Bayesian function-on-function regression for multilevel functional data

Abstract

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.

Keywords: Basis functions; Bayesian inference; Function-on-function regression; Functional data analysis; Functional mixed models; Functional testing; Principal components; Wavelet regression.

© 2015, The International Biometric Society.

Figures

Figure 1

Heat maps of the true surfaces for simulation study are above heat maps of a single estimated surfaces for each simulated scenario based on a sample size of n = 25 with two measure per subject, Ci = 2 ∀ i, for a total of N = 50 observations. Each surface has near average rMSE and is based on a total of 2000 MCMC chains with the first 1000 discarded.

Figure 2

On the left, raw profile curves are plotted in gray with the mean in red from electrode 129 under the cigarette image condition. On the right, are raw curves and the mean from electrode 129 under the neutral image condition.

Figure 3

The top row contains surface estimates for the association between electrodes 129 and 55. Posterior surfaces comparing electrodes 11 to 75 are in the second row. The estimated posterior surface of the difference between cigarette and neutral is found in the first column. Group specific surface estimates are in the second and third columns, Neutral and Cigarette respectively. ERP output from electrode 129 is the response and the output from electrode 55 is the predictor for the first model and electrode 75 is the predictor of electrode 11 in the second model.

Figure 4

Heat maps containing the posterior probabilities from the BFDR procedure using a δ intensity change of 0.05. Locations (υ, t) in white have a high probability of being greater than δ and thus likely to be included in ψ, the set of locations identified as significant. Black locations (υ, t) have a low probability of being greater than δ and are thus less likely to be identified as significant. The top row contains results from the model using electrodes 129 and 55 while the second row contains results from the model using electrodes 75 and 11.

Figure 5

Heat maps containing the SimBa scores for each surface of both models. The top row contains results from the model using electrodes 129 and 55 while the second row contains results from the model using electrodes 75 and 11. Scores are plotted on the log-scale with the color axis on the exponential scale. White regions represent coefficients with low SimBa scores, black regions represent coefficients with high SimBa scores.