tangentPCA {fdWasserstein}

Tangent space principal component analysis

Description

The function performs a standard PCA of K covariances after projecting them on the tangent space at their Wasserstein barycenter. Rationale and details are given in Masarotto, Panaretos & Zemel (2019, 2022)

Usage

tangentPCA(sigma, max.iter=30)

Arguments

sigma

An MxMxK array containing the K covariances.

max.iter

Maximum number of gradient descent iterations used to compute the Wasserstein barycenter of the covariances in sigma.

Value

A standard prcomp object with added a MxMxK array containing the eigenvectors projected back to the covariances space.

Author(s)

Valentina Masarotto

References

Masarotto, V., Panaretos, V.M. & Zemel, Y. (2022) "Transportation-Based Functional ANOVA and PCA for Covariance Operators", arXiv, https://arxiv.org/abs/2212.04797

See Also

gaussBary, prcomp

Examples

# Example taken from https://arxiv.org/abs/2212.04797  . 
data(phoneme)
# resampling the log-periodograms
# 12 sample covariances for each phoneme
# each estimated on 50 curves 
set.seed(12345)
nsubsamples <- 12
n <- 50
gg <- unique(Phoneme)
nphonemes <- length(gg)
K <- n*nsubsamples*nphonemes
M <- NCOL(logPeriodogram)
Sigma <- array(dim=c(M, M, nphonemes*nsubsamples))
r <- 0
for (l in gg) {
  for (i in 1:nsubsamples) {
      r <- r+1
      Sigma[,,r] <- cov(logPeriodogram[sample(which(Phoneme==l),n), ])
  }
}
pca <- tangentPCA(Sigma, max.iter=3)
summary(pca)
plot(pca)
# See https://arxiv.org/abs/2212.04797 for the interpretation
# of the figure
pairs(pca$x[,1:5], col=rep(1:nphonemes, rep(nsubsamples, nphonemes)))