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11
Functional additive mixed models
 Journal of Computational and Graphical Statistics, Published online
, 2014
"... We propose an extensive framework for additive regression models for correlated functional responses, allowing for multiple partially nested or crossed functional random effects with flexible correlation structures for, e.g., spatial, temporal, or longitudinal functional data. Additionally, our fr ..."
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We propose an extensive framework for additive regression models for correlated functional responses, allowing for multiple partially nested or crossed functional random effects with flexible correlation structures for, e.g., spatial, temporal, or longitudinal functional data. Additionally, our framework includes linear and nonlinear effects of functional and scalar covariates that may vary smoothly over the index of the functional response. It accommodates densely or sparsely observed functional responses and predictors which may be observed with additional error and includes both splinebased and functional principal componentbased terms. Estimation and inference in this framework is based on standard additive mixed models, allowing us to take advantage of established methods and robust, flexible algorithms. We provide easytouse open source software in the pffr() function for the Rpackage refund. Simulations show that the proposed method recovers relevant effects reliably, handles small sample sizes well and also scales to larger data sets. Applications with spatially and longitudinally observed functional data demonstrate the flexibility in modeling and interpretability of results of our approach.
Modeling multiple correlated functional outcomes with spatially heterogeneous shape characteristics
, 2013
"... Summary: We propose a copulabased approach for analyzing functional data with multiple outcomes exhibiting spatially heterogeneous shape characteristics. To accommodate the possibly large number of parameters in multiple outcome data, parameter estimation is performed in two steps: first, the param ..."
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Summary: We propose a copulabased approach for analyzing functional data with multiple outcomes exhibiting spatially heterogeneous shape characteristics. To accommodate the possibly large number of parameters in multiple outcome data, parameter estimation is performed in two steps: first, the parameters for the marginal distributions are estimated using the skew t family, and then the dependence structure both within and across outcomes is estimated using a Gaussian copula. We develop an estimation algorithm for the dependence parameters based on the KarhunenLoeve expansion and an EM algorithm that significantly reduces the dimension of the problem and is computationally efficient. We also demonstrate prediction of an unknown outcome when the other outcomes are known. We apply our methodology to diffusion tensor imaging (DTI) data for multiple sclerosis (MS) patients with three outcomes, and identify differences in both the marginal distributions and the dependence structure between the MS and control groups. ROC curves show that the crosscorrelations between DTI outcomes are predictive of MS status. Our proposed methodology is quite general and can be applied to other functional data with multiple outcomes in biology and other fields.
Classical Testing in Functional Linear Models
"... We extend four tests common in classical regression Wald, score, likelihood ratio and F tests to functional linear regression, for testing the null hypothesis, that there is no association between a scalar response and a functional covariate. Using functional principal component analysis we reexp ..."
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We extend four tests common in classical regression Wald, score, likelihood ratio and F tests to functional linear regression, for testing the null hypothesis, that there is no association between a scalar response and a functional covariate. Using functional principal component analysis we reexpress the functional linear model as a standard linear model, where the effect of the functional covariate can be approximated by a finite linear combination of the functional principal component scores. In this setting, we consider application of the four traditional tests. The proposed testing procedures are investigated theoretically when the number of principal components diverges, and for both densely and sparsely observed functional covariates. Using the theoretical distribution of the tests under the alternative hypothesis, we develop a procedure for sample size calculation in the context of functional linear regression. The four tests are further compared numerically in simulation experiments and using two real data applications.
A Two Sample DistributionFree Test for Functional Data with Application to a Diffusion Tensor Imaging Study of Multiple Sclerosis
"... Summary. Motivated by an imaging study, this paper develops a nonparametric testing procedure for testing the null hypothesis that two samples of curves observed at discrete grids and with noise have the same underlying distribution. The objective is to formally compare white matter tract profiles ..."
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Summary. Motivated by an imaging study, this paper develops a nonparametric testing procedure for testing the null hypothesis that two samples of curves observed at discrete grids and with noise have the same underlying distribution. The objective is to formally compare white matter tract profiles between healthy individuals and multiple sclerosis patients, as assessed by conventional diffusion tensor imaging measures. We propose to decompose the curves using functional principal component analysis of a mixture process, which we refer to as marginal functional principal component analysis. This approach reduces the dimension of the testing problem in a way that enables the use of traditional nonparametric univariate testing procedures. The procedure is computationally efficient and accommodates different sampling designs. Numerical studies are presented to validate the size and power properties of the test in many realistic scenarios. In these cases, the proposed test has been found to be more powerful than its primary competitor. Application to the diffusion tensor imaging data reveals that all the tracts studied are associated with multiple sclerosis and the choice of the diffusion tensor image measurement is important when assessing axonal disruption. 2 G.M. Pomann, A.M. Staicu, and S. Ghosh 1.
Variable Selection in Generalized Functional Linear Models
"... Modern research data, where a large number of functional predictors is collected on few subjects are becoming increasingly common. In this paper we propose a variable selection technique, when the predictors are functional and the response is scalar. Our approach is based on adopting a generalized f ..."
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Modern research data, where a large number of functional predictors is collected on few subjects are becoming increasingly common. In this paper we propose a variable selection technique, when the predictors are functional and the response is scalar. Our approach is based on adopting a generalized functional linear model framework and using a penalized likelihood method that simultaneously controls the sparsity of the model and the smoothness of the corresponding coefficient functions by adequate penalization. The methodology is characterized by high predictive accuracy, and yields interpretable models, while retaining computational efficiency. The proposed method is investigated numerically in finite samples, and applied to a diffusion tensor imaging tractography data set and a chemometric data set. Copyright c © 2013 John Wiley & Sons, Ltd.
Statistica Sinica (2013): Preprint 1 Optimal Prediction in an Additive Functional Model
"... Abstract: The functional generalized additive model (FGAM), also known as the continuous additive model (CAM), provides a more flexible functional regression model than the wellstudied functional linear regression model. This paper restricts attention to the FGAM with identity link and additive er ..."
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Abstract: The functional generalized additive model (FGAM), also known as the continuous additive model (CAM), provides a more flexible functional regression model than the wellstudied functional linear regression model. This paper restricts attention to the FGAM with identity link and additive errors, which we will call the additive functional model and is a generalization of the functional linear model. This paper studies the minimax rate of convergence of predictions from the additive functional model in the framework of reproducing kernel Hilbert space. It is shown that the optimal rate is determined by the decay rate of the eigenvalues of a certain kernel function, which in turn is determined by the reproducing kernel and the joint distribution of any two points in the random predictor function. In the special case of the functional linear model, this kernel function is jointly determined by the covariance function of the predictor function and the reproducing kernel. The easily implementable roughnessregularized predictor is shown to achieve the optimal rate of convergence. Numerical studies are carried out to illustrate the merits of the predictor. Our simulations and real data examples demonstrate a competitive performance against the existing approach. Key words and phrases: Functional regression, minimax rate of convergence, principal component analysis, reproducing kernel Hilbert space. 1
Bayesian Functional Generalized Additive Models with Sparsely Observed Covariates
, 2013
"... The functional generalized additive model (FGAM) was recently proposed in McLean et al. (2012) as a more flexible alternative to the common functional linear model (FLM) for regressing a scalar on functional covariates. In this paper, we develop a Bayesian version of FGAM for the case of Gaussian er ..."
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The functional generalized additive model (FGAM) was recently proposed in McLean et al. (2012) as a more flexible alternative to the common functional linear model (FLM) for regressing a scalar on functional covariates. In this paper, we develop a Bayesian version of FGAM for the case of Gaussian errors with identity link function. Our approach allows the functional covariates to be sparsely observed and measured with error, whereas the estimation procedure of McLean et al. (2012) required that they be noiselessly observed on a regular grid. We consider both Monte Carlo and variational Bayes methods for fitting the FGAM with sparsely observed covariates. Due to the complicated form of the model posterior distribution and full conditional distributions, standard Monte Carlo and variational Bayes algorithms cannot be used. The strategies we use to handle the updating of parameters without closedform full conditionals should be of independent interest to applied Bayesian statisticians working with nonconjugate models. Our numerical studies demonstrate the benefits of our algorithms over a twostep approach of first recovering the complete trajectories using standard techniques and then fitting a functional regression model. In a real data
Cox Regression Models with Functional Covariates for Survival Data
"... We extend the Cox proportional hazards model to cases when the exposure is a densely sampled functional process, measured at baseline. The fundamental idea is to combine penalized signal regression with methods developed for mixed effects proportional hazards models. The model is fit by maximizing t ..."
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We extend the Cox proportional hazards model to cases when the exposure is a densely sampled functional process, measured at baseline. The fundamental idea is to combine penalized signal regression with methods developed for mixed effects proportional hazards models. The model is fit by maximizing the penalized partial likelihood, with smoothing parameters estimated by a likelihoodbased criterion such as AIC or EPIC. The model may be extended to allow for multiple functional predictors, time varying coefficients, and missing or unequallyspaced data. Methods were inspired by and applied to a study of the association between time to death after hospital discharge and daily measures of disease severity collected in the intensive care unit, among survivors of acute respiratory distress syndrome. Keywords:
Printed in Great Britain Continuously additive models for nonlinear functional
"... We introduce continuously additive models, which can be viewed as extensions of additive regression models with vector predictors to the case of infinitedimensional predictors. This approach produces a class of flexible functional nonlinear regression models, where random predictor curves are coupl ..."
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We introduce continuously additive models, which can be viewed as extensions of additive regression models with vector predictors to the case of infinitedimensional predictors. This approach produces a class of flexible functional nonlinear regression models, where random predictor curves are coupled with scalar responses. In continuously additive modelling, integrals taken over a smooth surface along graphs of predictor functions relate the predictors to the responses in a nonlinear fashion. We use tensor product basis expansions to fit the smooth regression surface that characterizes the model. In a theoretical investigation, we show that the predictions obtained from fitting continuously additive models are consistent and asymptotically normal. We also consider extensions to generalized responses. The proposed class of models outperforms existing functional regression models in simulations and realdata examples.