Shape Analysis of Functional Data With Elastic Partial Matching

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. In practice, functional data often comes with unmatched boundaries. It happens, for example, 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 of functions with phase variability 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.

Curve Registration of Functional Data for Approximate Bayesian Computation

Approximate Bayesian computation is a likelihood-free inference method which relies on comparing model realisations to observed data with informative distance measures. We obtain functional data that are not only subject to noise along their y axis but also to a random warping along their x axis, which we refer to as the time axis. Conventional distances on functions, such as the L2 distance, are not informative under these conditions. The Fisher–Rao metric, previously generalised from the space of probability distributions to the space of functions, is an ideal objective function for aligning one function to another by warping the time axis. We assess the usefulness of alignment with the Fisher–Rao metric for approximate Bayesian computation with four examples: two simulation examples, an example about passenger flow at an international airport, and an example of hydrological flow modelling. We find that the Fisher–Rao metric works well as the objective function to minimise for alignment; however, once the functions are aligned, it is not necessarily the most informative distance for inference. This means that likelihood-free inference may require two distances: one for alignment and one for parameter inference.

Functional Data Analyses of Gait Data Measured Using In-Shoe Sensors

In studies of gait, continuous measurement of force exerted by the ground on a body, or ground reaction force (GRF), provides valuable insights into biomechanics, locomotion, and the possible presence of pathology. However, gold-standard measurement of GRF requires a costly in-lab observation obtained with sophisticated equipment and computer systems. Recently, in-shoe sensors have been pursued as a relatively inexpensive alternative to in-lab measurement. In this study, we explore the properties of continuous in-shoe sensor recordings using a functional data analysis approach. Our case study is based on measurements of three healthy subjects, with more than 300 stances (defined as the period between the foot striking and lifting from the ground) per subject. The sensor data show both phase and amplitude variabilities; we separate these sources via curve registration. We examine the correlation of phase shifts across sensors within a stance to evaluate the pattern of phase variability shared across sensors. Using the registered curves, we explore possible associations between in-shoe sensor recordings and GRF measurements to evaluate the in-shoe sensor recordings as a possible surrogate for in-lab GRF measurements.

Functional regression models: Some directions of future research

We congratulate the authors for their excellent work that provides a clear overview of the large and now mature field of regression models for functional data. We here complement their discussion indicating some directions of further research that we deem particularly important.

A general framework for functional regression modelling

Researchers are increasingly interested in regression models for functional data. This article discusses a comprehensive framework for additive (mixed) models for functional responses and/or functional covariates based on the guiding principle of reframing functional regression in terms of corresponding models for scalar data, allowing the adaptation of a large body of existing methods for these novel tasks. The framework encompasses many existing as well as new models. It includes regression for ‘generalized’ functional data, mean regression, quantile regression as well as generalized additive models for location, shape and scale (GAMLSS) for functional data. It admits many flexible linear, smooth or interaction terms of scalar and functional covariates as well as (functional) random effects and allows flexible choices of bases—particularly splines and functional principal components—and corresponding penalties for each term. It covers functional data observed on common (dense) or curve-specific (sparse) grids. Penalized-likelihood-based and gradient-boosting-based inference for these models are implemented in R packages refund and FDboost , respectively. We also discuss identifiability and computational complexity for the functional regression models covered. A running example on a longitudinal multiple sclerosis imaging study serves to illustrate the flexibility and utility of the proposed model class. Reproducible code for this case study is made available online.