arXiv subjects

Hans-Georg Müller

Publications and source records attributed to Hans-Georg Müller.

At least 55 records · Page 3Linked to original sources

arXiv2021

Conditional Wasserstein Barycenters and Interpolation/Extrapolation of Distributions

Increasingly complex data analysis tasks motivate the study of the dependency of distributions of multivariate continuous random variables on scalar or vector predictors. Statistical regression models for distributional responses so far have primarily been investigated for the case of one-dimensional response distributions. We investigate here the case of multivariate response distributions while adopting the 2-Wasserstein metric in the distribution space. The challenge is that unlike the situation in the univariate case, the optimal transports that correspond to geodesics in the space of distributions with the 2-Wasserstein metric do not have an explicit representation for multivariate distributions. We show that under some regularity assumptions the conditional Wasserstein barycenters constructed for a geodesic in the Euclidean predictor space form a corresponding geodesic in the Wasserstein distribution space and demonstrate how the notion of conditional barycenters can be harnessed to interpolate as well as extrapolate multivariate distributions. The utility of distributional inter- and extrapolation is explored in simulations and examples. We study both global parametric-like and local smoothing-like models to implement conditional Wasserstein barycenters and establish asymptotic convergence properties for the corresponding estimates. For algorithmic implementation we make use of a Sinkhorn entropy-penalized algorithm. Conditional Wasserstein barycenters and distribution extrapolation are illustrated with applications in climate science and studies of aging.

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arXiv2021

Wasserstein Regression

The analysis of samples of random objects that do not lie in a vector space is gaining increasing attention in statistics. An important class of such object data is univariate probability measures defined on the real line. Adopting the Wasserstein metric, we develop a class of regression models for such data, where random distributions serve as predictors and the responses are either also distributions or scalars. To define this regression model, we utilize the geometry of tangent bundles of the space of random measures endowed with the Wasserstein metric for mapping distributions to tangent spaces. The proposed distribution-to-distribution regression model provides an extension of multivariate linear regression for Euclidean data and function-to-function regression for Hilbert space valued data in functional data analysis. In simulations, it performs better than an alternative transformation approach where one maps distributions to a Hilbert space through the log quantile density transformation and then applies traditional functional regression. We derive asymptotic rates of convergence for the estimator of the regression operator and for predicted distributions and also study an extension to autoregressive models for distribution-valued time series. The proposed methods are illustrated with data on human mortality and distributional time series of house prices.

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arXiv2021

Uniform convergence of local Fréchet regression, with applications to locating extrema and time warping for metric space valued trajectories

Local Fréchet regression is a nonparametric regression method for metric space valued responses and Euclidean predictors, which can be utilized to obtain estimates of smooth trajectories taking values in general metric spaces from noisy metric space valued random objects. We derive uniform rates of convergence, which so far have eluded theoretical analysis of this method, for both fixed and random target trajectories, where we utilize tools from empirical processes. These results are shown to be widely applicable in metric space valued data analysis. In addition to simulations, we provide two pertinent examples where these results are important: The consistent estimation of the location of properly defined extrema in metric space valued trajectories, which we illustrate with the problem of locating the age of minimum brain connectivity as obtained from fMRI data; Time warping for metric space valued trajectories, illustrated with yearly age-at-death distributions for different countries.

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arXiv2021

Wasserstein Gradients for the Temporal Evolution of Probability Distributions

Many studies have been conducted on flows of probability measures, often in terms of gradient flows. We utilize a generalized notion of derivatives with respect to time to model the instantaneous evolution of empirically observed one-dimensional distributions that vary over time and develop consistent estimates for these derivatives. Employing local Fréchet regression and working in local tangent spaces with regard to the Wasserstein metric, we derive the rate of convergence of the proposed estimators. The resulting time dynamics are illustrated with time-varying distribution data that include yearly income distributions and the evolution of mortality over calendar years.

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arXiv2021

High-dimensional MANOVA via Bootstrapping and its Application to Functional and Sparse Count Data

We propose a new approach to the problem of high-dimensional multivariate ANOVA via bootstrapping max statistics that involve the differences of sample mean vectors. The proposed method proceeds via the construction of simultaneous confidence regions for the differences of population mean vectors. It is suited to simultaneously test the equality of several pairs of mean vectors of potentially more than two populations. By exploiting the variance decay property that is a natural feature in relevant applications, we are able to provide dimension-free and nearly-parametric convergence rates for Gaussian approximation, bootstrap approximation, and the size of the test. We demonstrate the proposed approach with ANOVA problems for functional data and sparse count data. The proposed methodology is shown to work well in simulations and several real data applications.

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arXiv2020

Additive Models for Symmetric Positive-Definite Matrices, Riemannian Manifolds and Lie groups

In this paper an additive regression model for a symmetric positive-definite matrix valued response and multiple scalar predictors is proposed. The model exploits the abelian group structure inherited from either the Log-Cholesky metric or the Log-Euclidean framework that turns the space of symmetric positive-definite matrices into a Riemannian manifold and further a bi-invariant Lie group. The additive model for responses in the space of symmetric positive-definite matrices with either of these metrics is shown to connect to an additive model on a tangent space. This connection not only entails an efficient algorithm to estimate the component functions but also allows to generalize the proposed additive model to general Riemannian manifolds that might not have a Lie group structure. Optimal asymptotic convergence rates and normality of the estimated component functions are also established. Numerical studies show that the proposed model enjoys superior numerical performance, especially when there are multiple predictors. The practical merits of the proposed model are demonstrated by analyzing diffusion tensor brain imaging data.

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arXiv2020

Cross-Component Registration for Multivariate Functional Data, With Application to Growth Curves

Multivariate functional data are becoming ubiquitous with advances in modern technology and are substantially more complex than univariate functional data. We propose and study a novel model for multivariate functional data where the component processes are subject to mutual time warping. That is, the component processes exhibit a similar shape but are subject to systematic phase variation across their time domains. To address this previously unconsidered mode of warping, we propose new registration methodology which is based on a shift-warping model. Our method differs from all existing registration methods for functional data in a fundamental way. Namely, instead of focusing on the traditional approach to warping, where one aims to recover individual-specific registration, we focus on shift registration across the components of a multivariate functional data vector on a population-wide level. Our proposed estimates for these shifts are identifiable, enjoy parametric rates of convergence and often have intuitive physical interpretations, all in contrast to traditional curve-specific registration approaches. We demonstrate the implementation and interpretation of the proposed method by applying our methodology to the Zürich Longitudinal Growth data and study its finite sample properties in simulations.

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arXiv2020

Rank Dynamics for Functional Data

The study of the dynamic behavior of cross-sectional ranks over time for functional data and the ranks of the observed curves at each time point and their temporal evolution can yield valuable insights into the time dynamics of functional data. This approach is of interest in various application areas. For the analysis of the dynamics of ranks, estimation of the cross-sectional ranks of functional data is a first step. Several statistics of interest for ranked functional data are proposed. To quantify the evolution of ranks over time, a model for rank derivatives is introduced, where rank dynamics are decomposed into two components. One component corresponds to population changes and the other to individual changes that both affect the rank trajectories of individuals. The joint asymptotic normality for suitable estimates of these two components is established. The proposed approaches are illustrated with simulations and three longitudinal data sets: Growth curves obtained from the Zürich Longitudinal Growth Study, monthly house price data in the US from 1996 to 2015 and Major League Baseball offensive data for the 2017 season.

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arXiv2020

Fréchet Change Point Detection

We propose a method to infer the presence and location of change-points in the distribution of a sequence of independent data taking values in a general metric space, where change-points are viewed as locations at which the distribution of the data sequence changes abruptly in terms of either its Fréchet mean or Fréchet variance or both. The proposed method is based on comparisons of Fréchet variances before and after putative change-point locations and does not require a tuning parameter except for the specification of cut-off intervals near the endpoints where change-points are assumed not to occur. Our results include theoretical guarantees for consistency of the test under contiguous alternatives when a change-point exists and also for consistency of the estimated location of the change-point if it exists, where under the null hypothesis of no change-point the limit distribution of the proposed scan function is the square of a standardized Brownian Bridge. These consistency results are applicable for a broad class of metric spaces under mild entropy conditions. Examples include the space of univariate probability distributions and the space of graph Laplacians for networks. Simulation studies demonstrate the effectiveness of the proposed methods, both for inferring the presence of a change-point and estimating its location. We also develop theory that justifies bootstrap-based inference and illustrate the new approach with sequences of maternal fertility distributions and communication networks.

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arXiv2019

Functional Models for Time-Varying Random Objects

In recent years, samples of time-varying object data such as time-varying networks that are not in a vector space have been increasingly collected. These data can be viewed as elements of a general metric space that lacks local or global linear structure and therefore common approaches that have been used with great success for the analysis of functional data, such as functional principal component analysis, cannot be applied. In this paper we propose metric covariance, a novel association measure for paired object data lying in a metric space $(Ω,d)$ that we use to define a metric auto-covariance function for a sample of random $Ω$-valued curves, where $Ω$ generally will not have a vector space or manifold structure. The proposed metric auto-covariance function is non-negative definite when the squared semimetric $d^2$ is of negative type. Then the eigenfunctions of the linear operator with the auto-covariance function as kernel can be used as building blocks for an object functional principal component analysis for $Ω$-valued functional data, including time-varying probability distributions, covariance matrices and time-dynamic networks. Analogues of functional principal components for time-varying objects are obtained by applying Fréchet means and projections of distance functions of the random object trajectories in the directions of the eigenfunctions, leading to real-valued Fréchet scores. Using the notion of generalized Fréchet integrals, we construct object functional principal components that lie in the metric space $Ω$.

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arXiv2019

Fréchet Analysis Of Variance For Random Objects

Fréchet mean and variance provide a way of obtaining mean and variance for general metric space valued random variables and can be used for statistical analysis of data objects that lie in abstract spaces devoid of algebraic structure and operations. Examples of such spaces include covariance matrices, graph Laplacians of networks and univariate probability distribution functions. We derive a central limit theorem for Fréchet variance under mild regularity conditions, utilizing empirical process theory, and also provide a consistent estimator of the asymptotic variance. These results lead to a test to compare k populations based on Fréchet variance for general metric space valued data objects, with emphasis on comparing means and variances. We examine the finite sample performance of this inference procedure through simulation studies for several special cases that include probability distributions and graph Laplacians, which leads to tests to compare populations of networks. The proposed methodology has good finite sample performance in simulations for different kinds of random objects. We illustrate the proposed methods with data on mortality profiles of various countries and resting state Functional Magnetic Resonance Imaging data.

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arXiv2018

Wasserstein Covariance for Multiple Random Densities

A common feature of methods for analyzing samples of probability density functions is that they respect the geometry inherent to the space of densities. Once a metric is specified for this space, the Fréchet mean is typically used to quantify and visualize the average density from the sample. For one-dimensional densities, the Wasserstein metric is popular due to its theoretical appeal and interpretive value as an optimal transport metric, leading to the Wasserstein-Fréchet mean or barycenter as the mean density. We extend the existing methodology for samples of densities in two key directions. First, motivated by applications in neuroimaging, we consider dependent density data, where a $p$-vector of univariate random densities is observed for each sampling unit. Second, we introduce a Wasserstein covariance measure and propose intuitively appealing estimators for both fixed and diverging $p$, where the latter corresponds to continuously-indexed densities. We also give theory demonstrating consistency and asymptotic normality, while accounting for errors introduced in the unavoidable preparatory density estimation step. The utility of the Wasserstein covariance matrix is demonstrated through applications to functional connectivity in the brain using functional magnetic resonance imaging data and to the secular evolution of mortality for various countries.

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arXiv2018

Modeling Longitudinal Data on Riemannian Manifolds

When considering functional principal component analysis for sparsely observed longitudinal data that take values on a nonlinear manifold, a major challenge is how to handle the sparse and irregular observations that are commonly encountered in longitudinal studies. Addressing this challenge, we provide theory and implementations for a manifold version of the principal analysis by conditional expectation (PACE) procedure that produces representations intrinsic to the manifold, extending a well-established version of functional principal component analysis targeting sparsely sampled longitudinal data in linear spaces. Key steps are local linear smoothing methods for the estimation of a Fréchet mean curve, mapping the observed manifold-valued longitudinal data to tangent spaces around the estimated mean curve, and applying smoothing methods to obtain the covariance structure of the mapped data. Dimension reduction is achieved via representations based on the first few leading principal components. A finitely truncated representation of the original manifold-valued data is then obtained by mapping these tangent space representations to the manifold. We show that the proposed estimates of mean curve and covariance structure achieve state-of-the-art convergence rates. For longitudinal emotional well-being data for unemployed workers as an example of time-dynamic compositional data that are located on a sphere, we demonstrate that our methods lead to interpretable eigenfunctions and principal component scores. In a second example, we analyze the body shapes of wallabies by mapping the relative size of their body parts onto a spherical pre-shape space. Compared to standard functional principal component analysis, which is based on Euclidean geometry, the proposed approach leads to improved trajectory recovery for sparsely sampled data on nonlinear manifolds.

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arXiv2018

Associating Growth in Infancy and Cognitive Performance in Early Childhood: A functional data analysis approach

Physical growth traits can be naturally represented by continuous functions. In a large dataset of infancy growth patterns, we develop a practical approach to infer statistical associations between growth-trajectories and IQ performance in early childhood. The main objective of this study is to show how to assess physical growth curves and detect if particular infancy growth patterns are associated with differences in IQ (Full-scale WASI scores) in later ages using a semi-parametric functional response model. Additionally, we investigate the association between different growth measurements in terms of their cross-correlation with each other, their correlation with later IQ, as well as their time-varying dynamics. This analysis framework can easily incorporate or learn population information in a non-parametric way, rendering the existence of prior population charts partially redundant.

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arXiv2018

Fréchet Estimation of Time-Varying Covariance Matrices From Sparse Data, With Application to the Regional Co-Evolution of Myelination in the Developing Brain

Assessing brain development for small infants is important for determining how the human brain grows during the early period of life when the rate of brain growth is at its peak. The development of MRI techniques has enabled the quantification of brain development. A key quantity that can be extracted from MRI measurements is the level of myelination, where myelin acts as an insulator around nerve fibers and its deployment makes nerve pulse propagation more efficient. The co-variation of myelin deployment across different brain regions provides insights into the co-development of brain regions and can be assessed as a correlation matrix that varies with age. Typically, available data for each child are very sparse, due to the cost and logistic difficulties of arranging MRI brain scans for infants. We showcase here a method where data per subject are limited to measurements taken at only one random age, while aiming at the time-varying dynamics. This situation is encountered more generally in cross-sectional studies where one observes $p$-dimensional vectors at one random time point per subject and is interested in the $p \times p$ correlation matrix function over the time domain. The challenge is that at each observation time one observes only a $p$-vector of measurements but not a covariance or correlation matrix. For such very sparse data, we develop a Fréchet estimation method. Given a metric on the space of covariance matrices, the proposed method generates a matrix function where at each time the matrix is a non-negative definite covariance matrix, for which we demonstrate consistency properties. We discuss how this approach can be applied to myelin data in the developing brain and what insights can be gained.

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arXiv2017

Principal Component Analysis for Functional Data on Riemannian Manifolds and Spheres

Functional data analysis on nonlinear manifolds has drawn recent interest. Sphere-valued functional data, which are encountered for example as movement trajectories on the surface of the earth, are an important special case. We consider an intrinsic principal component analysis for smooth Riemannian manifold-valued functional data and study its asymptotic properties. Riemannian functional principal component analysis (RFPCA) is carried out by first mapping the manifold-valued data through Riemannian logarithm maps to tangent spaces around the time-varying Fréchet mean function, and then performing a classical multivariate functional principal component analysis on the linear tangent spaces. Representations of the Riemannian manifold-valued functions and the eigenfunctions on the original manifold are then obtained with exponential maps. The tangent-space approximation through functional principal component analysis is shown to be well-behaved in terms of controlling the residual variation if the Riemannian manifold has nonnegative curvature. Specifically, we derive a central limit theorem for the mean function, as well as root-$n$ uniform convergence rates for other model components, including the covariance function, eigenfunctions, and functional principal component scores. Our applications include a novel framework for the analysis of longitudinal compositional data, achieved by mapping longitudinal compositional data to trajectories on the sphere, illustrated with longitudinal fruit fly behavior patterns. RFPCA is shown to be superior in terms of trajectory recovery in comparison to an unrestricted functional principal component analysis in applications and simulations and is also found to produce principal component scores that are better predictors for classification compared to traditional functional functional principal component scores.

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arXiv2017

Derivative Principal Component Analysis for Representing the Time Dynamics of Longitudinal and Functional Data

We propose a nonparametric method to explicitly model and represent the derivatives of smooth underlying trajectories for longitudinal data. This representation is based on a direct Karhunen--Loève expansion of the unobserved derivatives and leads to the notion of derivative principal component analysis, which complements functional principal component analysis, one of the most popular tools of functional data analysis. The proposed derivative principal component scores can be obtained for irregularly spaced and sparsely observed longitudinal data, as typically encountered in biomedical studies, as well as for functional data which are densely measured. Novel consistency results and asymptotic convergence rates for the proposed estimates of the derivative principal component scores and other components of the model are derived under a unified scheme for sparse or dense observations and mild conditions. We compare the proposed representations for derivatives with alternative approaches in simulation settings and also in a wallaby growth curve application. It emerges that representations using the proposed derivative principal component analysis recover the underlying derivatives more accurately compared to principal component analysis-based approaches especially in settings where the functional data are represented with only a very small number of components or are densely sampled. In a second wheat spectra classification example, derivative principal component scores were found to be more predictive for the protein content of wheat than the conventional functional principal component scores.

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arXiv2017

Dynamic Modeling with Conditional Quantile Trajectories for Longitudinal Snippet Data, with Application to Cognitive Decline of Alzheimer's Patients

Longitudinal data are often plagued with sparsity of time points where measurements are available. The functional data analysis perspective has been shown to provide an effective and flexible approach to address this problem for the case where measurements are sparse but their times are randomly distributed over an interval. Here we focus on a different scenario where available data can be characterized as snippets, which are very short stretches of longitudinal measurements. For each subject the stretch of available data is much shorter than the time frame of interest, a common occurrence in accelerated longitudinal studies. An added challenge is introduced if a time proxy that is basic for usual longitudinal modeling is not available. This situation arises in the case of Alzheimer's disease and comparable scenarios, where one is interested in time dynamics of declining performance, but the time of disease onset is unknown and the chronological age does not provide a meaningful time reference for longitudinal modeling. Our main methodological contribution is to address this problem with a novel approach. Key quantities for our approach are conditional quantile trajectories for monotonic processes that emerge as solutions of a dynamic system, and for which we obtain uniformly consistent estimates. These trajectories are shown to be useful to describe processes that quantify deterioration over time, such as hippocampal volumes in Alzheimer's patients.

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