
    eigvals[j] = max(base::eigen(spdmats[,,j])$values)
  }
  eig4[[i]] = eigvals
}
eigA   = unlist(eig4)
eiglim = c(min(eigA), max(eigA))

#  Step 4. Visualize
#  4-1. show distribution of subset posterior samples' eigenvalues
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,3))
for (i in 1:4){
  hist(eig4[[i]], main=paste("subset", i), xlab="largest eigenvalues", 
       prob=TRUE, xlim=eiglim, ylim=c(0,0.1))
  lines(stats::density(eig4[[i]]), lwd=1, col="red")
  lines(stats::density(eigA),      lwd=1, col="blue")
}

#  4-2. 250 median posterior samples via importance sampling
id250 = base::sample(1:length(eigA), 250, prob=myrun$med.weights, replace=TRUE)
sp250 = eigA[id250]
hist(sp250, main="median samples", xlab="largest eigenvalues", 
     prob=TRUE, xlim=eiglim, ylim=c(0,0.1))
lines(stats::density(sp250), lwd=1, col="red")
lines(stats::density(eigA),  lwd=1, col="blue")

#  4-3. convergence over iterations
matplot(myrun$weiszfeld.history, xlab="iteration", ylab="value",
        type="b", main="convergence of weights")
par(opar)