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105
Properties of principal component methods for functional and longitudinal data analysis
 Ann. Statist
, 2006
"... The use of principal component methods to analyze functional data is appropriate in a wide range of different settings. In studies of “functional data analysis, ” it has often been assumed that a sample of random functions is observed precisely, in the continuum and without noise. While this has bee ..."
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Cited by 73 (5 self)
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The use of principal component methods to analyze functional data is appropriate in a wide range of different settings. In studies of “functional data analysis, ” it has often been assumed that a sample of random functions is observed precisely, in the continuum and without noise. While this has been the traditional setting for functional data analysis, in the context of longitudinal data analysis a random function typically represents a patient, or subject, who is observed at only a small number of randomly distributed points, with nonnegligible measurement error. Nevertheless, essentially the same methods can be used in both these cases, as well as in the vast number of settings that lie between them. How is performance affected by the sampling plan? In this paper we answer that question. We show that if there is a sample of n functions, or subjects, then estimation of eigenvalues is a semiparametric problem, with rootn consistent estimators, even if only a few observations are made of each function,
Prediction in functional linear regression
, 2006
"... There has been substantial recent work on methods for estimating the slope function in linear regression for functional data analysis. However, as in the case of more conventional finitedimensional regression, much of the practical interest in the slope centers on its application for the purpose of ..."
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Cited by 72 (5 self)
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There has been substantial recent work on methods for estimating the slope function in linear regression for functional data analysis. However, as in the case of more conventional finitedimensional regression, much of the practical interest in the slope centers on its application for the purpose of prediction, rather than on its significance in its own right. We show that the problems of slopefunction estimation, and of prediction from an estimator of the slope function, have very different characteristics. While the former is intrinsically nonparametric, the latter can be either nonparametric or semiparametric. In particular, the optimal meansquare convergence rate of predictors is n −1, where n denotes sample size, if the predictand is a sufficiently smooth function. In other cases, convergence occurs at a polynomial rate that is strictly slower than n −1. At the boundary between these two regimes, the meansquare convergence rate is less than n −1 by only a logarithmic factor. More generally, the rate of convergence of the predicted value of the mean response in the regression model, given a particular value of the explanatory variable, is determined by a subtle interaction among the smoothness of the predictand, of the slope function in the model, and of the autocovariance function for the distribution of explanatory variables. 1. Introduction. In
Smoothing Splines Estimators in Functional Linear Regression with ErrorsinVariables
, 2006
"... This work deals with a generalization of the Total Least Squares method in the context of the functional linear model. We first propose a smoothing splines estimator of the functional coefficient of the model without noise in the covariates and we obtain an asymptotic result for this estimator. Then ..."
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Cited by 69 (3 self)
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This work deals with a generalization of the Total Least Squares method in the context of the functional linear model. We first propose a smoothing splines estimator of the functional coefficient of the model without noise in the covariates and we obtain an asymptotic result for this estimator. Then, we adapt this estimator to the case where the covariates are noisy and we also derive an upper bound for the convergence speed. Our estimation procedure is evaluated by means of simulations.
Functional linear regression analysis for longitudinal data
 Ann. of Statist
, 2005
"... We propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data, where both the predictor and response are functions of a covariate such as time. Predictor and response processes have smooth random trajectories, and the data consist of a small number ..."
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Cited by 69 (7 self)
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We propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data, where both the predictor and response are functions of a covariate such as time. Predictor and response processes have smooth random trajectories, and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects. In longitudinal studies, the number of repeated measurements per subject is often small and may be modeled as a discrete random number and, accordingly, only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit. We propose a functional regression approach for this situation, using functional principal component analysis, where we estimate the functional principal component scores through conditional expectations. This allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory. The resulting technique is flexible
Functional adaptive model estimation
 J. Amer
, 2005
"... 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 (GLM), generalized additive models (GAM) and proj ..."
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Cited by 39 (8 self)
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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 (GLM), generalized additive models (GAM) 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. A functional principal components decomposition of the predictor functions is used 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. The FAME approach is illustrated on simulated data as well as on the prediction of arthritis based on bone shape. We end with a discussion of the relationships between standard regression approaches, their extensions to functional data and FAME.
Estimation in generalized linear models for functional data via penalized likelihood
 Journal of Multivariate Analysis
, 2005
"... We analyze in a regression setting the link between a scalar response and a functional predictor by means of a Functional Generalized Linear Model. We first give a theoretical framework and then discuss identifiability of the model. The functional coefficient of the model is estimated via penalized ..."
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Cited by 39 (0 self)
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We analyze in a regression setting the link between a scalar response and a functional predictor by means of a Functional Generalized Linear Model. We first give a theoretical framework and then discuss identifiability of the model. The functional coefficient of the model is estimated via penalized likelihood with spline approximation. The L² rate of convergence of this estimator is given under smoothness assumption on the functional coefficient. Heuristic arguments show how these rates may be improved for some particular frameworks.
2012, ‘Augmented sparse principal component analysis for high dimensional data’. arXiv preprint arXiv:1202.1242
"... Principal components analysis (PCA) has been a widely used technique in reducing dimensionality of multivariate data. A traditional setting where PCA is applicable is when one has repeated observations from a multivariate population that can be described reasonably well by its first two moments. Wh ..."
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Cited by 38 (6 self)
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Principal components analysis (PCA) has been a widely used technique in reducing dimensionality of multivariate data. A traditional setting where PCA is applicable is when one has repeated observations from a multivariate population that can be described reasonably well by its first two moments. When the dimension of sample observations, is fixed, distributional
Functional generalized additive models
 Journal of Computational and Graphical Statistics
, 2014
"... We introduce the functional generalized additive model (FGAM), a novel regression model for association studies between a scalar response and a functional predictor. We model the linktransformed mean response as the integral with respect to t of F{X(t), t} where F (·, ·) is an unknown regression fu ..."
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Cited by 12 (3 self)
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We introduce the functional generalized additive model (FGAM), a novel regression model for association studies between a scalar response and a functional predictor. We model the linktransformed mean response as the integral with respect to t of F{X(t), t} where F (·, ·) is an unknown regression function and X(t) is a functional covariate. Rather than having an additive model in a finite number of principal components as in Müller and Yao (2008), our model incorporates the functional predictor directly and thus our model can be viewed as the natural functional extension of generalized additive models. We estimate F (·, ·) using tensorproduct Bsplines with roughness penalties. A pointwise quantile transformation of the functional predictor is also considered to ensure each tensorproduct Bspline has observed data on its support. The methods are evaluated using simulated data and their predictive performance is compared with other competing scalaronfunction regression alternatives. We illustrate the usefulness of our approach through an application to brain tractography, where X(t) is a signal from diffusion tensor imaging at position, t, along a tract in the brain. In one example, the response is diseasestatus (case or control) and in a second example, it
doi:http://dx.doi.org/10.5705/ss.2010.034 A SIMULTANEOUS CONFIDENCE BAND FOR SPARSE LONGITUDINAL REGRESSION
"... Abstract: Functional data analysis has received considerable recent attention and a number of successful applications have been reported. In this paper, asymptotically simultaneous confidence bands are obtained for the mean function of the functional regression model, using piecewise constant spline ..."
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Cited by 9 (3 self)
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Abstract: Functional data analysis has received considerable recent attention and a number of successful applications have been reported. In this paper, asymptotically simultaneous confidence bands are obtained for the mean function of the functional regression model, using piecewise constant spline estimation. Simulation experiments corroborate the asymptotic theory. The confidence band procedure is illustrated by analyzing CD4 cell counts of HIV infected patients. Key words and phrases: B spline, confidence band, functional data, KarhunenLoève L 2 representation, knots, longitudinal data, strong approximation. 1.
CLT in functional linear regression models
 Probab. Theory Related Fields
, 2007
"... We propose in this work to derive a CLT in the functional linear regression model. The main difficulty is due to the fact that estimation of the functional parameter leads to a kind of illposed inverse problem. We consider estimators that belong to a large class of regularizing methods and we first ..."
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Cited by 9 (1 self)
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We propose in this work to derive a CLT in the functional linear regression model. The main difficulty is due to the fact that estimation of the functional parameter leads to a kind of illposed inverse problem. We consider estimators that belong to a large class of regularizing methods and we first show that, contrary to the multivariate case, it is not possible to state a CLT in the topology of the considered functional space. However, we show that we can get a CLT for the weak topology under mild hypotheses and in particular without assuming any strong assumptions on the decay of the eigenvalues of the covariance operator. Rates of convergence depend on the smoothness of the functional coefficient and on the point in which the prediction is made.