Bayesian function‐on‐function regression for multilevel functional data

Abstract

Summary Medical and public health research increasingly involves the collection of complex and high dimensional data. In particular, functional data—where the unit of observation is a curve or set of curves that are finely sampled over a grid—is frequently obtained. Moreover, researchers often sample multiple curves per person resulting in repeated functional measures. A common question is how to analyze the relationship between two functional variables. We propose a general function-on-function regression model for repeatedly sampled functional data on a fine grid, presenting a simple model as well as a more extensive mixed model framework, and introducing various functional Bayesian inferential procedures that account for multiple testing. We examine these models via simulation and a data analysis with data from a study that used event-related potentials to examine how the brain processes various types of images.

A Unified Analysis of Structured Sonar-Terrain Data Using Bayesian Functional Mixed Models

“…Depending on characteristics of the data, other basis such as splines and eigen-functions can also be used for dual space modeling as discussed in Morris et al (2011) and Meyer et al (2015).…”

“…This calculation follows the approach of Ruppert et al (2003, page 142). The SCB is calculated for a range of α values, and the SimBaS is defined at each t as the smallest α for which the 100(1 − α )% SCB excludes zero (Meyer et al, 2015). …”

Nonlinear function on function additive model with multiple predictor curves

“…(Meyer et al, 2015): we pool all the pairs of hourly concentration values of NO 2 and NO x , and calculate the correlation coefficient using all the pairs with the concentration value of NO 2 less than 7 and the correlation coefficient using all the pairs with the concentration value of NO 2 greater than or equal to 7, respectively. The two correlation coefficients are -0.49 and -0.26, respectively, implying a great change of the correlation between NO 2 and NO x with the increase of concentration level of NO x .…”

“…Much effort has been made for linear regression models with functional predictors, such as Ramsay and Dalzell (1991), Cardot et al (1999), Brown et al (2001, Ratcliffe et al (2002), Reiss and Ogden (2007), Goldsmith et al (2012) and Delaigle et al (2012) for linear scalar-on-function regression models, and Ramsay and Silverman (2005), Yao et al (2005), Ivanescu et al (2014), Meyer et al (2015), Chiou et al (2016), Luo and Qi (2017) and Luo et al (2016) for linear function-on-function regression. There have also been numerous studies on nonlinear scalar-on-function regression models.…”

Comparison and contrast of two general functional regression modelling frameworks

“…Meyer, et al (2015) showed how this framework could allow functional predictors X ia ( s ) for s ∈ 𝒮 by introducing function-on-function regression terms…”