Varying-Coefficient Functional Linear Regression Models

This article considers a generalization of the functional linear regression in which an additional real variable influences smoothly the functional coefficient. We thus define a varying-coefficient regression model for functional data. We propose two estimators based, respectively, on conditional functional principal regression and on local penalized regression splines and prove their pointwise consistency. We check, with the prediction one day ahead of ozone concentration in the city of Toulouse, the ability of such nonlinear functional approaches to produce competitive estimations.     

A diagnostic statistic for functional-coefficient autoregressive models

In this paper, functional coefficient autoregressive (FAR) models proposed by Chen and Tsay (1993) are considered. We propose a diagnostic statistic for FAR models constructed by comparing between parametric and nonparametric estimators of the functional form of the FAR models. We show asymptotic properties of our statistic mathematically and it can be applied to the estimation of the delay parameter and the specification of the functional form of FAR models.     

Recent History Functional Linear Models for Sparse Longitudinal Data

We consider the recent history functional linear models, relating a longitudinal response to a longitudinal predictor where the predictor process only in a sliding window into the recent past has an effect on the response value at the current time. We propose an estimation procedure for recent history functional linear models that is geared towards sparse longitudinal data, where the observation times across subjects are irregular and total number of measurements per subject is small. The proposed estimation procedure builds upon recent developments in literature for estimation of functional linear models with sparse data and utilizes connections between the recent history functional linear models and varying coefficient models. We establish uniform consistency of the proposed estimators, propose prediction of the response trajectories and derive their asymptotic distribution leading to asymptotic point-wise confidence bands. We include a real data application and simulation studies to demonstrate the efficacy of the proposed methodology.     

Multivariate functional response low‐rank regression with an application to brain imaging data

We propose a multivariate functional response low‐rank regression model with possible high‐dimensional functional responses and scalar covariates. By expanding the slope functions on a set of sieve bases, we reconstruct the basis coefficients as a matrix. To estimate these coefficients, we propose an efficient procedure using nuclear norm regularization. We also derive error bounds for our estimates and evaluate our method using simulations. We further apply our method to the Human Connectome Project neuroimaging data to predict cortical surface motor task‐evoked functional magnetic resonance imaging signals using various clinical covariates to illustrate the usefulness of our results.     

Testing in mixed-effects FANOVA models

We consider the testing problem in the mixed-effects functional analysis of variance models. We develop asymptotically optimal (minimax) testing procedures for testing the significance of functional global trend and the functional fixed effects based on the empirical wavelet coefficients of the data. Wavelet decompositions allow one to characterize various types of assumed smoothness conditions on the response function under the nonparametric alternatives. The distribution of the functional random-effects component is defined in the wavelet domain and captures the sparseness of wavelet representation for a wide variety of functions. The simulation study presented in the paper demonstrates the finite sample properties of the proposed testing procedures. We also applied them to the real data from the physiological experiments.     

Residual Kriging for Functional Spatial Prediction of Salinity Curves

Recently, several methodologies to perform geostatistical analysis of functional data have been proposed. All of them assume that the spatial functional process considered is stationary. However, in practice, we often have nonstationary functional data because there exists an explicit spatial trend in the mean. Here, we propose a methodology to extend kriging predictors for functional data to the case where the mean function is not constant through the region of interest. We consider an approach based on the classical residual kriging method used in univariate geostatistics. We propose a three steps procedure. Initially, a functional regression model is used to detrend the mean. Then we apply kriging methods for functional data to the regression residuals to predict a residual curve at a non-data location. Finally, the prediction curve is obtained as the sum of the trend and the residual prediction. We apply the methodology to salinity data corresponding to 21 salinity curves recorded at the Ciénaga Grande de Santa Marta estuary, located in the Caribbean coast of Colombia. A cross-validation analysis was carried out to track the performance of the proposed methodology.     

NONPARAMETRIC REGRESSION ON FUNCTIONAL DATA: INFERENCE AND PRACTICAL ASPECTS

We consider the problem of predicting a real random variable from a functional explanatory variable. The problem is tackled using a nonparametric kernel approach, which has been recently adapted to this functional context. We derive theoretical results from a deep asymptotic analysis of the behaviour of the estimate, including mean squared convergence (with rates and precise evaluation of the constant terms) as well as asymptotic distribution. Practical use of these results relies on the ability to estimate these constants. Some perspectives in this direction are discussed. In particular, a functional version of wild bootstrapping ideas is proposed and used both on simulated and real functional datasets.     

DESCRIBING DISABILITY THROUGH INDIVIDUAL-LEVEL MIXTURE MODELS FOR MULTIVARIATE BINARY DATA

Data on functional disability are of widespread policy interest in the United States, especially with respect to planning for Medicare and Social Security for a growing population of elderly adults. We consider an extract of functional disability data from the National Long Term Care Survey (NLTCS) and attempt to develop disability profiles using variations of the Grade of Membership (GoM) model. We first describe GoM as an individual-level mixture model that allows individuals to have partial membership in several mixture components simultaneously. We then prove the equivalence between individual-level and population-level mixture models, and use this property to develop a Markov Chain Monte Carlo algorithm for Bayesian estimation of the model. We use our approach to analyze functional disability data from the NLTCS.     

Functional Analysis of Variance,Discriminant Analysis,and Clustering in a Manifold of Elastic Curves

We develop functional data analysis techniques using the differential geometry of a manifold of smooth elastic functions on an interval in which the functions are represented by a log-speed function and an angle function. The manifold's geometry provides a method for computing a sample mean function and principal components on tangent spaces. Using tangent principal component analysis, we estimate probability models for functional data and apply them to functional analysis of variance, discriminant analysis, and clustering. We demonstrate these tasks using a collection of growth curves from children from ages 1–18.     

Optimal design for classification of functional data

We study the design problem for the optimal classification of functional data. The goal is to select sampling time points so that functional data observed at these time points can be classified accurately. We propose optimal designs that are applicable to either dense or sparse functional data. Using linear discriminant analysis, we formulate our design objectives as explicit functions of the sampling points. We study the theoretical properties of the proposed design objectives and provide a practical implementation. The performance of the proposed design is evaluated through simulations and real data applications. The Canadian Journal of Statistics 48: 285–307; 2020 © 2019 Statistical Society of Canada     

Functional calibration estimation by the maximum entropy on the mean principle

We extend the problem of obtaining an estimator for the finite population mean parameter incorporating complete auxiliary information through calibration estimation in survey sampling under a functional data framework. The functional calibration sampling weights of the estimator are obtained by matching the calibration estimation problem with the maximum entropy on the mean – MEM – principle. In particular, the calibration estimation is viewed as an infinite-dimensional linear inverse problem following the structure of the MEM approach. We give a precise theoretical setting and estimate the functional calibration weights assuming, as prior measures, the centred Gaussian and compound Poisson random measures. Additionally, through a simple simulation study, we show that the proposed functional calibration estimator improves its accuracy compared with the Horvitz–Thompson one.     

Functional and dynamic magnetic resonance imaging using vector adaptive weights smoothing

We consider the problem of statistical inference for functional and dynamic magnetic resonance imaging (MRI). A new approach is proposed which extends the adaptive weights smoothing procedure of Polzehl and Spokoiny that was originally designed for image denoising. We demonstrate how the adaptive weights smoothing method can be applied to time series of images, which typically occur in functional and dynamic MRI. It is shown how signal detection in functional MRI and the analysis of dynamic MRI can benefit from spatially adaptive smoothing. The performance of the procedure is illustrated by using real and simulated data.     

Forecasting functional time series

We propose forecasting functional time series using weighted functional principal component regression and weighted functional partial least squares regression. These approaches allow for smooth functions, assign higher weights to more recent data, and provide a modeling scheme that is easily adapted to allow for constraints and other information. We illustrate our approaches using age-specific French female mortality rates from 1816 to 2006 and age-specific Australian fertility rates from 1921 to 2006, and show that these weighted methods improve forecast accuracy in comparison to their unweighted counterparts. We also propose two new bootstrap methods to construct prediction intervals, and evaluate and compare their empirical coverage probabilities.     

Wavelet modeling of functional random effects with application to human vision data

In modern statistical practice, it is increasingly common to observe a set of curves or images, often measured with noise, and to use these as the basis of analysis (functional data analysis). We consider a functional data model consisting of measurement error and functional random effects motivated by data from a study of human vision. By transforming the data into the wavelet domain we are able to exploit the expected sparse representation of the underlying function and the mechanism generating the random effects. We propose simple fitting procedures and illustrate the methods on the vision data.     

OLS and IV estimation of regression models including endogenous interaction terms

We analyze a class of linear regression models including interactions of endogenous regressors and exogenous covariates. We show how to generate instrumental variables using the nonlinear functional form of the structural equation when traditional excluded instruments are unknown. We propose to use these instruments with identification robust IV inference. We furthermore show that, whenever functional form identification is not valid, the ordinary least squares (OLS) estimator of the coefficient of the interaction term is consistent and standard OLS inference applies. Using our alternative empirical methods we confirm recent empirical findings on the nonlinear causal relation between financial development and economic growth.     

Generalized linear models with functional predictors

Summary. We present a technique for extending generalized linear models to the situation where some of the predictor variables are observations from a curve or function. The technique is particularly useful when only fragments of each curve have been observed. We demonstrate, on both simulated and real data sets, how this approach can be used to perform linear, logistic and censored regression with functional predictors. In addition, we show how functional principal components can be used to gain insight into the relationship between the response and functional predictors. Finally, we extend the methodology to apply generalized linear models and principal components to standard missing data problems.     

Minimax powerful functional analysis of covariance tests with application to longitudinal genome-wide association studies

We model the Alzheimer's disease-related phenotype response variables observed on irregular time points in longitudinal Genome-Wide Association Studies as sparse functional data and propose nonparametric test procedures to detect functional genotype effects while controlling the confounding effects of environmental covariates. Our new functional analysis of covariance tests are based on a seemingly unrelated kernel smoother, which takes into account the within-subject temporal correlations, and thus enjoy improved power over existing functional tests. We show that the proposed test combined with a uniformly consistent nonparametric covariance function estimator enjoys the Wilks phenomenon and is minimax most powerful. Data used in the preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative database, where an application of the proposed test lead to the discovery of new genes that may be related to Alzheimer's disease.     

Empirical likelihood confidence intervals for nonparametric functional data analysis

We consider the problem of constructing confidence intervals for nonparametric functional data analysis using empirical likelihood. In this doubly infinite-dimensional context, we demonstrate the Wilk's phenomenon and propose a bias-corrected construction that requires neither undersmoothing nor direct bias estimation. We also extend our results to partially linear regression models involving functional data. Our numerical results demonstrate improved performance of the empirical likelihood methods over normal approximation-based methods.     

Sieve Estimation of Time-Varying Panel Data Models With Latent Structures

We propose a heterogeneous time-varying panel data model with a latent group structure that allows the coefficients to vary over both individuals and time. We assume that the coefficients change smoothly over time and form different unobserved groups. When treated as smooth functions of time, the individual functional coefficients are heterogeneous across groups but homogeneous within a group. We propose a penalized-sieve-estimation-based classifier-Lasso (C-Lasso) procedure to identify the individuals’ membership and to estimate the group-specific functional coefficients in a single step. The classification exhibits the desirable property of uniform consistency. The C-Lasso estimators and their post-Lasso versions achieve the oracle property so that the group-specific functional coefficients can be estimated as well as if the individuals’ membership were known. Several extensions are discussed. Simulations demonstrate excellent finite sample performance of the approach in both classification and estimation. We apply our method to study the heterogeneous trending behavior of GDP per capita across 91 countries for the period 1960–2012 and find four latent groups.     

Modelling price paths in on-line auctions: smoothing sparse and unevenly sampled curves by using semiparametric mixed models

Summary.  On-line auctions pose many challenges for the empirical researcher, one of which is the effective and reliable modelling of price paths. We propose a novel way of modelling price paths in eBay's on-line auctions by using functional data analysis. One of the practical challenges is that the functional objects are sampled only very sparsely and unevenly. Most approaches rely on smoothing to recover the underlying functional object from the data, which can be difficult if the data are irregularly distributed. We present a new approach that can overcome this challenge. The approach is based on the ideas of mixed models. Specifically, we propose a semiparametric mixed model with boosting to recover the functional object. As well as being able to handle sparse and unevenly distributed data, the model also results in conceptually more meaningful functional objects. In particular, we motivate our method within the framework of eBay's on-line auctions. On-line auctions produce monotonic increasing price curves that are often correlated across auctions. The semiparametric mixed model accounts for this correlation in a parsimonious way. It also manages to capture the underlying monotonic trend in the data without imposing model constraints. Our application shows that the resulting functional objects are conceptually more appealing. Moreover, when used to forecast the outcome of an on-line auction, our approach also results in more accurate price predictions compared with standard approaches. We illustrate our model on a set of 183 closed auctions for Palm M515 personal digital assistants.