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26
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 ..."
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Cited by 12 (3 self)
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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
Properties of designbased functional principal components analysis
 J. Statist. Plann. Inference
, 2010
"... This work aims at performing Functional Principal Components Analysis (FPCA) with HorvitzThompson estimators when the observations are curves collected with survey sampling techniques. One important motivation for this study is that FPCA is a dimension reduction tool which is the first step to deve ..."
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Cited by 11 (4 self)
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This work aims at performing Functional Principal Components Analysis (FPCA) with HorvitzThompson estimators when the observations are curves collected with survey sampling techniques. One important motivation for this study is that FPCA is a dimension reduction tool which is the first step to develop model assisted approaches that can take auxiliary information into account. FPCA relies on the estimation of the eigenelements of the covariance operator which can be seen as nonlinear functionals. Adapting to our functional context the linearization technique based on the influence function developed by Deville (1999), we prove that these estimators are asymptotically design unbiased and consistent. Under mild assumptions, asymptotic variances are derived for the FPCA ’ estimators and consistent estimators of them are proposed. Our approach is illustrated with a simulation study and we check the good properties of the proposed estimators of the eigenelements as well as their variance estimators obtained with the linearization approach.
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 ..."
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Cited by 9 (3 self)
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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.
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 ..."
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Cited by 6 (0 self)
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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.
Continuously additive models for nonlinear functional regression
 Biometrika
, 2012
"... We introduce continuously additive models, which can be motivated as extensions of additive regression models with vector predictors to the case of infinitedimensional predictors. This approach provides a class of flexible functional nonlinear regression models, where random predictor curves are c ..."
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Cited by 5 (0 self)
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We introduce continuously additive models, which can be motivated as extensions of additive regression models with vector predictors to the case of infinitedimensional predictors. This approach provides a class of flexible functional nonlinear regression models, where random predictor curves are coupled with scalar responses. In continuously additive modeling, 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 approach outperforms existing functional regression models in simulations and data illustrations.
Nonlinear Manifold Representations for Functional Data
"... For functional data lying on an unknown nonlinear lowdimensional space, we study manifold learning and introduce the notions of manifold mean, manifold modes of functional variation and of functional manifold components. These constitute nonlinear representations of functional data that complement ..."
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Cited by 5 (0 self)
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For functional data lying on an unknown nonlinear lowdimensional space, we study manifold learning and introduce the notions of manifold mean, manifold modes of functional variation and of functional manifold components. These constitute nonlinear representations of functional data that complement classical linear representations such as eigenfunctions and functional principal components. Our manifold learning procedures borrow ideas from existing nonlinear dimension reduction methods, which we modify to address functional data settings. In simulations and applications, we study examples of functional data which lie on a manifold and validate the superior behavior of manifold mean and functional manifold components over traditional crosssectional mean and functional principal components. We also include consistency proofs for our estimators under certain assumptions. Key words and phrases: functional data analysis, modes of functional variation, functional manifold components, dimension reduction, smoothing.
Linear manifold modeling of multivariate functional data
, 2013
"... Multivariate functional data are increasingly encountered in data analysis, while statistical models for such data are not well developed yet. Motivated by a case study where one aims to quantify the relationship between various longitudinally recorded behavior intensities for Drosophila flies, we p ..."
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Cited by 3 (0 self)
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Multivariate functional data are increasingly encountered in data analysis, while statistical models for such data are not well developed yet. Motivated by a case study where one aims to quantify the relationship between various longitudinally recorded behavior intensities for Drosophila flies, we propose a functional linear manifold model. This model reflects the functional dependency between the components of multivariate random processes and is defined through datadetermined linear combinations of the multivariate component trajectories, which are characterized by a set of varying coefficient functions. The timevarying linear relationships that govern the components of multivariate random functions yield insights about the underlying processes and also lead to noisereduced representations of the multivariate component trajectories. The proposed functional linear manifold model is put to the task for an analysis of longitudinally observed behavioral patterns of flying, feeding, walking and resting over the lifespan of Drosophila flies and is also investigated in simulations.
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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Cited by 3 (0 self)
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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.
Functional Data Analysis for Point Processes with Rare Events 1
, 2011
"... In various applications, one encounters samples of objects, where each object consists of a small number of repeated event times observed over a fixed time interval. For such rare events data there are no flexible methods available that can be applied when the shapes of the intensity functions that ..."
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Cited by 3 (0 self)
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In various applications, one encounters samples of objects, where each object consists of a small number of repeated event times observed over a fixed time interval. For such rare events data there are no flexible methods available that can be applied when the shapes of the intensity functions that generate the observed event times are not known or vary substantially between objects. We model the underlying intensity functions as nonparametric objectspecific random functions. Applying a novel functional method to obtain the covariance structure of the associated random densities, we reconstruct objectspecific density functions that reflect the distribution of event times. We demonstrate in simulations that the proposed functional approach is superior to conventional nonparametric methods, as it borrows strength from the entire sample of objects, rather than aiming at the estimation of each object’s density separately. Our method is based on a key relationship that allows to reduce the covariance estimation problem for random densities to the simpler problem of estimating a nonrandom joint density from pooled event times. We describe an application to model bid arrivals for a sample of online auctions and also include asymptotic justifications of the methodology.
Waveletbased scalaronfunction finite mixture regression models. Preprint, available at: arXiv:1312.0652
"... Classical finite mixture regression is useful for modeling the relationship between scalar predictors and scalar responses arising from subpopulations defined by the differing associations between those predictors and responses. Here we extend the classical finite mixture regression model to incor ..."
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Cited by 2 (1 self)
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Classical finite mixture regression is useful for modeling the relationship between scalar predictors and scalar responses arising from subpopulations defined by the differing associations between those predictors and responses. Here we extend the classical finite mixture regression model to incorporate functional predictors by taking a waveletbased approach in which we represent both the functional predictors and the componentspecific coefficient functions in terms of an appropriate wavelet basis. In the wavelet representation of the model, the coefficients corresponding to the functional covariates become the predictors. In this setting, we typically have many more predictors than observations. Hence we use a lassotype penalization to perform variable selection and estimation. We also consider an adaptive version of our waveletbased model. We discuss the specification of the model, provide a fitting algorithm, and apply and evaluate our method using both simulations and a real data set from a study of the relationship between cognitive ability and diffusion tensor imaging measures in subjects with multiple sclerosis.