elements of a function analytic approach to probability we first provide a detailed motivation for using probability theory as a mathematical context in which to analyze engineering and scientific systems that possess uncertainties we then present introductory notes on the function analytic approach to probabilistic analysis emphasizing the connections to various classical deterministic mathematical analysis elements lastly we describe how to use the approach as a means to augment deterministic analysis methods in a particular hilbert space context and thus enable a rigorous framework for commingling deterministic and probabilistic analysis tools in an application setting

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

symmetric simple exclusion process with free boundaries we consider the one dimensional symmetric simple exclusion process with additional births and deaths restricted to a subset of configurations where there is a leftmost hole and a rightmost particle at a fixed rate birth of particles occur at the position of the leftmost hole and at the same rate independently the rightmost particle dies we prove convergence to a hydrodynamic limit and discuss its relation with a free boundary problem

on consistency and sparsity for principal components analysis in high dimensions principal components analysis pca is a classic method for the reduction of dimensionality of data in the form of n observations or cases of a vector with p variables contemporary datasets often have p comparable with or even much larger than n our main assertions in such settings are a that some initial reduction in dimensionality is desirable before applying any pca type search for principal modes and b the initial reduction in dimensionality is best achieved by working in a basis in which the signals have a sparse representation we describe a simple asymptotic model in which the estimate of the leading principal component vector via standard pca is consistent if and only if p n n 0 we provide a simple algorithm for selecting a subset of coordinates with largest sample variances and show that if pca is done on the selected subset then consistency is recovered even if p n n

existence of positive solutions of halflinear delay differential equation abstract in this paper we consider the halflinear delay differential equation x t 1 x t i 1 n p i t x t i t 1 x t i t 0 for t t 0 and 0 using the method of generalized riccati equation we give conditions for the existence of positive solutions

identification of random shapes from images through polynomial chaos expansion of random level set functions in this paper an efficient method is proposed for the identification of random shapes in a form suitable for numerical simulation within the extended stochastic finite element method x sfem the method starts from a collection of images representing different outcomes of the random shape to identify the key point of the method is to represent the random geometry in an implicit manner using the level set technique in this context the problem of random geometry identification is equivalent to the identification of a random level set function which is a random field this random field is represented on a polynomial chaos pc basis and various efficient numerical strategies are proposed in order to identify the coefficients of its pc decomposition the performance of these strategies is evaluated through some manufactured problems and useful conclusions are provided the propagation of geometrical uncertainties in structural analysis using the x sfem is finally examined copyright 2009 john wiley sons ltd

basis adaptation and domain decomposition for steady state partial differential equations with random coefficients abstract we present a novel approach for solving steady state stochastic partial differential equations in high dimensional random parameter space the proposed approach combines spatial domain decomposition with basis adaptation for each subdomain the basis adaptation is used to address the curse of dimensionality by constructing an accurate low dimensional representation of the stochastic pde solution probability density function and or its leading statistical moments in each subdomain restricting the basis adaptation to a specific subdomain affords finding a locally accurate solution then the solutions from all of the subdomains are stitched together to provide a global solution we support our construction with numerical experiments for a steady state diffusion equation with a random spatially dependent coefficient our results show that accurate global solutions can be obtained with significantly reduced computational costs

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

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

oscillation behavior of second order nonlinear neutral differential equations with deviating arguments oscillation criteria are established for second order nonlinear neutral differential equations with deviating arguments of the form r t z t a 1z t f t x s t 0 t t 0 where a 0 and z t x t p t x t t our results improve and extend some known results in the literature some illustrating examples are also provided to show the importance of our results

oscillation criteria for certain even order quasi linear neutral differential equations with deviating arguments abstract oscillation criteria for nth order neutral differential equations with deviating arguments of the form r t x t p t x t n 1 1 x t p t x t n 1 q t f x t 0 t t 0 n even are established where p q c t 0 r f c r r and 0 0 are constants the results obtained essentially improve and extend those of dzurina and stavroulakis j dzurina i p stavroulakis oscillation criteria for second order delay differential equations appl math comp 140 2003 445 453 sun and meng y g sun f w meng note on the paper of dzurina and stavroulakis appl math comput 174 2006 1634 1641 xu and meng r xu f meng some new oscillation criteria for second order quasi linear neutral delay differential equations appl math comput 182 2006 797 803

nonparametric density estimation from covariate information an increasing number of statistical problems arise in connection with functional calibration in each case inexpensive indirect data in a particular context are combined with direct expensive to acquire data from different but related settings to estimate quantities in the former case sometimes e g in chemometrics problems where spectroscopic calibration is used the indirect data are functional but more commonly they are scalar or vector valued and the functional component is the quantity that we wish to estimate the problem treated here is of the latter type we observe data that give us access to the distribution of u given v and from these and data on u we wish to estimate the density of v the motivating real datasets are of age and covariate information in fish populations we suggest two methodologies each of which is based on transforming the problem to one involving inversion of a symmetric linear operator our techniques have connections to methods for functional data analysis and fo

functional principal component regression and functional partial least squares regression of a scalar response on signal predictors such as near infrared nir spectra of chemical samples presents a major challenge when as is typically the case the dimension of the signals far exceeds their number most solutions to this problem reduce the dimension of the predictors either by regressing on components e g principal component regression pcr and partial least squares pls or by smoothing methods which restrict the coefficient function to the span of a spline basis this article introduces functional versions of pcr and pls which combine both of the foregoing dimension reduction approaches two versions of functional pcr are developed both using b splines and roughness penalties the regularized components version applies such a penalty to the construction of the principal components i e it uses functional principal components whereas the regularized regression version incorporates a penalty in the regression for the latter form of functional pcr the penalty parame

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

oscillation criteria for second order quasilinear neutral delay differential equations abstract oscillation criteria for second order quasilinear delay differential equation r t x t z t 1 z t q t f x t 0 t t 0 where z t x t p t x t and 0 are established under the conditions t 0 r 1 s ds or t 0 r 1 s ds respectively the results obtained here improve and extend some known results in literature

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

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

estimating derivatives for samples of sparsely observed functions with application to online auction dynamics it is often of interest to recover derivatives of a sample of random functions from sparse and noise contaminated measurements especially when the dynamics of underlying processes is of interest we propose a novel approach based on estimating derivatives of eigenfunctions and expansions of random functions into their eigenfunctions to obtain a representation for derivatives in combination with estimates for functional principal component scores for sparse data this leads to a viable solution of the challenging problem to recover derivatives for sparsely observed functions we establish consistency results and demonstrate in simulations that the method is superior to alternative approaches derivative estimation with random effects models based on b spline bases kernel smoothing smoothing splines or p splines our study is motivated by an analysis of bidding histories for ebay auctions for which bids are typically very sparse in the middle and somewhat more frequent near the beginning and end of a

dynamic retrospective regression for functional data samples of curves or functional data usually present phase variability in addition to amplitude variability existing functional regression methods do not handle phase variability in an efficient way in this article we propose a functional regression method that incorporates phase synchronization as an intrinsic part of the model and then attains better predictive power than ordinary linear regression in a simple and parsimonious way the finite sample properties of the estimators are studied by simulation as an example of application we analyze neuromotor data arising from a study of human lip movement this article has supplementary materials online

stochastic model reduction for chaos representations abstract this paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic galerkin schemes for the solution of partial differential equations with stochastic coefficients in particular an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation spde in the hilbert space spanned by wiener chaos a description of the stochastic aspect of the problem on two well separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale with such algorithms at hand the solution of spde s becomes both computationally manageable and efficient moreover a solid foundation is thus provided for the adaptive error control in stochastic galerkin procedures different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics

assessing the finite dimensionality of functional data summary if a problem in functional data analysis is low dimensional then the methodology for its solution can often be reduced to relatively conventional techniques in multivariate analysis hence there is intrinsic interest in assessing the finite dimensionality of functional data we show that this problem has several unique features from some viewpoints the problem is trivial in the sense that continuously distributed functional data which are exactly finite dimensional are immediately recognizable as such if the sample size is sufficiently large however in practice functional data are almost always observed with noise for example resulting from rounding or experimental error then the problem is almost insolubly difficult in such cases a part of the average noise variance is confounded with the true signal and is not identifiable however it is possible to define the unconfounded part of the noise variance this represents the best possible lower bound to all potential values of average noise variance and is estimable in low noise settings moreover bootstrap methods can be used to describe the reliability of estimates of unconfounded noise variance under the assumption that the signal is finite dimensional motivated by these ideas we suggest techniques for assessing the finiteness of dimensionality in particular we show how to construct a critical point 9q such that if the distribution of our functional data has fewer than q 1 degrees of freedom then we should be willing to assume that the average variance of the added noise is at least 9q if this level seems too high then we must conclude that the dimension is at least q 1 we show that simpler more conventional techniques based on hypothesis testing are generally not effective

interval criteria for oscillation of second order functional differential equations by using averaging functions new interval oscillation criteria are established for the second order functional differential equation r t x t a 1x t f t x t x t t x t x t t 0 t t 0that are different from most known ones in the sense that they are based on information only on a sequence of subintervals of t 0 rather than on the whole half line our results can be applied to three cases ordinary delay and advance differential equations in the case of half linear functional differential equations our criteria implies that the t t t delay and gt t t advance cases do not affect the oscillation in particular several examples are given to illustrate the importance of our results

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

assessing extrema of empirical principal component functions the difficulties of estimating and representing the distributions of functional data mean that principal component methods play a substantially greater role in functional data analysis than in more conventional finite dimensional settings local maxima and minima in principal component functions are of direct importance they indicate places in the domain of a random function where influence on the function value tends to be relatively strong but of opposite sign we explore statistical properties of the relationship between extrema of empirical principal component functions and their counterparts for the true principal component functions it is shown that empirical principal component funcions have relatively little trouble capturing conventional extrema but can experience difficulty distinguishing a shoulder in a curve from a small bump for example when the true principal component function has a shoulder the probability that the empirical principal component function has instead a bump is approximately equal to 1 2 we suggest and describe the performance of bootstrap methods for assessing the strength of extrema it is shown that the subsample bootstrap is more effective than the standard bootstrap in this regard a bootstrap likelihood is proposed for measuring extremum strength exploratory numerical methods are suggested

nonequilibrium measures which exhibit a temperature gradient study of a model we give some rules to define measures which could describe heat flow in homogeneous crystals we then study a particular model which is explicitly solvable the one dimensional nearest neighborhood ising model we analyze two cases in the first one the spins at the two boundaries interact with reservoirs at different temperatures in the thermodynamical limit the measure we introduce converges locally to gibbs measures and a temperature profile is so derived we obtain an explicit expression for the thermal conductivity coefficient which depends on the temperature in the second case we study the asymptotic behavior starting from an initial state in which each half of the space is at a different temperature we find again a temperature profile which asymptotically obeys the heat equation with the thermal conductivity coefficient previously derived from a mathematical point of view the analysis of the invariant measure is made possible by studying a time reversed process related to a graphical representation of an associated process this provides us with an explicit formula for then fold correlation function and we study the limiting behavior using both this representation for proving an exchangeability result and a donsker type spacetime renormalization procedure

sampled forms of functional pca in reproducing kernel hilbert spaces space of smooth functions this model includes time and frequency sampling as special cases in contrast to classical approach in fpca in which access to entire functions is assumed having a limited number m of functional samples places limitations on the performance of statistical procedures we study these effects by analyzing the rate of convergence of an m estimator for the subspace spanned by the leading components in a multi spiked covariance model the estimator takes the form of regularized pca and hence is computationally attractive we analyze the behavior of this estimator within a nonasymptotic framework and provide bounds that hold with high probability as a function of the number of statistical samples n and the number of functional samples m we also derive lower bounds showing that the rates obtained are minimax optimal

robust kernel principal component analysis this letter discusses the robustness issue of kernel principal component analysis a class of new robust procedures is proposed based on eigenvalue decomposition of weighted covariance the proposed procedures will place less weight on deviant patterns and thus be more resistant to data contamination and model deviation theoretical influence functions are derived and numerical examples are presented as well both theoretical and numerical results indicate that the proposed robust method outperforms the conventional approach in the sense of being less sensitive to outliers our robust method and results also apply to functional principal component analysis

some new oscillation criteria for second order quasi linear neutral delay differential equations in this paper we will establish some new oscillation criteria for second order quasi linear neutral delay differential equations of the form r t x t p t x email protected a 1 x t p t x email protected q t f x s t 0 the results obtained essentially improve and extend those of dzurina and stavroulakis j dzurina i p stavroulakis oscillation criteria for second order delay differential equations appl math comput 140 2003 445 453 sun and meng y g sun f w meng note on the paper of dzurina and stavroulakis appl math comput 174 2006 1634 1641

variable selection in canonical discriminant analysis for family studies summary in family studies canonical discriminant analysis can be used to find linear combinations of phenotypes that exhibit high ratios of between family to within family variabilities but with large numbers of phenotypes canonical discriminant analysis may overfit to estimate the predicted ratios associated with the coefficients obtained from canonical discriminant analysis two methods are developed one is based on bias correction and the other based on cross validation because the cross validation is computationally intensive an approximation to the cross validation is also developed furthermore these methods can be applied to perform variable selection in canonical discriminant analysis the proposed methods are illustrated with simulation studies and applications to two real examples

robust function on function regression functional linear regression is a widely used approach to model functional responses with respect to functional inputs however classical functional linear regression models can be severely affect

new oscillation criteria for second order nonlinear neutral delay differential equations in this paper several new oscillation criteria for the second order nonlinear neutral delay differential equation r t x t p t x s t m 1 x t p t x s t q t f x t t 0 t t 0 are established these oscillation criteria extend and improve some known results an interesting example illustrating the importance of our results is also provided

the analysis of two way functional data using two way regularized singular value decompositions two way functional data consist of a data matrix whose row and column domains are both structured for example temporally or spatially as when the data are time series collected at different locations in space we extend one way functional principal component analysis pca to two way functional data by introducing regularization of both left and right singular vectors in the singular value decomposition svd of the data matrix we focus on a penalization approach and solve the nontrivial problem of constructing proper two way penalties from one way regression penalties we introduce conditional cross validated smoothing parameter selection whereby left singular vectors are cross validated conditional on right singular vectors and vice versa the concept can be realized as part of an alternating optimization algorithm in addition to the penalization approach we briefly consider two way regularization with basis expansion the proposed methods are illustrated with one simulated and two real data exam

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

oscillation criteria for second order nonlinear neutral differential equations by refining the standard integral averaging technique we obtain new oscillation criteria for a class of second order nonlinear neutral differential equations of the form r t x t p t x email protected q t f x t x s t 0 assumptions in our theorems are less restrictive whereas the proofs are significantly simpler compared to those in the recent paper by shi and wang 18 and related contributions to the subject examples are provided to illustrate the relevance of new theorems

oscillation theorems for nonlinear second order differential equations in this paper some new oscillation criteria are given for the second order nonlinear differential equation r t x t x t c t x t 0 t t0 where r c t0 0 c c t0 r c r r and r r is defined by s s p 2s with p 1 a fixed real number these criteria involve the use of averaging functions

an rkhs formulation of the inverse regression dimension reduction problem suppose that y is a scalar and x is a second order stochastic process where y and x are conditionally independent given the random variables 1 p which belong to the closed span l2 of x this paper investigates a unified framework for the inverse regression dimension reduction problem it is found that the identification of l2 with the reproducing kernel hilbert space of x provides a platform for a seamless extension from the finite to infinite dimensional settings it also facilitates convenient computational algorithms that can be applied to a variety of models

function on function linear regression by signal compression abstractwe consider functional linear regression models with a functional response and multiple functional predictors with the goal of finding the best finite dimensional approximation to the signal part of the response function defining the integrated squared correlation coefficient between a random variable and a random function we propose to solve a penalized generalized functional eigenvalue problem whose solutions satisfy that projections on the original predictors generate new scalar uncorrelated variables and these variables have the largest integrated squared correlation coefficient with the signal function with these new variables we transform the original function on function regression model to a function on scalar regression model whose predictors are uncorrelated and estimate the model by penalized least square method this method is also extended to models with both multiple functional and scalar predictors we provide the asymptotic consistency and the corresponding convergence rates

current reservoirs in the simple exclusion process we consider the symmetric simple exclusion process in the interval n n with additional birth and death processes respectively on n k n k 0 and n n k the exclusion is speeded up by a factor n2 births and deaths by a factor n assuming propagation of chaos a property proved in a companion paper de masi et al http arxiv org abs 1104 3447 we prove convergence in the limit n to the linear heat equation with dirichlet condition on the boundaries the boundary conditions however are not known a priori they are obtained by solving a non linear equation the model simulates mass transport with current reservoirs at the boundaries and the fourier law is proved to hold

a semiparametric change point regression model for longitudinal observations many longitudinal studies involve relating an outcome process to a set of possibly time varying covariates giving rise to the usual regression models for longitudinal data when the purpose of the study is to investigate the covariate effects when experimental environment undergoes abrupt changes or to locate the periods with different levels of covariate effects a simple and easy to interpret approach is to introduce change points in regression coefficients in this connection we propose a semiparametric change point regression model in which the error process stochastic component is nonparametric and the baseline mean function functional part is completely unspecified the observation times are allowed to be subject specific and the number locations and magnitudes of change points are unknown and need to be estimated we further develop an estimation procedure that combines the recent advance in semiparametric analysis based on counting process argument and multiple change points inference an

forecasting age specific breast cancer mortality using functional data models accurate estimates of future age specific incidence and mortality are critical for allocation of resources to breast cancer control programmes and evaluation of screening programmes the purpose of this study is to apply functional data analysis techniques to model age specific breast cancer mortality time trends and forecast entire age specific mortality functions using a state space approach we use annual unadjusted breast cancer mortality rates in australia from 1921 to 2001 in 5 year age groups 45 to 85 we use functional data analysis techniques where mortality and incidence are modelled as curves with age as a functional covariate varying by time data are smoothed using non parametric smoothing methods then decomposed using principal components analysis to estimate basis functions that represent the functional curve period effects from the fitted coefficients are forecast then multiplied by the basis functions resulting in a forecast mortality curve with prediction intervals to forecast we adopt a state space approach and an automatic modelling framework for selecting among exponential smoothing methods overall breast cancer mortality rates in australia remained relatively stable from 1960 to the late 1990s but have declined over the last few years a set of four basis functions minimized the mean integrated squared forecasting error and account for 99 3 per cent of variation around the mean mortality curve twenty year forecasts suggest a continuing decline but at a slower rate and stabilizing beyond 2010 forecasts show a decline in all age groups with the greatest decline in older women the proposed methods have the potential to incorporate important covariates such as hormone replacement therapy and interventions to represent mammographic screening this would be particularly useful for evaluating the impact of screening on mortality and incidence from breast cancer

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

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

remarks on the paper by sun and meng appl math comput 174 2006 in the paper the second order half linear delay differential equation r t u t 1 u t p t u t 1 u t 0 1 is studied under the condition r 1 t d t the oscillation criterion proved by sun and meng 2006 theorem 2 2 is carefully examined and improved by simplifying and sharpening the estimates used in the original proof and removing unnecessary assumptions among others using elementary arguments we show that an arbitrary positive nondecreasing function from this criterion can be safely and with no loss of generality replaced by a constant function an extension to other related equations such as neutral equations is also provided

oscillation theorems for second order nonlinear neutral differential equations the purpose of this paper is to study the oscillation of the second order neutral differential equations of the form a t z t c q t x b s t 0 where z t x t p t x t t we explore properties of given equations by examining those of associated first order delay equations new comparison theorems essentially simplify the examination of the equations studied as they allow us to deduce the oscillation of the second order delay differential equation by applying the oscillation criteria obtained to the first order delay equations the results obtained are easy to verify

penalized spline models for functional principal component analysis summary we propose an iterative estimation procedure for performing functional principal component analysis the procedure aims at functional or longitudinal data where the repeated measurements from the same subject are correlated an increasingly popular smoothing approach penalized spline regression is used to represent the mean function this allows straightforward incorporation of covariates and simple implementation of approximate inference procedures for coefficients for the handling of the within subject correlation we develop an iterative procedure which reduces the dependence between the repeated measurements that are made for the same subject the resulting data after iteration are theoretically shown to be asymptotically equivalent in probability to a set of independent data this suggests that the general theory of penalized spline regression that has been developed for independent data can also be applied to functional data the effectiveness of the proposed procedure is demonstrated via a simulation study and an application to yeast cell cycle gene expression data

oscillation and asymptotic analysis on a new generalized emden fowler equation abstract in this work we analyze the new generalized emden fowler equation with neutral type delays r t z t 1 z t q t x t 1 x t 0 where z t x t p t x t 0 by use of averaging technique and specific analytical skills some easily accessible oscillation and asymptotic criteria are established which have extended the results in the cited literature

localized functional principal component analysis we propose localized functional principal component analysis lfpca looking for orthogonal basis functions with localized support regions that explain most of the variability of a random process the lfpca is formulated as a convex optimization problem through a novel deflated fantope localization method and is implemented through an efficient algorithm to obtain the global optimum we prove that the proposed lfpca converges to the original functional principal component analysis fpca when the tuning parameters are chosen appropriately simulation shows that the proposed lfpca with tuning parameters chosen by cross validation can almost perfectly recover the true eigenfunctions and significantly improve the estimation accuracy when the eigenfunctions are truly supported on some subdomains in the scenario that the original eigenfunctions are not localized the proposed lfpca also serves as a nice tool in finding orthogonal basis functions that balance between interpretability and the capability of exp

oscillation analysis of numerical solution in the θ methods for equation x t ax t a1x t 1 0 abstract the paper deals with the oscillation analysis of numerical solution in the methods for equation x t ax t a 1 x t 1 0 the conditions of the oscillation for the method are obtained it is proved that the oscillation of the analytic solution is preserved by the method it turns out that the zeros of the linear interpolation function of the numerical solution can converge to the zeros of the analytic solution with the order of accuracy 1 1 2 and 2 1 2 some numerical experiments are given

canonical correlation analysis when the data are curves summary it is not immediately straightforward to extend canonical correlation analysis to the context of functional data analysis where the data are themselves curves or functions the obvious approach breaks down and it is necessary to use a method involving smoothing in some way such a method is introduced and discussed with reference to a data set on human gait the breakdown of the unsmoothed method is illustrated in a practical context and is demonstrated theoretically a consistency theorem for the smoothed method is proved in an increasing number of problems in a wide range of fields the data observed are not the univariate or multivariate observations of classical statistics but are func tions observed continuously ramsay and dalzell 1991 give the name functional data analysis to the analysis of data of this kind in most cases the observations will be functions of time or a closely related variable but there are clearly applications where the functions are surfaces observed over two or three dimensional space the motivating example for the present paper is a set of data on children s gait collected by the motion analysis laboratory at children s hospital san diego california see olshen et al 1989 for full details for each of a number of children several angles made by the child s joints knee hip etc are observed during the child s gait cycle one aim of the study is to gain understanding of the gait cycle of a normal child to make comparisons with children suffering from walking diffi culties these data motivated rice and silverman 1991 to discuss the extension of principal component analysis to the functional setting and to explain how smoothing can be incorporated into the analysis in a natural way in a substantial complemen tary paper ramsay and dalzell 1991 discuss different approaches to regression and principal component analysis in functional data analysis illustrating their work by a meteorological example the insights of functional data analysis may be helpful in chemometrics for instance a standard chemometric problem is the analysis of spectra observed in chromatography and these are to all intents and purposes functional observations another obvious area of potential relevance is the analysis of growth curve data the focus in the present paper will be on canonical correlation analysis cca in the gait example one might ask how variability in the knee angle cycle is related to

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

oscillation theorems for second order neutral differential equations the aim of this paper is to study the oscillation of the second order neutral differential equations e r t x t p t x t t q t x s t 0 the obtained results are based on the new comparison theorems that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation the obtained comparison principles essentially simplify the examination of the studied equations

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

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

functional singular component analysis summary aiming at quantifying the dependence of pairs of functional data x y we develop the concept of functional singular value decomposition for covariance and functional singular component analysis building on the concept of canonical expansion of compact operators in functional analysis we demonstrate the estimation of the resulting singular values functions and components for the practically relevant case of sparse and noise contaminated longitudinal data and provide asymptotic consistency results expanding bivariate functional data into singular functions emerges as a natural extension of the popular functional principal component analysis for single processes to the case of paired processes a natural application of the functional singular value decomposition is a measure of functional correlation owing to the involvement of an inverse operation most previously considered functional correlation measures are plagued by numerical instabilities and strong sensitivity to the choice of smoothing parameters these problems are exacerbated for the case of sparse longitudinal data on which we focus the functional correlation measure that is derived from the functional singular value decomposition behaves well with respect to numerical stability and statistical error as we demonstrate in a simulation study practical feasibility for applications to longitudinal data is illustrated with examples from a study on aging and on line auctions

reduced rank mixed effects models for spatially correlated hierarchical functional data hierarchical functional data are widely seen in complex studies where subunits are nested within units which in turn are nested within treatment groups we propose a general framework of functional mixed effects model for such data within unit and within subunit variations are modeled through two separate sets of principal components the subunit level functions are allowed to be correlated penalized splines are used to model both the mean functions and the principal components functions where roughness penalties are used to regularize the spline fit an expectation maximization em algorithm is developed to fit the model while the specific covariance structure of the model is utilized for computational efficiency to avoid storage and inversion of large matrices our dimension reduction with principal components provides an effective solution to the difficult tasks of modeling the covariance kernel of a random function and modeling the correlation between functions the proposed methodology is illus

interval oscillation of second order half linear functional differential equations by using an inequality due to hardy littlewood and polya and averaging functions new interval oscillation criteria are established for the half linear functional differential equationr t y t a 1y t q t y t t a 1y t t 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results extend and improve some previous oscillation criteria and handle the cases which are not covered by known results and implies that the delay t t t t does not effect the oscillation where t 0 is a constant in particular several examples that dwell upon the sharp conditions of our results are also included

free knot spline smoothing for functional data summary the paper introduces free knot regression spline estimators for the mean and the variance components of a sample of curves the asymptotic distribution of the mean estimator is derived and asymptotic confidence bands are constructed a comparative simulation study shows that free knot splines estimate salient features of the functions such as sharp peaks more accurately than smoothing splines this adaptive behaviour is also illustrated by an analysis of weather data

oscillation criteria for second order delay differential equations the aim of this paper is to establish some new oscillation criteria for the second order retarded differential equation r t u t a 1u t p t u t t a 1u t t 0 the results obtained essentially improve known results in the literature

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

current large deviations for asymmetric exclusion processes with open boundaries we study the large deviation functional of the current for the weakly asymmetric simple exclusion process in contact with two reservoirs we compare this functional in the large drift limit to the one of the totally asymmetric simple exclusion process in particular to the jensen varadhan functional conjectures for generalizing the jensen varadhan functional to open systems are also stated

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

functional samples and bootstrap for predicting sulfur dioxide levels in this article we give enhancements of several functional techniques to forecast sulfur dioxide levels near a power plant the data are considered as a time series of curves assuming a lag one dependence the predictions are computed using the functional kernel with local bandwith and the linear autoregressive hilbertian model we carry out the estimation with a so called historical matrix which is a subsample that emphasizes uncommon shapes we introduce a bootstrap method to evaluate the range of the forecasts which uses fraiman and muniz s order for functional data finally we compare our functional techniques with neural networks and semiparametric methods and find that the former models are often more effective

kamenev type and interval oscillation criteria for second order linear differential equations on a measure chain abstract we establish kamenev type criteria and interval criteria for oscillation of the second order scalar differential equation p t x t q t x t 0 on a measure chain our results cover those for differential equations and provide new oscillation criteria for difference equations several examples are given to show the significance of the results

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

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

robust functional principal components a projection pursuit approach in many situations data are recorded over a period of time and may be regarded as realizations of a stochastic process in this paper robust estimators for the principal components are considered by adapting the projection pursuit approach to the functional data setting our approach combines robust projection pursuit with different smoothing methods consistency of the estimators are shown under mild assumptions the performance of the classical and robust procedures are compared in a simulation study under different contamination schemes 1 introduction analogous to classical principal components analysis pca the projection pursuit approach to robust pca is based on finding projections of the data which have maximal dispersion instead of using the variance as a measure of dispersion a robust scale estimator sn is used for the maximization problem this approach was introduced by li and chen 1985 who proposed estimators based on maximizing or minimizing a robust scale in this way given a sample xi r d 1 i n the first robust principal component vector is defined as

generalized spectral decomposition method for solving stochastic finite element equations invariant subspace problem and dedicated algorithms stochastic galerkin methods have become a significant tool for the resolution of stochastic partial differential equations spde however they suffer from prohibitive computational times and memory requirements when dealing with large scale applications and high stochastic dimensionality some alternative techniques based on the construction of suitable reduced deterministic or stochastic bases have been proposed in order to reduce these computational costs recently a new approach based on the concept of generalized spectral decomposition gsd has been introduced for the definition and the automatic construction of reduced bases in this paper the concept of gsd initially introduced for a class of linear elliptic spde is extended to a wider class of stochastic problems the proposed definition of the gsd leads to the resolution of an invariant subspace problem which is interpreted as an eigen like problem this interpretation allows the construction of efficient numerical algorithms for building optimal reduced bases which are associated with dominant generalized eigenspaces the proposed algorithms by separating the resolution of reduced stochastic and deterministic problems lead to drastic computational savings their efficiency is illustrated on several examples where they are compared to classical resolution techniques

large deviation of the density profile in the steady state of the open symmetric simple exclusion process we consider an open one dimensional lattice gas on sites i 1 n with particles jumping independently with rate 1 to neighboring interior empty sites the simple symmetric exclusion process the particle fluxes at the left and right boundaries corresponding to exchanges with reservoirs at different chemical potentials create a stationary nonequilibrium state sns with a steady flux of particles through the system the mean density profile in this state which is linear describes the typical behavior of a macroscopic system i e this profile occurs with probability 1 when n the probability of microscopic configurations corresponding to some other profile x x i n has the asymptotic form exp n f f is the large deviation functional in contrast to equilibrium systems for which f eq is just the integral of the appropriately normalized local free energy density the f we find here for the nonequilibrium system is a nonlocal function of this gives rise to the long range correlations in the sns predicted by fluctuating hydrodynamics and suggests similar non local behavior of f in general sns where the long range correlations have been observed experimentally

non equilibrium current fluctuations in stochastic lattice gases we study current fluctuations in lattice gases in the macroscopic limit extending the dynamic approach for density fluctuations developed in previous articles more precisely we establish a large deviation principle for a space time fluctuation j of the empirical current with a rate functional i j we then estimate the probability of a fluctuation of the average current over a large time interval this probability can be obtained by solving a variational problem for the functional i we discuss several possible scenarios interpreted as dynamical phase transitions for this variational problem they actually occur in specific models we finally discuss the time reversal properties of i and derive a fluctuation relationship akin to the gallavotti cohen theorem for the entropy production