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We propose the first comprehensive approach for modeling and analyzing the spatiotemporal shape variability in tree-like 4D objects, i.e., 3D objects whose shapes bend, stretch, and change in their branching structure over time as they deform, grow, and interact with their environment. Our key contribution is the representation of tree-like 3D shapes using Square Root Velocity Function Trees (SRVFT). By solving the spatial registration in the SRVFT space, which is equipped with an L2 metric, 4D tree-shaped structures become time-parameterized trajectories in this space. This reduces the problem of modeling and analyzing 4D tree-like shapes to that of modeling and analyzing elastic trajectories in the SRVFT space, where elasticity refers to time warping. In this paper, we propose a novel mathematical representation of the shape space of such trajectories, a Riemannian metric on that space, and computational tools for fast and accurate spatiotemporal registration and geodesics computation between 4D tree-shaped structures. Leveraging these building blocks, we develop a full framework for modelling the spatiotemporal variability using statistical models and generating novel 4D tree-like structures from a set of exemplars. We demonstrate and validate the proposed framework using real 4D plant data.

This paper focuses on the statistical analysis of shapes of data objects called shape graphs, a set of nodes connected by articulated curves with arbitrary shapes. A critical need here is a constrained registration of points (nodes to nodes, edges to edges) across objects. This, in turn, requires optimization over the permutation group, made challenging by differences in nodes (in terms of numbers, locations) and edges (in terms of shapes, placements, and sizes) across objects. This paper tackles this registration problem using a novel neural-network architecture and involves an unsupervised loss function developed using the elastic shape metric for curves. This architecture results in (1) state-of-the-art matching performance and (2) an order of magnitude reduction in the computational cost relative to baseline approaches. We demonstrate the effectiveness of the proposed approach using both simulated data and real-world 2D and 3D shape graphs. Code and data will be made publicly available after review to foster research.

This paper investigates the challenge of learning image manifolds, specifically pose manifolds, of 3D objects using limited training data. It proposes a DNN approach to manifold learning and for predicting images of objects for novel, continuous 3D rotations. The approach uses two distinct concepts: (1) Geometric Style-GAN (Geom-SGAN), which maps images to low-dimensional latent representations and maintains the (first-order) manifold geometry. That is, it seeks to preserve the pairwise distances between base points and their tangent spaces, and (2) uses Euler's elastica to smoothly interpolate between directed points (points + tangent directions) in the low-dimensional latent space. When mapped back to the larger image space, the resulting interpolations resemble videos of rotating objects. Extensive experiments establish the superiority of this framework in learning paths on rotation manifolds, both visually and quantitatively, relative to state-of-the-art GANs and VAEs.

Estimating signals underlying noisy data is a significant problem in statistics and engineering. Numerous estimators are available in the literature, depending on the observation model and estimation criterion. This paper introduces a framework that estimates the shape of the unknown signal and the signal itself. The approach utilizes a peak-persistence diagram (PPD), a novel tool that explores the dominant peaks in the potential solutions and estimates the function's shape, which includes the number of internal peaks and valleys. It then imposes this shape constraint on the search space and estimates the signal from partially-aligned data. This approach balances two previous solutions: averaging without alignment and averaging with complete elastic alignment. From a statistical viewpoint, it achieves an optimal estimator under a model with both additive noise and phase or warping noise. We also present a computationally-efficient procedure for implementing this solution and demonstrate its effectiveness on several simulated and real examples. Notably, this geometric approach outperforms the current state-of-the-art in the field.

This paper provides theoretical and computational developments in statistical shape analysis of shape graphs, and demonstrates them using analysis of complex data from retinal blood-vessel (RBV) networks. The shape graphs are represented by a set of nodes and edges (planar articulated curves) connecting some of these nodes. The goals are to utilize shapes of edges and connectivities and locations of nodes to: (1) characterize full shapes, (2) quantify shape differences, and (3) model statistical variability. We develop a mathematical representation, elastic Riemannian shape metrics, and associated tools for such statistical analysis. Specifically, we derive tools for shape graph registration, geodesics, summaries, and shape modeling. Geodesics are convenient for visualizing optimal deformations, and PCA helps in dimension reduction and statistical modeling. One key challenge here is comparisons of shape graphs with vastly different complexities (in number of nodes and edges). This paper introduces a novel multi-scale representation of shape graphs to handle this challenge. Using the notions of (1) ``effective resistance" to cluster nodes and (2) elastic shape averaging of edge curves, one can reduce shape graph complexity while maintaining overall structures. This way, we can compare shape graphs by bringing them to similar complexity. We demonstrate these ideas on Retinal Blood Vessel (RBV) networks taken from the STARE and DRIVE databases.

Alignment or registration of functions is a fundamental problem in statistical analysis of functions and shapes. While there are several approaches available, a more recent approach based on Fisher-Rao metric and square-root velocity functions (SRVFs) has been shown to have good performance. However, this SRVF method has two limitations: (1) it is susceptible to over alignment, i.e., alignment of noise as well as the signal, and (2) in case there is additional information in form of landmarks, the original formulation does not prescribe a way to incorporate that information. In this paper we propose an extension that allows for incorporation of landmark information to seek a compromise between matching curves and landmarks. This results in a soft landmark alignment that pushes landmarks closer, without requiring their exact overlays to finds a compromise between contributions from functions and landmarks. The proposed method is demonstrated to be superior in certain practical scenarios.

How can one analyze detailed 3D biological objects, such as neurons and botanical trees, that exhibit complex geometrical and topological variation? In this paper, we develop a novel mathematical framework for representing, comparing, and computing geodesic deformations between the shapes of such tree-like 3D objects. A hierarchical organization of subtrees characterizes these objects -- each subtree has the main branch with some side branches attached -- and one needs to match these structures across objects for meaningful comparisons. We propose a novel representation that extends the Square-Root Velocity Function (SRVF), initially developed for Euclidean curves, to tree-shaped 3D objects. We then define a new metric that quantifies the bending, stretching, and branch sliding needed to deform one tree-shaped object into the other. Compared to the current metrics, such as the Quotient Euclidean Distance (QED) and the Tree Edit Distance (TED), the proposed representation and metric capture the full elasticity of the branches (i.e., bending and stretching) as well as the topological variations (i.e., branch death/birth and sliding). It completely avoids the shrinkage that results from the edge collapse and node split operations of the QED and TED metrics. We demonstrate the utility of this framework in comparing, matching, and computing geodesics between biological objects such as neurons and botanical trees. The framework is also applied to various shape analysis tasks: (i) symmetry analysis and symmetrization of tree-shaped 3D objects, (ii) computing summary statistics (means and modes of variations) of populations of tree-shaped 3D objects, (iii) fitting parametric probability distributions to such populations, and (iv) finally synthesizing novel tree-shaped 3D objects through random sampling from estimated probability distributions.

The study of multidimensional discriminator (critic) output for Generative Adversarial Networks has been underexplored in the literature. In this paper, we generalize the Wasserstein GAN framework to take advantage of multidimensional critic output and explore its properties. We also introduce a square-root velocity transformation (SRVT) block which favors training in the multidimensional setting. Proofs of properties are based on our proposed maximal p-centrality discrepancy, which is bounded above by p-Wasserstein distance and fits the Wasserstein GAN framework with multidimensional critic output n. Especially when n = 1 and p = 1, the proposed discrepancy equals 1-Wasserstein distance. Theoretical analysis and empirical evidence show that high-dimensional critic output has its advantage on distinguishing real and fake distributions, and benefits faster convergence and diversity of results., Comment: Frontiers in Adversarial Machine Learning ICML 2022

This paper develops a generative statistical model for representing, modeling, and comparing the morphological evolution of biological cells undergoing motility. It uses the elastic shape analysis to separate cell kinematics (overall location, rotation, speed, etc.) from its morphology and represents morphological changes using transported square-root vector fields (TSRVFs). This TSRVF representation, followed by a PCA-based dimension reduction, provides a convenient mathematical representation of a shape sequence in the form of a Euclidean time series. Fitting a vector auto-regressive (VAR) model to this TSRVF-PCA time series leads to statistical modeling of the overall shape dynamics. We use the parameters of the fitted VAR model to characterize morphological evolution. We validate VAR models through model comparisons, synthesis, and sequence classifications. For classification, we use the VAR parameters in conjunction with different classifiers: SVM, Random Forest, and CNN, and obtain high classification rates. Extensive experiments presented here demonstrate the success of the proposed pipeline. These results are the first of the kind in classifying cell migration videos using shape dynamics.

There is increasing evidence on the importance of brain morphology in predicting and classifying mental disorders. However, the vast majority of current shape approaches rely heavily on vertex-wise analysis that may not successfully capture complexities of subcortical structures. Additionally, the past works do not include interactions between these structures and exposure factors. Predictive modeling with such interactions is of paramount interest in heterogeneous mental disorders such as PTSD, where trauma exposure interacts with brain shape changes to influence behavior. We propose a comprehensive framework that overcomes these limitations by representing brain substructures as continuous parameterized surfaces and quantifying their shape differences using elastic shape metrics. Using the elastic shape metric, we compute shape summaries of subcortical data and represent individual shapes by their principal scores. These representations allow visualization tools that help localize changes when these PCs are varied. Subsequently, these PCs, the auxiliary exposure variables, and their interactions are used for regression modeling. We apply our method to data from the Grady Trauma Project, where the goal is to predict clinical measures of PTSD using shapes of brain substructures. Our analysis revealed considerably greater predictive power under the elastic shape analysis than widely used approaches such as vertex-wise shape analysis and even volumetric analysis. It helped identify local deformations in brain shapes related to change in PTSD severity. To our knowledge, this is one of the first brain shape analysis approaches that can seamlessly integrate the pre-processing steps under one umbrella for improved accuracy and are naturally able to account for interactions between brain shape and additional covariates to yield superior predictive performance when modeling clinical outcomes., Comment: 33 pages; Supplementary Materials and interactive visualizations are available in https://www.dropbox.com/home/Paper%20Interactive%20Figures

Elastic Riemannian metrics have been used successfully in the past for statistical treatments of functional and curve shape data. However, this usage has suffered from an important restriction: the function boundaries are assumed fixed and matched. Functional data exhibiting unmatched boundaries typically arise from dynamical systems with variable evolution rates such as COVID-19 infection rate curves associated with different geographical regions. In this case, it is more natural to model such data with sliding boundaries and use partial matching, i.e., only a part of a function is matched to another function. Here, we develop a comprehensive Riemannian framework that allows for partial matching, comparing, and clustering of functions under both phase variability and uncertain boundaries. We extend past work by: (1) Forming a joint action of the time-warping and time-scaling groups; (2) Introducing a metric that is invariant to this joint action, allowing for a gradient-based approach to elastic partial matching; and (3) Presenting a modification that, while losing the metric property, allows one to control relative influence of the two groups. This framework is illustrated for registering and clustering shapes of COVID-19 rate curves, identifying essential patterns, minimizing mismatch errors, and reducing variability within clusters compared to previous methods., Comment: Submitted to IEEE Transactions on Pattern Analysis and Machine Intelligence

We propose a novel framework to learn the spatiotemporal variability in longitudinal 3D shape data sets, which contain observations of objects that evolve and deform over time. This problem is challenging since surfaces come with arbitrary parameterizations and thus, they need to be spatially registered. Also, different deforming objects, also called 4D surfaces, evolve at different speeds and thus they need to be temporally aligned. We solve this spatiotemporal registration problem using a Riemannian approach. We treat a 3D surface as a point in a shape space equipped with an elastic Riemannian metric that measures the amount of bending and stretching that the surfaces undergo. A 4D surface can then be seen as a trajectory in this space. With this formulation, the statistical analysis of 4D surfaces can be cast as the problem of analyzing trajectories embedded in a nonlinear Riemannian manifold. However, performing the spatiotemporal registration, and subsequently computing statistics, on such nonlinear spaces is not straightforward as they rely on complex nonlinear optimizations. Our core contribution is the mapping of the surfaces to the space of Square-Root Normal Fields where the L2 metric is equivalent to the partial elastic metric in the space of surfaces. Thus, by solving the spatial registration in the SRNF space, the problem of analyzing 4D surfaces becomes the problem of analyzing trajectories embedded in the SRNF space, which has a Euclidean structure. In this paper, we develop the building blocks that enable such analysis. These include: (1) the spatiotemporal registration of arbitrarily parameterized 4D surfaces in the presence of large elastic deformations and large variations in their execution rates; (2) the computation of geodesics between 4D surfaces; (3) the computation of statistical summaries; and (4) the synthesis of random 4D surfaces.

The motion-and-time analysis has been a popular research topic in operations research, especially for analyzing work performances in manufacturing and service operations. It is regaining attention as continuous improvement tools for lean manufacturing and smart factory. This paper develops a framework for data-driven analysis of work motions and studies their correlations to work speeds or execution rates, using data collected from modern motion sensors. The past analyses largely relied on manual steps involving time-consuming stop-watching and video-taping, followed by manual data analysis. While modern sensing devices have automated the collection of motion data, the motion analytics that transform the new data into knowledge are largely underdeveloped. Unsolved technical questions include: How the motion and time information can be extracted from the motion sensor data, how work motions and execution rates are statistically modeled and compared, and what are the statistical correlations of motions to the rates? In this paper, we develop a novel mathematical framework for motion and time analysis with motion sensor data, by defining new mathematical representation spaces of human motions and execution rates and by developing statistical tools on these new spaces. This methodological research is demonstrated using five use cases applied to manufacturing motion data., Comment: Keywords: motion and time study, motion sensors, Riemannian manifold, probability distribution on manifold, temporal evolution of probability distributions

Structures of brain arterial networks (BANs) - that are complex arrangements of individual arteries, their branching patterns, and inter-connectivities - play an important role in characterizing and understanding brain physiology. One would like tools for statistically analyzing the shapes of BANs, i.e. quantify shape differences, compare population of subjects, and study the effects of covariates on these shapes. This paper mathematically represents and statistically analyzes BAN shapes as elastic shape graphs. Each elastic shape graph is made up of nodes that are connected by a number of 3D curves, and edges, with arbitrary shapes. We develop a mathematical representation, a Riemannian metric and other geometrical tools, such as computations of geodesics, means and covariances, and PCA for analyzing elastic graphs and BANs. This analysis is applied to BANs after separating them into four components -- top, bottom, left, and right. This framework is then used to generate shape summaries of BANs from 92 subjects, and to study the effects of age and gender on shapes of BAN components. We conclude that while gender effects require further investigation, the age has a clear, quantifiable effect on BAN shapes. Specifically, we find an increased variance in BAN shapes as age increases., Comment: arXiv admin note: substantial text overlap with arXiv:2003.00287

This paper develops an agent-level simulation model, termed ALPS, for simulating the spread of an infectious disease in a confined community. The mechanism of transmission is agent-to-agent contact, using parameters reported for Corona COVID-19 pandemic. The main goal of the ALPS simulation is analyze effects of preventive measures -- imposition and lifting of lockdown norms -- on the rates of infections, fatalities and recoveries. The model assumptions and choices represent a balance between competing demands of being realistic and being efficient for real-time inferences. The model provides quantification of gains in reducing casualties by imposition and maintenance of restrictive measures in place.

Past approaches for statistical shape analysis of objects have focused mainly on objects within the same topological classes, e.g., scalar functions, Euclidean curves, or surfaces, etc. For objects that differ in more complex ways, the current literature offers only topological methods. This paper introduces a far-reaching geometric approach for analyzing shapes of graphical objects, such as road networks, blood vessels, brain fiber tracts, etc. It represents such objects, exhibiting differences in both geometries and topologies, as graphs made of curves with arbitrary shapes (edges) and connected at arbitrary junctions (nodes). To perform statistical analyses, one needs mathematical representations, metrics and other geometrical tools, such as geodesics, means, and covariances. This paper utilizes a quotient structure to develop efficient algorithms for computing these quantities, leading to useful statistical tools, including principal component analysis and analytical statistical testing and modeling of graphical shapes. The efficacy of this framework is demonstrated using various simulated as well as the real data from neurons and brain arterial networks., Comment: Visualization improved

Complex analyses involving multiple, dependent random quantities often lead to graphical models - a set of nodes denoting variables of interest, and corresponding edges denoting statistical interactions between nodes. To develop statistical analyses for graphical data, especially towards generative modeling, one needs mathematical representations and metrics for matching and comparing graphs, and subsequent tools, such as geodesics, means, and covariances. This paper utilizes a quotient structure to develop efficient algorithms for computing these quantities, leading to useful statistical tools, including principal component analysis, statistical testing, and modeling. We demonstrate the efficacy of this framework using datasets taken from several problem areas, including letters, biochemical structures, and social networks.

Intensity estimation for Poisson processes is a classical problem and has been extensively studied over the past few decades. Practical observations, however, often contain compositional noise, i.e. a nonlinear shift along the time axis, which makes standard methods not directly applicable. The key challenge is that these observations are not "aligned", and registration procedures are required for successful estimation. In this paper, we propose an alignment-based framework for positive intensity estimation. We first show that the intensity function is area-preserved with respect to compositional noise. Such a property implies that the time warping is only encoded in the normalized intensity, or density, function. Then, we decompose the estimation of the intensity by the product of the estimated total intensity and estimated density. The estimation of the density relies on a metric which measures the phase difference between two density functions. An asymptotic study shows that the proposed estimation algorithm provides a consistent estimator for the normalized intensity. We then extend the framework to estimating non-negative intensity functions. The success of the proposed estimation algorithms is illustrated using two simulations. Finally, we apply the new framework in a real data set of neural spike trains, and find that the newly estimated intensities provide better classification accuracy than previous methods.

Functional variables are often used as predictors in regression problems. A commonly-used parametric approach, called {\it scalar-on-function regression}, uses the $\ltwo$ inner product to map functional predictors into scalar responses. This method can perform poorly when predictor functions contain undesired phase variability, causing phases to have disproportionately large influence on the response variable. One past solution has been to perform phase-amplitude separation (as a pre-processing step) and then use only the amplitudes in the regression model. Here we propose a more integrated approach, termed elastic functional regression model (EFRM), where phase-separation is performed inside the regression model, rather than as a pre-processing step. This approach generalizes the notion of phase in functional data, and is based on the norm-preserving time warping of predictors. Due to its invariance properties, this representation provides robustness to predictor phase variability and results in improved predictions of the response variable over traditional models. We demonstrate this framework using a number of datasets involving gait signals, NMR data, and stock market prices., Comment: 30 pages

The root is an important organ of a plant since it is responsible for water and nutrient uptake. Analyzing and modelling 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.

This paper studies change-points in human brain functional connectivity (FC) and seeks patterns that are common across multiple subjects under identical external stimulus. FC relates to the similarity of fMRI responses across different brain regions when the brain is simply resting or performing a task. While the dynamic nature of FC is well accepted, this paper develops a formal statistical test for finding {\it change-points} in times series associated with FC. It represents short-term connectivity by a symmetric positive-definite matrix, and uses a Riemannian metric on this space to develop a graphical method for detecting change-points in a time series of such matrices. It also provides a graphical representation of estimated FC for stationary subintervals in between the detected change-points. Furthermore, it uses a temporal alignment of the test statistic, viewed as a real-valued function over time, to remove inter-subject variability and to discover common change-point patterns across subjects. This method is illustrated using data from Human Connectome Project (HCP) database for multiple subjects and tasks.

Human brain functional connectivity (FC) is often measured as the similarity of functional MRI responses across brain regions when a brain is either resting or performing a task. This paper aims to statistically analyze the dynamic nature of FC by representing the collective time-series data, over a set of brain regions, as a trajectory on the space of covariance matrices, or symmetric-positive definite matrices (SPDMs). We use a recently developed metric on the space of SPDMs for quantifying differences across FC observations, and for clustering and classification of FC trajectories. To facilitate large scale and high-dimensional data analysis, we propose a novel, metric-based dimensionality reduction technique to reduce data from large SPDMs to small SPDMs. We illustrate this comprehensive framework using data from the Human Connectome Project (HCP) database for multiple subjects and tasks, with task classification rates that match or outperform state-of-the-art techniques., Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible

We study regression using functional predictors in situations where these functions contain both phase and amplitude variability. In other words, the functions are misaligned due to errors in time measurements, and these errors can significantly degrade both model estimation and prediction performance. The current techniques either ignore the phase variability, or handle it via pre-processing, i.e., use an off-the-shelf technique for functional alignment and phase removal. We develop a functional principal component regression model which has comprehensive approach in handling phase and amplitude variability. The model utilizes a mathematical representation of the data known as the square-root slope function. These functions preserve the $\mathbf{L}^2$ norm under warping and are ideally suited for simultaneous estimation of regression and warping parameters. Using both simulated and real-world data sets, we demonstrate our approach and evaluate its prediction performance relative to current models. In addition, we propose an extension to functional logistic and multinomial logistic regression

While spherical data arises in many contexts, including in directional statistics, the current tools for density estimation and population comparison on spheres are quite limited. Popular approaches for comparing populations (on Euclidean domains) mostly involvea two-step procedure: (1) estimate probability density functions (pdfs) from their respective samples, most commonly using the kernel density estimator, and, (2) compare pdfs using a metric such as the L2 norm. However, both the estimated pdfs and their differences depend heavily on the chosen kernels, bandwidths, and sample sizes. Here we develop a framework for comparing spherical populations that is robust to these choices. Essentially, we characterize pdfs on spherical domains by quantifying their smoothness. Our framework uses a spectral representation, with densities represented by their coefficients with respect to the eigenfunctions of the Laplacian operator on a sphere. The change in smoothness, akin to using different kernel bandwidths, is controlled by exponential decays in coefficient values. Then we derive a proper distance for comparing pdf coefficients while equalizing smoothness levels, negating influences of sample size and bandwidth. This signifies a fair and meaningful comparisons of populations, despite vastly different sample sizes, and leads to a robust and improved performance. We demonstrate this framework using examples of variables on S1 and S2, and evaluate its performance using a number of simulations and real data experiments.

While the problem of estimating a probability density function (pdf) from its observations is classical, the estimation under additional shape constraints is both important and challenging. We introduce an efficient, geometric approach for estimating pdfs given the number of its modes. This approach explores the space of constrained pdf's using an action of the diffeomorphism group that preserves their shapes. It starts with an initial template, with the desired number of modes and arbitrarily chosen heights at the critical points, and transforms it via: (1) composition by diffeomorphisms and (2) normalization to obtain the final density estimate. The search for optimal diffeomorphism is performed under the maximum-likelihood criterion and is accomplished by mapping diffeomorphisms to the tangent space of a Hilbert sphere, a vector space whose elements can be expressed using an orthogonal basis. This framework is first applied to shape-constrained univariate, unconditional pdf estimation and then extended to conditional pdf estimation. We derive asymptotic convergence rates of the estimator and demonstrate this approach using a synthetic dataset involving speed distribution for different traffic flow on Californian driveways., Comment: 31 pages, Initial version. Presented at IISA 2017, Shape Constrained Methods Workshp, BIRS,2018

The problem of estimating trend and seasonal variation in time-series data has been studied over several decades, although mostly using single time series. This paper studies the problem of estimating these components from functional data, i.e. multiple time series, in situations where seasonal effects exhibit arbitrary time warpings or phase variability across different observations. Rather than ignoring the phase variability, or using an off-the-shelf alignment method to remove phase, we take a model-based approach and seek MLEs of the trend and the seasonal effects, while performing alignments over the seasonal effects at the same time. The MLEs of trend, seasonality, and phase are computed using a coordinate-descent based optimization method. We use bootstrap replication for computing confidence bands and for testing hypothesis about the estimated components. We also utilize log-likelihood for selecting the trend subspace, and for comparisons with other candidate models. This framework is demonstrated using experiments involving synthetic data and three real data (Berkeley Growth Velocity, U.S. electricity price, and USD exchange fluctuation)., Comment: 18 pages, 19 figures

Spherical regression explores relationships between variables on spherical domains. We develop a nonparametric model that uses a diffeomorphic map from a sphere to itself. The restriction of this mapping to diffeomorphisms is natural in several settings. The model is estimated in a penalized maximum-likelihood framework using gradient-based optimization. Towards that goal, we specify a first-order roughness penalty using the Jacobian of diffeomorphisms. We compare the prediction performance of the proposed model with state-of-the-art methods using simulated and real data involving cloud deformations, wind directions, and vector-cardiograms. This model is found to outperform others in capturing relationships between spherical variables.

We introduce a novel two-step approach for estimating a probability density function (pdf) given its samples, with the second and important step coming from a geometric formulation. The procedure involves obtaining an initial estimate of the pdf and then transforming it via a warping function to reach the final estimate. The initial estimate is intended to be computationally fast, albeit suboptimal, but its warping creates a larger, flexible class of density functions, resulting in substantially improved estimation. The search for optimal warping is accomplished by mapping diffeomorphic functions to the tangent space of a Hilbert sphere, a vector space whose elements can be expressed using an orthogonal basis. Using a truncated basis expansion, we estimate the optimal warping under a (penalized) likelihood criterion and, thus, the optimal density estimate. This framework is introduced for univariate, unconditional pdf estimation and then extended to conditional pdf estimation. The approach avoids many of the computational pitfalls associated with classical conditional-density estimation methods, without losing on estimation performance. We derive asymptotic convergence rates of the density estimator and demonstrate this approach using both synthetic datasets and real data, the latter relating to the association of a toxic metabolite on preterm birth., Comment: Submitted to a journal currently. Sections rewritten and arranged differently from previous version. Some errors in proofs and notations corrected. Section on Bivariate density estimation and Irrelevant predictors removed. Performance of conditional density estimation compared to a different package with different simulation examples

Recent developments in elastic shape analysis (ESA) are motivated by the fact that it provides comprehensive frameworks for simultaneous registration, deformation, and comparison of shapes. These methods achieve computational efficiency using certain square-root representations that transform invariant elastic metrics into Euclidean metrics, allowing for applications of standard algorithms and statistical tools. For analyzing shapes of embeddings of $\mathbb{S}^2$ in $\mathbb{R}^3$, Jermyn et al. introduced square-root normal fields (SRNFs) that transformed an elastic metric, with desirable invariant properties, into the $\mathbb{L}^2$ metric. These SRNFs are essentially surface normals scaled by square-roots of infinitesimal area elements. A critical need in shape analysis is to invert solutions (deformations, averages, modes of variations, etc) computed in the SRNF space, back to the original surface space for visualizations and inferences. Due to the lack of theory for understanding SRNFs maps and their inverses, we take a numerical approach and derive an efficient multiresolution algorithm, based on solving an optimization problem in the surface space, that estimates surfaces corresponding to given SRNFs. This solution is found effective, even for complex shapes, e.g. human bodies and animals, that undergo significant deformations including bending and stretching. Specifically, we use this inversion for computing elastic shape deformations, transferring deformations, summarizing shapes, and for finding modes of variability in a given collection, while simultaneously registering the surfaces. We demonstrate the proposed algorithms using a statistical analysis of human body shapes, classification of generic surfaces and analysis of brain structures.