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High dimensional statistical inference and random matrices
 IN: PROCEEDINGS OF INTERNATIONAL CONGRESS OF MATHEMATICIANS
, 2006
"... Multivariate statistical analysis is concerned with observations on several variables which are thought to possess some degree of interdependence. Driven by problems in genetics and the social sciences, it first flowered in the earlier half of the last century. Subsequently, random matrix theory ..."
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Cited by 49 (1 self)
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Multivariate statistical analysis is concerned with observations on several variables which are thought to possess some degree of interdependence. Driven by problems in genetics and the social sciences, it first flowered in the earlier half of the last century. Subsequently, random matrix theory (RMT) developed, initially within physics, and more recently widely in mathematics. While some of the central objects of study in RMT are identical to those of multivariate statistics, statistical theory was slow to exploit the connection. However, with vast data collection ever more common, data sets now often have as many or more variables than the number of individuals observed. In such contexts, the techniques and results of RMT have much to offer multivariate statistics. The paper reviews some of the progress to date.
Di: Generalized Multilevel Functional Regression 1561
 Journal of the Royal Statistical Society, Ser. B
, 2006
"... We introduce Generalized Multilevel Functional Linear Models (GMFLMs), a novel statistical framework for regression models where exposure has a multilevel functional structure. We show that GMFLMs are, in fact, generalized multilevel mixed models. Thus, GMFLMs can be analyzed using the mixed effects ..."
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Cited by 18 (9 self)
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We introduce Generalized Multilevel Functional Linear Models (GMFLMs), a novel statistical framework for regression models where exposure has a multilevel functional structure. We show that GMFLMs are, in fact, generalized multilevel mixed models. Thus, GMFLMs can be analyzed using the mixed effects inferential machinery and can be generalized within a wellresearched statistical framework. We propose and compare two methods for inference: (1) a twostage frequentist approach; and (2) a joint Bayesian analysis. Our methods are motivated by and applied to the Sleep Heart Health Study, the largest community cohort study of sleep. However, our methods are general and easy to apply to a wide spectrum of emerging biological and medical datasets. Supplemental materials for this article are available online.
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.
Optimal estimation of the mean function based on discretely sampled funcitonal data: phase transition
 Ann. Statist
, 2011
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Bivariate splines for spatial functional regression models
 J. Nonparametr. Stat
, 2010
"... We consider the functional linear regression model where the explanatory variable is a random surface and the response is a real random variable, with bounded or normal noise. Bivariate splines over triangulations represent the random surfaces. We use this representation to construct least square ..."
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Cited by 7 (5 self)
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We consider the functional linear regression model where the explanatory variable is a random surface and the response is a real random variable, with bounded or normal noise. Bivariate splines over triangulations represent the random surfaces. We use this representation to construct least squares estimators of the regression function with or without a penalization term. Under the assumptions that the regressors in the sample are bounded and span a large enough space of functions, bivariate splines approximation properties yield the consistency of the estimators. Simulations demonstrate the quality of the asymptotic properties on a realistic domain. We also carry out an application to ozone concentration forecasting over the US that illustrates the predictive skills of the method. 1
CONSISTENCY OF RESTRICTED MAXIMUM LIKELIHOOD ESTIMATORS OF PRINCIPAL COMPONENTS
"... In this paper we consider two closely related problems: estimation of eigenvalues and eigenfunctions of the covariance kernel of functional data based on (possibly) irregular measurements, and the problem of estimating the eigenvalues and eigenvectors of the covariance matrix for highdimensional Gau ..."
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Cited by 6 (4 self)
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In this paper we consider two closely related problems: estimation of eigenvalues and eigenfunctions of the covariance kernel of functional data based on (possibly) irregular measurements, and the problem of estimating the eigenvalues and eigenvectors of the covariance matrix for highdimensional Gaussian vectors. In [A geometric approach to maximum likelihood estimation of covariance kernel from sparse irregular longitudinal data (2007)], a restricted maximum likelihood (REML) approach has been developed to deal with the first problem. In this paper, we establish consistency and derive rate of convergence of the REML estimator for the functional data case, under appropriate smoothness conditions. Moreover, we prove that when the number of measurements per sample curve is bounded, under squarederror loss, the rate of convergence of the REML estimators of eigenfunctions is nearoptimal. In the case of Gaussian vectors, asymptotic consistency and an efficient score representation of the estimators are obtained under the assumption that the effective dimension grows at a rate slower than the sample size. These results are derived through an explicit utilization of the intrinsic geometry of the parameter space, which is nonEuclidean. Moreover, the results derived in this paper suggest an asymptotic equivalence between the inference on functional data with dense measurements and that of the highdimensional Gaussian vectors. 1. Introduction. Analysis of functional data
1 Nonparametric time series forecasting with dynamic updating
"... Abstract: We present a nonparametric method to forecast a seasonal time series, and propose four dynamic updating methods to improve point forecast accuracy. Our forecasting and dynamic updating methods are datadriven and computationally fast, and they are thus feasible to be applied in practice. W ..."
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Cited by 4 (3 self)
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Abstract: We present a nonparametric method to forecast a seasonal time series, and propose four dynamic updating methods to improve point forecast accuracy. Our forecasting and dynamic updating methods are datadriven and computationally fast, and they are thus feasible to be applied in practice. We will demonstrate the effectiveness of these methods using monthly El Niño time series from 1950 to 2008
NONLINEAR PRINCIPAL COMPONENTS AND LONGRUN IMPLICATIONS OF MULTIVARIATE DIFFUSIONS
, 2009
"... We investigate a method for extracting nonlinear principal components (NPCs). These NPCs maximize variation subject to smoothness and orthogonality constraints; but we allow for a general class of constraints and multivariate probability densities, including densities without compact support and eve ..."
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Cited by 3 (1 self)
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We investigate a method for extracting nonlinear principal components (NPCs). These NPCs maximize variation subject to smoothness and orthogonality constraints; but we allow for a general class of constraints and multivariate probability densities, including densities without compact support and even densities with algebraic tails. We provide primitive sufficient conditions for the existence of these NPCs. By exploiting the theory of continuoustime, reversible Markov diffusion processes, we give a different interpretation of these NPCs and the smoothness constraints. When the diffusion matrix is used to enforce smoothness, the NPCs maximize longrun variation relative to the overall variation subject to orthogonality constraints. Moreover, the NPCs behave as scalar autoregressions with heteroskedastic innovations; this supports semiparametric identification and estimation of a multivariate reversible diffusion process and tests of the overidentifying restrictions implied by such a process from lowfrequency data. We also explore implications for stationary, possibly nonreversible diffusion processes. Finally, we suggest a sieve method to estimate the NPCs from discretelysampled data.
Functional Modeling of Longitudinal Data
, 2006
"... Functional data analysis provides an inherently nonparametric approach for the analysis of data which consist of samples of time courses or random trajectories. It is a relatively young field aiming at modeling and data exploration under very flexible model assumptions with no or few parametric comp ..."
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Cited by 3 (0 self)
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Functional data analysis provides an inherently nonparametric approach for the analysis of data which consist of samples of time courses or random trajectories. It is a relatively young field aiming at modeling and data exploration under very flexible model assumptions with no or few parametric components. Basic tools of functional data analysis are smoothing, functional principal components, functional linear models and timewarping. Warping or curve registration aims at adjusting for random time distortions. While in the usual functional data analysis paradigm the sample functions were considered as continuously observed, in longitudinal data analysis one mostly deals with sparsely and irregularly observed data that also are corrupted with noise. Adjustments of functional data analysis techniques which take these particular features into account are needed to use them to advantage for longitudinal data. We review some techniques that have been recently proposed to connect functional data analysis methodology with longitudinal data. The extension of functional data analysis towards longitudinal data is a fairly recent undertaking that presents a promising avenue for future research. This article provides a review of some of the recent developments.
Generalized functional linear models with semiparametric singleindex interactions
 Journal of the American Statistical Association
, 2010
"... We introduce a new class of functional generalized linear models, where the response is a scalar and some of the covariates are functional. We assume that the response depends on multiple covariates, a finite number of latent features in the functional predictor, and interaction between the two. To ..."
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Cited by 3 (1 self)
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We introduce a new class of functional generalized linear models, where the response is a scalar and some of the covariates are functional. We assume that the response depends on multiple covariates, a finite number of latent features in the functional predictor, and interaction between the two. To achieve parsimony, the interaction between the multiple covariates and the functional predictor is modeled semiparametrically with a singleindex structure. We propose a two step estimation procedure based on local estimating equations, and investigate two situations: (a) when the basis functions are predetermined, e.g., Fourier or wavelet basis functions and the functional features of interest are known; and (b) when the basis functions are data driven, such as with functional principal components. Asymptotic properties are developed. Notably, we show that when the functional features are data driven, the parameter estimates have an increased asymptotic variance, due to the estimation error of the basis functions. Our methods are illustrated with a simulation study and applied to an empirical data set, where a previously unknown interaction is detected. Technical proofs of our theoretical results are provided in the online supplemental materials.