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Functional Modeling and Classification of Longitudinal Data
"... We review and extend some statistical tools that have proved useful for analyzing functional data. Functional data analysis primarily is designed for the analysis of random trajectories and infinitedimensional data, and there exists a need for the development of adequate statistical estimation and ..."
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Cited by 40 (11 self)
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We review and extend some statistical tools that have proved useful for analyzing functional data. Functional data analysis primarily is designed for the analysis of random trajectories and infinitedimensional data, and there exists a need for the development of adequate statistical estimation and inference techniques. While this field is in flux, some methods have proven useful. These include warping methods, functional principal component analysis, and conditioning under Gaussian assumptions for the case of sparse data. The latter is a recent development that may provide a bridge between functional and more classical longitudinal data analysis. Besides presenting a brief review of functional principal components and functional regression, we develop some concepts for estimating functional principal component scores in the sparse situation. An extension of the socalled generalized functional linear model to the case of sparse longitudinal predictors is proposed. This extension includes functional binary regression models for longitudinal data and is illustrated with data on primary biliary cirrhosis.
Di: Generalized Multilevel Functional Regression 1561
 Journal of the Royal Statistical Society, Ser. B
, 2006
"... We introduce Generalized Multilevel Functional Linear Models (GMFLMs), a novel statistical framework for regression models where exposure has a multilevel functional structure. We show that GMFLMs are, in fact, generalized multilevel mixed models. Thus, GMFLMs can be analyzed using the mixed effects ..."
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Cited by 18 (9 self)
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We introduce Generalized Multilevel Functional Linear Models (GMFLMs), a novel statistical framework for regression models where exposure has a multilevel functional structure. We show that GMFLMs are, in fact, generalized multilevel mixed models. Thus, GMFLMs can be analyzed using the mixed effects inferential machinery and can be generalized within a wellresearched statistical framework. We propose and compare two methods for inference: (1) a twostage frequentist approach; and (2) a joint Bayesian analysis. Our methods are motivated by and applied to the Sleep Heart Health Study, the largest community cohort study of sleep. However, our methods are general and easy to apply to a wide spectrum of emerging biological and medical datasets. Supplemental materials for this article are available online.
Detecting changes in the mean of functional observations
"... Principal component analysis (PCA) has become a fundamental tool of functional data analysis. It represents the functional data as Xi(t) = µ(t) + 1≤`< ∞ ηi,`v`(t), where µ is the common mean, v ` are the eigenfunctions of the covariance operator, and the ηi, ` are the scores. Inferential procedu ..."
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Cited by 10 (4 self)
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Principal component analysis (PCA) has become a fundamental tool of functional data analysis. It represents the functional data as Xi(t) = µ(t) + 1≤`< ∞ ηi,`v`(t), where µ is the common mean, v ` are the eigenfunctions of the covariance operator, and the ηi, ` are the scores. Inferential procedures assume that the mean function µ(t) is the same for all values of i. If, in fact, the observations do not come from one population, but rather their mean changes at some point(s), the results of PCA are confounded by the change(s). It is therefore important to develop a methodology to test the assumption of a common functional mean. We develop such a test using quantities which can be readily computed in the R package fda. The null distribution of the test statistic is asymptotically pivotal with a wellknown asymptotic distribution. The asymptotic test has excellent finite sample performance. Its application is illustrated on temperature data from Prague, England and Greenland.
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.
Asymptotic equivalence of functional linear regression and a white noise inverse problem
 Ann. Statist
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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 tw ..."
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Cited by 4 (0 self)
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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 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 illustrated using simulations and an empirical data set from a colon carcinogenesis study. Supplemental
Local linear regression for functional data, submitted
, 2008
"... We study a non linear regression model with functional data as inputs and scalar response. We propose a pointwise estimate of the regression function that maps a Hilbert space onto the real line by a local linear method. We provide the asymptotic mean square error. Computations involve a linear inve ..."
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Cited by 4 (2 self)
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We study a non linear regression model with functional data as inputs and scalar response. We propose a pointwise estimate of the regression function that maps a Hilbert space onto the real line by a local linear method. We provide the asymptotic mean square error. Computations involve a linear inverse problem as well as a representation of the small ball probability of the data and are based on recent advances in this area. The rate of convergence of our estimate outperforms those already obtained in the literature on this model.
Using basis expansions for estimating functional PLS regression. Applications with chemometric data
, 2010
"... There are many chemometric applications, such as spectroscopy, where the objective is to explain a scalar response from a functional variable (the spectrum) whose observations are functions of wavelengths rather than vectors. In this paper, PLS regression is considered for estimating the linear mode ..."
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Cited by 4 (0 self)
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There are many chemometric applications, such as spectroscopy, where the objective is to explain a scalar response from a functional variable (the spectrum) whose observations are functions of wavelengths rather than vectors. In this paper, PLS regression is considered for estimating the linear model when the predictor is a functional random variable. Due to the infinite dimension of the space to which the predictor observations belong, they are usually approximated by curves/functions within a finite dimensional space spanned by a basis of functions. We show that PLS regression with a functional predictor is equivalent to finite multivariate PLS regression using expansion basis coefficients as the predictor, in the sense that, at each step of the PLS iteration, the same prediction is obtained. In addition, from the linear model estimated using the basis coefficients, we derive the expression of the PLS estimate of the regression coefficient function from the model with a functional predictor. The results provided by this functional PLS approach are compared with those given by functional PCR and discrete PLS and PCR using different sets of simulated and spectrometric data.
Functional Modeling of Longitudinal Data
, 2006
"... Functional data analysis provides an inherently nonparametric approach for the analysis of data which consist of samples of time courses or random trajectories. It is a relatively young field aiming at modeling and data exploration under very flexible model assumptions with no or few parametric comp ..."
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Cited by 3 (0 self)
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Functional data analysis provides an inherently nonparametric approach for the analysis of data which consist of samples of time courses or random trajectories. It is a relatively young field aiming at modeling and data exploration under very flexible model assumptions with no or few parametric components. Basic tools of functional data analysis are smoothing, functional principal components, functional linear models and timewarping. Warping or curve registration aims at adjusting for random time distortions. While in the usual functional data analysis paradigm the sample functions were considered as continuously observed, in longitudinal data analysis one mostly deals with sparsely and irregularly observed data that also are corrupted with noise. Adjustments of functional data analysis techniques which take these particular features into account are needed to use them to advantage for longitudinal data. We review some techniques that have been recently proposed to connect functional data analysis methodology with longitudinal data. The extension of functional data analysis towards longitudinal data is a fairly recent undertaking that presents a promising avenue for future research. This article provides a review of some of the recent developments.
Designadaptive minimax local linear regression for longitudinal/clustered data
 Statist. Sin
, 2008
"... Abstract: This paper studies a weighted local linear regression smoother for longitudinal/clustered data, which takes a form similar to the classical weighted least squares estimate. As a hybrid of the methods of Chen and Jin (2005) and Wang (2003), the proposed local linear smoother maintains the a ..."
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Cited by 2 (0 self)
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Abstract: This paper studies a weighted local linear regression smoother for longitudinal/clustered data, which takes a form similar to the classical weighted least squares estimate. As a hybrid of the methods of Chen and Jin (2005) and Wang (2003), the proposed local linear smoother maintains the advantages of both methods in computational and theoretical simplicity, variance minimization and bias reduction. Moreover, the proposed smoother is optimal in the sense that it attains linear minimax efficiency when the withincluster correlation is correctly specified. In the special case that the joint density of covariates in a cluster exists and is continuous, any working withincluster correlation would lead to linear minimax efficiency for the proposed method. Key words and phrases: Asymptotic bias, generalized estimating equations, kernel function, linear minimax efficiency, mean squared error, nonparametric curve estimation. 1.