  Distributional (or distribution-valued) data are a new type of data arising
from several sources and are considered as realizations of distributional
variables. A new set of fuzzy c-means algorithms for data described by
distributional variables is proposed.
  The algorithms use the $L2$ Wasserstein distance between distributions as
dissimilarity measures. Beside the extension of the fuzzy c-means algorithm for
distributional data, and considering a decomposition of the squared $L2$
Wasserstein distance, we propose a set of algorithms using different automatic
way to compute the weights associated with the variables as well as with their
components, globally or cluster-wise. The relevance weights are computed in the
clustering process introducing product-to-one constraints.
  The relevance weights induce adaptive distances expressing the importance of
each variable or of each component in the clustering process, acting also as a
variable selection method in clustering. We have tested the proposed algorithms
on artificial and real-world data. Results confirm that the proposed methods
are able to better take into account the cluster structure of the data with
respect to the standard fuzzy c-means, with non-adaptive distances.
