combining registration and fitting for functional models a registration method can be defined as a process of aligning features of a sample of curves by monotone transformations of their domain the aligned curves exhibit only amplitude variation and the domain transformations called warping functions capture the phase variation in the original curves in this article we precisely define a new type of registration process in which the warping functions optimize the fit of a principal components decomposition to the aligned curves the principal components are effectively the features that this process aligns we discuss the relationship of registration to closure of a function space under convex operations and define consistency for registration methods we define an explicit decomposition of functional variation into amplitude and phase partitions and develop an algorithm for combining registration with principal components analysis and apply it to simulated and real data

locally adaptive bandwidth choice for kernel regression estimators abstract kernel estimators with a global bandwidth are commonly used to estimate regression functions on the other hand it is obvious that the choice of a local bandwidth can lead to better results because a larger class of kernel estimators is available evidently this may in turn affect variability the optimal bandwidths depend essentially on the regression function itself and on the residual variance and it is desirable to estimate them from the data in this article a local bandwidth estimator is studied a comparison with its global bandwidth equivalent is performed both in theory and in simulations as the main result it is shown that the possible gain in mean integrated squared error of the resulting regression estimator must be paid for by a larger variability of the estimator this may lead to worse results if the sample size is small an algorithm has been devised that puts special weight on stability aspects our simulation study shows that improvements over a global bandwidth estimator

functional convex averaging and synchronization for time warped random curves data that can be best described as a sample of curves are now fairly common in science and engineering when the dynamics of development growth or response over time are at issue subjects or experimental units may experience events at different temporal paces for functional data where trajectories may be individually time transformed it is usually inadequate to use commonly used sample statistics such as the cross sectional mean or median or the cross sectional sample variance if one observes time warped curve data i e random curves or random trajectories that exhibit random transformations of the time scale then the usual l2 norm and metric typically are inadequate one may then consider subjecting each observed curve to a time transformation in an attempt to reverse the warping of the time scale before further statistical analysis dynamic time warping alignment curve registration and landmark based methods have been put forward with the goal of finding adequate empirical time transforma

registration for exponential family functional data we consider the problem of aligning curves from exponential family distributions the approach is based on the combination of alignment and functional principal components analysis and is facilitated by recent extensions of fpca to non gaussian settings our work is motivated by the study of physical activity using accelerometers wearable devices that provide around the clock monitoring of activity and produce non gaussian measurements we apply the proposed methods to activity counts using a poisson distribution and to a binary active vs inactive indicator using a binomial distribution after alignment the trajectories show clear peaks of activity in the morning and afternoon with a dip in the middle of the day

stability analyticity and almost best approximation in maximum norm for parabolic finite element equations we consider semidiscrete solutions in quasi uniform finite element spaces of order o hr of the initial boundary value problem with neumann boundary conditions for a second order parabolic differential equation with time independent coefficients in a bounded domain in r n we show that the semigroup on l defined by the semidiscrete solution of the homogeneous equation is bounded and analytic uniformly in h we also show that the semidiscrete solution of the inhomogeneous equation is bounded in the space time l norm modulo a logarithmic factor for r 2 and we give a corresponding almost best approximation property

residual variance and residual pattern in nonlinear regression summary a nonparametric estimator of residual variance in nonlinear regression is proposed it is based on local linear fitting asymptotically the estimator has a small bias but a larger variance compared with the parametric estimator in linear regression finite sample properties are investigated in a simulation study including a comparison with other nonparametric estimators the method is also useful for spotting heteroscedasticity and outliers in the residuals at an early stage of the data analysis a further application is checking the fit of parametric models this is illustrated for longitudinal growth data

searching for structure in curve samples abstract the shape of a regression curve can to a large extent be characterized by the succession of structural features like extrema inflection points and so on when analyzing a sample of regression curves it is often important to know at an early stage of data analysis which structural features are occurring consistently in each curve of the sample such a definition is usually not easy due to substantial interindividual variation both in the x and the y axis and due to the influence of noise a method is proposed for identifying typical features without relying on an a priori specified functional model for the curves the approach is based on the frequencies of occurrence of structural features as for example maxima in the curve sample along the x axis important tools are nonparametric regression and differentiation and kernel density estimation apart from a theoretical foundation the usefulness of the method is documented by application to two interesting biomedical areas growth and develop

a flexible and fast method for automatic smoothing abstract the choice of smoothing parameter or bandwidth is crucial when applying nonparametric regression estimators such as kernel estimators the optimal choice depends on the data at hand a data driven bandwidth selection close to the optimal one would make these curve estimators objective more reliable and easier to use minimizing residual mean squared error criteria such as cross validation have been frequently proposed to estimate the optimal smoothing parameter empirical and theoretical evidence indicates that cross validation rules and related methods lead to rather variable estimated optimal bandwidths the method presented here builds on estimating the asymptotically optimal bandwidth from the data since estimators for the residual variance and for an asymptotic expression for the bias are plugged into the asymptotic formula such selection rules are called plug in estimators the functional that quantifies bias is approximated by the integrated squared second derivative of the regre

bandwidth choice for nonparametric regression this paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression we analyze a tapered fourier series estimate and discuss the relationship of this estimate to a kernel estimate we first consider a method based on an unbiased estimate of mean square error and show that the bandwidth thus chosen is asymptotically optimal other methods are examined as well and are shown to be asymptotically equivalent a small simulation shows however that for small or moderate sample size the methods perform quite differently

statistical tools to analyze data representing a sample of curves the paper is concerned with data representing a sample of smooth curves which can be considered as independent realizations of an underlying biological chemical ldots process such samples of curves often possess the following features there is a typical structural pattern common to all curves of the sample on the other hand individual realizations of the typical shape show different dynamics and intensity in particular typical peaks are shifted from individual to individual differences in dynamics complicate the analysis of samples of curves for example the cross sectional average usually does not reflect an average pattern due to shifts structure is smeared or might even disappear our approach consists in synchronizing the individual curves before determining the average or any further statistics pointwise averaging of the synchronized curves then leads to an average curve which represents the common structure with average dynamics and average intensity the method requires the introduction of new statistical objects they are defined mathematically their properties are discussed and possible estimators are proposed the asymptotic bias and variance of the estimators are derived an application to visually evoked brain potentials illustrates the approach

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

amplitude and phase variation of point processes we develop a canonical framework for the study of the problem of registration of multiple point processes subjected to warping known as the problem of separation of amplitude and phase variation the amplitude variation of a real random function y x x is an element of 0 1 corresponds to its random oscillations in the y axis typically encapsulated by its co variation around a mean level in contrast its phase variation refers to fluctuations in the x axis often caused by random time changes we formalise similar notions for a point process and nonparametrically separate them based on realisations of i i d copies pi i of the phase varying point process a key element in our approach is to demonstrate that when the classical phase variation assumptions of functional data analysis fda are applied to the point process case they become equivalent to conditions interpretable through the prism of the theory of optimal transportation of measure we demonstrate that these induce a natural wasserstein geometry tailored to the warping problem including a formal notion of bias expressing over registration within this framework we construct nonparametric estimators that tend to avoid over registration in finite samples we show that they consistently estimate the warp maps consistently estimate the structural mean and consistently register the warped point processes even in a sparse sampling regime we also establish convergence rates and derivev root n consistency and a central limit theorem in the cox process case under dense sampling showing rate optimality of our structural mean estimator in that case

second order comparison of gaussian random functions and the geometry of dna minicircles given two samples of continuous zero mean iid gaussian processes on 0 1 we consider the problem of testing whether they share the same covariance structure our study is motivated by the problem of determining whether the mechanical properties of short strands of dna are significantly affected by their base pair sequence though expected to be true had so far not been observed in three dimensional electron microscopy data the testing problem is seen to involve aspects of ill posed inverse problems and a test based on a karhunen loeve approximation of the hilbert schmidt distance of the empirical covariance operators is proposed and investigated when applied to a dataset of dna minicircles obtained through the electron microscope our test seems to suggest potential sequence effects on dna shape supplemental material available online

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

pairwise curve synchronization for functional data summary data collected by scientists are increasingly in the form of trajectories or curves often these can be viewed as realizations of a composite process driven by both amplitude and time variation we consider the situation in which functional variation is dominated by time variation and develop a curve synchronization method that uses every trajectory in the sample as a reference to obtain pairwise warping functions in the first step these initial pairwise warping functions are then used to create improved estimators of the underlying individual warping functions in the second step a truncated averaging process is used to obtain robust estimation of individual warping functions the method compares well with other available time synchronization approaches and is illustrated with berkeley growth data and gene expression data for multiple sclerosis

some statistical methods for random process data from seismology and neurophysiology examples are presented of statistical techniques for the analysis of random process data and of their uses in the substantive fields of seismology and neurophysiology the problems addressed include frequency estimation for decaying cosinusoids signal estimation association measurement causal connection assessment estimation of speed and direction and structural modeling the techniques employed include complex demodulation nonlinear regression probit analysis deconvolution maximum likelihood singular value decomposition fourier analysis and averaging

kernel smoothing of data with correlated errors abstract kernel smoothing is a common method of estimating the mean function in the nonparametric regression model y f x e where f x is a smooth deterministic mean function and e is an error process with mean zero in this article the mean squared error of kernel estimators is computed for processes with correlated errors and the estimators are shown to be consistent when the sequence of error processes converges to a mixing sequence the standard techniques for bandwidth selection such as cross validation and generalized cross validation are shown to perform very badly when the errors are correlated standard selection techniques are shown to favor undersmoothing when the correlations are predominantly positive and oversmoothing when negative the selection criteria can however be adjusted to correct for the effect of correlation in simulations the standard selection criteria are shown to behave as predicted the corrected criteria are shown to be very effective when the correlation functi

semiparametric curve alignment and shift density estimation for biological data we observe a large number of signals all of them with identical although unknown shape but with a different random shift the objective is to estimate the individual time shifts and their distribution such an objective appears in several biological applications like neuroscience or ecg signal processing in which the estimation of the distribution of the elapsed time between repetitive pulses with a possibly low signal noise ratio and without a knowledge of the pulse shape is of interest we suggest an m estimator leading to a three stage algorithm we first split our data set in blocks then the shift estimation in each block is done by minimizing a cost function based on the periodogram the estimated shifts are eventually plugged into a standard density estimator we show that under mild regularity assumptions the density estimate converges weakly to the true shift distribution the theory is applied both to simulations and to alignment of real ecg signals the proposed approach outperforms the standard methods for curve alignment and shift density estimation even in the case of low signal to noise ratio and is robust to numerous perturbations common in ecg signals

asymptotically optimal difference based estimation of variance in nonparametric regression summary we define and compute asymptotically optimal difference sequences for estimating error variance in homoscedastic nonparametric regression our optimal difference sequences do not depend on unknowns such as the mean function and provide substantial improvements over the suboptimal sequences commonly used in practice for example in the case of normal data the usual variance estimator based on symmetric second order differences is only 64 efficient relative to the estimator based on optimal second order differences the efficiency of an optimal mth order difference estimator relative to the error sample variance is 2m 2m 1 again this is for normal data and increases as the tails of the error distribution become heavier

a method for the evaluation of radiation q based on modal approach a new formula for the evaluation of the modal radiation q factor is derived the total q of selected structures is to be calculated from the set of eigenmodes with associated eigen energies and eigen powers thanks to the analytical expression of these quantities the procedure is highly accurate respecting arbitrary current densities flowing along the radiating device the electric field integral equation delaunay triangulation method of moments rao wilton glisson basis function and the theory of characteristic modes constitute the underlying theoretical background in terms of the modal radiation q all necessary relations are presented and the essential points of implementation are discussed calculation of the modal energies and q factors enable us to study the effect of the radiating shape separately to the feeding this approach can be very helpful in antenna design a few examples are given including a thin strip dipole two coupled dipoles a bowtie antenna and an electrically small meander folded dipole results are compared with prior estimates and some observations are discussed good agreement is observed for different methods

integral shape averaging and structural average estimation a comparative study classical signal averaging is an important operation in signal processing applications but is meaningless if shape or time variability is present among the signals being averaged structural average estimation sae and integral shape averaging isa are recent methods that deal with these types of variability by providing a structural average for sae and a mean shape signal for isa that can act as a complement to the classical average in signal analysis in this paper these two techniques are regarded from the perspective of the time warping problem we approach isa using a new formalism that emphasizes the theoretical connections between the two techniques a simulation study is then carried out to compare the two methods in three kinds of time fluctuation two sae algorithms are employed namely dynamic time warping dtw and self modeling registration smr the discussion of the results obtained in this comparison offers specific guidelines for the application dependent use of each approach

choice of bandwidth for kernel regression when residuals are correlated summary standard techniques for selecting the bandwidth of a kernel estimator from the data in a nonparametric regression model perform badly when the errors are correlated in this paper we propose a modified version of an existing plug in bandwidth selector the method is generalized to stationary error variables by estimating a functional of the residual covariance function the proposed bandwidth selector shows good properties in asymptotic theory and in simulations without assuming a parametric model for the error process

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