optimal eigen expansions and uniform bounds let x k k in mathbb z in mathbb l 2 mathcal t be a stationary process with associated lag operators varvec mathcal c h uniform asymptotic expansions of the corresponding empirical eigenvalues and eigenfunctions are established under almost optimal conditions on the lag operators in terms of the eigenvalues spectral gap in addition the underlying dependence assumptions are optimal in a certain sense including both short and long memory processes this allows us to study the relative maximum deviation of the empirical eigenvalues under very general conditions among other things convergence to an extreme value distribution is shown we also discuss how the asymptotic expansions transfer to the long run covariance operator varvec mathcal g in a general framework

modeling conditional distributions for functional responses with application to traffic monitoring via gps enabled mobile phones motivated by problems involving a traffic monitoring system in which trajectory data are obtained from global positioning system enabled mobile phones we propose a novel approach to functional regression modeling where instead of the usual mean regression the entire distribution of functional responses is modeled conditionally on predictors an approach that sensibly balances flexibility and stability is obtained by assuming that the response functions are drawn from a gaussian process the mean and covariance function of which depend on predictors the dependence of the mean function and covariance function of the response on the predictors is modeled additively we demonstrate the proposed methods by constructing predicted curves and corresponding prediction regions for traffic velocity trajectories for a future time period using current traffic velocity fields as predictor functions the proposed functional regression and conditional distribution approach is of general interest for functional respon

random walk percolation based modeling of two phase flow in porous media breakthrough time and net to gross ratio estimation abstract fluid flow modeling in porous media has many applications in waste treatment hydrology and petroleum engineering in any geological model flow behavior is controlled by multiple properties these properties must be known in advance of common flow simulations when uncertainties are present deterministic modeling often produces poor results percolation and random walk rw methods have recently been used in flow modeling their stochastic basis is useful in dealing with uncertainty problems they are also useful in finding the relationship between porous media descriptions and flow behavior this paper employs a simple methodology based on random walk and percolation techniques the method is applied to a well defined model reservoir in which the breakthrough time distributions are estimated the results of this method and the conventional simulation are then compared the effect of the net to gross ratio on the breakthrough time distribution is studied in terms of shannon entropy use of the entropy plot allows one to assign the appropriate net to gross ratio to any porous medium

properties of nodal domains in a pseudointegrable barrier billiard we investigate experimentally the scaling properties of the number of nodal domains of the wave functions and the distribution of the nodal domain areas in a pseudointegrable barrier billiard the number of nodal domains is smaller than expected for billiards whose dynamics is chaotic or regular this reduction is explained by the appearance of superscars in the barrier billiard i e regular structures which are embedded into the underlying chaotic wave functions furthermore the area distribution of nodal domains follows the predictions of a percolation model for chaotic systems

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

dispersion operators and resistant second order functional data analysis inferences related to the second order properties of functional data as expressed by covariance structure can become unreliable when the data are non gaussian or contain unusual observations in the functional setting it is often difficult to identify atypical observations as their distinguishing characteristics can be manifold but subtle in this paper we introduce the notion of a dispersion operator investigate its use in probing the second order structure of functional data and develop a test for comparing the second order characteristics of two functional samples that is resistant to atypical observations and departures from normality the proposed test is a regularized m test based on a spectrally truncated version of the hilbert schmidt norm of a score operator defined via the dispersion operator we derive the asymptotic distribution of the test statistic investigate the behaviour of the test in a simulation study and illustrate the method on a structural biology dataset

detecting and localizing differences in functional time series dynamics a case study in molecular biophysics abstractmotivated by the problem of inferring the molecular dynamics of dna in solution and linking them with its base pair composition we consider the problem of comparing the dynamics of functional time series fts and of localizing any inferred differences in frequency and along curvelength the approach we take is one of fourier analysis where the complete second order structure of the fts is encoded by its spectral density operator indexed by frequency and curvelength the comparison is broken down to a hierarchy of stages at a global level we compare the spectral density operators of the two fts across frequencies and curvelength based on a hilbert schmidt criterion then we localize any differences to specific frequencies and finally we further localize any differences along the length of the random curves that is in physical space a hierarchical multiple testing approach guarantees control of the averaged false discovery rate over the selected frequencies in this sense we are ab

effect of anisotropy on finite size scaling in percolation theory we investigate the effects of anisotropy on finite size scaling of site percolation in two dimensions we consider a lattice of size n x ifmmode times else texttimes fi n y we define an aspect ratio ensuremath omega n x n y and consider the mean connected fraction p averaged over the realizations as a function of the site occupancy probability p the system size n x and this aspect ratio it is clear that there is an easy direction for percolation which is in the short direction i e y if ensuremath omega g1 and a difficult direction which is along the long axis we define an apparent percolation threshold in each direction as the value of p when 50 of realizations connect in that direction we show that standard finite size scaling applies if we use this apparent threshold we also find a finite size scaling for the fluctuations about this mean connected fraction

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

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

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

adaptive inference for the mean of a gaussian process in functional data summary we propose and analyse fully data driven methods for inference about the mean function of a gaussian process from a sample of independent trajectories of the process observed at random time points and corrupted by additive random error our methods are based on thresholded least squares estimators relative to an approximating function basis the variable threshold levels are determined from the data and the resulting estimates adapt to the unknown sparsity of the mean function relative to the approximating basis these results are obtained via novel oracle inequalities which are further used to derive the rates of convergence of our mean estimates in addition we construct confidence balls that adapt to the unknown regularity of the mean and covariance function of the stochastic process they are easy to compute since they do not require explicit estimation of the covariance operator of the process a simulation study shows that the new method performs very well in practice and is robust against large variations that may be introduced by the random error terms

bootstrap based inference on the difference in the means of two correlated functional processes we propose nonparametric inference methods on the mean difference between two correlated functional processes we compare methods that 1 incorporate different levels of smoothing of the mean and covariance 2 preserve the sampling design and 3 use parametric and nonparametric estimation of the mean functions we apply our method to estimating the mean difference between average normalized power of sleep electroencephalograms for 51 subjects with severe sleep apnea and 51 matched controls in the first 4 h after sleep onset we obtain data from the sleep heart health study the largest community cohort study of sleep although methods are applied to a single case study they can be applied to a large number of studies that have correlated functional data copyright 2012 john wiley sons ltd

unifying amplitude and phase analysis a compositional data approach to functional multivariate mixed effects modeling of mandarin chinese mandarin chinese is characterized by being a tonal language the pitch or f0 of its utterances carries considerable linguistic information however speech samples from different individuals are subject to changes in amplitude and phase which must be accounted for in any analysis that attempts to provide a linguistically meaningful description of the language a joint model for amplitude phase and duration is presented which combines elements from functional data analysis compositional data analysis and linear mixed effects models by decomposing functions via a functional principal component analysis and connecting registration functions to compositional data analysis a joint multivariate mixed effect model can be formulated which gives insights into the relationship between the different modes of variation as well as their dependence on linguistic and nonlinguistic covariates the model is applied to the cospro 1 dataset a comprehensive database of spoken taiwanese mandarin containing appro

bootstrap based testing of equality of mean functions or equality of covariance operators for functional data we investigate the properties of a simple bootstrap method for testing the equality of mean functions or of covariance operators in functional data theoretical size and power results are derived for certain test statistics whose limiting distributions depend on unknown infinite dimensional parameters simulations demonstrate good size and power of the bootstrap based functional tests

identifying the finite dimensionality of curve time series the curve time series framework provides a convenient vehicle to accommodate some nonstationary features into a stationary setup we propose a new method to identify the dimensionality of curve time series based on the dynamical dependence across different curves the practical implementation of our method boils down to an eigenanalysis of a finite dimensional matrix furthermore the determination of the dimensionality is equivalent to the identification of the nonzero eigenvalues of the matrix which we carry out in terms of some bootstrap tests asymptotic properties of the proposed method are investigated in particular our estimators for zero eigenvalues enjoy the fast convergence rate n while the estimators for nonzero eigenvalues converge at the standard n rate the proposed methodology is illustrated with both simulated and real data sets

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

detecting changes in the mean of functional observations principal component analysis has become a fundamental tool of functional data analysis it represents tne functional data as x i t t 1 i l v l t wnere is the common mean v l are the eigenfunctions of the covariance operator and the i l are the scores inferential procedures assume that the mean function t is the same for all values of i if in fact the observations do not come from one population but rather their mean changes at some point s the results of principal component analysis are confounded by the change s it is therefore important to develop a methodology to test the assumption of a common functional mean we develop such a test using quantities which can be readily computed in the r package fda the null distribution of the test statistic is asymptotically pivotal with a well known asymptotic distribution the asymptotic test has excellent finite sample performance its application is illustrated on temperature data from england

theoretical investigation of finite size effects at dna melting we investigated how the finiteness of the length of a sequence affects the phase transition that takes place at the dna melting temperature for this purpose we modified the transfer integral method to adapt it to the calculation of both extensive partition function entropy specific heat etc and nonextensive order parameter and average separation between paired bases thermodynamic quantities of finite sequences with open boundary conditions and applied the modified procedure to two different dynamical models we characterized in some detail the three effects that take place when the length of the sequence is decreased namely i the decrease of the critical temperature ii the decrease of the peak values of all quantities that diverge at the thermodynamic limit but remain finite for finite sequences like the specific heat and the correlation length and iii the broadening of the temperature range over which the transition affects the dynamics of the system we also performed a finite size scaling analysis of the two models and showed that the singular part of the free energy can indeed be expressed in terms of a homogeneous function however josephson s identity is satisfied for none of the investigated models so that the derivation of the characteristic exponents which appear for example in the expression of the specific heat requires some care

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

outlier detection in functional observations with applications to profile monitoring the presence of outliers has serious adverse effects on the modeling and forecasting of functional data therefore outlier detection aiming at identifying abnormal functional curves from a dataset is quite important this article proposes a new testing procedure based on functional principal component analysis under mild conditions the null distribution of the test statistic is shown to be asymptotically pivotal with a well known asymptotic distribution simulation results demonstrate good finite sample performance of the asymptotic test and detection procedure finally by illustrating the connection between profile monitoring in statistical process control and outlier detection in functional data we apply the proposed approach to a real data example from a manufacturing process and show that it performs quite well in detecting outlying profiles supplementary material for this article is posted online on the journal web site

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

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

effect of anisotropy on the scaling of connectivity and conductivity in continuum percolation theory we investigate the effects of anisotropy on the finite size scaling of connectivity and conductivity of continuum percolation in three dimensions we consider a system of size x ifmmode times else texttimes fi y ifmmode times else texttimes fi z in which cubic bodies of size a ifmmode times else texttimes fi b ifmmode times else texttimes fi c are placed randomly we define two aspect ratios to request anisotropy then we expect that the displacement of average connected fraction p averaged over the realizations about the isotropic universal curves will be a function of the two aspect ratios this is accounted by considering an apparent percolation threshold in each direction which leads to 50 of realizations connecting in that direction we find the aspect ratios dependency of the apparent threshold and investigate the finite size scaling transformations for the mean connected fraction and its associated fluctuations moreover we apply a single phase pressure solver to determine the conductivity of various realizations of the system finally we apply the same idea to account for the effect of anisotropy on the conductivity scaling

nonlinear principal components and long run implications of multivariate diffusions we investigate a method for extracting nonlinear principal components npcs these npcs maximize variation subject to smoothness and orthogonality constraints but we allow for a general class of constraints and multivariate probability densities including densities without compact support and even densities with algebraic tails we provide primitive sufficient conditions for the existence of these npcs by exploiting the theory of continuous time reversible markov diffusion processes we give a different interpretation of these npcs and the smoothness constraints when the diffusion matrix is used to enforce smoothness the npcs maximize long run variation relative to the overall variation subject to orthogonality constraints moreover the npcs behave as scalar autoregressions with heteroskedastic innovations this supports semiparametric identification and estimation of a multivariate reversible diffusion process and tests of the overidentifying restrictions implied by such a process from low frequency data we also explore implications for stationary possibly nonreversible diffusion processes finally we suggest a sieve method to estimate the npcs from discretely sampled data

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

common functional principal components functional principal component analysis fpca based on the karhunen loeve decomposition has been successfully applied in many applications mainly for one sample problems in this paper we con sider common functional principal components for two sample prob lems our research is motivated not only by the theoretical challenge of this data situation but also by the actual question of dynamics of implied volatility iv functions for different maturities the log returns of ivs are samples of smooth random functions and the methods proposed here study the similarities of their stochastic be havior first we present a new method for estimation of functional principal components from discrete noisy data next we present the two sample inference for fpca and develop the two sample theory we propose bootstrap tests for testing the equality of eigenvalues eigenfunctions and mean functions of two functional samples illus trate the test properties by simulation study and apply the method to the iv analysis

a computer assisted study of periodic parabolic systems both theoretical and computational studies of long term behavior for a general class of parabolic systems with asymptotically time periodic coefficients are presented the results are applied to the lotka volterra systems describing interaction of two competing species