Search for: All records

Elastic Riemannian metrics have been used successfully for statistical treatments of functional and curve shape data. However, this usage suffers from a significant restriction: the function boundaries are assumed to be fixed and matched. Functional data often comes with unmatched boundaries, {\it e.g.}, in dynamical systems with variable evolution rates, such as COVID-19 infection rate curves associated with different geographical regions. Here, we develop a Riemannian framework that allows for partial matching, comparing, and clustering functions under phase variability {\it and} uncertain boundaries. We extend past work by (1) Defining a new diffeomorphism group G over the positive reals that is the semidirect product of a time-warping group and a time-scaling group; (2) Introducing a metric that is invariant to the action of G; (3) Imposing a Riemannian Lie group structure on G to allow for an efficient gradient-based optimization for elastic partial matching; and (4) Presenting a modification that, while losing the metric property, allows one to control the amount of boundary disparity in the registration. We illustrate this framework by registering and clustering shapes of COVID-19 rate curves, identifying basic patterns, minimizing mismatch errors, and reducing variability within clusters compared to previous methods. 

Despite enormous structural variability exhibited in 3D chromosomal conformations at a global scale, there is a significant commonality of structures visible at smaller, local levels. We hypothesize that chromosomal conformations are representable as concatenations of a handful of prototypical shapelets, termed shape letters. This is akin to expressing complicated sentences in a language using only a small set of letters. Our goal is to organize the vast variability of 3D chromosomal conformation by constructing a set of predominant shape letters, termed a shape alphabet, using statistical shape analysis of curvelets taken from training conformations. This paper utilizes conformations generated from Integrative Genome Modeling to develop a shape alphabet as follows: it first segments 3D conformations into curvelets according to their Topologically Associated Domains. It then clusters these segments, estimates mean shapes, and refines and reorders these shapes into a Chromosome Shape Alphabet. The paper demonstrates effectiveness of this construction by successfully representing independent test conformations taken from IGM and other methods such as SIMBA3D, both symbolically and structurally, using the constructed alphabet.