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123
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 root-n consistent estimators, even if only a few observations are made of each function,
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
Smoothing Splines Estimators in Functional Linear Regression with Errors-in-Variables
, 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 67 (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 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 infinite-dimensional data, and there exists a need for the development of adequate statistical estimation and ..."
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Cited by 41 (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 infinite-dimensional 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 so-called 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.
Analysis of Longitudinal Data With Semiparametric Estimation of Covariance Function
, 2005
"... Improving efficiency for regression coefficients and predicting trajectories of individuals are two important aspects in the analysis of longitudinal data. Both involve estimation of the covariance function. Yet challenges arise in estimating the covariance function of longitudinal data collected at ..."
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Cited by 29 (5 self)
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Improving efficiency for regression coefficients and predicting trajectories of individuals are two important aspects in the analysis of longitudinal data. Both involve estimation of the covariance function. Yet challenges arise in estimating the covariance function of longitudinal data collected at irregular time points. A class of semiparametric models for the covariance function by that imposes a parametric correlation structure while allowing a nonparametric variance function is proposed. A kernel estimator for estimating the nonparametric variance function is developed. Two methods for estimating parameters in the correlation structure—a quasi-likelihood approach and a minimum generalized variance method—are proposed. A semiparametric varying coefficient partially linear model for longitudinal data is introduced, and an estimation procedure for model coefficients using a profile weighted least squares approach is proposed. Sampling properties of the proposed estimation procedures are studied, and asymptotic normality of the resulting estimators is established. Finite-sample performance of the proposed procedures is assessed by Monte Carlo simulation studies. The proposed methodology is illustrated with an analysis of a real data example. KEY WORDS: Kernel regression; Local linear regression; Profile weighted least squares; Semiparametric varying coefficient model.
F: Functional additive models
- J Am Stat Assoc
"... In commonly used functional regression models, the regression of a scalar or functional response on the functional predictor is assumed to be linear. This means the response is a linear function of the functional principal component scores of the predictor process. We relax the linearity assumption ..."
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Cited by 29 (8 self)
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In commonly used functional regression models, the regression of a scalar or functional response on the functional predictor is assumed to be linear. This means the response is a linear function of the functional principal component scores of the predictor process. We relax the linearity assumption and propose to replace it by an additive structure. This leads to a more widely applicable and much more flexible framework for functional regression models. The proposed functional additive regression models are suitable for both scalar and functional responses. The regularization needed for effective estimation of the regression parameter function is implemented through a projection on the eigenbasis of the covariance operator of the functional components in the model. The utilization of functional principal components in an additive rather than linear way leads to substantial broadening of the scope of functional regression models and emerges as a natural approach, as the uncorrelatedness of the functional principal components is shown to lead to a straightforward implementation of the functional additive model, just based on a sequence of one-dimensional smoothing steps and without need for backfitting. This facilitates the theoretical analysis, and we establish asymptotic
Modeling Sparse Generalized Longitudinal Observations With Latent Gaussian Processes
, 2007
"... SUMMARY. In longitudinal data analysis one frequently encounters non-Gaussian data that are repeatedly collected for a sample of individuals over time. The repeated observations could be binomial, Poisson or of another discrete type or could be continuous. The timings of the repeated measurements ar ..."
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Cited by 18 (1 self)
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SUMMARY. In longitudinal data analysis one frequently encounters non-Gaussian data that are repeatedly collected for a sample of individuals over time. The repeated observations could be binomial, Poisson or of another discrete type or could be continuous. The timings of the repeated measurements are often sparse and irregular. We introduce a latent Gaussian process model for such data, establishing a connection to functional data analysis. The proposed functional methods are nonparametric and computationally straightforward as they do not involve a likelihood. We develop functional principal components analysis for this situation and demonstrate the prediction of individual trajectories from sparse observations. This method can handle missing data and leads to predictions of the functional principal component scores which serve as random effects in this model. These scores can then be used for further statistical analysis, such as inference, regression, discriminant analysis or clustering. We illustrate these nonparametric methods with longitudinal data on primary biliary cirrhosis and show in simulations that they are competitive in comparisons with Generalized Estimating Equations (GEE) and Generalized Linear Mixed Models (GLMM).
Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data
- Ann. Statist
, 2010
"... We consider nonparametric estimation of the mean and covariance func-tions for functional/longitudinal data. Strong uniform convergence rates are developed for estimators that are local-linear smoothers. Our results are ob-tained in a unified framework in which the number of observations within each ..."
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Cited by 15 (3 self)
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We consider nonparametric estimation of the mean and covariance func-tions for functional/longitudinal data. Strong uniform convergence rates are developed for estimators that are local-linear smoothers. Our results are ob-tained in a unified framework in which the number of observations within each curve/cluster can be of any rate relative to the sample size. We show that the convergence rates for the procedures depend on both the number of sample curves and the number of observations on each curve. For sparse functional data, these rates are equivalent to the optimal rates in nonparamet-ric regression. For dense functional data, root-n rates of convergence can be achieved with proper choices of bandwidths. We further derive almost sure rates of convergence for principal component analysis using the estimated covariance function. The results are illustrated with simulation studies. 1. Introduction. Estimating
Conditional functional principal components analysis
- Scandinavian Journal of Statistics
, 2006
"... This work proposes an extension of the functional principal com-ponents analysis, or Karhunen-Loève expansion, which can take into account non-parametrically the effects of an additional covariate. Such models can also be interpreted as non-parametric mixed effects models for functional data. We pr ..."
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Cited by 13 (3 self)
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This work proposes an extension of the functional principal com-ponents analysis, or Karhunen-Loève expansion, which can take into account non-parametrically the effects of an additional covariate. Such models can also be interpreted as non-parametric mixed effects models for functional data. We propose estimators based on kernel smoothers and a data-driven selection procedure of the smoothing parameters based on a two-steps cross-validation criterion. The conditional func-tional principal components analysis is illustrated with the analysis of a data set consisting of egg laying curves for female fruit flies. Conver-gence rates are given for estimators of the conditional mean function and the conditional covariance operator when the entire curves are col-lected. Almost sure convergence is also proven when one only observes discretized noisy sample paths. A simulation study allows us to check the good behavior of the estimators.
