  Bike sharing systems (BSSs) have become a means of sustainable intermodal
transport and are now proposed in many cities worldwide. Most BSSs also provide
open access to their data, particularly to real-time status reports on their
bike stations. The analysis of the mass of data generated by such systems is of
particular interest to BSS providers to update system structures and policies.
This work was motivated by interest in analyzing and comparing several European
BSSs to identify common operating patterns in BSSs and to propose practical
solutions to avoid potential issues. Our approach relies on the identification
of common patterns between and within systems. To this end, a model-based
clustering method, called FunFEM, for time series (or more generally functional
data) is developed. It is based on a functional mixture model that allows the
clustering of the data in a discriminative functional subspace. This model
presents the advantage in this context to be parsimonious and to allow the
visualization of the clustered systems. Numerical experiments confirm the good
behavior of FunFEM, particularly compared to state-of-the-art methods. The
application of FunFEM to BSS data from JCDecaux and the Transport for London
Initiative allows us to identify 10 general patterns, including pathological
ones, and to propose practical improvement strategies based on the system
comparison. The visualization of the clustered data within the discriminative
subspace turns out to be particularly informative regarding the system
efficiency. The proposed methodology is implemented in a package for the R
software, named funFEM, which is available on the CRAN. The package also
provides a subset of the data analyzed in this work.
