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45
Generalized functional linear models
 Ann. Statist
, 2005
"... 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 expe ..."
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Cited by 40 (6 self)
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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 quasilikelihood. 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 quasilikelihood procedure. An essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated KarhunenLoève 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
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 26 (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.
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 26 (3 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
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 24 (7 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.
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 12 (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
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 11 (5 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
Principal component estimation of functional logistic regression: discussion of two different approaches
 Journal of Nonparametric Statistics
, 2004
"... Over the last few years many methods have been developed for analyzing functional data with different objectives. The purpose of this paper is to predict a binary response variable in terms of a functional variable whose sample information is given by a set of curves measured without error. In order ..."
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Cited by 11 (1 self)
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Over the last few years many methods have been developed for analyzing functional data with different objectives. The purpose of this paper is to predict a binary response variable in terms of a functional variable whose sample information is given by a set of curves measured without error. In order to solve this problem we formulate a functional logistic regression model and propose its estimation by approximating the sample paths in a finite dimensional space generated by a basis. Then, the problem is reduced to a multiple logistic regression model with highly correlated covariates. In order to reduce dimension and to avoid multicollinearity, two different approaches of functional principal component analysis of the sample paths are proposed. Finally, a simulation study for evaluating the estimating performance of the proposed principal component approaches is developed.
Dynamic Profiling of Online Auctions Using Curve Clustering”, Working
, 2003
"... Electronic commerce, and in particular online auctions, have received an extreme surge of popularity in recent years. While auction theory has been studied for a long time from a gametheory perspective, the electronic implementation of the auction mechanism poses new and challenging research questi ..."
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Cited by 10 (6 self)
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Electronic commerce, and in particular online auctions, have received an extreme surge of popularity in recent years. While auction theory has been studied for a long time from a gametheory perspective, the electronic implementation of the auction mechanism poses new and challenging research questions. Although the body of empirical research on online auctions is growing, there is a lack of treatment of these data from a modern statistical point of view. In this work, we present a new source of rich auction data and introduce an innovative way of modelling and analyzing online bidding behavior. In particular, we use functional data analysis to investigate and scrutinize online auction dynamics. We describe the structure of such data and suggest suitable methods, including data smoothing and curve clustering, that allow one to profile online auctions and display different bidding behavior. We illustrate the methods on a set of eBay auction data and tie our results to the existing literature on online auctions. Key words and phrases: functional data analysis, smoothing, penalized splines, clustering, unsupervised
FUNCTIONAL LINEAR REGRESSION THAT’S INTERPRETABLE 1
"... 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, β(t), with Y related to X(t) through β(t)X(t)dt. Regions where β(t) ̸ = 0 correspond to places where there is a relationshi ..."
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Cited by 8 (3 self)
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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, β(t), with Y related to X(t) through β(t)X(t)dt. Regions where β(t) ̸ = 0 correspond to places where there is a relationship between X(t) and Y. Alternatively, points where β(t) = 0indicate no relationship. Hence, for interpretation purposes, it is desirable for a regression procedure to be capable of producing estimates of β(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 β(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 β(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 realworld data sets. In addition, nonasymptotic theoretical bounds on the estimation error are presented. The bounds provide strong theoretical motivation for our approach.
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 8 (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