Clustering and visualization of single-cell RNA-seq data using path metrics.

Recent advances in single-cell technologies have enabled high-resolution characterization of tissue and cancer compositions. Although numerous tools for dimension reduction and clustering are available for single-cell data analyses, these methods often fail to simultaneously preserve local cluster structure and global data geometry. To address these challenges, we developed a novel analyses framework, Single-Cell Path Metrics Profiling (scPMP), using power-weighted path metrics, which measure distances between cells in a data-driven way. Unlike Euclidean distance and other commonly used distance metrics, path metrics are density sensitive and respect the underlying data geometry. By combining path metrics with multidimensional scaling, a low dimensional embedding of the data is obtained which preserves both the global data geometry and cluster structure. We evaluate the method both for clustering quality and geometric fidelity, and it outperforms current scRNAseq clustering algorithms on a wide range of benchmarking data sets.

An analysis of classical multidimensional scaling with applications to clustering.

Classical multidimensional scaling is a widely used dimension reduction technique. Yet few theoretical results characterizing its statistical performance exist. This paper provides a theoretical framework for analyzing the quality of embedded samples produced by classical multidimensional scaling. This lays a foundation for various downstream statistical analyses, and we focus on clustering noisy data. Our results provide scaling conditions on the signal-to-noise ratio under which classical multidimensional scaling followed by a distance-based clustering algorithm can recover the cluster labels of all samples. Simulation studies confirm these scaling conditions are sharp. Applications to the cancer gene-expression data, the single-cell RNA sequencing data and the natural language data lend strong support to the methodology and theory.

Wavelet invariants for statistically robust multi-reference alignment.

We propose a nonlinear, wavelet-based signal representation that is translation invariant and robust to both additive noise and random dilations. Motivated by the multi-reference alignment problem and generalizations thereof, we analyze the statistical properties of this representation given a large number of independent corruptions of a target signal. We prove the nonlinear wavelet-based representation uniquely defines the power spectrum but allows for an unbiasing procedure that cannot be directly applied to the power spectrum. After unbiasing the representation to remove the effects of the additive noise and random dilations, we recover an approximation of the power spectrum by solving a convex optimization problem, and thus reduce to a phase retrieval problem. Extensive numerical experiments demonstrate the statistical robustness of this approximation procedure.

Interpreting measures of risk: Translating evidence into practice.

Advanced practice registered nurses must have a working knowledge of statistical principles in order to provide high-quality, evidence-based care. This article presents basic concepts of risk indexes and case study examples illustrating how these measures can inform practice.