frechet regression for random objects with euclidean predictors increasingly statisticians are faced with the task of analyzing complex data that are non euclidean and specifically do not lie in a vector space to address the need for statistical methods for such data we introduce the concept of frechet regression this is a general approach to regression when responses are complex random objects in a metric space and predictors are in mathcal r p achieved by extending the classical concept of a frechet mean to the notion of a conditional frechet mean we develop generalized versions of both global least squares regression and local weighted least squares smoothing the target quantities are appropriately defined population versions of global and local regression for response objects in a metric space we derive asymptotic rates of convergence for the corresponding fitted regressions using observed data to the population targets under suitable regularity conditions by applying empirical process methods for the special case of random objects that reside in a hilbert space such as regression models with vector predictors and functional data as responses we obtain a limit distribution the proposed methods have broad applicability illustrative examples include responses that consist of probability distributions and correlation matrices and we demonstrate both global and local frechet regression for demographic and brain imaging data local frechet regression is also illustrated via a simulation with response data which lie on the sphere

functional variance processes we introduce the notion of a functional variance process to quantify variation in functional data the functional data are modeled as samples of smooth random trajectories observed under additive noise the noise is assumed to be composed of white noise and a smooth random process the functional variance process which gives rise to smooth random trajectories of variance the functional variance process is a tool for analyzing stochastic time trends in noise variance as a smooth random process it can be characterized by the eigenfunctions and eigenvalues of its autocovariance operator we develop methods to estimate these characteristics from the data applying concepts from functional data analysis to the residuals obtained after an initial smoothing step asymptotic justifications for the proposed estimates are provided the proposed functional variance process extends the concept of a variance function an established tool in nonparametric and semiparametric regression analysis to the case of functio

modeling repeated functional observations we introduce a new methodological framework for repeatedly observed and thus dependent functional data aiming at situations where curves are recorded repeatedly for each subject in a sample our methodology covers the case where the recordings of the curves are scheduled on a regular and dense grid and also situations more typical for longitudinal studies where the timing of recordings is often sparse and random the proposed models lead to an interpretable and straightforward decomposition of the inherent variation in repeatedly observed functional data and are implemented through a straightforward two step functional principal component analysis we provide consistency results and asymptotic convergence rates for the estimated model components we compare the proposed model with an alternative approach via a two dimensional karhunen loeve expansion and illustrate it through the analysis of longitudinal mortality data from period lifetables that are repeatedly observed for a sample of countries over ma

functional clustering by bayesian wavelet methods summary we propose a nonparametric bayes wavelet model for clustering of functional data the wavelet based methodology is aimed at the resolution of generic global and local features during clustering and is suitable for clustering high dimensional data based on the dirichlet process the nonparametric bayes model extends the scope of traditional bayes wavelet methods to functional clustering and allows the elicitation of prior belief about the regularity of the functions and the number of clusters by suitably mixing the dirichlet processes posterior inference is carried out by gibbs sampling with conjugate priors which makes the computation straightforward we use simulated as well as real data sets to illustrate the suitability of the approach over other alternatives

frechet integration and adaptive metric selection for interpretable covariances of multivariate functional data summary for multivariate functional data recorded for a sample of subjects on a common domain one 10 is often interested in the covariance between pairs of the component functions extending the notion of a covariance matrix for multivariate data to the functional case a straightforward approach is to integrate the pointwise covariance matrices over the functional time domain we generalize this approach by defining the frintegral which depends on the metric chosen for the space of covariance matrices and demonstrate that the ordinary integration is a special 15 case when the frobenius metric is used as the space of covariance matrices is nonlinear we propose a class of power metrics as alternatives to the frobenius metric for any such power metric the calculation of frintegrals is equivalent to transforming the covariance matrices with the chosen power applying the classical riemann integral to the transformed matrices and finally applying the inverse transformation to return to the original scale we also propose data 20 adaptive metric selection with respect to a user specified target criterion for example fastest decline of the eigenvalues establish consistency of the proposed procedures and demonstrate their effectiveness in a simulation the proposed functional covariance approach through fr echet integration is illustrated in a comparison of connectivity between brain voxels for normal subjects and alzheimer s patients based on fmri data 25

joint modelling of paired sparse functional data using principal components we propose a modelling framework to study the relationship between two paired longitudinally observed variables the data for each variable are viewed as smooth curves measured at discrete time points plus random errors while the curves for each variable are summarized using a few important principal components the association of the two longitudinal variables is modelled through the association of the principal component scores we use penalized splines to model the mean curves and the principal component curves and cast the proposed model into a mixed effects model framework for model fitting prediction and inference the proposed method can be applied in the difficult case in which the measurement times are irregular and sparse and may differ widely across individuals use of functional principal components enhances model interpretation and improves statistical and numerical stability of the parameter estimates

geodesic regression and the theory of least squares on riemannian manifolds this paper develops the theory of geodesic regression and least squares estimation on riemannian manifolds geodesic regression is a method for finding the relationship between a real valued independent variable and a manifold valued dependent random variable where this relationship is modeled as a geodesic curve on the manifold least squares estimation is formulated intrinsically as a minimization of the sum of squared geodesic distances of the data to the estimated model geodesic regression is a direct generalization of linear regression to the manifold setting and it provides a simple parameterization of the estimated relationship as an initial point and velocity analogous to the intercept and slope a nonparametric permutation test for determining the significance of the trend is also given for the case of symmetric spaces two main theoretical results are established first conditions for existence and uniqueness of the least squares problem are provided second a maximum likelihood criteria is developed for a suitable definition of gaussian errors on the manifold while the method can be generally applied to data on any manifold specific examples are given for a set of synthetically generated rotation data and an application to analyzing shape changes in the corpus callosum due to age

measuring and testing dependence by correlation of distances distance correlation is a new measure of dependence between random vectors distance covariance and distance correlation are analogous to product moment covariance and correlation but unlike the classical definition of correlation distance correlation is zero only if the random vectors are independent the empirical distance dependence measures are based on certain euclidean distances between sample elements rather than sample moments yet have a compact representation analogous to the classical covariance and correlation asymptotic properties and applications in testing independence are discussed implementation of the test and monte carlo results are also presented

elastic functional coding of riemannian trajectories visual observations of dynamic phenomena such as human actions are often represented as sequences of smoothly varying features in cases where the feature spaces can be structured as riemannian manifolds the corresponding representations become trajectories on manifolds analysis of these trajectories is challenging due to non linearity of underlying spaces and high dimensionality of trajectories in vision problems given the nature of physical systems involved these phenomena are better characterized on a low dimensional manifold compared to the space of riemannian trajectories for instance if one does not impose physical constraints of the human body in data involving human action analysis the resulting representation space will have highly redundant features learning an effective low dimensional embedding for action representations will have a huge impact in the areas of search and retrieval visualization learning and recognition traditional manifold learning addresses this problem for static points in the euclidean space but its extension to riemannian trajectories is non trivial and remains unexplored the difficulty lies in inherent non linearity of the domain and temporal variability of actions that can distort any traditional metric between trajectories to overcome these issues we use the framework based on transported square root velocity fields tsrvf this framework has several desirable properties including a rate invariant metric and vector space representations we propose to learn an embedding such that each action trajectory is mapped to a single point in a low dimensional euclidean space and the trajectories that differ only in temporal rates map to the same point we utilize the tsrvf representation and accompanying statistical summaries of riemannian trajectories to extend existing coding methods such as pca ksvd and label consistent ksvd to riemannian trajectories or more generally to riemannian functions we show that such coding efficiently captures trajectories in applications such as action recognition stroke rehabilitation visual speech recognition clustering and diverse sequence sampling using this framework we obtain state of the art recognition results while reducing the dimensionality complexity by a factor of 100 250 times since these mappings and codes are invertible they can also be used to interactively visualize riemannian trajectories and synthesize actions

a test of weak separability for multi way functional data with application to brain connectivity studies this paper concerns the modeling of multi way functional data where double or multiple indices are involved we introduce a concept of weak separability the weakly separable structure supports the use of factorization methods that decompose the signal into its spatial and temporal components the analysis reveals interesting connections to the usual strongly separable covariance structure and provides insights into tensor methods for multi way functional data we propose a formal test for the weak separability hypothesis where the asymptotic null distribution of the test statistic is a chi square type mixture the method is applied to study brain functional connectivity derived from source localized magnetoencephalography signals during motor tasks

weak curvature conditions and functional inequalities we give sufficient conditions for a measured length space x d to admit local and global poincare inequalities along with a sobolev inequality we first introduce a condition dm on x d d ef ined in terms of transport of measures we show that dm together with a doubling condition on implies a scaleinvariant local poincare inequality we show that if x d has nonnegative n ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies dm with constant 2 n t he condition dm is preserved by measured gromov hausdorff limits we then prove a sobolev inequality for measured length spaces with n ricci curvature bounded below by k 0 finally we derive a sharp global poincare inequality

multivariate functional halfspace depth this article defines and studies a depth for multivariate functional data by the multivariate nature and by including a weight function it acknowledges important characteristics of functional data namely differences in the amount of local amplitude shape and phase variation we study both population and finite sample versions the multivariate sample of curves may include warping functions derivatives and integrals of the original curves for a better overall representation of the functional data via the depth we present a simulation study and data examples that confirm the good performance of this depth function supplementary materials for this article are available online

pseudomonotone and copositive star matrices abstract we study general and complementarity properties of matrices which are either pseudomonotone or copositive star on a closed convex cone in particular we show that if a matrix is pseudomonotone on the nonnegative orthant of r n then it belongs to p 0 q 0 we also show that if a matrix t is copositive star on a closed convex cone k then the linear complementarity problem lcp t k q is solvable for all q in r n iff zero is the only solution of the homogeneous problem lcp t k 0 we characterize pseudomonotonicity of a matrix t by the nonnegativity of the quadratic form tx x on a certain set

regression models on riemannian symmetric spaces the aim of this paper is to develop a general regression framework for the analysis of manifold valued response in a riemannian symmetric space rss and its association with multiple covariates of interest such as age or gender in euclidean space such rss valued data arises frequently in medical imaging surface modeling and computer vision among many others we develop an intrinsic regression model solely based on an intrinsic conditional moment assumption avoiding specifying any parametric distribution in rss we propose various link functions to map from the euclidean space of multiple covariates to the rss of responses we develop a two stage procedure to calculate the parameter estimates and determine their asymptotic distributions we construct the wald and geodesic test statistics to test hypotheses of unknown parameters we systematically investigate the geometric invariant property of these estimates and test statistics simulation studies and a real data analysis are used to evaluate the finite sample properties of our methods

optimal bayes classifiers for functional data and density ratios bayes classifiers for functional data pose a challenge this is because probability density functions do not exist for functional data as a consequence the classical bayes classifier using density quotients needs to be modified we propose to use density ratios of projections on a sequence of eigenfunctions that are common to the groups to be classified the density ratios can then be factored into density ratios of individual functional principal components whence the classification problem is reduced to a sequence of nonparametric one dimensional density estimates this is an extension to functional data of some of the very earliest nonparametric bayes classifiers that were based on simple density ratios in the one dimensional case by means of the factorization of the density quotients the curse of dimensionality that would otherwise severely affect bayes classifiers for functional data can be avoided we demonstrate that in the case of gaussian functional data the proposed functional bayes classifier reduces to a functional version of the classical quadratic discriminant a study of the asymptotic behavior of the proposed classifiers in the large sample limit shows that under certain conditions the misclassification rate converges to zero a phenomenon that has been referred to as perfect classification the proposed classifiers also perform favorably in finite sample applications as we demonstrate in comparisons with other functional classifiers in simulations and various data applications including wine spectral data functional magnetic resonance imaging fmri data for attention deficit hyperactivity disorder adhd patients and yeast gene expression data

extrinsic local regression on manifold valued data we propose an extrinsic regression framework for modeling data with manifold valued responses and euclidean predictors regression with manifold responses has wide applications in shape analysis neuroscience medical imaging and many other areas our approach embeds the manifold where the responses lie onto a higher dimensional euclidean space obtains a local regression estimate in that space and then projects this estimate back onto the image of the manifold outside the regression setting both intrinsic and extrinsic approaches have been proposed for modeling i i d manifold valued data however to our knowledge our work is the first to take an extrinsic approach to the regression problem the proposed extrinsic regression framework is general computationally efficient and theoretically appealing asymptotic distributions and convergence rates of the extrinsic regression estimates are derived and a large class of examples are considered indicating the wide applicability of our approach

a geometric approach to confidence regions and bands for functional parameters summary functional data analysis is now a well established discipline of statistics with its core concepts and perspectives in place despite this there are still fundamental statistical questions which have received relatively little attention one of these is the systematic construction of confidence regions for functional parameters this work is concerned with developing understanding and visualizing such regions we provide a general strategy for constructing confidence regions in a real separable hilbert space by using hyperellipsoids and hyper rectangles we then propose specific implementations which work especially well in practice they provide powerful hypothesis tests and useful visualization tools without relying on simulation we also demonstrate the negative result that nearly all regions including our own have zero coverage when working with empirical covariances to overcome this challenge we propose a new paradigm for evaluating confidence regions by showing that the distance between an estimated region and the desired region with proper coverage tends to 0 faster than the regions shrink to a point we call this phenomena ghosting and refer to the empirical regions as ghost regions we illustrate the proposed methods in a simulation study and an application to fractional anisotropy tract profile data

tests for separability in nonparametric covariance operators of random surfaces we consider the problem of testing for separability in nonparametric covariance operators of multidimensional functional data is considered we cast the problem in a tensor product of hilbert space framework where the role of the partial trace operator is emphasized and the tests proposed are computationally tractable an applications to acoustic phonetic data is also presented

a qualitative mathematical analysis of a class of linear variational inequalities via semi complementarity problems applications in electronics the main object of this paper is to present a general mathematical theory applicable to the study of a large class of linear variational inequalities arising in electronics our approach uses recession tools so as to define a new class of problems that we call semi complementarity problems then we show that the study of semi complementarity problems can be used to prove new qualitative results applicable to the study of linear variational inequalities of the second kind

functional data analysis for sparse longitudinal data we propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data the method aims at irregularly spaced longitudinal data where the number of repeated measurements available per subject is small in contrast classical functional data analysis requires a large number of regularly spaced measurements per subject we assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random subject specific trajectory plus measurement errors basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories and the estimation of the variance of the measurement errors the eigenfunction basis is estimated from the data and functional principal components score estimates are obtained by a conditioning step this conditional estimation method is conceptually simple and straightforward to implement

distances and inference for covariance operators a framework is developed for inference concerning the covariance operator of a functional random process where the covariance operator itself is an object of interest for statistical analysis distances for comparing positive definite covariance matrices are either extended or shown to be inapplicable to functional data in particular an infinite dimensional analogue of the procrustes size and shape distance is developed convergence of finite dimensional approximations to the infinite dimensional distance metrics is also shown for inference a frechet estimator of both the covariance operator itself and the average covariance operator is introduced a permutation procedure to test the equality of the covariance operators between two groups is also considered additionally the use of such distances for extrapolation to make predictions is explored as an example of the proposed methodology the use of covariance operators has been suggested in a philological study of cross linguistic dependence as a way to incorporate quantitative phonetic information it is shown that distances between languages derived from phonetic covariance functions can provide insight into the relationships between the romance languages

functional limit theorems for u processes un processus u est une collection de statistiques u indicees par une famille de noyaux symetriques on utilise la theorie limite classique pour demontrer pour le processus u des analogues du theoreme central limite pour les processus empiriques

strong consistency of k means clustering a random sample is divided into the k clusters that minimise the within cluster sum of squares conditions are found that ensure the almost sure convergence as the sample size increases of the set of means of the k clusters the result is proved for a more general clustering criterion

properties of principal component methods for functional and longitudinal data analysis the use of principal component methods to analyze functional data is appropriate in a wide range of different settings in studies of functional data analysis it has often been assumed that a sample of random functions is observed precisely in the continuum and without noise while this has been the traditional setting for functional data analysis in the context of longitudinal data analysis a random function typically represents a patient or subject who is observed at only a small number of randomly distributed points with nonnegligible measurement error nevertheless essentially the same methods can be used in both these cases as well as in the vast number of settings that lie between them how is performance affected by the sampling plan in this paper we answer that question we show that if there is a sample of n functions or subjects then estimation of eigenvalues is a semiparametric problem with root n consistent estimators even if only a few observations are made of each function and if each observation is encumbered by noise however estimation of eigenfunctions becomes a nonparametric problem when observations are sparse the optimal convergence rates in this case are those which pertain to more familiar function estimation settings we also describe the effects of sampling at regularly spaced points as opposed to random points in particular it is shown that there are often advantages in sampling randomly however even in the case of noisy data there is a threshold sampling rate depending on the number of functions treated above which the rate of sampling either randomly or regularly has negligible impact on estimator performance no matter whether eigenfunctions or eigenvectors are being estimated

functional linear regression analysis for longitudinal data we propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data where both the predictor and response are functions of a covariate such as time predictor and response processes have smooth random trajectories and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects in longitudinal studies the number of repeated measurements per subject is often small and may be modeled as a discrete random number and accordingly only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit we propose a functional regression approach for this situation using functional principal component analysis where we estimate the functional principal component scores through conditional expectations this allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory the resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories asymptotic properties for a sample of n subjects are investigated under mild conditions as n oo and we obtain consistent estimation for the regression function besides convergence results for the components of functional linear regression such as the regression parameter function we construct asymptotic pointwise confidence bands for the predicted trajectories a functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced extending the standard r 2 to the functional case the proposed methods are illustrated with a simulation study longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index

functional models for time varying random objects functional data analysis provides a popular toolbox of functional models for the analysis of samples of random functions that are real valued 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 we propose metric covariance a novel association measure for paired object data lying in a metric space d that we use to define a metric autocovariance function for a sample of random valued curves where generally will not have a vector space or manifold structure the proposed metric autocovariance function is non negative definite when the squared semimetric d2 is of negative type then the eigenfunctions of the linear operator with the autocovariance function as kernel can be used as building blocks for an object functional principal component analysis for 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 frechet means and projections of distance functions of the random object trajectories in the directions of the eigenfunctions leading to real valued frechet scores using the notion of generalized frechet integrals we construct object functional principal components that lie in the metric space we establish asymptotic consistency of the sample based estimators for the corresponding population targets under mild metric entropy conditions on and continuity of the valued random curves these concepts are illustrated with samples of time varying probability distributions for human mortality time varying covariance matrices derived from trading patterns and time varying networks that arise from new york taxi trips

piecewise smoothness local invertibility and parametric analysis of normal maps this paper is concerned with properties of the euclidean projection map onto a convex set defined by finitely many smooth convex inequalities and affine equalities under a constant rank constraint qualification we show that the projection map is piecewise smooth pc1 hence bouligand differentiable or directionally differentiable and a relatively simple formula is given for the b derivative these properties of the projection map are used to obtain inverse and implicit function theorems for associated normal maps using a new characterization of invertibility of a pc1 function in terms of its b derivative an extension of the implicit function theorem which does not require local uniqueness is also presented degree theory plays a major role in the analysis of both the locally unique case and its extension

dynamical correlation for multivariate longitudinal data nonparametric methodology for longitudinal data analysis is becoming increasingly popular the analysis of multivariate longitudinal data where data on several time courses are recorded for each subject has received considerably less attention despite its importance for practical data analysis in particular there is a need for measures and estimates to capture dependency between the components of vector valued longitudinal data we propose and analyze a simple and effective nonparametric method to quantify the covariation of components of multivariate longitudinal observations which are viewed as realizations of a random process this includes the notion of a correlation between derivatives and time shifted versions the concept of dynamical correlation is based on a scalar product obtained from pairs of standardized smoothed curves the proposed method can be used when observation times are irregular and not matching between subjects or between responses within a subject for higher dimensional dat

clustering for sparsely sampled functional data we develop a flexible model based procedure for clustering functional data the technique can be applied to all types of curve data but is particularly useful when individuals are observed at a sparse set of time points in addition to producing final cluster assignments the procedure generates predictions and confidence intervals for missing portions of curves our approach also provides many useful tools for evaluating the resulting models clustering can be assessed visually via low dimensional representations of the curves and the regions of greatest separation between clusters can be determined using a discriminant function finally we extend the model to handle multiple functional and finite dimensional covariates and show how it can be applied to standard finite dimensional clustering problems involving missing data

comparing the shapes of regression functions summary does a regression function follow a specified shape do two regression functions have the same shape how can regression functions be grouped based on shape these questions can occur when investigating monotonicity when counting local maxima or when studying variation in families of curves one can address these questions by considering the rank correlation coefficient between two functions this correlation is a generalisation of the rank correlation between two finite sets of numbers and is equal to one if and only if the two functions have the same shape a sample rank correlation based on smoothed estimates of the regression functions consistently estimates the true correlation this sample rank correlation can be used as a measure of similarity between functions in cluster analysis and as a measure of monotonicity or modality

complementarity problems over locally compact cones this paper proves an existence result for a complementarily problem on a locally convex space where the mapping is copositive positive homogeneous and of monotone type on a locally compact cone a perturbation theorem is proved that extends a result of mangasarian and doverspike proved for n times n matrices on the nonnegative orthant

limit theorems for u processes a u process is a collection of u statistics concretely a u process over a family h of kernels h of m variables based on a probability measure p on s s is the collection u n m h p h in mathcal h of u statistics this chapter is devoted to the asymptotic theory of u processes we are interested in finding conditions on h and p ensuring that the law of large numbers or the central limit theorem or the law of the iterated logarithm hold for u n m h p uniformly in h h the theory of empirical processes deals with the same questions for the case m 1 and u process theory is patterned after it this is a relatively new subject at least in the generality presented here h being an arbitrary collection of kernels which are defined on a general measurable space there is therefore a need to indicate its usefulness to this effect a section on applications is added at the end of the chapter section 5 5 there we illustrate the use of each of the main theorems in this chapter by deriving properties of certain multidimensional generalizations of the median and in general of m estimators and by studying estimators of the cumulative hazard and distribution functions of a random variable based on truncated data

a cone programming approach to the bilinear matrix inequality problem and its geometry we discuss an approach for solving the bilinear matrix inequality bmi based on its connections with certain problems defined over matrix cones these problems are among others the cone generalization of the linear programming lp and the linear complementarity problem lcp referred to as the cone lp and the cone lcp respectively specifically we show that solving a given bmi is equivalent to examining the solution set of a suitably constructed cone lp or cone lcp this approach facilitates our understanding of the geometry of the bmi and opens up new avenues for the development of the computational procedures for its solution

the linear complementarity problem under asymptotic analysis in this work we study the classical linear complementarity problem lcp by describing the asymptotic behavior of the approximate solutions to its variational inequality formulation thus some properties satisfied by the directions which are limits of the normalized unbounded approximate solutions will be established based on this analysis various equivalent conditions guaranteeing the existence of solutions to lcp are given in particular the sufficient condition of gowda and pang expressed in terms of the solutions to augmented linear complementarity problems is written in a way that is more easily verifiable our approach allows us to deal with garca a matrices semimonotone copositive q pseudomonotone matrices among others in a unified framework furthermore we introduce a larger class of matrices for which many of the results including a sensitivity one due to gowda and pang are still valid in addition some conditions ensuring the boundedness of the solution set are also provided and some estimates for the asymptotic cone of the solution set for different classes of matrices are given as well hence the present approach sheds new light and offers an alternative to view classical results

clustering high dimensional time series based on parallelism this article considers the problem of clustering high dimensional time series based on trend parallelism the underlying process is modeled as a nonparametric trend function contaminated by locally stationary errors a special class of nonstationary processes for each group where the parallelism holds i semiparametrically estimate its representative trend function and vertical shifts of group members and establish their central limit theorems an information criterion consisting of in group similarities and number of groups is then proposed for the purpose of clustering i prove its theoretical consistency and propose a splitting coalescence algorithm to reduce the computational burden in practice the method is illustrated by both simulation and a real data example

on the volume growth of alexandrov spaces with nonnegative curvature abstract recently zhang and zhu introduced a new definition for ricci curvature bounded below on alexandrov spaces in this paper we give a volume growth lower bound estimate of alexandrov spaces with nonnegative ricci curvature in the sense of zhang zhu which extends yau s result from riemannian manifolds to alexandrov spaces

u processes rates of convergence on introduit un nouveau processus stochastique une collection de statistiques u indicees par une famille de noyaux symetriques on obtient des conditions pour la convergence presque sure uniforme d une suite de tels processus

functional quasi likelihood regression models with smooth random effects summary we propose a class of semiparametric functional regression models to describe the influence of vector valued covariates on a sample of response curves each observed curve is viewed as the realization of a random process composed of an overall mean function and random components the finite dimensional covariates influence the random components of the eigenfunction expansion through single index models that include unknown smooth link and variance functions the parametric components of the single index models are estimated via quasi score estimating equations with link and variance functions being estimated nonparametrically we obtain several basic asymptotic results the functional regression models proposed are illustrated with the analysis of a data set consisting of egg laying curves for 1000 female mediterranean fruit flies medflies

local polynomial regression for symmetric positive definite matrices local polynomial regression has received extensive attention for the nonparametric estimation of regression functions when both the response and the covariate are in euclidean space however little has been done when the response is in a riemannian manifold we develop an intrinsic local polynomial regression estimate for the analysis of symmetric positive definite spd matrices as responses that lie in a riemannian manifold with covariate in euclidean space the primary motivation and application of the proposed methodology is in computer vision and medical imaging we examine two commonly used metrics including the trace metric and the log euclidean metric on the space of spd matrices for each metric we develop a cross validation bandwidth selection method derive the asymptotic bias variance and normality of the intrinsic local constant and local linear estimators and compare their asymptotic mean square errors simulation studies are further used to compare the estimators under the two metrics and to examine their finite sample performance we use our method to detect diagnostic differences between diffusion tensors along fiber tracts in a study of human immunodeficiency virus

functional data analysis for density functions by transformation to a hilbert space functional data that are nonnegative and have a constrained integral can be considered as samples of one dimensional density functions such data are ubiquitous due to the inherent constraints densities do not live in a vector space and therefore commonly used hilbert space based methods of functional data analysis are not applicable to address this problem we introduce a transformation approach mapping probability densities to a hilbert space of functions through a continuous and invertible map basic methods of functional data analysis such as the construction of functional modes of variation functional regression or classification are then implemented by using representations of the densities in this linear space representations of the densities themselves are obtained by applying the inverse map from the linear functional space to the density space transformations of interest include log quantile density and log hazard transformations among others rates of convergence are derived for the representations that are obtained for a general class of transformations under certain structural properties if the subject specific densities need to be estimated from data these rates correspond to the optimal rates of convergence for density estimation the proposed methods are illustrated through simulations and applications in brain imaging

stability analysis of variational inequalities and nonlinear complementarity problems via the mixed linear complementarity problem and degree theory this paper is concerned with the mixed linear complementarity problem and the role it and its variants play in the stability analysis of the nonlinear complementarity problem and the karush kuhn tucker system of a variational inequality problem under a nonsingular assumption the mixed linear complementarity problem can be converted to the standard problem in this case the rich theory of the latter can be directly applied to the former in this work we employ degree theory to derive some sufficient conditions for the existence of a solution to the mixed linear complementarity problem in the absence of the nonsingularity property next we extend this existence theory to the mixed nonlinear complementarity problem and establish a main stability result under a certain degree theoretic assumption concerning the linearized problem we then specialize this stability result and its consequences to the parametric variational inequality problem under the assumption of a unique set of multipliers finally we

cats clustering after transformation and smoothing cats clustering after transformation and smoothing is a technique for nonparametrically estimating and clustering a large number of curves our motivating example is a genetic microarray experiment but the method is very general the method includes transformation and smoothing multiple curves multiple nonparametric testing for screening out flat curves clustering curves with similar shape and nonparametrically inferring the clustering estimation error rate

principal modes of variation for processes with continuous sample curves analysis of a process with continuous sample curves can be carried out in a manner similar to principal components analysis of vector processes by appropriate definition of a best linear model in the continuous case we show that principal modes of variation consist of eigenfunctions of the process covariance function c s t procedures for estimation of these eigenfunctions from a finite sample of observed curves are given and results are compared with principal components analysis of the same data

distance covariance in metric spaces we extend the theory of distance brownian covariance from euclidean spaces where it was introduced by szekely rizzo and bakirov to general metric spaces we show that for testing independence it is necessary and sufficient that the metric space be of strong negative type in particular we show that this holds for separable hilbert spaces which answers a question of kosorok instead of the manipulations of fourier transforms used in the original work we use elementary inequalities for metric spaces and embeddings in hilbert spaces

frechet analysis of variance for random objects summaryfrechet mean and variance provide a way of obtaining a mean and variance for 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 data objects include covariance matrices graph laplacians of networks and univariate probability distribution functions we derive a central limit theorem for the frechet variance under mild regularity conditions using empirical process theory and also provide a consistent estimator of the asymptotic variance these results lead to a test for comparing k populations of metric space valued data objects in terms of frechet means and variances we examine the finite sample performance of this novel inference procedure through simulation studies on several special cases that include probability distributions and graph laplacians leading to a test for comparing populations of networks the proposed approach has good finite sample performance in simulations for different kinds of random objects we illustrate the proposed methods by analysing data on mortality profiles of various countries and resting state functional magnetic resonance imaging data

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

functional clustering and identifying substructures of longitudinal data summary a functional clustering fc method k centres fc for longitudinal data is proposed the k centres fc approach accounts for both the means and the modes of variation differentials between clusters by predicting cluster membership with a reclassification step the cluster membership predictions are based on a non parametric random effect model of the truncated karhunen loeve expansion coupled with a non parametric iterative mean and covariance updating scheme we show that under the identifiability conditions derived the k centres fc method proposed can greatly improve cluster quality as compared with conventional clustering algorithms moreover by exploring the mean and covariance functions of each cluster thek centres fc method provides an additional insight into cluster structures which facilitates functional cluster analysis practical performance of the k centres fc method is demonstrated through simulation studies and data applications including growth curve and gene expression profile data

principal component analysis for functional data on riemannian manifolds and spheres author s dai xiongtao muller hans georg abstract functional data analysis on nonlinear manifolds has drawn recent interest we propose an intrinsic principal component analysis for smooth riemannian manifold valued functional data and study its asymptotic properties the proposed 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 fr x27 echet 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 by mapping back 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 we derive uniform convergence rates for the model components including the mean function covariance function eigenfunctions and functional principal component scores and apply the proposed methodology to obtain a novel framework for the analysis of longitudinal compositional data this is achieved by mapping longitudinal compositional data to trajectories on the sphere and we illustrate this approach with longitudinal fruit fly behavior patterns the proposed riemannian functional principal component analysis is shown to be superior in terms of trajectory recovery in comparison to an unrestricted functional principal component analysis in applications that include flight trajectories as well as in simulations rfpca is also found to produce principal component scores that are better predictors for classification compared to traditional functional functional principal component scores

modelling function valued stochastic processes with applications to fertility dynamics summary we introduce a simple and interpretable model for functional data analysis for situations where the observations at each location are functional rather than scalar this new approach is based on a tensor product representation of the function valued process and utilizes eigenfunctions of marginal kernels the resulting marginal principal components and product principal components are shown to have nice properties given a sample of independent realizations of the underlying function valued stochastic process we propose straightforward fitting methods to obtain the components of this model and to establish asymptotic consistency and rates of convergence for the estimates proposed the methods are illustrated by modelling the dynamics of annual fertility profile functions for 17 countries this analysis demonstrates that the approach proposed leads to insightful interpretations of the model components and interesting conclusions

testing separability of space time functional processes summaryseparability is a common simplifying assumption on the covariance structure of spatiotemporal functional data we present three tests of separability one a functional extension of the monte carlo likelihood method of mitchell et al 2006 and two based on quadratic forms our tests are based on asymptotic distributions of maximum likelihood estimators and do not require monte carlo simulation the main theoretical contribution of this paper is the specification of the joint asymptotic distribution of these estimators which can be used to derive many other tests the main methodological finding is that one of the quadratic form methods which we call a norm approach emerges as a clear winner in terms of finite sample performance in nearly every setting we considered this approach focuses directly on the frobenius distance between the spatiotemporal covariance function and its separable approximation we demonstrate the efficacy of our methods via simulations and application to irish wind data

alexandrov spaces with large volume growth abstract let x d be an n dimensional alexandrov space whose hausdorff measure h n satisfies a condition giving the metric measure space x d h n a notion of having nonnegative ricci curvature we examine the influence of large volume growth on these spaces and generalize some classical arguments from riemannian geometry showing that when the volume growth is sufficiently large then x d h n has finite topological type

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

equivalence of distance based and rkhs based statistics in hypothesis testing we provide a unifying framework linking two classes of statistics used in two sample and independence testing on the one hand the energy distances and distance covariances from the statistics literature on the other maximum mean discrepancies mmd that is distances between embeddings of distributions to reproducing kernel hilbert spaces rkhs as established in machine learning in the case where the energy distance is computed with a semimetric of negative type a positive definite kernel termed distance kernel may be defined such that the mmd corresponds exactly to the energy distance conversely for any positive definite kernel we can interpret the mmd as energy distance with respect to some negative type semimetric this equivalence readily extends to distance covariance using kernels on the product space we determine the class of probability distributions for which the test statistics are consistent against all alternatives finally we investigate the performance of the family of distance kernels in two sample and independence tests we show in particular that the energy distance most commonly employed in statistics is just one member of a parametric family of kernels and that other choices from this family can yield more powerful tests