Results 1  10
of
39
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 ..."
Abstract

Cited by 40 (11 self)
 Add to MetaCart
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 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 ..."
Abstract

Cited by 38 (4 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 29 (8 self)
 Add to MetaCart
(Show Context)
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
Curve alignment by moments
 Annals of Applied Statistics
"... A significant problem with most functional data analyses is that of misaligned curves. Without adjustment, even an analysis as simple as estimation of the mean will fail. One common method to synchronize a set of curves involves equating “landmarks ” such as peaks or troughs. The landmarks method ca ..."
Abstract

Cited by 19 (1 self)
 Add to MetaCart
A significant problem with most functional data analyses is that of misaligned curves. Without adjustment, even an analysis as simple as estimation of the mean will fail. One common method to synchronize a set of curves involves equating “landmarks ” such as peaks or troughs. The landmarks method can work well but will fail if marker events can not be identified or are missing from some curves. An alternative approach, the “continuous monotone registration ” method, works by transforming the curves so that they are as close as possible to a target function. This method can also perform well but is highly dependent on identifying an accurate target function. We develop an alignment method based on equating the “moments ” of a given set of curves. These moments are intended to capture the locations of important features which may represent local behavior, such as maximums and minimums, or more global characteristics, such as the slope of the curve averaged over time. Our method works by equating the moments of the curves while also shrinking towards a common shape. This allows us to capture the advantages of both the landmark and continuous monotone registration approaches. The method is illustrated on several data sets and a simulation study is performed. 1
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 ..."
Abstract

Cited by 12 (3 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
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.
Estimation of Functional Derivatives
, 2009
"... Situations of a functional predictor paired with a scalar response are increasingly encountered in data analysis. Predictors are often appropriately modeled as square integrable smooth random functions. Imposing minimal assumptions on the nature of the functional relationship, we aim to estimate the ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
Situations of a functional predictor paired with a scalar response are increasingly encountered in data analysis. Predictors are often appropriately modeled as square integrable smooth random functions. Imposing minimal assumptions on the nature of the functional relationship, we aim to estimate the directional derivatives and gradients of the response with respect to the predictor functions. In statistical applications and data analysis, functional derivatives provide a quantitative measure of the often intricate relationship between changes in predictor trajectories and those in scalar responses. This approach provides a natural extension of classical gradient fields in vector space and provides directions of steepest descent. We suggest a kernelbased method for the nonparametric estimation of functional derivatives that utilizes the decomposition of the random predictor functions into their eigenfunctions. These eigenfunctions define a canonical set of directions into which the gradient field is expanded. The proposed method is shown to lead to asymptotically consistent estimates of functional derivatives and is illustrated in an application to growth curves.
Logistic regression with Brownianlike predictors
, 2009
"... help with the gene expression data. This article introduces a new type of logistic regression model involving functional predictors of binary responses, along with an extension of the approach to generalized linear models. The predictors are trajectories that have certain samplepath properties in ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
help with the gene expression data. This article introduces a new type of logistic regression model involving functional predictors of binary responses, along with an extension of the approach to generalized linear models. The predictors are trajectories that have certain samplepath properties in common with Brownian motion. Time points are treated as parameters of interest, and confidence intervals developed under prospective and retrospective (casecontrol) sampling designs. In an application to fMRI data, signals from individual subjects are used to find the portion of the time course that is most predictive of the response. This allows the identification of sensitive time points, specific to a brain region and associated with a certain task, that can be used to distinguish between responses. A second application concerns gene expression data in a casecontrol study involving breast cancer, where the aim is to identify genetic loci along a chromosome that best discriminate between cases and controls.
Single and multiple index functional regression models with nonparametric link Annals of Statistics 39
 Probability Theory: Independence, Interchangeability, Martingales (3rd ed
, 2011
"... ar ..."
(Show Context)
Functional Additive Regression
, 2011
"... We suggest a new method, called “Functional Additive Regression”, or FAR, for efficiently performing high dimensional functional regression. FAR extends the usual linear regression model involving a functional predictor, X(t), and a scalar response, Y, in two key respects. First, FAR uses a penalize ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
We suggest a new method, called “Functional Additive Regression”, or FAR, for efficiently performing high dimensional functional regression. FAR extends the usual linear regression model involving a functional predictor, X(t), and a scalar response, Y, in two key respects. First, FAR uses a penalized least squares optimization approach to efficiently deal with high dimensional problems involving a large number of different functional predictors. Second, FAR extends beyond the standard linear regression setting to fit general nonlinear additive models. We demonstrate that FAR can be implemented with a wide range of penalty functions using a highly efficient coordinate descent algorithm. Theoretical results are developed which provide motivation for the FAR optimization criterion. Finally, we show through simulations and two real data sets that FAR can significantly outperform competing methods.