two step estimation of functional linear models with applications to longitudinal data functional linear models are useful in longitudinal data analysis they include many classical and recently proposed statistical models for longitudinal data and other functional data recently smoothing spline and kernel methods have been proposed for estimating their coefficient functions nonparametrically but these methods are either intensive in computation or inefficient in performance to overcome these drawbacks in this paper a simple and powerful two step alternative is proposed in particular the implementation of the proposed approach via local polynomial smoothing is discussed methods for estimating standard deviations of estimated coefficient functions are also proposed some asymptotic results for the local polynomial estimators are established two longitudinal data sets one of which involves time dependent covariates are used to demonstrate the approach proposed simulation studies show that our two step approach improves the kernel method proposed by hoover and co workers in several aspects such as accuracy computational time and visual appeal of the estimators

adaptive global testing for functional linear models this article studies global testing of the slope function in functional linear regression models a major challenge in functional global testing is to choose the dimension of projection when approximating the functional regression model by a finite dimensional multivariate linear regression model we develop a new method that simultaneously tests the slope vectors in a sequence of functional principal components regression models the sequence of models being tested is determined by the sample size and is an integral part of the testing procedure our theoretical analysis shows that the proposed method is uniformly powerful over a class of smooth alternatives when the signal to noise ratio exceeds the detection boundary the methods and results reflect the deep connection between the functional linear regression model and the gaussian sequence model we also present an extensive simulation study and a real data example to illustrate the finite sample performance of our method supplementary materials for

estimation of functional derivatives situations of a functional predictor paired with a scalar response are increasingly encountered in data analysis predictors are often appropriately modeled as square integrable smooth random functions imposing minimal assumptions on the nature of the functional relationship we aim to estimate the directional derivatives and gradients of the response with respect to the predictor functions in statistical applications and data analysis functional derivatives provide a quantitative measure of the often intricate relationship between changes in predictor trajectories and those in scalar responses this approach provides a natural extension of classical gradient fields in vector space and provides directions of steepest descent we suggest a kernel based method for the nonparametric estimation of functional derivatives that utilizes the decomposition of the random predictor functions into their eigenfunctions these eigenfunctions define a canonical set of directions into which the gradient field is expanded the proposed method is shown to lead to asymptotically consistent estimates of functional derivatives and is illustrated in an application to growth curves

generalized gaussian process regression model for non gaussian functional data in this article we propose a generalized gaussian process concurrent regression model for functional data where the functional response variable has a binomial poisson or other non gaussian distribution from an exponential family while the covariates are mixed functional and scalar variables the proposed model offers a nonparametric generalized concurrent regression method for functional data with multidimensional covariates and provides a natural framework on modeling common mean structure and covariance structure simultaneously for repeatedly observed functional data the mean structure provides overall information about the observations while the covariance structure can be used to catch up the characteristic of each individual batch the prior specification of covariance kernel enables us to accommodate a wide class of nonlinear models the definition of the model the inference and the implementation as well as its asymptotic properties are discussed several numerical examples with differen

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

necessary conditions for existence of non degenerate hamiltonian structures the necessary criteria are pointed out for the exisence of hamiltonian and bi hamiltonian non degenerate structures for a nonlinear system of partial differential equations of first order the results are formulated in terms of the new invariants of the intrinsic geometry introduced in this paper connected with the nijenhuis and haantjes tensors of a 1 1 tensor field

efficient estimation of the nonparametric mean and covariance functions for longitudinal and sparse functional data abstractwe consider the estimation of mean and covariance functions for longitudinal and sparse functional data by using the full quasi likelihood coupling a modification of the local kernel smooth

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

doubly functional graphical models in high dimensions we consider estimating a functional graphical model from multivariate functional observations in functional data analysis the classical assumption is that each function has been measured over a densely sampled grid however in practice the functions have often been observed with measurement error at a relatively small number of points we propose a class of doubly functional graphical models to capture the evolving conditional dependence relationship among a large number of sparsely or densely sampled functions our approach first implements a nonparametric smoother to perform functional principal components analysis for each curve then estimates a functional covariance matrix and finally computes sparse precision matrices which in turn provide the doubly functional graphical model we derive some novel concentration bounds uniform convergence rates and model selection properties of our estimator for both sparsely and densely sampled functional data in the high dimensional large p small n regime we demonstrate via simulations that the proposed method significantly outperforms possible competitors our proposed method is applied to a brain imaging dataset

approximating fragmented functional data by segments of markov chains we consider curve extension and linear prediction for functional data observed only on a part of their domain in the form of fragments we suggest an approach based on a combination of markov chains and nonparametric smoothing techniques which enables us to extend the observed fragments and construct approximated prediction intervals around them construct mean and covariance function estimators and derive a linear predictor the procedure is illustrated on real and simulated data

functional varying coefficient models for longitudinal data the proposed functional varying coefficient model provides a versatile and flexible analysis tool for relating longitudinal responses to longitudinal predictors specifically this approach provides a novel representation of varying coefficient functions through suitable auto and cross covariances of the underlying stochastic processes which is particularly advantageous for sparse and irregular designs as often encountered in longitudinal studies existing methodology for varying coefficient models is not adapted to such data the proposed approach extends the customary varying coefficient models to a more general setting in which not only current but also recent past values of the predictor time course may have an impact on the current value of the response time course the influence of past predictor values is modeled by a smooth history index function while the effects on the response are described by smooth varying coefficient functions the resulting estimators for varying coefficient and history

interpretable functional principal component analysis functional principal component analysis fpca is a popular approach to explore major sources of variation in a sample of random curves these major sources of variation are represented by functional principal components fpcs the intervals where the values of fpcs are significant are interpreted as where sample curves have major variations however these intervals are often hard for naive users to identify because of the vague definition of significant values in this article we develop a novel penalty based method to derive fpcs that are only nonzero precisely in the intervals where the values of fpcs are significant whence the derived fpcs possess better interpretability than the fpcs derived from existing methods to compute the proposed fpcs we devise an efficient algorithm based on projection deflation techniques we show that the proposed interpretable fpcs are strongly consistent and asymptotically normal under mild conditions simulation studies confirm that with a competitive performance in explaining variations of sample curves the proposed fpcs are more interpretable than the traditional counterparts this advantage is demonstrated by analyzing two real datasets namely electroencephalography data and canadian weather data

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

dynamics of polynomial mappings of c 2 we study the dynamics of polynomial self mappings f of c 2 we construct for a large class of mappings an invariant measure which is mixing and of maximal entropy h f max log d t f log 1 f where d t f is the topological degree of f and 1 f its first dynamical degree to achieve this we look at the meromorphic extensions of f to smooth minimal compactifications of c 2 when a good compactification is found we construct an f invariant green current t which contains many dynamical informations when d t f 1 f 1 the measure is obtained as p dd c vt where v is a partial green function defined on the support of t when 1 t t where t is a globally defined f invariant current

nested hierarchical functional data modeling and inference for the analysis of functional plant phenotypes abstractin a plant science root image study the process of seedling roots bending in response to gravity is recorded using digital cameras and the bending rates are modeled as functional plant phenotype data the functional phenotypes are collected from seeds representing a large variety of genotypes and have a three level nested hierarchical structure with seeds nested in groups nested in genotypes the seeds are imaged on different days of the lunar cycle and an important scientific question is whether there are lunar effects on root bending we allow the mean function of the bending rate to depend on the lunar day and model the phenotypic variation between genotypes groups of seeds imaged together and individual seeds by hierarchical functional random effects we estimate the covariance functions of the functional random effects by a fast penalized tensor product spline approach perform multi level functional principal component analysis fpca using the best linear unbiased predictor of the pri

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

schouten tensor and bi hamiltonian systems of hydrodynamic type the necessary conditions are derived for existence of a bi hamiltonian structure for a given hydrodynamic type system one of the conditions is the vanishing of its haantjes tensor a theorem is proved on the canonical forms of the generic bi hamiltonian systems the schouten 2 1 tensor sijk is connected with any hamiltonian system of hydrodynamic type the complete symmetry of the 3 0 tensor sijk is demonstrated the necessary conditions for existence of a single nondegenerate hamiltonian structure are obtained in terms of the special differential k forms p1 pk u1 uk

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

nijenhuis g manifolds and lenard bicomplexes a new approach to kp systems we suggest a method to extend the theory of recursion operators to integrable hamiltonian systems in two space dimensions like kp systems the approach aims to stress the conceptual unity of the theories in one and two space dimensions a sound explanation of the appearance of bilocal operators is also given

reduction techniques for infinite dimensional hamiltonian systems some ideas and applications in the language of tensor analysis on differentiable manifolds we present a reduction method of integrability structures and apply it to recover some well known hierarchies of integrable nonlinear evolution equations

structured functional additive regression in reproducing kernel hilbert spaces summary functional additive models provide a flexible yet simple framework for regressions involving functional predictors the utilization of a data driven basis in an additive rather than linear structure naturally extends the classical functional linear model however the critical issue of selecting non linear additive components has been less studied in this work we propose a new regularization framework for structure estimation in the context of reproducing kernel hilbert spaces the approach proposed takes advantage of functional principal components which greatly facilitates implementation and theoretical analysis the selection and estimation are achieved by penalized least squares using a penalty which encourages the sparse structure of the additive components theoretical properties such as the rate of convergence are investigated the empirical performance is demonstrated through simulation studies and a real data application

generalized quasi likelihood ratio tests for semiparametric analysis of covariance models in longitudinal data abstractwe model generalized longitudinal data from multiple treatment groups by a class of semiparametric analysis of covariance models which take into account the parametric effects of time dependent covariates and the nonparametric time effects in these models the treatment effects are represented by nonparametric functions of time and we propose a generalized quasi likelihood ratio test procedure to test if these functions are identical our estimation procedure is based on profile estimating equations combined with local linear smoothers we find that the much celebrated wilks phenomenon which is well established for independent data still holds for longitudinal data if a working independence correlation structure is assumed in the test statistic however this property does not hold in general especially when the working variance function is misspecified our empirical study also shows that incorporating correlation into the test statistic does not necessarily improve the power of the test the

joint modeling and clustering paired generalized longitudinal trajectories with application to cocaine abuse treatment data in a cocaine dependence treatment study we have paired binary longitudinal trajectories that record the cocaine use patterns of each patient before and after a treatment to better understand the drug using behaviors among the patients we propose a general framework based on functional data analysis to jointly model and cluster these paired non gaussian longitudinal trajectories our approach assumes that the response variables follow distributions from the exponential family with the canonical parameters determined by some latent gaussian processes to reduce the dimensionality of the latent processes we express them by a truncated karhunen loeve kl expansion allowing the mean and covariance functions to be different across clusters we further represent the mean and eigenfunctions functions by flexible spline bases and determine the orders of the truncated kl expansions using data driven methods by treating the cluster membership as a missing value we cluster the cocaine use trajectories by a l

from sparse to dense functional data and beyond nonparametric estimation of mean and covariance functions is important in functional data analysis we investigate the performance of local linear smoothers for both mean and covariance functions with a general weighing scheme which includes two commonly used schemes equal weight per observation obs and equal weight per subject subj as two special cases we provide a comprehensive analysis of their asymptotic properties on a unified platform for all types of sampling plan be it dense sparse or neither three types of asymptotic properties are investigated in this paper asymptotic normality l 2 convergence and uniform convergence the asymptotic theories are unified on two aspects 1 the weighing scheme is very general 2 the magnitude of the number n i of measurements for the i th subject relative to the sample size n can vary freely based on the relative order of n i to n functional data are partitioned into three types non dense dense and ultra dense functional data for the obs and subj schemes these two weighing schemes are compared both theoretically and numerically we also propose a new class of weighing schemes in terms of a mixture of the obs and subj weights of which theoretical and numerical performances are examined and compared

optimal penalized function on function regression under a reproducing kernel hilbert space framework many scientific studies collect data where the response and predictor variables are both functions of time location or some other covariate understanding the relationship between these functiona

flcrm functional linear cox regression model ummary we consider a functional linear cox regression model for characterizing the association between time to event data and a set of functional and scalar predictors the functional linear cox regression model incorporates a functional principal component analysis for modeling the functional predictors and a high dimensional cox regression model to characterize the joint effects of both functional and scalar predictors on the time to event data we develop an algorithm to calculate the maximum approximate partial likelihood estimates of unknown finite and infinite dimensional parameters we also systematically investigate the rate of convergence of the maximum approximate partial likelihood estimates and a score test statistic for testing the nullity of the slope function associated with the functional predictors we demonstrate our estimation and testing procedures by using simulations and the analysis of the alzheimer s disease neuroimaging initiative adni data our real data analyses show that high dimensional hippocampus surface data may be an important marker for predicting time to conversion to alzheimer s disease data used in the preparation of this article were obtained from the adni database adni loni usc edu

unified empirical likelihood ratio tests for functional concurrent linear models and the phase transition from sparse to dense functional data summary we consider the problem of testing functional constraints in a class of functional concurrent linear models where both the predictors and the response are functional data measured at discrete time points we propose test procedures based on the empirical likelihood with bias corrected estimating equations to conduct both pointwise and simultaneous inferences the asymptotic distributions of the test statistics are derived under the null and local alternative hypotheses where sparse and dense functional data are considered in a unified framework we find a phase transition in the asymptotic null distributions and the orders of detectable alternatives from sparse to dense functional data specifically the tests proposed can detect alternatives of n order when the number of repeated measurements per curve is of an order larger than n 0 with n being the number of curves the transition points 0 for pointwise and simultaneous tests are different and both are smaller than the transition point in the estimation problem simulation studies and real data analyses are conducted to demonstrate the methods proposed

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

tolerance bands for functional data often the object of inference in biomedical applications is a range that brackets a given fraction of individual observations in a population a classical estimate of this range for univariate measurements is a tolerance interval this article develops its natural extension for functional measurements a tolerance band and proposes a methodology for constructing its pointwise and simultaneous versions that incorporates both sparse and dense functional data assuming that the measurements are observed with noise the methodology uses functional principal component analysis in a mixed model framework to represent the measurements and employs bootstrapping to approximate the tolerance factors needed for the bands the proposed bands also account for uncertainty in the principal components decomposition simulations show that the methodology has generally acceptable performance unless the data are quite sparse and unbalanced in which case the bands may be somewhat liberal the methodology is illustrated using two real datasets a sparse dataset involving cd4 cell counts and a dense dataset involving core body temperatures

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

partially linear functional additive models for multivariate functional data we investigate a class of partially linear functional additive models plfam that predicts a scalar response by both parametric effects of a multivariate predictor and nonparametric effects of a m

nonparametric regression analysis of longitudinal data abstract nonparametric methods are developed for estimating the dose effect when a response consists of correlated observations over time measured in a dose response experiment the methods can also be applied to data collected from a completely randomized design experiment methods are developed for the detection and description of the effects of dose time and their interaction the methods allow for individual variation in the timing and number of observations a generalization allowing baseline covariates to be incorporated is addressed these results may be used in an exploratory fashion in the process of building a random effects model for longitudinal data

modelling sparse generalized longitudinal observations with latent gaussian processes summary in longitudinal data analysis one frequently encounters non gaussian data that are repeatedly collected for a sample of individuals over time the repeated observations could be binomial poisson or of another discrete type or could be continuous the timings of the repeated measurements are often sparse and irregular we introduce a latent gaussian process model for such data establishing a connection to functional data analysis the functional methods proposed are non parametric and computationally straightforward as they do not involve a likelihood we develop functional principal components analysis for this situation and demonstrate the prediction of individual trajectories from sparse observations this method can handle missing data and leads to predictions of the functional principal component scores which serve as random effects in this model these scores can then be used for further statistical analysis such as inference regression discriminant analysis or clustering we illustrate these non parametric methods with longitudinal data on primary biliary cirrhosis and show in simulations that they are competitive in comparisons with generalized estimating equations and generalized linear mixed models

a smoothing based goodness of fit test of covariance for functional data functional data methods are often applied to longitudinal data as they provide a more flexible way to capture dependence across repeated observations however there is no formal testing procedure to determine if functional methods are actually necessary we propose a goodness of fit test for comparing parametric covariance functions against general nonparametric alternatives for both irregularly observed longitudinal data and densely observed functional data we consider a smoothing based test statistic and approximate its null distribution using a bootstrap procedure we focus on testing a quadratic polynomial covariance induced by a linear mixed effects model and the method can be used to test any smooth parametric covariance function performance and versatility of the proposed test is illustrated through a simulation study and three data applications

nonparametric estimation of common regressors for similar curve data the paper is concerned with data from a collection of different but related regression curves m j j 1 n n much greater than 1 in statistical practice analysis of such data is most frequently based on low dimensional linear models it is then assumed that each regression curve mj is a linear combination of a small number l much less than n of common functions g 1 g l for example if all m j s are straight lines this holds with l 2 g 1 1 and g 2 x x in this paper the assumption of a prespecified model is dropped a nonparametric method is presented which allows estimation of the smallest l and corresponding functions g 1 g l from the data the procedure combines smoothing techniques with ideas related to principal component analysis an asymptotic theory is presented which yields detailed insight into properties of the resulting estimators an application to household expenditure data illustrates the approach

stringing high dimensional data for functional analysis we propose stringing a class of methods where one views high dimensional observations as functional data stringing takes advantage of the high dimension by representing such data as discretized and noisy observations that originate from a hidden smooth stochastic process assuming that the observations result from scrambling the original ordering of the observations of the process stringing reorders the components of the high dimensional vectors followed by transforming the high dimensional vector observations into functional data established techniques from functional data analysis can be applied for further statistical analysis once an underlying stochastic process and the corresponding random trajectory for each subject have been identified stringing of high dimensional data is implemented with distance based metric multidimensional scaling mapping high dimensional data to locations on a real interval such that predictors that are close in a suitable sample metric also are located close to each

unified inference for sparse and dense longitudinal models in longitudinal data analysis statistical inference for sparse data and dense data could be substantially different for kernel smoothing estimate of the mean function the convergence rates and limiting variance functions are different under the two scenarios the latter phenomenon poses challenges for statistical inference as a subjective choice between the sparse and dense cases may lead to wrong conclusions we develop self normalization based methods that can adapt to the sparse and dense cases in a unified framework simulations show that the proposed methods outperform some existing methods

optimal estimation of the mean function based on discretely sampled functional data phase transition the problem of estimating the mean of random functions based on discretely sampled data arises naturally in functional data analysis in this paper we study optimal estimation of the mean function under both common and independent designs minimax rates of convergence are established and easily implementable rate optimal estimators are introduced the analysis reveals interesting and different phase transition phenomena in the two cases under the common design the sampling frequency solely determines the optimal rate of convergence when it is relatively small and the sampling frequency has no effect on the optimal rate when it is large on the other hand under the independent design the optimal rate of convergence is determined jointly by the sampling frequency and the number of curves when the sampling frequency is relatively small when it is large the sampling frequency has no effect on the optimal rate another interesting contrast between the two settings is that smoothing is necessary under the independent design while somewhat surprisingly it is not essential under the common design

analysis of long period variable stars with nonparametric tests for trend detection in astronomy the study of variable stars that is stars characterized by showing significant variation in their brightness over time has made crucial contributions to our understanding of many phenomena from stellar birth and evolution to the calibration of the extragalactic distance scale in this article we develop a method for analyzing multiple pseudo periodic time series with the goal of detecting temporal trends in their periods we allow for nonstationary noise and for clustering among the various time series we apply this method to the long standing astronomical problem of identifying variable stars whose regular brightness fluctuations have periods that change over time the results of our analysis show that such changes can be substantial raising the possibility that astronomers estimates of galactic distances can be refined two significant contributions of our approach relative to existing methods for this problem are as follows 1 the method is nonparametric making minimal assumpt

principal components analysis of sampled functions this paper describes a technique for principal components analysis of data consisting ofn functions each observed atp argument values this problem arises particularly in the analysis of longitudinal data in which some behavior of a number of subjects is measured at a number of points in time in such cases information about the behavior of one or more derivatives of the function being sampled can often be very useful as for example in the analysis of growth or learning curves it is shown that the use of derivative information is equivalent to a change of metric for the row space in classical principal components analysis the reproducing kernel for the hilbert space of functions plays a central role and defines the best interpolating functions which are generalized spline functions an example is offered of how sensitivity to derivative information can reveal interesting aspects of the data

block diagonalizability problem for hydrodynamic type systems the necessary and sufficient conditions are derived for the reducibility of a hydrodynamic type system into a block diagonal form with k mutually interacting blocks applications to the perturbations of the benney system and to the hamiltonian systems of partial differential equations are presented the algebraic identities connecting the nijenhuis tensors nb a u v and na u v and haantjes tensors hb a u v and ha u v are discovered

semiparametric gee analysis in partially linear single index models for longitudinal data in this article we study a partially linear single index model for longitudinal data under a general framework which includes both the sparse and dense longitudinal data cases a semiparametric estimation method based on a combination of the local linear smoothing and generalized estimation equations gee is introduced to estimate the two parameter vectors as well as the unknown link function under some mild conditions we derive the asymptotic properties of the proposed parametric and nonparametric estimators in different scenarios from which we find that the convergence rates and asymptotic variances of the proposed estimators for sparse longitudinal data would be substantially different from those for dense longitudinal data we also discuss the estimation of the covariance or weight matrices involved in the semiparametric gee method furthermore we provide some numerical studies including monte carlo simulation and an empirical application to illustrate our methodology and theory

from multiple gaussian sequences to functional data and beyond a stein estimation approach summary we expand the notion of gaussian sequence models to n experiments and propose a stein estimation strategy which relies on pooling information across experiments an oracle inequality is established to assess conditional risks given the underlying effects based on which we can quantify the size of relative error and obtain a tuning free recovery strategy that is easy to compute produces model parsimony and extends to unknown variance we show that the simultaneous recovery is adaptive to an oracle strategy which also enjoys a robustness guarantee in a minimax sense a connection to functional data is established via le cam theory for fixed and random designs under general regularity settings we further extend the model projection to general bases with mild conditions on correlation structure and conclude with potential application to other statistical problems simulated and real data examples are provided to lend empirical support to the methodology proposed and to illustrate the potential for substantial computational savings

functional graphical models graphical models have attracted increasing attention in recent years especially in settings involving high dimensional data in particular gaussian graphical models are used to model the conditio

on the decoupling problem of general quasilinear first order systems in two independent variables abstract the paper deals with the decoupling problem of general quasilinear first order systems in two independent variables we consider either the case of homogeneous and autonomous systems or the one of nonhomogeneous and or nonautonomous systems necessary and sufficient conditions for the partial or full decoupling of the systems at hand are provided the conditions involve the properties of eigenvalues and eigenvectors of the coefficient matrix and provide the differential constraints whose integration leads to the decoupling transformation some applications of physical interest are also given

an accelerated time model for response curves abstract we propose a nonparametric method of data analysis for situations where the data consist of a sample of curves and a covariate is present we assume the observed curves to be time accelerated versions of a basic underlying stochastic process and assume the time acceleration factor for each observed process determining the eigenzeit to be a smooth function of the covariate we discuss an iterative procedure to estimate the components of the model nonparametrically using cross validation to determine smoothing parameters using a leave one curve out technique our example concerns time courses of mortality for a sample of cohorts of fruit flies

mixture inner product spaces and their application to functional data analysis we introduce the concept of mixture inner product spaces associated with a given separable hilbert space which feature an infinite dimensional mixture of finite dimensional vector spaces and are dense in the underlying hilbert space any hilbert valued random element can be arbitrarily closely approximated by mixture inner product space valued random elements while this concept can be applied to data in any infinite dimensional hilbert space the case of functional data that are random elements in the l 2 space of square integrable functions is of special interest for functional data mixture inner product spaces provide a new perspective where each realization of the underlying stochastic process falls into one of the component spaces and is represented by a finite number of basis functions the number of which corresponds to the dimension of the component space in the mixture representation of functional data the number of included mixture components used to represent a given random element in l 2 is specifically adapted to each random trajectory and may be arbitrarily large key benefits of this novel approach are first that it provides a new perspective on the construction of a probability density in function space under mild regularity conditions and second that individual trajectories possess a trajectory specific dimension that corresponds to a latent random variable making it possible to use a larger number of components for less smooth and a smaller number for smoother trajectories this enables flexible and parsimonious modeling of heterogeneous trajectory shapes we establish estimation consistency of the functional mixture density and introduce an algorithm for fitting the functional mixture model based on a modified expectation maximization algorithm simulations confirm that in comparison to traditional functional principal component analysis the proposed method achieves similar or better data recovery while using fewer components on average its practical merits are also demonstrated in an analysis of egg laying trajectories for medflies

inference for density families using functional principal component analysis we consider t 1 t samples of iid observations x1t xntt from unknown population densities ft to characterize differences and similarities of ft we assume their expansions into the first l principal components from the given observations xit we study inference on the components and on their required number l a detailed asymptotic theory is presented our method is applied in the analysis of yearly cross sectional samples of british households interpretation of the estimated principal components and their scores provides new insights into the evolution and interplay of household income and age distributions from 1968 1988 from estimating their required numbersl we draw conclusions on the dimensionality of mixture models for describing the densities

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

functional principal component model for high dimensional brain imaging we explore a connection between the singular value decomposition svd and functional principal component analysis fpca models in high dimensional brain imaging applications we formally link right singular vectors to principal scores of fpca this combined with the fact that left singular vectors estimate principal components allows us to deploy the numerical efficiency of svd to fully estimate the components of fpca even for extremely high dimensional functional objects such as brain images as an example a fpca model is fit to high resolution morphometric ravens images the main directions of morphometric variation in brain volumes are identified and discussed

penalized pca approaches for b spline expansions of smooth functional data functional principal component analysis fpca is a dimension reduction technique that explains the dependence structure of a functional data set in terms of uncorrelated variables in many applications the data are a set of smooth functions observed with error in these cases the principal components are difficult to interpret because the estimated weight functions have a lot of variability and lack of smoothness the most common way to solve this problem is based on penalizing the roughness of a function by its integrated squared d order derivative two alternative forms of penalized fpca based on b spline basis expansions of sample curves and a simpler discrete penalty that measures the roughness of a function by summing squared d order differences between adjacent b spline coefficients p spline penalty are proposed in this paper the main difference between both smoothed fpca approaches is that the first uses the p spline penalty in the least squares approximation of the sample curves in terms of a b spline basis meanwhile the second introduces the p spline penalty in the orthonormality constraint of the algorithm that computes the principal components leave one out cross validation is adapted to select the smoothing parameter for these two smoothed fpca approaches a simulation study and an application with chemometric functional data are developed to test the performance of the proposed smoothed approaches and to compare the results with non penalized fpca and regularized fpca

smoothed functional principal components analysis by choice of norm the principal components analysis of functional data is often enhanced by the use of smoothing it is shown that an attractive method of incorporating smoothing is to replace the usual l 2 orthonormality constraint on the principal components by orthonormality with respect to an inner product that takes account of the roughness of the functions the method is easily implemented in practice by making use of appropriate function transforms fourier transforms for periodic data and standard principal components analysis programs several alternative possible interpretations of the smoothed principal components as obtained by the method are presented some theoretical properties of the method are discussed the estimates are shown to be consistent under appropriate conditions and asymptotic expansion techniques are used to investigate their bias and variance properties these indicate that the form of smoothing proposed is advantageous under mild conditions indeed milder than those for existing methods of smoothed functional principal components analysis the choice of smoothing parameter by cross validation is discussed the methodology of the paper is illustrated by an application to a biomechanical data set obtained in the study of the behaviour of the human thumb forefinger system

an optimal selection of regression variables summary an asymptotically optimal selection of regression variables is proposed the key assumption is that the number of control variables is infinite or increases with the sample size it is also shown that mallows s qp akaike s fpe and aic methods are all asymptotically equivalent to this method

nonparametric function estimation for clustered data when the predictor is measured without with error abstract we consider local polynomial kernel regression with a single covariate for clustered data using estimating equations we assume that at most m observations are available on each cluster in the case of random regressors with no measurement error in the predictor we show that it is generally the best strategy to ignore entirely the correlation structure within each cluster and instead pretend that all observations are independent in the further special case of longitudinal data on individuals with fixed common observation times we show that equivalent to the pooled data approach is the strategy of fitting separate nonparametric regressions at each observation time and constructing an optimal weighted average we also consider what happens when the predictor is measured with error using the simex approach to correct for measurement error we construct an asymptotic theory for both the pooled and the weighted average estimators surprisingly for the same amount of smoothing the weighted a

linear hypothesis testing with functional data in real data analysis it is often interesting to consider a general linear hypothesis testing glht problem for functional data which includes the one way anova post hoc or contrast analysis a

geometric properties of upper level sets of lelong numbers on projective spaces let t be a positive closed current of unit mass on the complex projective space mathbb p n for certain values alpha 1 we prove geometric properties of the set of points in mathbb p n where the lelong number of t exceeds alpha we also consider the case of positive closed currents of bidimension 1 1 on multiprojective spaces

decoupling problem for systems of quasi linear pde s the necessary and sufficient conditions for the decoupling of a quasi linear system of partial differential equations into k non interacting subsystems are derived several necessary conditions for the decoupling are found and applied to the benney system

partially functional linear regression in high dimensions summary in modern experiments functional and nonfunctional data are often encountered simultaneously when observations are sampled from random processes and high dimensional scalar covariates it is difficult to apply existing methods for model selection and estimation we propose a new class of partially functional linear models to characterize the regression between a scalar response and covariates of both functional and scalar types the new approach provides a unified and flexible framework that simultaneously takes into account multiple functional and ultrahigh dimensional scalar predictors enables us to identify important features and offers improved interpretability of the estimators the underlying processes of the functional predictors are considered to be infinite dimensional and one of our contributions is to characterize the effects of regularization on the resulting estimators we establish the consistency and oracle properties of the proposed method under mild conditions demonstrate its performance with simulation studies and illustrate its application using air pollution data

nonparametric mixed effects models for unequally sampled noisy curves we propose a method of analyzing collections of related curves in which the individual curves are modeled as spline functions with random coefficients the method is applicable when the individual curves are sampled at variable and irregularly spaced points this produces a low rank low frequency approximation to the covariance structure which can be estimated naturally by the em algorithm smooth curves for individual trajectories are constructed as best linear unbiased predictor blup estimates combining data from that individual and the entire collection this framework leads naturally to methods for examining the effects of covariates on the shapes of the curves we use model selection techniques akaike information criterion aic bayesian information criterion bic and cross validation to select the number of breakpoints for the spline approximation we believe that the methodology we propose provides a simple flexible and computationally efficient means of functional data analysis