Functional Data Analysis: An Introduction and Recent Developments

Functional data analysis (FDA) is a statistical framework that allows for the analysis of curves, images, or functions on higher dimensional domains. The goals of FDA, such as descriptive analyses, classification, and regression, are generally the same as for statistical analyses of scalar-valued or multivariate data, but FDA brings additional challenges due to the high- and infinite dimensionality of observations and parameters, respectively. This paper provides an introduction to FDA, including a description of the most common statistical analysis techniques, their respective software implementations, and some recent developments in the field. The paper covers fundamental concepts such as descriptives and outliers, smoothing, amplitude and phase variation, and functional principal component analysis. It also discusses functional regression, statistical inference with functional data, functional classification and clustering, and machine learning approaches for functional data analysis. The methods discussed in this paper are widely applicable in fields such as medicine, biophysics, neuroscience, and chemistry and are increasingly relevant due to the widespread use of technologies that allow for the collection of functional data. Sparse functional data methods are also relevant for longitudinal data analysis. All presented methods are demonstrated using available software in R by analyzing a dataset on human motion and motor control. To facilitate the understanding of the methods, their implementation, and hands-on application, the code for these practical examples is made available through a code and data supplement and on GitHub.

Mitochondrial transport in symmetric and asymmetric axons with multiple branching junctions: a computational study

Mitochondrial aging has been proposed to be involved in a variety of neurodegenerative disorders, such as Parkinson's disease. Here, we explore the impact of multiple branching junctions in axons on the mean age of mitochondria and their age density distributions in demand sites. The study examined mitochondrial concentration, mean age, and age density distribution in relation to the distance from the soma. We developed models for a symmetric axon containing 14 demand sites and an asymmetric axon containing 10 demand sites. We investigated how the concentration of mitochondria changes when an axon splits into two branches at the branching junction. Additionally, we studied whether mitochondrial concentrations in the branches are affected by what proportion of mitochondrial flux enters the upper branch versus the lower branch. Furthermore, we explored whether the distributions of mitochondrial mean age and age density in branching axons are affected by how the mitochondrial flux splits at the branching junction. When the mitochondrial flux is unevenly split at the branching junction of an asymmetric axon, with a greater proportion of the flux entering the longer branch, the average age of mitochondria (system age) in the axon increases. Our findings elucidate the effects of axonal branching on the mitochondrial age.

RV-ESA: A novel computer-aided elastic shape analysis system for retinal vessels in diabetic retinopathy

Diabetic retinopathy (DR), one of the most common and serious complications of diabetes, has become one of the main blindness diseases. The retinal vasculature is the only part of the human circulatory system that allows direct noninvasive visualization of the body's microvasculature, which provides the opportunity to detect the structural and functional changes before DR becomes unable to intervene. For decades, as the fundamental step in computer-assisted analysis of retinopathy, retinal vascular extraction methods have been largely developed. However, further research focusing on retinal vascular analysis is still in its infancy. Meanwhile, due to the complexity of retinal vascular structure, the relationship between vascular geometry and DR has never been concluded. This paper aims to provide a novel computer-aided shape analysis system for retinal vessels. To perform retinal vascular shape analysis, a mathematical geometric representation is firstly generated by utilizing the proposed shape modeling method. Then, several useful statistical tools (e.g. Graph Mean, Graph PCA) are adopted to quantitatively analyze the vascular shape. Besides, in order to visualize the changes in vascular shape in the progression of DR, a geodesic tool is used to display the deformation process for ophthalmologists to observe. The efficacy of this analysis system is demonstrated in the EyePACS dataset and the subsequent visit records of 98 patients from the proprietary dataset. The experimental results show that there is a certain correlation between the variation of retinal vascular shape and DR progression, and the Graph PCA scores of retinal vessels are negatively correlated with DR grades. The code of our RV-ESA system can be publicly available at github.com/XiaolingLuo/RV-ESA.

Fitting Splines to Axonal Arbors Quantifies Relationship Between Branch Order and Geometry

Neuromorphology is crucial to identifying neuronal subtypes and understanding learning. It is also implicated in neurological disease. However, standard morphological analysis focuses on macroscopic features such as branching frequency and connectivity between regions, and often neglects the internal geometry of neurons. In this work, we treat neuron trace points as a sampling of differentiable curves and fit them with a set of branching B-splines. We designed our representation with the Frenet-Serret formulas from differential geometry in mind. The Frenet-Serret formulas completely characterize smooth curves, and involve two parameters, curvature and torsion. Our representation makes it possible to compute these parameters from neuron traces in closed form. These parameters are defined continuously along the curve, in contrast to other parameters like tortuosity which depend on start and end points. We applied our method to a dataset of cortical projection neurons traced in two mouse brains, and found that the parameters are distributed differently between primary, collateral, and terminal axon branches, thus quantifying geometric differences between different components of an axonal arbor. The results agreed in both brains, further validating our representation. The code used in this work can be readily applied to neuron traces in SWC format and is available in our open-source Python package brainlit: http://brainlit.neurodata.io/.

Toward a Three-Dimensional Chromosome Shape Alphabet

The study of the three-dimensional (3D) structure of chromosomes-the largest macromolecules in biology-is one of the most challenging to date in structural biology. Here, we develop a novel representation of 3D chromosome structures, as sequences of shape letters from a finite shape alphabet, which provides a compact and efficient way to analyze ensembles of chromosome shape data, akin to the analysis of texts in a language by using letters. We construct a Chromosome Shape Alphabet from an ensemble of chromosome 3D structures inferred from Hi-C data-via SIMBA3D or other methods-by segmenting curves based on topologically associating domains (TADs) boundaries, and by clustering all TADs' 3D structures into groups of similar shapes. The median shapes of these groups, with some pruning and processing, form the Chromosome Shape Letters (CSLs) of the alphabet. We provide a proof of concept for these CSLs by reconstructing independent test curves by using only CSLs (and corresponding transformations) and comparing these reconstructions with the original curves. Finally, we demonstrate how CSLs can be used to summarize shapes in an ensemble of chromosome 3D structures by using generalized sequence logos.

Simplifying Transforms for General Elastic Metrics on the Space of Plane Curves

In the shape analysis approach to computer vision problems, one treats shapes as points in an infinite-dimensional Riemannian manifold, thereby facilitating algorithms for statistical calculations such as geodesic distance between shapes and averaging of a collection of shapes. The performance of these algorithms depends heavily on the choice of the Riemannian metric. In the setting of plane curve shapes, attention has largely been focused on a two-parameter family of first order Sobolev metrics, referred to as elastic metrics. They are particularly useful due to the existence of simplifying coordinate transformations for particular parameter values, such as the well-known square-root velocity transform. In this paper, we extend the transformations appearing in the existing literature to a family of isometries, which take any elastic metric to the flat L ² metric. We also extend the transforms to treat piecewise linear curves and demonstrate the existence of optimal matchings over the diffeomorphism group in this setting. We conclude the paper with multiple examples of shape geodesics for open and closed curves. We also show the benefits of our approach in a simple classification experiment.

Statistical analysis and modeling of the geometry and topology of plant roots

The root is an important organ of a plant since it is responsible for water and nutrient uptake. Analyzing and modeling variabilities in the geometry and topology of roots can help in assessing the plant's health, understanding its growth patterns, and modeling relations between plant species and between plants and their environment. In this article, we develop a framework for the statistical analysis and modeling of the geometry and topology of plant roots. We represent root structures as points in a tree-shape space equipped with a metric that quantifies geometric and topological differences between pairs of roots. We then use these building blocks to compute geodesics, i.e., optimal deformations under the metric between root structures, and to perform statistical analysis on root populations. We demonstrate the utility of the proposed framework through an application to a dataset of wheat roots grown in different environmental conditions. We also show that the framework can be used in various applications including classification and regression.