Modeling Persistent Trends in Distributions

TL;DR

This paper introduces a nonparametric regression framework for modeling sequences of probability distributions that evolve with a persistent trend, particularly useful for analyzing biological data like single-cell RNA sequencing over time.

Contribution

It proposes a novel Wasserstein-trend regression model for distributions with an efficient algorithm, addressing the need to distinguish progression effects from noise in sequential distribution data.

Findings

Effective in simulations demonstrating trend detection.

Successful application to single-cell gene expression data.

Outperforms classical methods in capturing distributional changes.

Abstract

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.