estimation of functional regression model via functional dimension reduction abstract the functional linear regression model corresponding to a scalar response and a functional predictor is becoming increasingly common since the predictor is infinite dimensional some form of dimension reduction is essential there are two popular dimension reduction methods such as expanding the functional predictor or regression parameter function on the functional principal component basis or on a fixed bases such as b spline wavelet in the present paper we estimate the functional linear model by using the functional sufficient dimension reduction fsdr basis compared to the existing methods the proposed method is appealing because the fsdr basis is related to both the functional predictor and the response variable whereas the functional principal component basis is only related to the functional predictor and the fixed basis b spline wavelet is independent from both the functional predictor and the response variable our techniques involve methods for giving a new expansion for the predictor for giving a specific expression for the regression parameter function for estimating the fsdr space by a new method and some asymptotical properties about the regression parameter function and the prediction for the test samples numerical studies including both simulation studies and applications on real life data are presented to demonstrate the accuracy of the proposed method

discriminant analysis by gaussian mixtures fisher rao linear discriminant analysis lda is a valuable tool for multigroup classification lda is equivalent to maximum likelihood classification assuming gaussian distributions for each class in this paper we fit gaussian mixtures to each class to facilitate effective classification in non normal settings especially when the classes are clustered low dimensional views are an important by product of lda our new techniques inherit this feature we can control the within class spread of the subclass centres relative to the between class spread our technique for fitting these models permits a natural blend with nonparametric versions of lda

unexpected properties of bandwidth choice when smoothing discrete data for constructing a functional data classifier the data functions that are studied in the course of functional data analysis are assembled from discrete data and the level of smoothing that is used is generally that which is appropriate for accurate approximation of the conceptually smooth functions that were not actually observed existing literature shows that this approach is effective and even optimal when using functional data methods for prediction or hypothesis testing however in the present paper we show that this approach is not effective in classification problems there a useful rule of thumb is that undersmoothing is often desirable but there are several surprising qualifications to that approach first the effect of smoothing the training data can be more significant than that of smoothing the new data set to be classified second undersmoothing is not always the right approach and in fact in some cases using a relatively large bandwidth can be more effective and third these perverse results are the consequence of very unusual properties of error rates expressed as functions of smoothing parameters for example the orders of magnitude of optimal smoothing parameter choices depend on the signs and sizes of terms in an expansion of error rate and those signs and sizes can vary dramatically from one setting to another even for the same classifier

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

classification of non parametric regression functions in longitudinal data models summary we investigate a longitudinal data model with non parametric regression functions that may vary across the observed individuals in a variety of applications it is natural to impose a group structure on the regression curves specifically we may suppose that the observed individuals can be grouped into a number of classes whose members all share the same regression function we develop a statistical procedure to estimate the unknown group structure from the data moreover we derive the asymptotic properties of the procedure and investigate its finite sample performance by means of a simulation study and a real data example

correlation based functional clustering via subspace projection a correlation based functional clustering method is proposed for grouping curves with similar shapes a correlation between two random functions defined through the functional inner product is used as a similarity measure curves with similar shapes are embedded in the cluster subspace spanned by a mean shape function and eigenfunctions of the covariance kernel the cluster membership prediction for each curve attempts to maximize the functional correlation between the observed and predicted curves via shape standardization and subspace projection among all possible clusters the proposed method accounts for shape differentials through the functional multiplicative random effects shape function model for each cluster which regards random scales and intercept shifts as a nuisance a consistent estimate is proposed for the random scale effect whose sample variance estimate is also consistent the derived identifiability conditions for the clustering procedure unravel the predictability of cluster membersh

third order differential equations with integral boundary conditions abstract in this paper we study nonlinear third order differential equations with integral boundary conditions we provide sufficient conditions on the nonlinearity and the functions appearing in the boundary conditions that guarantee the existence of at least one solution to our problem our technique is based on a priori bounds and fixed point theorems

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

on properties of functional principal components analysis summary functional data analysis is intrinsically infinite dimensional functional principal component analysis reduces dimension to a finite level and points to the most significant components of the data however although this technique is often discussed its properties are not as well understood as they might be we show how the properties of functional principal component analysis can be elucidated through stochastic expansions and related results our approach quantifies the errors that arise through statistical approximation in successive terms of orders n 1 2 n 1 n 3 2 where n denotes sample size the expansions show how spacings among eigenvalues impact on statistical performance the term of size n 1 2 illustrates first order properties and leads directly to limit theory which describes the dominant effect of spacings thus for example spacings are seen to have an immediate first order effect on properties of eigenfunction estimators but only a second order effect on eigenvalue estimators our results can be used to explore properties of existing methods and also to suggest new techniques in particular we suggest bootstrap methods for constructing simultaneous confidence regions for an infinite number of eigenvalues and also for individual eigenvalues and eigenvectors

classification of functional fragments by regularized linear classifiers with domain selection we consider classification of functional data into two groups by linear classifiers based on one dimensional projections of functions we reformulate the task of finding the best classifier as an optimization problem and solve it by the conjugate gradient method with early stopping the principal component method and the ridge method we study the empirical version with finite training samples consisting of incomplete functions observed on different subsets of the domain and show that the optimal possibly zero misclassification probability can be achieved in the limit along a possibly nonconvergent empirical regularization path we propose a domain extension and selection procedure that finds the best domain beyond the common observation domain of all curves in a simulation study we compare the different regularization methods and investigate the performance of domain selection our method is illustrated on a medical dataset where we observe a substantial improvement of classification accuracy due to domain extension

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

defining probability density for a distribution of random functions the notion of probability density for a random function is not as straightforward as in finite dimensional cases while a probability density function generally does not exist for functional data we show that it is possible to develop the notion of density when functional data are considered in the space determined by the eigenfunctions of principal component analysis this leads to a transparent and meaningful surrogate for density defined in terms of the average value of the logarithms of the densities of the distributions of principal components for a given dimension this density approximation is estimable readily from data it accurately represents in a monotone way key features of small ball approximations to density our results on estimators of the densities of principal component scores are also of independent interest they reveal interesting shape differences that have not previously been considered the statistical implications of these results and properties are identified and discussed and practical ramifications are illustrated in numerical work

most predictive design points for functional data predictors we suggest a way of reducing the very high dimension of a functional predictor x to a low number of dimensions chosen so as to give the best predictive performance specifically if x is observed on a fine grid of design points t 1 t r we propose a method for choosing a small subset of these say t i1 t ik to optimize the prediction of a response variable y the values t ij are referred to as the most predictive design points or covariates for a given value of k and are computed using information contained in a set of independent observations x i y i of x y the algorithm is based on local linear regression and calculations can be accelerated using linear regression to preselect the design points boosting can be employed to further improve the predictive performance we illustrate the usefulness of our ideas through simulations and examples drawn from chemometrics and we develop theoretical arguments showing that the methodology can be applied successfully in a range of settings

clustering functional data into groups by using projections we show that in the functional data context by appropriately exploiting the functional nature of the data it is possible to cluster the observations asymptotically perfectly we demonstrate that this level of performance can sometimes be achieved by the k means algorithm as long as the data are projected on a carefully chosen finite dimensional space in general the notion of an ideal cluster is not clearly defined we derive our results in the setting where the data come from two populations whose distributions differ at least in terms of means and where an ideal cluster corresponds to one of these two populations we propose an iterative algorithm to choose the projection functions in a way that optimizes clustering performance where to avoid peculiar solutions we use a weighted least squares criterion we apply our iterative clustering procedure on simulated and real data where we show that it works well

multiscale clustering of nonparametric regression curves abstract in a wide range of modern applications one observes a large number of time series rather than only a single one it is often natural to suppose that there is some group structure in the observed time series when each time series is modeled by a nonparametric regression equation one may in particular assume that the observed time series can be partitioned into a small number of groups whose members share the same nonparametric regression function we develop a bandwidth free clustering method to estimate the unknown group structure from the data more precisely speaking we construct multiscale estimators of the unknown groups and their unknown number which are free of classical bandwidth or smoothing parameters in the theoretical part of the paper we analyze the statistical properties of our estimators our theoretical results are derived under general conditions which allow the data to be dependent both in time series direction and across different time series the technical analysis of the paper is complemented by simulated and real data examples

on the use of reproducing kernel hilbert spaces in functional classification the hajek feldman dichotomy establishes that two gaussian measures are either mutually absolutely continuous with respect to each other and hence there is a radon nikodym density for each measure

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

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

shape descriptors for classification of functional data curve discrimination is an important task in engineering and other sciences we propose several shape descriptors for classifying functional data inspired by form analysis from the image analysis field statistical moments coefficients of the components of independent component analysis and two mathematical morphology descriptors morphological covariance and spatial size distributions these are applied to three problems an artificial problem a speech recognition problem and a biomechanical application shape descriptors are compared with other methods in the literature with better or similar performance obtaining

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

functional classification in hilbert spaces let x be a random variable taking values in a separable hilbert space x with label y spl isin 0 1 we establish universal weak consistency of a nearest neighbor type classifier based on n independent copies x sub i y sub i of the pair x y extending the classical result of stone to infinite dimensional hilbert spaces under a mild condition on the distribution of x we also prove strong consistency we reduce the infinite dimension of x by considering only the first d coefficients of a fourier series expansion of each x sub i and then we perform k nearest neighbor classification in spl ropf sup d both the dimension and the number of neighbors are automatically selected from the data using a simple data splitting device an application of this technique to a signal discrimination problem involving speech recordings is presented

componentwise classification and clustering of functional data the infinite dimension of functional data can challenge conventional methods for classification and clustering a variety of techniques have been introduced to address this problem particularly in the case of prediction but the structural models that they involve can be too inaccurate or too abstract or too difficult to interpret for practitioners in this paper we develop approaches to adaptively choose components enabling classification and clustering to be reduced to finite dimensional problems we explore and discuss properties of these methodologies our techniques involve methods for estimating classifier error rate and cluster tightness and for choosing both the number of components and their locations to optimize these quantities a major attraction of this approach is that it allows identification of parts of the function domain that convey important information for classification and clustering it also permits us to determine regions that are relevant to one of these analyses but not the other copyright 2012 oxford university press

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

deconvolution when classifying noisy data involving transformations in the present study we consider the problem of classifying spatial data distorted by a linear transformation or convolution and contaminated by additive random noise in this setting we show that classifier performance can be improved if we carefully invert the data before the classifier is applied however the inverse transformation is not constructed so as to recover the original signal and in fact we show that taking the latter approach is generally inadvisable we introduce a fully data driven procedure based on cross validation and use several classifiers to illustrate numerical properties of our approach theoretical arguments are given in support of our claims our procedure is applied to data generated by light detection and ranging lidar technology where we improve on earlier approaches to classifying aerosols this article has supplementary materials online

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

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

existence of nontrivial periodic solutions for a nonlinear second order periodic boundary value problem abstract in this paper we study the existence of nontrivial periodic solutions to the following nonlinear differential equation u t a t u t f t u t t r u 0 u u 0 u where a r r is an periodic continuous function with a t 0 f r r r is continuous may take negative values and can be sign changing without making any nonnegative assumption on nonlinearity by using the first eigenvalue corresponding to the relevant linear operator and the topological degree the existence of nontrivial periodic solutions to the above periodic boundary value problem is established finally three examples are given to demonstrate the validity of our main results

optimal designs for longitudinal and functional data summary we propose novel optimal designs for longitudinal data for the common situation where the resources for longitudinal data collection are limited by determining the optimal locations in time where measurements should be taken as for all optimal designs some prior information is needed to implement the optimal designs proposed we demonstrate that this prior information may come from a pilot longitudinal study that has irregularly measured and noisy measurements where for each subject one has available a small random number of repeated measurements that are randomly located on the domain a second possibility of interest is that a pilot study consists of densely measured functional data and one intends to take only a few measurements at strategically placed locations in the domain for the future collection of similar data we construct optimal designs by targeting two criteria optimal designs to recover the unknown underlying smooth random trajectory for each subject from a few optimally placed measurements such that squared prediction errors are minimized optimal designs that minimize prediction errors for functional linear regression with functional or longitudinal predictors and scalar responses again from a few optimally placed measurements the optimal designs proposed address the need for sparse data collection when planning longitudinal studies by taking advantage of the close connections between longitudinal and functional data analysis we demonstrate in simulations that the designs perform considerably better than randomly chosen design points and include a motivating data example from the baltimore longitudinal study of aging the designs are shown to have an asymptotic optimality property

functional linear discriminant analysis for irregularly sampled curves we introduce a technique for extending the classical method of linear discriminant analysis lda to data sets where the predictor variables are curves or functions this procedure which we call functional linear discriminant analysis flda is particularly useful when only fragments of the curves are observed all the techniques associated with lda can be extended for use with flda in particular flda can be used to produce classifications on new test curves give an estimate of the discriminant function between classes and provide a one or two dimensional pictorial representation of a set of curves we also extend this procedure to provide generalizations of quadratic and regularized discriminant analysis

a functional data analytic approach to signal discrimination motivated by specific problems involving radar range profiles we suggest techniques for real time discrimination in the context of signal analysis the key to our approach is to regard the signals as curves in the continuum and employ a functional data analytic fda method for dimension reduction based on the fda technique for principal coordinates analysis this has the advantage relative to competing methods such as canonical variates analysis of providing a signal approximation that is best possible in an l2 sense for a given dimension as a result it produces particularly good discrimination we explore the use of both nonparametric and gaussian based discriminators applied to the dimensionreduced data

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

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

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

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

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

the mahalanobis distance for functional data with applications to classification this article presents a new semidistance for functional observations that generalizes the mahalanobis distance for multivariate datasets the main characteristics of the functional mahalanobis semidistance are shown to illustrate the applicability of this measure of proximity between functional observations new versions of several well known functional classification procedures are developed using the functional mahalanobis semidistance a monte carlo study and the analysis of two real examples indicate that the classification methods used in conjunction with the functional mahalanobis semidistance give better results than other well known functional classification procedures this article has supplementary material online

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

multilevel functional clustering analysis in this article we investigate clustering methods for multilevel functional data which consist of repeated random functions observed for a large number of units e g genes at multiple subunits e g bacteria types to describe the within and between variability induced by the hierarchical structure in the data we take a multilevel functional principal component analysis mfpca approach we develop and compare a hard clustering method applied to the scores derived from the mfpca and a soft clustering method using an mfpca decomposition in a simulation study we assess the estimation accuracy of the clustering membership and the cluster patterns under a series of settings small versus moderate number of time points various noise levels and varying number of subunits per unit we demonstrate the applicability of the clustering analysis to a real data set consisting of expression profiles from genes activated by immunity system cells prevalent response patterns are identified by clustering the expression profiles using our multilevel clustering analysis

restricted weak type endpoint estimates for k spherical maximal functions in this paper we use the approximation formula for the fourier transform of the solution set of lattice points on k spheres and methods of bourgain and ionescu to refine the ell p mathbb z d boundedness results for discrete k spherical maximal functions to a restricted weak type result at the endpoint we introduce a density parameter which may be viewed as a discrete version of minkowski dimension used in related works on the continuous analgoue in order to exploit recent progress of wooley and bourgain demeter guth on the vinogradov mean value conjectures via a novel approximation formula for a single average and obtain improved bounds for lacunary discrete k spherical maximal functions when k ge 3

achieving near perfect classification for functional data summary we show that in functional data classification problems perfect asymptotic classification is often possible making use of the intrinsic very high dimensional nature of functional data this performance is often achieved by linear methods which are optimal in important cases these results point to a marked contrast between classification for functional data and its counterpart in conventional multivariate analysis where the dimension is kept fixed as the sample size diverges in the latter setting linear methods can sometimes be quite inefficient and there are no prospects for asymptotically perfect classification except in pathological cases where for example a variance vanishes by way of contrast in finite samples of functional data good performance can be achieved by truncated versions of linear methods truncation can be implemented by partial least squares or projection onto a finite number of principal components using in both cases cross validation to determine the truncation point we establish consistency of the cross validation procedure

variable selection for the prediction of c 0 1 valued autoregressive processes using reproducing kernel hilbert spaces a model for the prediction of functional time series is introduced where observations are assumed to be continuous random functions we model the dependence of the data with a nonstandard autoregr

classification using censored functional data we consider classification of functional data when the training curves are not observed on the same interval different types of classifier are suggested one of which involves a new curve extension procedure our approach enables us to exploit the information contained in the endpoints of these intervals by incorporating it in an explicit but flexible way we study asymptotic properties of our classifiers and show that in a variety of settings they can even produce asymptotically perfect classification the performance of our techniques is illustrated in applications to real and simulated data supplementary materials for this article are available online

on the existence of periodic solutions for certain differential equations here we are concerned with the problem of the existence of periodic solution for certain second and third order nonlinear differential equations our method here is to consider the problem as an eigenvalue problem and treat it by the topological degree theory in particular we establish the conditions of the existence of periodic solution first for a simpler system which is homotopic to the original system and then generalize the obtained results for the focal system the method employed here is applicable also for a system of nonlinear differential equations just with simple modifications finally we present some specific examples numerically to show that the results are valid and realizable

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

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

modeling and forecasting daily electricity load curves a hybrid approach we propose a hybrid approach for the modeling and the short term forecasting of electricity loads two building blocks of our approach are 1 modeling the overall trend and seasonality by fitting a generalized additive model to the weekly averages of the load and 2 modeling the dependence structure across consecutive daily loads via curve linear regression for the latter a new methodology is proposed for linear regression with both curve response and curve regressors the key idea behind the proposed methodology is dimension reduction based on a singular value decomposition in a hilbert space which reduces the curve regression problem to several ordinary i e scalar linear regression problems we illustrate the hybrid method using french electricity loads between 1996 and 2009 on which we also compare our method with other available models including the electricite de france operational model supplementary materials for this article are available online

clustering multivariate functional data with phase variation when functional data come as multiple curves per subject characterizing the source of variations is not a trivial problem the complexity of the problem goes deeper when there is phase variation in addition to amplitude variation we consider clustering problem with multivariate functional data that have phase variations among the functional variables we propose a conditional subject specific warping framework in order to extract relevant features for clustering using multivariate growth curves of various parts of the body as a motivating example we demonstrate the effectiveness of the proposed approach the found clusters have individuals who show different relative growth patterns among different parts of the body

bayesian curve classification using wavelets we propose classification models for binary and multicategory data where the predictor is a random function we use bayesian modeling with wavelet basis functions that have nice approximation properties over a large class of functional spaces and can accommodate a wide variety of functional forms observed in real life applications we develop an unified hierarchical model to encompass both the adaptive wavelet based function estimation model and the logistic classification model we couple together these two models are to borrow strengths from each other in a unified hierarchical framework the use of gibbs sampling with conjugate priors for posterior inference makes the method computationally feasible we compare the performance of the proposed model with other classification methods such as the existing naive plug in methods by analyzing simulated and real data sets

on the use of fuzzy constraints in semisupervised clustering this paper introduces fuzzy hss a semisupervised hierarchical clustering approach that uses fuzzy instance level constraints these constraints are external information on the shape of fuzzy must link and fuzzy cannot link restrictions they allow uncertainty when indicating whether two instances of a dataset belong to the same group fuzzy must link constraints give a degree of belief of two instances belonging to the same group analogously fuzzy cannot link constraints indicate the degree of belief of two instances not belonging to the same group these constraints have been introduced in a hierarchical clustering process allowing us to obtain the optimal number of groups in a dendrogram when the number of clusters is not known the optimal amount of constraints needed in the process is determined by means of fuzzy entropy an extensive experimental study is provided by comparing this fuzzy semisupervised approach with classic unsupervised methods as well as a crisp semisupervised alternative

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