  In 2019, Yoshida et al. introduced a notion of tropical principal component
analysis (PCA). The output is a tropical polytope with a fixed number of
vertices that best fits the data. We here apply tropical PCA to dimension
reduction and visualization of data sampled from the space of phylogenetic
trees. Our main results are twofold: the existence of a tropical cell
decomposition into regions of fixed tree topology and the development of a
stochastic optimization method to estimate the tropical PCA using a Markov
Chain Monte Carlo (MCMC) approach. This method performs well with simulation
studies, and it is applied to three empirical datasets: Apicomplexa and African
coelacanth genomes as well as sequences of hemagglutinin for influenza from New
York.
