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Functional data analysis for sparse longitudinal data.
 Journal of the American Statistical Association
, 2005
"... 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 a ..."
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Cited by 123 (24 self)
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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 (subjectspecific) 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. A key step is the derivation of asymptotic consistency and distribution results under mild conditions, using tools from functional analysis. Functional data analysis for sparse longitudinal data enables prediction of individual smooth trajectories even if only one or few measurements are available for a subject. Asymptotic pointwise and simultaneous confidence bands are obtained for predicted individual trajectories, based on asymptotic distributions, for simultaneous bands under the assumption of a finite number of components. Model selection techniques, such as the Akaike information criterion, are used to choose the model dimension corresponding to the number of eigenfunctions in the model. The methods are illustrated with a simulation study, longitudinal CD4 data for a sample of AIDS patients, and timecourse gene expression data for the yeast cell cycle.
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,
Multilevel functional principal component analysis
 Annals of Applied Statistics
, 2009
"... The Sleep Heart Health Study (SHHS) is a comprehensive landmark study of sleep and its impacts on health outcomes. A primary metric of the SHHS is the inhome polysomnogram, which includes two electroencephalographic (EEG) channels for each subject, at two visits. The volume and importance of this d ..."
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Cited by 44 (10 self)
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The Sleep Heart Health Study (SHHS) is a comprehensive landmark study of sleep and its impacts on health outcomes. A primary metric of the SHHS is the inhome polysomnogram, which includes two electroencephalographic (EEG) channels for each subject, at two visits. The volume and importance of this data presents enormous challenges for analysis. To address these challenges, we introduce multilevel functional principal component analysis (MFPCA), a novel statistical methodology designed to extract core intra and intersubject geometric components of multilevel functional data. Though motivated by the SHHS, the proposed methodology is generally applicable, with potential relevance to many modern scientific studies of hierarchical or longitudinal functional outcomes. Notably, using MFPCA, we identify and quantify associations between EEG activity during sleep and adverse cardiovascular outcomes. 1. Introduction.
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 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 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.
Functional and longitudinal data analysis: perspectives on smoothing
 Statist. Sin
, 2004
"... Abstract: The perspectives and methods of functional data analysis and longitudinal data analysis for smoothing are contrasted and compared. Topics include kernel methods and random effects models for smoothing, basis function methods, and examination of correlates of curve shapes. Some directions ..."
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Cited by 34 (0 self)
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Abstract: The perspectives and methods of functional data analysis and longitudinal data analysis for smoothing are contrasted and compared. Topics include kernel methods and random effects models for smoothing, basis function methods, and examination of correlates of curve shapes. Some directions in which methodology might advance are identified. Key words and phrases: Functional data analysis, longitudinal data analysis, nonparametric curve estimation. 1.
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 onedimensional smoothing steps and without need for backfitting. This facilitates the theoretical analysis, and we establish asymptotic
Functional response models
 Statistica Sinica
, 2004
"... 1 Abstract: We review functional regression models and discuss in more detail the situation where the predictor is a vector or scalar such as a dose and the response is a random trajectory. These models incorporate the influence of the predictor either through the mean response function, through the ..."
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Cited by 27 (6 self)
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1 Abstract: We review functional regression models and discuss in more detail the situation where the predictor is a vector or scalar such as a dose and the response is a random trajectory. These models incorporate the influence of the predictor either through the mean response function, through the random components of a KarhunenLoève or functional principal components expansion, or by means of a combination of both. In a case study, we analyze doseresponse data with functional responses from an experiment on the agespecific reproduction of medflies. Daily egglaying was recorded for a sample of 874 medflies in response to dietary dose provided to the flies. We compare several functional response models for these data. A useful criterion to evaluate models is a model’s ability to predict the response at a new dose. We quantify this notion by means of a conditional prediction error that is obtained through a leaveonedoseout technique. Key words and phrases: Doseresponse, eigenfunctions, functional data analysis, functional regression, multiplicative modeling, principal components, smoothing. 2
Timevarying functional regression for predicting remaining lifetime distributions from longitudinal trajectories
 Biometrics
, 2005
"... A recurring objective in longitudinal studies on aging and longevity has been the investigation of the relationship between ageatdeath and current values of a longitudinal covariate trajectory that quantifies reproductive or other behavioral activity. We propose a novel technique for predicting ag ..."
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Cited by 18 (9 self)
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A recurring objective in longitudinal studies on aging and longevity has been the investigation of the relationship between ageatdeath and current values of a longitudinal covariate trajectory that quantifies reproductive or other behavioral activity. We propose a novel technique for predicting ageatdeath distributions for situations where an entire covariate history is included in the predictor. The predictor trajectories up to current time are represented by timevarying functional principal component scores, which are continuously updated as time progresses and are considered to be timevarying predictor variables that are entered into a class of timevarying functional regression models that we propose. We demonstrate for biodemographic data how these methods can be applied to obtain predictions for ageatdeath and estimates of remaining lifetime distributions, including estimates of quantiles and of prediction intervals for remaining lifetime. Estimates and predictions are obtained for individual subjects, based on their observed behavioral trajectories, and include a dimensionreduction step that is implemented by projecting on a single index. The proposed techniques are illustrated with data on longitudinal daily egglaying for female medflies, predicting remaining lifetime and ageatdeath distributions from individual event histories observed up to current time. 1
Modeling Sparse Generalized Longitudinal Observations With Latent Gaussian Processes
, 2007
"... SUMMARY. In longitudinal data analysis one frequently encounters nonGaussian 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 nonGaussian 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).
Functional data analysis for sparse auction data
 In Statistical Methods in eCommerce Research
, 2008
"... Bid arrivals of eBay auctions often exhibit “bid sniping”, a phenomenon where “snipers ” place their bids at the last moments of an auction. This is one reason why bid histories for eBay auctions tend to have sparse data in the middle and denser data both in the beginning and at the end of the aucti ..."
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Cited by 14 (5 self)
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Bid arrivals of eBay auctions often exhibit “bid sniping”, a phenomenon where “snipers ” place their bids at the last moments of an auction. This is one reason why bid histories for eBay auctions tend to have sparse data in the middle and denser data both in the beginning and at the end of the auction. Time spacing of the bids is thus irregular and sparse. For nearly identical products that are auctioned repeatedly, one may view the price history of each of these auctions as realization of an underlying smooth stochastic process, the price process. While the traditional Functional Data Analysis (FDA) approach requires that entire trajectories of the underlying process are observed without noise, this assumption is not satisfied for typical auction data. We provide a review of a recently developed version of functional principal component analysis (Yao et al., 2005), which is geared towards sparse, irregularly observed and noisy data, the principal analysis through conditional expectation (PACE) method. The PACE method borrows and pools information from the sparse data in all auctions. This allows the recovery of the price process even in situations where only few bids are observed. In a modified approach, we adapt PACE to summarize the bid history for varying current times during an ongoing auction through timevarying principal component scores. These scores then serve as timevarying predictors for the closing price. We study the resulting timevarying predictions using both linear regression and generalized additive modelling, with current scores as predictors. These methods will be illustrated with a case study for 157 Palm M515 PDA auctions from eBay, and the proposed methods are seen to work reasonably well. Other related issues will also be discussed. 1 1