  We present local biplots, a an extension of the classic principal components
biplot to multi-dimensional scaling. Noticing that principal components biplots
have an interpretation as the Jacobian of a map from data space to the
principal subspace, we define local biplots as the Jacobian of the analogous
map for multi-dimensional scaling. In the process, we show a close relationship
between our local biplot axes, generalized Euclidean distances, and generalized
principal components. In simulations and real data we show how local biplots
can shed light on what variables or combinations of variables are important for
the low-dimensional embedding provided by multi-dimensional scaling. They give
particular insight into a class of phylogenetically-informed distances commonly
used in the analysis of microbiome data, showing that different variants of
these distances can be interpreted as implicitly smoothing the data along the
phylogenetic tree and that the extent of this smoothing is variable.
