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The estimation of large functional and longitudinal data, which refers to the estimation of mean function, estimation of covariance function, and prediction of individual trajectory, is one of the most challenging problems in the field of high-dimensional statistics. Functional Principal Components Analysis (FPCA) and Functional Linear Mixed Model (FLMM) are two major statistical tools used to address the estimation of large functional and longitudinal data; however, the former suffers from a dramatically increasing computational burden while the latter does not have clear asymptotic properties. In this paper, we propose a computationally effective estimator of large functional and longitudinal data within the framework of FLMM, in which all the parameters can be automatically estimated. Under certain regularity assumptions, we prove that the mean function estimation and individual trajectory prediction reach the minimax lower bounds of all nonparametric estimations. Through numerous simulations and real data analysis, we show that our new estimator outperforms the traditional FPCA in terms of mean function estimation, individual trajectory prediction, variance estimation, covariance function estimation, and computational effectiveness.

This paper deals with the problem of predicting the real-valued response variable using explanatory variables containing both multivariate random variable and random curve. The proposed functional partial linear single-index model treats the multivariate random variable as linear part and the random curve as functional single-index part, respectively. To estimate the non-parametric link function, the functional single-index and the parameters in the linear part, a two-stage estimation procedure is proposed. Compared with existing semi-parametric methods, the proposed approach requires no initial estimation and iteration. Asymptotical properties are established for both the parameters in the linear part and the functional single-index. The convergence rate for the non-parametric link function is also given. In addition, asymptotical normality of the error variance is obtained that facilitates the construction of confidence region and hypothesis testing for the unknown parameter. Numerical experiments including simulation studies and a real-data analysis are conducted to evaluate the empirical performance of the proposed method.

This work deals with statistical modeling and forecasting of telecommunications data. Main mobile traffic events (SMS, Voice calls, Mobile data) are smoothed using B-spline functions and later analyzed in a functional framework. Functional linear auto-regression models are fitted using both bottom-up and topdown design methodologies. The advantages and disadvantages of both approaches for the prediction of mobile telephone users’ habits are discussed.

Quantifying heritability is the first step in understanding the contribution of genetic variation to the risk architecture of complex human diseases and traits. Heritability can be estimated for univariate phenotypes from nonfamily data using linear mixed effects models. There is, however, no fully developed methodology for defining or estimating heritability from longitudinal studies. By examining longitudinal studies, researchers have the opportunity to better understand the genetic influence on the temporal development of diseases, which can be vital for populations with rapidly changing phenotypes such as children or the elderly. To define and estimate heritability for longitudinally measured phenotypes, we present a framework based on functional data analysis, FDA. While our procedures have important genetic consequences, they also represent a substantial development for FDA. In particular, we present a very general methodology for constructing optimal, unbiased estimates of variance components in functional linear models. Such a problem is challenging as likelihoods and densities do not readily generalize to infinite-dimensional settings. Our procedure can be viewed as a functional generalization of the minimum norm quadratic unbiased estimation procedure, MINQUE, presented by C. R. Rao, and is equivalent to residual maximum likelihood, REML, in univariate settings. We apply our methodology to the Childhood Asthma Management Program, CAMP, a 4-year longitudinal study examining the long term effects of daily asthma medications on children.

Existing approaches for multivariate functional principal component analysis are restricted to data on the same one-dimensional interval. The presented approach focuses on multivariate functional data on different domains that may differ in dimension, such as functions and images. The theoretical basis for multivariate functional principal component analysis is given in terms of a Karhunen-Loève Theorem. For the practically relevant case of a finite Karhunen-Loève representation, a relationship between univariate and multivariate functional principal component analysis is established. This offers an estimation strategy to calculate multivariate functional principal components and scores based on their univariate counterparts. For the resulting estimators, asymptotic results are derived. The approach can be extended to finite univariate expansions in general, not necessarily orthonormal bases. It is also applicable for sparse functional data or data with measurement error. A flexible R implementation is available on CRAN. The new method is shown to be competitive to existing approaches for data observed on a common one-dimensional domain. The motivating application is a neuroimaging study, where the goal is to explore how longitudinal trajectories of a neuropsychological test score covary with FDG-PET brain scans at baseline. Supplementary material, including detailed proofs, additional simulation results, and software is available online.

Functional data analysis on nonlinear manifolds has drawn recent interest. Sphere-valued functional data, which are encountered, for example, as movement trajectories on the surface of the earth are an important special case. We consider an intrinsic principal component analysis for smooth Riemannian manifold-valued functional data and study its asymptotic properties. Riemannian functional principal component analysis (RFPCA) is carried out by first mapping the manifold-valued data through Riemannian logarithm maps to tangent spaces around the Fréchet mean function, and then performing a classical functional principal component analysis (FPCA) on the linear tangent spaces. Representations of the Riemannian manifold-valued functions and the eigenfunctions on the original manifold are then obtained with exponential maps. The tangent-space approximation yields upper bounds to residual variances if the Riemannian manifold has nonnegative curvature. We derive a central limit theorem for the mean function, as well as root-n uniform convergence rates for other model components. Our applications include a novel framework for the analysis of longitudinal compositional data, achieved by mapping longitudinal compositional data to trajectories on the sphere, illustrated with longitudinal fruit fly behavior patterns. RFPCA is shown to outperform an unrestricted FPCA in terms of trajectory recovery and prediction in applications and simulations.

Functional near-infrared spectroscopy (fNIRS) research articles show a large heterogeneity in the analysis approaches and pre-processing procedures. Additionally, there is often a lack of a complete description of the methods applied, necessary for study replication or for results comparison. The aims of this paper were (i) to review and investigate which information is generally included in published fNIRS papers, and (ii) to define a signal pre-processing procedure to set a common ground for standardization guidelines. To this goal, we have reviewed 110 fNIRS articles published in 2016 in the field of cognitive neuroscience, and performed a simulation analysis with synthetic fNIRS data to optimize the signal filtering step before applying the GLM method for statistical inference. Our results highlight the fact that many papers lack important information, and there is a large variability in the filtering methods used. Our simulations demonstrated that the optimal approach to remove noise and recover the hemodynamic response from fNIRS data in a GLM framework is to use a 1000th order band-pass Finite Impulse Response filter. Based on these results, we give preliminary recommendations as to the first step toward improving the analysis of fNIRS data and dissemination of the results.

As crowdfunding success relies heavily on designing campaigns that attempt to influence potential backers within a fixed time frame, this study leverages signaling theory to explore the time-varying direct and interactive effects of rhetorical (i.e., emotional tone, cognitive tone, communal language style, and linguistic style match) and substantive (i.e., backer support) signals employed in project descriptions on funding formation. Using linguistic inquiry and word count (LIWC) and functional data analysis (FDA), we analyze data on 1245 projects from a rewards-based crowdfunding platform (Kickstarter) and demonstrate that (a) the four rhetorical signals are positively related to funding formation, (b) emotional and cognitive tone exert stronger effects in early phases of projects, whereas linguistic style match and communal language style exert stronger effects in later phases, and (c) the dynamic effects of rhetorical signals on funding formation are greater when the number of backers increases. We conclude with implications for crowdfunding research and insights for improving campaign success.

We address the problem of dimension reduction for time series of functional data (Xt:t∈Z). Such functional time series frequently arise, for example, when a continuous time process is segmented into some smaller natural units, such as days. Then each Xₜrepresents one intraday curve. We argue that functional principal component analysis, though a key technique in the field and a benchmark for any competitor, does not provide an adequate dimension reduction in a time series setting. Functional principal component analysis indeed is a static procedure which ignores the essential information that is provided by the serial dependence structure of the functional data under study. Therefore, inspired by Brillinger's theory of dynamic principal components, we propose a dynamic version of functional principal component analysis which is based on a frequency domain approach. By means of a simulation study and an empirical illustration, we show the considerable improvement that the dynamic approach entails when compared with the usual static procedure.

Interest in functional time series has spiked in the recent past with papers covering both methodology and applications being published at a much increased pace. This article contributes to the research in this area by proposing a new stationarity test for functional time series based on frequency domain methods. The proposed test statistics is based on joint dimension reduction via functional principal components analysis across the spectral density operators at all Fourier frequencies, explicitly allowing for frequency-dependent levels of truncation to adapt to the dynamics of the underlying functional time series. The properties of the test are derived both under the null hypothesis of stationary functional time series and under the smooth alternative of locally stationary functional time series. The methodology is theoretically justified through asymptotic results. Evidence from simulation studies and an application to annual temperature curves suggests that the test works well in finite samples.