  Phylogenetic PCA (p-PCA) is a version of PCA for observations that are leaf
nodes of a phylogenetic tree. P-PCA accounts for the fact that such
observations are not independent, due to shared evolutionary history. The
method works on Euclidean data, but in evolutionary biology there is a need for
applying it to data on manifolds, particularly shapes. We provide a
generalization of p-PCA to data lying on Riemannian manifolds, called Tangent
p-PCA. Tangent p-PCA thus makes it possible to perform dimension reduction on a
data set of shapes, taking into account both the non-linear structure of the
shape space as well as phylogenetic covariance. We show simulation results on
the sphere, demonstrating well-behaved error distributions and fast convergence
of estimators. Furthermore, we apply the method to a data set of mammal jaws,
represented as points on a landmark manifold equipped with the LDDMM metric.
