  We present a nonparametric framework to model a short sequence of probability
distributions that vary both due to underlying effects of sequential
progression and confounding noise. To distinguish between these two types of
variation and estimate the sequential-progression effects, our approach
leverages an assumption that these effects follow a persistent trend. This work
is motivated by the recent rise of single-cell RNA-sequencing experiments over
a brief time course, which aim to identify genes relevant to the progression of
a particular biological process across diverse cell populations. While
classical statistical tools focus on scalar-response regression or
order-agnostic differences between distributions, it is desirable in this
setting to consider both the full distributions as well as the structure
imposed by their ordering. We introduce a new regression model for ordinal
covariates where responses are univariate distributions and the underlying
relationship reflects consistent changes in the distributions over increasing
levels of the covariate. This concept is formalized as a "trend" in
distributions, which we define as an evolution that is linear under the
Wasserstein metric. Implemented via a fast alternating projections algorithm,
our method exhibits numerous strengths in simulations and analyses of
single-cell gene expression data.
