two step estimation of functional linear models with applications to longitudinal data functional linear models are useful in longitudinal data analysis they include many classical and recently proposed statistical models for longitudinal data and other functional data recently smoothing spline and kernel methods have been proposed for estimating their coefficient functions nonparametrically but these methods are either intensive in computation or inefficient in performance to overcome these drawbacks in this paper a simple and powerful two step alternative is proposed in particular the implementation of the proposed approach via local polynomial smoothing is discussed methods for estimating standard deviations of estimated coefficient functions are also proposed some asymptotic results for the local polynomial estimators are established two longitudinal data sets one of which involves time dependent covariates are used to demonstrate the approach proposed simulation studies show that our two step approach improves the kernel method proposed by hoover and co workers in several aspects such as accuracy computational time and visual appeal of the estimators

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

truncated linear models for functional data summary a conventional linear model for functional data involves expressing a response variable y in terms of the explanatory function x t via the model where a is a scalar b is an unknown function and is a compact interval however in some problems the support of b or x say is a proper and unknown subset of and is a quantity of particular practical interest motivated by a real data example involving particulate emissions we develop methods for estimating we give particular emphasis to the case where 0 and suggest two methods for estimating a b and jointly we introduce techniques for selecting tuning parameters and we explore properties of our methodology by using both simulation and the real data example mentioned above additionally we derive theoretical properties of the methodology and discuss implications of the theory our theoretical arguments give particular emphasis to the problem of identifiability

functional adaptive model estimation in this article we are interested in modeling the relationship between a scalar y and a functional predictor x t we introduce a highly flexible approach called functional adaptive model estimation fame which extends generalized linear models glms generalized additive models gams and projection pursuit regression ppr to handle functional predictors the fame approach can model any of the standard exponential family of response distributions that are assumed for glm or gam while maintaining the flexibility of ppr for example standard linear or logistic regression with functional predictors as well as far more complicated models can easily be applied using this approach we use a functional principal components decomposition of the predictor functions to aid visualization of the relationship between x t and y we also show how the fame procedure can be extended to deal with multiple functional and standard finite dimensional predictors possibly with missing data we illustrate the fame

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

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

variable domain functional regression for modeling icu data we introduce a class of scalar on function regression models with subject specific functional predictor domains the fundamental idea is to consider a bivariate functional parameter that depends both on the functional argument and on the width of the functional predictor domain both parametric and nonparametric models are introduced to fit the functional coefficient the nonparametric model is theoretically and practically invariant to functional support transformation or support registration methods were motivated by and applied to a study of association between daily measures of the intensive care unit icu sequential organ failure assessment sofa score and two outcomes in hospital mortality and physical impairment at hospital discharge among survivors methods are generally applicable to a large number of new studies that record a continuous variables over unequal domains supplementary materials for this article are available online

logistic regression with brownian like predictors this article introduces a new type of logistic regression model involving functional predictors of binary responses and provides an extension of this approach to generalized linear models the predictors are trajectories that have certain sample path properties in common with brownian motion time points are treated as parameters of interest and confidence intervals are developed under prospective and retrospective case control sampling designs in an application to functional magnetic resonance imaging data signals from individual subjects are used to find the portion of the time course that is most predictive of the response this allows the identification of sensitive time points specific to a brain region and associated with a certain task which can be used to distinguish between responses a second application concerns gene expression data in a case control study involving breast cancer where the aim is to identify genetic loci along a chromosome that best discriminate between cases and controls

structured functional additive regression in reproducing kernel hilbert spaces summary functional additive models provide a flexible yet simple framework for regressions involving functional predictors the utilization of a data driven basis in an additive rather than linear structure naturally extends the classical functional linear model however the critical issue of selecting non linear additive components has been less studied in this work we propose a new regularization framework for structure estimation in the context of reproducing kernel hilbert spaces the approach proposed takes advantage of functional principal components which greatly facilitates implementation and theoretical analysis the selection and estimation are achieved by penalized least squares using a penalty which encourages the sparse structure of the additive components theoretical properties such as the rate of convergence are investigated the empirical performance is demonstrated through simulation studies and a real data application

instrumental variable approach to estimating the scalar on function regression model with measurement error with application to energy expenditure assessment in childhood obesity wearable device technology allows continuous monitoring of biological markers and thereby enables study of time dependent relationships for example in this paper we are interested in the impact of daily energy expenditure over a period of time on subsequent progression toward obesity among children data from these devices appear as either sparsely or densely observed functional data and methods of functional regression are often used for their statistical analyses we study the scalar on function regression model with imprecisely measured values of the predictor function in this setting we have a scalar valued response and a function valued covariate that are both collected at a single time period we propose a generalized method of moments based approach for estimation while an instrumental variable belonging in the same time space as the imprecisely measured covariate is used for model identification additionally no distributional assumptions regarding the measurement errors are assumed while complex covariance structures are allowed for the measurement errors in the implementation of our proposed methods we demonstrate that our proposed estimator is l2 consistent and enjoys the optimal rate of convergence for univariate nonparametric functions in a simulation study we illustrate that ignoring measurement error leads to biased estimations of the functional coefficient the simulation studies also confirm our ability to consistently estimate the function valued coefficient when compared to approaches that ignore potential measurement errors our proposed methods are applied to our motivating example to assess the impact of baseline levels of energy expenditure on body mass index among elementary school aged children

structural condition assessment from white noise excitation and covariance of covariance matrix the sensitivity of covariance of covariancematrix with respect to a local stiffness change in the structural system is derived analytically based on the auto cross correlation function of the acceleration response and the original covariance of covariance formula is improved for use with any number of selected structure modes this paper presents the approach using the improved covariance of covariance matrix of acceleration response for structural condition assessment under white noise support excitation a simply supported 31 bar plane truss with and without 10 measurement noise is studied numerically results show that the sensitivity approach with the covariance of covariance matrix is accurate and when including the finite impulse response forward and backward filtering technique themethod would be suitable for industrial application the covariance of covariancemethod is shown to be insensitive to measurement noise when the measurement time duration is sufficiently long

variable inclusion and shrinkage algorithms the lasso is a popular and computationally efficient procedure for automatically performing both variable selection and coefficient shrinkage on linear regression models one limitation of the lasso is that the same tuning parameter is used for both variable selection and shrinkage as a result it typically ends up selecting a model with too many variables to prevent overshrinkage of the regression coefficients we suggest an improved class of methods called variable inclusion and shrinkage algorithms visa our approach is capable of selecting sparse models while avoiding overshrinkage problems and uses a path algorithm and so also is computationally efficient we show through extensive simulations that visa significantly outperforms the lasso and also provides improvements over more recent procedures such as the dantzig selector relaxed lasso and adaptive lasso in addition we provide theoretical justification for visa in terms of nonasymptotic bounds on the estimation error that suggest it shoul

generalized linear models with functional predictors summary we present a technique for extending generalized linear models to the situation where some of the predictor variables are observations from a curve or function the technique is particularly useful when only fragments of each curve have been observed we demonstrate on both simulated and real data sets how this approach can be used to perform linear logistic and censored regression with functional predictors in addition we show how functional principal components can be used to gain insight into the relationship between the response and functional predictors finally we extend the methodology to apply generalized linear models and principal components to standard missing data problems

semiparametric and nonparametric regression analysis of longitudinal data this article deals with the regression analysis of repeated measurements taken at irregular and possibly subject specific time points the proposed semiparametric and nonparametric models postulate that the marginal distribution for the repeatedly measured response variable y at time t is related to the vector of possibly time varying covariates x through the equations e y t x t 0 t 0x t and e y t x t 0 t 0 t x t where 0 t is an arbitrary function of t 0 is a vector of constant regression coefficients and 0 t is a vector of time varying regression coefficients the stochastic structure of the process y is completely unspecified we develop a class of least squares type estimators for 0 which is proven to be n consistent and asymptotically normal with simple variance estimators furthermore we develop a closed form estimator for a cumulative function of 0 t which is shown to be n consistent and on proper normalization converges weakly to a zero mean gaussian pr

a square root method for the density matrix and its applications to lindblad operators abstract the evolution of open systems subject to both hamiltonian and dissipative forces is studied by writing the nm element of the time t dependent density matrix in the form nm t 1 a 1 a n t m t the so called square root factors the t s are non square matrices and are averaged over a systems of the ensemble this square root description is exact evolution equations are then postulated for the t factors such as to reduce to the lindblad type evolution equations for the diagonal terms in the density matrix for the off diagonal terms they differ from the lindblad equations the square root factors t are not unique and the equations for the t s depend on the specific representation chosen two criteria can be suggested for fixing the choice of t s one is simplicity of the resulting equations and the other has to do with the reduction of the difference between the t formalism and the lindblad equations when the method is tested on cases which have been previously treated by other methods our results agree with them examples chosen are i molecular systems such that are either periodically driven near level degeneracies for which we calculate the decoherence occurring in multiple landau zener transitions or else when undergoing descent around conical intersections in the potential surfaces ii formal dissipative systems with lindblad type operators representing either a non markovian process or a two state system coupled to bosons attractive features of the present factorization method are complete positivity the no higher than linear increase of the implementation effort with the number of states involved and the introduction of randomness only at the start of the process

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

restricted likelihood ratio tests for functional effects in the functional linear model the goal of our article is to provide a transparent robust and computationally feasible statistical approach for testing in the context of scalar on function linear regression models assuming linearity between response and predictors we are interested in testing for the necessity of functional effects our methods are motivated by and applied to a large longitudinal study involving diffusion tensor imaging of intracranial white matter tracts in a susceptible cohort in the context of this study we conduct hypothesis tests that are motivated by anatomical knowledge and support recent findings regarding the relationship between cognitive impairment and white matter demyelination r code and data are in the examples of refund rlrt pfr supplementary materials for this article are available online

generalized functional linear models we propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function a linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function and the expected value of the response is related to this linear predictor via a link function if in addition a variance function is specified this leads to a functional estimating equation which corresponds to maximizing a functional quasi likelihood this general approach includes the special cases of the functional linear model as well as functional poisson regression and functional binomial regression the latter leads to procedures for classification and discrimination of stochastic processes and functional data we also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data using a semiparametric quasi likelihood procedure an essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated karhunen loeve expansion we develop asymptotic inference for the proposed class of generalized regression models in the proposed asymptotic approach the truncation parameter increases with sample size and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics we establish asymptotic normality for a properly scaled distance between estimated and true functions that corresponds to a suitable l 2 metric and is defined through a generalized covariance operator as a consequence we obtain asymptotic tests and simultaneous confidence bands for the parameter function that determines the model the proposed estimation inference and classification procedures and variants with unknown link and variance functions are investigated in a simulation study we find that the practical selection of the number of components works well with the aic criterion and this finding is supported by theoretical considerations we include an application to the classification of medflies regarding their remaining longevity status based on the observed initial egg laying curve for each of 534 female medflies

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

nonparametric smoothing estimates of time varying coefficient models with longitudinal data this paper considers nonparametric estimation in a varying coefficient model with repeated measurements y ij x ij t ij for i 1 n and j 1 n i where x ij x ij x ijk t and y ij x ij t ij denote the jth outcome covariate and time design points respectively of the ith subject the model considered here is y ij x ij t t ij e i t ij where t 0 t k t t for k 0 are smooth nonparametric functions of interest and e i t is a zero mean stochastic process the measurements are assumed to be independent for different subjects but can be correlated at different time points within each subject two nonparametric estimators of t namely a smoothing spline and a locally weighted polynomial are derived for such repeatedly measured data a crossvalidation criterion is proposed for the selection of the corresponding smoothing parameters asymptotic properties such as consistency rates of convergence and asymptotic mean squared errors are established for kernel estimators a special case of the local polynomials these asymptotic results give useful insights into the reliability of our general estimation methods an example of predicting the growth of children born to hiv infected mothers based on gender hiv status and maternal vitamin a levels shows that this model and the corresponding nonparametric estimators are useful in epidemiological studies

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

a generalized dantzig selector with shrinkage tuning summary the dantzig selector performs variable selection and model fitting in linear regression it uses an l 1 penalty to shrink the regression coefficients towards zero in a similar fashion to the lasso while both the lasso and dantzig selector potentially do a good job of selecting the correct variables they tend to overshrink the final coefficients this results in an unfortunate trade off one can either select a high shrinkage tuning parameter that produces an accurate model but poor coefficient estimates or a low shrinkage parameter that produces more accurate coefficients but includes many irrelevant variables we extend the dantzig selector to fit generalized linear models while eliminating overshrinkage of the coefficient estimates and develop a computationally efficient algorithm similar in nature to least angle regression to compute the entire path of coefficient estimates a simulation study illustrates the advantages of our approach relative to others we apply the methodology to two datasets

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

interpretable support vector machines for functional data support vector machines svm has been shown to be a powerful nonparametric classication technique even for high dimensional data although predictive ability is important obtaining an easy to interpret classier

warped functional regression a characteristic feature of functional data is the presence of phase variability in addition to amplitude variability existing functional regression methods do not handle time variability in an explicit and efficient way in this paper we introduce a functional regression method that incorporates time warping as an intrinsic part of the model the method achieves good predictive power in a parsimonious way and allows unified statistical inference about phase and amplitude components the asymptotic distribution of the estimators is derived and their finite sample properties are studied by simulation an application involving ground level ozone trajectories is presented

functional linear regression that s interpretable regression models to relate a scalar y to a functional predictor x t are becoming increasingly common work in this area has concentrated on estimating a coefficient function t with y related to x t through f t x t dt regions where t 0 correspond to places where there is a relationship between x t and y alternatively points where t 0 indicate no relationship hence for interpretation purposes it is desirable for a regression procedure to be capable of producing estimates of t that are exactly zero over regions with no apparent relationship and have simple structures over the remaining regions unfortunately most fitting procedures result in an estimate for t that is rarely exactly zero and has unnatural wiggles making the curve hard to interpret in this article we introduce a new approach which uses variable selection ideas applied to various derivatives of t to produce estimates that are both interpretable flexible and accurate we call our method functional linear regression that s interpretable flirti and demonstrate it on simulated and real world data sets in addition non asymptotic theoretical bounds on the estimation error are presented the bounds provide strong theoretical motivation for our approach

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures