generalized functional extended redundancy analysis functional extended redundancy analysis fera was recently developed to integrate data reduction into functional linear models this technique extracts a component from each of multiple sets of predictor data in such a way that the component accounts for the maximum variance of response data moreover it permits predictor and or response data to be functional fera can be of use in describing overall characteristics of each set of predictor data and in summarizing the relationships between predictor and response data in this paper we extend fera into the framework of generalized linear models glm so that it can deal with response data generated from a variety of distributions specifically the proposed method reduces each set of predictor functions to a component and uses the component for explaining exponential family responses as in glm we specify the random systematic and link function parts of the proposed method we develop an iterative algorithm to maximize a penalized log likelihood criterion that is derived in combination with a basis function expansion approach we conduct two simulation studies to investigate the performance of the proposed method based on synthetic data in addition we apply the proposed method to two examples to demonstrate its empirical usefulness

splines in statistics abstract this is a survey article that attempts to synthesize a broad variety of work on splines in statistics splines are presented as a nonparametric function estimating technique after a general introduction to the theory of interpolating and smoothing splines splines are treated in the nonparametric regression setting the method of cross validation for choosing the smoothing parameter is discussed and the general multivariate regression surface estimation problem is addressed an extensive discussion of splines as nonparametric density estimators is followed by a discussion of their role in time series analysis a comparison of the spline and isotonic regression methodologies leads to a formulation of a hybrid estimator the closing section provides a brief overall summary and formulates a number of open unsolved problems relating to splines in statistics

temporal process regression summary we consider regression for response and covariates which are temporal processes observed over intervals a functional generalised linear model is proposed which includes extensions of standard models in multi state survival analysis simple nonparametric esti mators of time indexed parameters are developed using working independence estimating equations and are shown to be uniformly consistent and to converge weakly to gaussian processes the procedure does not require smoothing or a markov assumption unlike approaches based on transition intensities the usual definition of optimal estimating equations for parametric models is then generalised to the functional model and the optimum is identified in a class of functional generalised estimating equations simulations demonstrate large efficiency gains relative to working independence at times where censoring is heavy the estimators are the basis for new tests of the covariate effects and for the estimation of models in which greater structure is imposed on the parameters providing novel goodness of fit tests the methodology s practical utility is illustrated in a data analysis

bayesian function on function regression for multilevel functional data over a grid is frequently obtained moreover researchers often sample multiple curves per person resulting in repeated functional measures a common question is how to analyze the relationship between two functional variables we propose a general function on function regression model for repeatedly sampled functional data on a ne grid presenting a simple model as well as a more extensive mixed model framework and introducing various functional bayesian inferential procedures that account for multiple testing we examine these models via simulation and a data analysis with data from a study that used event related potentials to examine how the brain processes various types of images

regression analysis for a functional response functional responses are encountered when units are observed over time although the whole function itself is not observed a sufficiently large number of evaluations as is common with modern recording equipment are assumed to be available functional regression analysis relates the smooth functional response y t to known covariates x by a linear combination of parameter functions t which are to be estimated the model takes the standard form y t x t t t this approach provides an alternative to standard longitudinal data methods used in the biological sciences where less and noisier data necessitate parametric modeling the methodology is illustrated by an application in ergonomics

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

incorporating parametric effects into functional principal components analysis summary the ideas of functional principal component analysis are extended to deal with data that are hybrids of functional and parametric effects the parametric effects may be more general than just the addition of a multiple of a given function a detailed development is given in the case of shifts of the time axis for functions observed on a periodic interval and some remarks are made for the extension to a far more general case given data a procrustes fitting method can be used to estimate the parametric effects several possible ways of treating the estimated parameter values are discussed the methods are illustrated by reference to temperature data at 35 canadian weather stations

plurality and resemblance in fmri data analysis abstract we apply nine analytic methods employed currently in imaging neuroscience to simulated and actual bold fmri signals and compare their performances under each signal type starting with baseline time series generated by a resting subject during a null hypothesis study we compare method performance with embedded focal activity in these series of three different types whose magnitudes and time courses are simple convolved with spatially varying hemodynamic responses and highly spatially interactive we then apply these same nine methods to bold fmri time series from contralateral primary motor cortex and ipsilateral cerebellum collected during a sequential finger opposition study paired comparisons of results across methods include a voxel specific concordance correlation coefficient for reproducibility and a resemblance measure that accommodates spatial autocorrelation of differences in activity surfaces receiver operating characteristic curves show considerable model differences in ranges less than 10 significance level false positives and greater than 80 power true positives concordance and resemblance measures reveal significant differences between activity surfaces in both data sets these measures can assist researchers by identifying groups of models producing similar and dissimilar results and thereby help to validate consolidate and simplify reports of statistical findings a pluralistic strategy for fmri data analysis can uncover invariant and highly interactive relationships between local activity foci and serve as a basis for further discovery of organizational principles of the brain results also suggest that a pluralistic empirical strategy coupled formally with substantive prior knowledge can help to uncover new brain behavior relationships that may remain hidden if only a single method is employed

nonparametric empirical bayes growth curve analysis abstract nonparametric and semiparametric regression have been suggested as alternatives to parametric models for growth curve analysis in this article we demonstrate that empirical bayes estimation can be used to improve linear smoothing estimates when multiple curves are available

dynamic functional principal components summary we address the problem of dimension reduction for time series of functional data xt t z such functional time series frequently arise for example when a continuous time process is segmented into some smaller natural units such as days then each xt represents one intraday curve we argue that functional principal component analysis though a key technique in the field and a benchmark for any competitor does not provide an adequate dimension reduction in a time series setting functional principal component analysis indeed is a static procedure which ignores the essential information that is provided by the serial dependence structure of the functional data under study therefore inspired by brillinger s theory of dynamic principal components we propose a dynamic version of functional principal component analysis which is based on a frequency domain approach by means of a simulation study and an empirical illustration we show the considerable improvement that the dynamic approach entails when compared with the usual static procedure

on the performance of anisotropic mesh adaptation for scroll wave turbulence dynamics in reaction diffusion systems abstract nonlinear reaction diffusion systems are widely employed to study the spatio temporal chaotic behavior that occurs in excitable media such as cardiac tissue where sufficiently strong perturbations can excite nonlinear propagating waves which can form spiral waves in two dimensions or scroll waves in three dimensions the numerical simulation of these waves calls for grids that are extremely fine over the whole computational domain to accurately predict the trajectories and multiplication of wave fronts and therefore leads to huge computational challenges mesh adaptation methods can reduce the number of degrees of freedom required for a given accuracy but they also have a cost and it is not clear if they are competitive with respect to very fine uniform meshes previous mesh adaptation techniques applied to spatio temporal chaotic behavior have been mostly limited to the two dimensional case the purpose of this paper is to explore the efficiency of a three dimensional anisotropic finite element mesh adaptivity for simulating scroll wave turbulence the computational efficiency of the proposed method is assessed using reference solutions obtained on a uniform refined mesh with more than 44 millions degrees of freedom the proposed method reduces significantly the number of elements leading to huge saving in memory as well as in computational time examples of the dynamics of ventricular fibrillation in cardiac tissue will be presented illustrating the performance of the overall methodology

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

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

testing the parametric null model hypothesis in semiparametric partial and generalized spline models on etend les resultats de cox et koh 1986 aux modeles generalises de wahba 1985 et a divers autres modeles splines partiels on montre que la statistique de test a une relation etroite avec le comportement de la fonction de validation croisee generalisee

hotelling s theorem on the volume of tubes some illustrations in simultaneous inference and data analysis we illustrate with contemporary examples hotelling s geometric approach to simultaneous probability calculations hotelling reduces the evaluation of certain normal theory significance probabilities to finding the volume of a tube about a curve in a hypersphere and shows that this volume is often exactly given by length times cross sectional area we review hotelling s result together with some recent complements and then use the approach to set simultaneous prediction regions for some data from gait analysis to study andrews plots in multivariate data analysis and to construct significance tests for projection pursuit regression a by product is a numerical criterion for tube self overlap relevant for example to uniqueness of certain nonlinear least squares estimates

functional association models for multivariate survival processes we consider multivariate temporal processes that are continuously observed within overlapping time windows the intended application is in censored multistate and multivariate survival settings where point processes are continuously observed these data differ from other functional data like longitudinal data which are discretely observed at irregular times existing functional approaches to survival processes use intensity models which require smoothing and depend critically on the choice of smoothing parameters similarly to discretely observed data in this article we study functional mean and association regression models for the point processes with unspecified time varying coefficients the continuous observation scheme is exploited the coefficients may be estimated nonparametrically by extending generalized estimating equations to continuously observed data the estimators automatically converge at the parametric rate without smoothing unlike with discretely observed data uniform consistenc

classifying non uniformly sampled vector valued curves we propose a novel means of classifying vector valued curves using functional principal components this uses cross validation to select curve components the degree of smoothing and scores associated with the best classification performance our approach is well suited to data generated by sensors of different modalities and time varying sampling frequencies experimental comparisons show this approach has significant advantages over conventional techniques using non uniformly sampled data our approach also generates novel forms of derivative analysis a widely used technique for classifying spectral functions from contaminated data

asymptotic confidence regions for kernel smoothing of a varying coefficient model with longitudinal data abstract we consider the estimation of the k 1 dimensional nonparametric component t of the varying coefficient model y t x t t t e t based on longitudinal observations yij x i tij tij i 1 n j 1 ni where tij is the jth observed design time point t of the ith subject and yij and x i tij are the real valued outcome and r k 1 valued covariate vectors of the ith subject at tij the subjects are independently selected but the repeated measurements within subject are possibly correlated asymptotic distributions are established for a kernel estimate of t that minimizes a local least squares criterion these asymptotic distributions are used to construct a class of approximate pointwise and simultaneous confidence regions for t applying these methods to an epidemiological study we show that our procedures are useful for predicting cd4 t helper lymphocytes cell changes among hiv human immunodeficiency virus infected persons the finite sample properties of o

functional data analysis with application to periodically stimulated foetal heart rate data ii functional logistic regression functional regression is used to model longitudinal data where the number of measurements on the functional covariate is much greater than the number of subjects in the study thus functional regression can be thought of as singular longitudinal analysis we have modified existing functional regression techniques to the case of a functional covariate with a repeated stimulus we applied this modified functional regression to periodically stimulated foetal heart rates the heart rate tracings were used as a predictor of the child s psychomotor development at approximately 18 months of age in the past this type of data has been analysed using the partially subjective concept of habituation by using the entire heart rate tracings through functional regression we have the advantage that habituation does not need to be defined and all available information is used to predict later child development

coherent states and bayesian duality we demonstrate how large classes of discrete and continuous statistical distributions can be incorporated into coherent states using the concept of a reproducing kernel hilbert space each family of coherent states is shown to contain in a sort of duality which resembles an analogous duality in bayesian statistics a discrete probability distribution and a discretely parametrized family of continuous distributions it turns out that nonlinear coherent states of the type widely studied in quantum optics are a particularly useful class of coherent states from this point of view in that they contain many of the standard statistical distributions we also look at vector coherent states and multidimensional coherent states as carriers of mixtures of probability distributions and joint probability distributions

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

alignment of curves by dynamic time warping when studying some process or development in different subjects or units be it biological chemical or physical we usually see a typical pattern common to all curves yet there is variation both in amplitude and dynamics between curves following some ideas of structural analysis introduced by kneip and gasser we study a method dynamic time warping with a proper cost function for estimating the shift or warping function from one curve to another to align the two functions for some models this method can identify the true shift functions if the data are noise free noisy data are smoothed by a nonparametric function estimate such as a kernel estimate it is shown that the proposed estimator is asymptotically normal and converges to the true shift function as the sample size per subject goes to infinity some simulation results are presented to illustrate the performance of this method

some tools for functional data analysis multivariate data analysis permits the study of observations which are finite sets of numbers but modern data collection situations can involve data or the processes giving rise to them which are functions functional data analysis involves infinite dimensional processes and or data the paper shows how the theory of l splines can support generalizations of linear modelling and principal components analysis to samples drawn from random functions spline smoothing rests on a partition of a function space into two orthogonal subspaces one of which contains the obvious or structural components of variation among a set of observed functions and the other of which contains residual components this partitioning is achieved through the use of a linear differential operator and we show how the theory of polynomial splines can be applied more generally with an arbitrary operator and associated boundary constraints these data analysis tools are illustrated by a study of variation in temperature precipitation patterns among some canadian weather stations

testing the goodness of fit of a linear model via nonparametric regression techniques abstract this article investigates the use of nonparametric regression methodology to test the adequacy of a parametric linear model the large sample properties of parametric goodness of fit tests for linearity are considered the inadequacies of such tests lead to the proposal of new tests that are constructed from nonparametric regression fits to the residuals from linear regression large sample theory is derived for two variants of this type of statistic the results demonstrate that such tests are consistent against all fixed smooth alternatives to linearity but are incapable of detecting local alternatives converging to a linear model at the parametric rate n 1 2 simulation experiments involving a test based on fitting cubic smoothing splines to residuals reveals that this test has good power properties against several reasonable alternatives

hypothesis testing in functional linear models functional data arise frequently in biomedical studies where it is often of interest to investigate the association between functional predictors and a scalar response variable while functional linear models flm are widely used to address these questions hypothesis testing for the functional association in the flm framework remains challenging a popular approach to testing the functional effects is through dimension reduction by functional principal component pc analysis however its power performance depends on the choice of the number of pcs and is not systematically studied in this article we first investigate the power performance of the wald type test with varying thresholds in selecting the number of pcs for the functional covariates and show that the power is sensitive to the choice of thresholds to circumvent the issue we propose a new method of ordering and selecting principal components to construct test statistics the proposed method takes into account both the association with the response and the variation along each eigenfunction we establish its theoretical properties and assess the finite sample properties through simulations our simulation results show that the proposed test is more robust against the choice of threshold while being as powerful as and often more powerful than the existing method we then apply the proposed method to the cerebral white matter tracts data obtained from a diffusion tensor imaging tractography study

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

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

a bayesian model for sparse functional data we propose a method for analyzing data which consist of curves on multiple individuals i e longitudinal or functional data we use a bayesian model where curves are expressed as linear combinations of b splines with random coefficients the curves are estimated as posterior means obtained via markov chain monte carlo mcmc methods which automatically select the local level of smoothing the method is applicable to situations where curves are sampled sparsely and or at irregular time points we construct posterior credible intervals for the mean curve and for the individual curves this methodology provides unified efficient and flexible means for smoothing functional data

smoothing spline models for the analysis of nested and crossed samples of curves abstract we introduce a class of models for an additive decomposition of groups of curves stratified by crossed and nested factors generalizing smoothing splines to such samples by associating them with a corresponding mixed effects model the models are also useful for imputation of missing data and exploratory analysis of variance we prove that the best linear unbiased predictors blups from the extended mixed effects model correspond to solutions of a generalized penalized regression where smoothing parameters are directly related to variance components and we show that these solutions are natural cubic splines the model parameters are estimated using a highly efficient implementation of the em algorithm for restricted maximum likelihood reml estimation based on a preliminary eigenvector decomposition variability of computed estimates can be assessed with asymptotic techniques or with a novel hierarchical bootstrap resampling scheme for nested mixed effects models our methods are applied to me

penalized discriminant analysis fisher s linear discriminant analysis lda is a popular data analytic tool for studying the relationship between a set of predictors and a categorical response in this paper we describe a penalized version of lda it is designed for situations in which there are many highly correlated predictors such as those obtained by discretizing a function or the grey scale values of the pixels in a series of images in cases such as these it is natural efficient and sometimes essential to impose a spatial smoothness constraint on the coefficients both for improved prediction performance and interpretability we cast the classification problem into a regression framework via optimal scoring using this our proposal facilitates the use of any penalized regression technique in the classification setting the technique is illustrated with examples in speech recognition and handwritten character recognition

principal differential analysis data reduction by differential operators functional data are observations that are either themselves functions or are naturally representable as functions when these functions can be considered smooth it is natural to use their derivatives in exploring their variation principal differential analysis pda identifies a linear differential operator l w 0 i w 1 d w m 1 d m 1 d m that comes as close as possible to annihilating a sample of functions convenient procedures for estimating the m weighting functions w j are developed the estimated differential operator l is analogous to the projection operator used as the data annihilator in principal components analysis and thus can be viewed as a type of data reduction or exploration tool the corresponding linear differential equation may also have a useful substantive interpretation modelling and regularization features can also be incorporated into pda

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

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics