Latent group detection in functional partially linear regression models

In this paper, we propose a functional partially linear regression model with latent group structures to accommodate the heterogeneous relationship between a scalar response and functional covariates. The proposed model is motivated by a salinity tolerance study of barley families, whose main objective is to detect salinity tolerant barley plants. Our model is flexible, allowing for heterogeneous functional coefficients while being efficient by pooling information within a group for estimation. We develop an algorithm in the spirit of the K-means clustering to identify latent groups of the subjects under study. We establish the consistency of the proposed estimator, derive the convergence rate and the asymptotic distribution, and develop inference procedures. We show by simulation studies that the proposed method has higher accuracy for recovering latent groups and for estimating the functional coefficients than existing methods. The analysis of the barley data shows that the proposed method can help identify groups of barley families with different salinity tolerant abilities.

Bayesian Wavelet-packet Historical Functional Linear Models

Historical Functional Linear Models (HFLM) quantify associations between a functional predictor and functional outcome where the predictor is an exposure variable that occurs before, or at least concurrently with, the outcome. Prior work on the HFLM has largely focused on estimation of a surface that represents a time-varying association between the functional outcome and the functional exposure. This existing work has employed frequentist and spline-based estimation methods, with little attention paid to formal inference or adjustment for multiple testing and no approaches that implement wavelet-bases. In this work, we propose a new functional regression model that estimates the time-varying, lagged association between a functional outcome and a functional exposure. Building off of recently developed function-on-function regression methods, the model employs a novel use the wavelet-packet decomposition of the exposure and outcome functions that allows us to strictly enforce the temporal ordering of exposure and outcome, which is not possible with existing wavelet-based functional models. Using a fully Bayesian approach, we conduct formal inference on the time-varying lagged association, while adjusting for multiple testing. We investigate the operating characteristics of our wavelet-packet HFLM and compare them to those of two existing estimation procedures in simulation. We also assess several inference techniques and use the model to analyze data on the impact of lagged exposure to particulate matter finer than 2.5μg, or PM_2.5, on heart rate variability in a cohort of journeyman boilermakers during the morning of a typical day's shift.

Longitudinal dynamic functional regression

The paper develops a parsimonious modelling framework to study the time-varying association between scalar outcomes and functional predictors observed at many instances, in longitudinal studies. The methods enable us to reconstruct the full trajectory of the response and are applicable to Gaussian and non-Gaussian responses. The idea is to model the time-varying functional predictors by using orthogonal basis functions and to expand the time-varying regression coefficient by using the same basis. Numerical investigation through simulation studies and data analysis show excellent performance in terms of accurate prediction and efficient computations, when compared with existing alternatives. The methods are inspired and applied to an animal science application, where of interest is to study the association between the feed intake of lactating sows and the minute-by-minute temperature throughout the 21 days of their lactation period. R code and an R illustration are provided.

The impact of model assumptions in scalar-on-image regression

Complex statistical models such as scalar-on-image regression often require strong assumptions to overcome the issue of nonidentifiability. While in theory, it is well understood that model assumptions can strongly influence the results, this seems to be underappreciated, or played down, in practice. This article gives a systematic overview of the main approaches for scalar-on-image regression with a special focus on their assumptions. We categorize the assumptions and develop measures to quantify the degree to which they are met. The impact of model assumptions and the practical usage of the proposed measures are illustrated in a simulation study and in an application to neuroimaging data. The results show that different assumptions indeed lead to quite different estimates with similar predictive ability, raising the question of their interpretability. We give recommendations for making modeling and interpretation decisions in practice based on the new measures and simulations using hypothetic coefficient images and the observed data.

Optimal Penalized Function-on-Function Regression under a Reproducing Kernel Hilbert Space Framework

Many scientific studies collect data where the response and predictor variables are both functions of time, location, or some other covariate. Understanding the relationship between these functional variables is a common goal in these studies. Motivated from two real-life examples, we present in this paper a function-on-function regression model that can be used to analyze such kind of functional data. Our estimator of the 2D coefficient function is the optimizer of a form of penalized least squares where the penalty enforces a certain level of smoothness on the estimator. Our first result is the Representer Theorem which states that the exact optimizer of the penalized least squares actually resides in a data-adaptive finite dimensional subspace although the optimization problem is defined on a function space of infinite dimensions. This theorem then allows us an easy incorporation of the Gaussian quadrature into the optimization of the penalized least squares, which can be carried out through standard numerical procedures. We also show that our estimator achieves the minimax convergence rate in mean prediction under the framework of function-on-function regression. Extensive simulation studies demonstrate the numerical advantages of our method over the existing ones, where a sparse functional data extension is also introduced. The proposed method is then applied to our motivating examples of the benchmark Canadian weather data and a histone regulation study.

On the estimation of functional random effects

Functional regression modelling has become one of the most vibrant areas of research in the last years. This discussion provides some alternative approaches to one of the key issues of functional data analysis: the basis representation of curves, and in particular, of functional random effects. First, we propose the estimation of functional principal components by penalizing the norm, and as an alternative, we provide an efficient and unified approach based on B-spline basis and quadratic penalties.

A general framework for functional regression modelling

Researchers are increasingly interested in regression models for functional data. This article discusses a comprehensive framework for additive (mixed) models for functional responses and/or functional covariates based on the guiding principle of reframing functional regression in terms of corresponding models for scalar data, allowing the adaptation of a large body of existing methods for these novel tasks. The framework encompasses many existing as well as new models. It includes regression for ‘generalized’ functional data, mean regression, quantile regression as well as generalized additive models for location, shape and scale (GAMLSS) for functional data. It admits many flexible linear, smooth or interaction terms of scalar and functional covariates as well as (functional) random effects and allows flexible choices of bases—particularly splines and functional principal components—and corresponding penalties for each term. It covers functional data observed on common (dense) or curve-specific (sparse) grids. Penalized-likelihood-based and gradient-boosting-based inference for these models are implemented in R packages refund and FDboost , respectively. We also discuss identifiability and computational complexity for the functional regression models covered. A running example on a longitudinal multiple sclerosis imaging study serves to illustrate the flexibility and utility of the proposed model class. Reproducible code for this case study is made available online.

Comparison and Contrast of Two General Functional Regression Modeling Frameworks

In this article, Greven and Scheipl describe an impressively general framework for performing functional regression that builds upon the generalized additive modeling framework. Over the past number of years, my collaborators and I have also been developing a general framework for functional regression, functional mixed models, which shares many similarities with this framework, but has many differences as well. In this discussion, I compare and contrast these two frameworks, to hopefully illuminate characteristics of each, highlighting their respecitve strengths and weaknesses, and providing recommendations regarding the settings in which each approach might be preferable.

A LAG FUNCTIONAL LINEAR MODEL FOR PREDICTION OF MAGNETIZATION TRANSFER RATIO IN MULTIPLE SCLEROSIS LESIONS

We propose a lag functional linear model to predict a response using multiple functional predictors observed at discrete grids with noise. Two procedures are proposed to estimate the regression parameter functions: (1) an approach that ensures smoothness for each value of time using generalized cross-validation; and (2) a global smoothing approach using a restricted maximum likelihood framework. Numerical studies are presented to analyze predictive accuracy in many realistic scenarios. The methods are employed to estimate a magnetic resonance imaging (MRI)-based measure of tissue damage (the magnetization transfer ratio, or MTR) in multiple sclerosis (MS) lesions, a disease that causes damage to the myelin sheaths around axons in the central nervous system. Our method of estimation of MTR within lesions is useful retrospectively in research applications where MTR was not acquired, as well as in clinical practice settings where acquiring MTR is not currently part of the standard of care. The model facilitates the use of commonly acquired imaging modalities to estimate MTR within lesions, and outperforms cross-sectional models that do not account for temporal patterns of lesion development and repair.