Robust and Gaussian spatial functional regression models for analysis of event-related potentials

Abstract

Event-related potentials (ERPs) summarize electrophysiological brain response to specific stimuli. They can be considered as correlated functions of time with both spatial correlation across electrodes and nested correlations within subjects. Commonly used analytical methods for ERPs often focus on pre-determined extracted components and/or ignore the correlation among electrodes or subjects, which can miss important insights, and tend to be sensitive to outlying subjects, time points or electrodes. Motivated by ERP data in a smoking cessation study, we introduce a Bayesian spatial functional regression framework that models the entire ERPs as spatially correlated functional responses and the stimulus types as covariates. This novel framework relies on mixed models to characterize the effects of stimuli while simultaneously accounting for the multilevel correlation structure. The spatial correlation among the ERP profiles is captured through basis-space Matérn assumptions that allow either separable or nonseparable spatial correlations over time. We induce both adaptive regularization over time and spatial smoothness across electrodes via a correlated normal-exponential-gamma (CNEG) prior on the fixed effect coefficient functions. Our proposed framework includes both Gaussian models as well as robust models using heavier-tailed distributions to make the regression automatically robust to outliers. We introduce predictive methods to select among Gaussian vs. robust models and models with separable vs. non-separable spatiotemporal correlation structures. Our proposed analysis produces global tests for stimuli effects across entire time (or time-frequency) and electrode domains, plus multiplicity-adjusted pointwise inference based on experiment-wise error rate or false discovery rate to flag spatiotemporal (or spatio-temporal-frequency) regions that characterize stimuli differences, and can also produce inference for any prespecified waveform components. Our analysis of the smoking cessation ERP data set reveals numerous effects across different types of visual stimuli.

Ultra-Fast Approximate Inference Using Variational Functional Mixed Models

“…As we have highlighted in Section 1, the current framework can be directly applied to FMM under different basis choices as well as several extensions of FMM without any major modifications. For other extensions, such as the robust FMM with heavier tailed distributions (Zhu et al, 2011) and FMM with spatial-temporal correlations in residuals (Zhang et al, 2016;Zhu et al, 2018), modifications are needed in order to estimate parameters induced by assuming heavier-tailed distribution or between-function correlation.…”

Multivariate functional response regression, with application to fluorescence spectroscopy in a cervical pre-cancer study

“…Extensions that incorporate heavier tailed distributions can be performed following the work of Zhu et al (2011). Furthermore, in addition to the multi-level structure, we can incorporate between-function spatial or temporal correlations in the residual terms and/or the fixed effect coefficient functions, following strategies similar to Zhang et al (2014) and Zhu et al (2016b).…”

Bayesian Semiparametric Functional Mixed Models for Serially Correlated Functional Data, With Application to Glaucoma Data

“…In this case, the random effect functions U hm ( t ) are iid mean zero Gaussian Processes with intrafunctional covariance cov{ U hm ( t 1 ) , U hm ( t 2 )} = Q h ( t 1 , t 2 ) and the residual error functions E i ( t ) are iid mean zero Gaussian Processes with intrafunctional covariance cov{ E i ( t 1 ) ,E i ( t 2 )} = S ( t 1 , t 2 ). Other extensions of this framework allow the option of conditional autoregressive (CAR) (Zhang et al, 2014) or Matern spatial covariance or AR( p ) temporal interfunctional correlation structures in the residual errors (Zhu et al, 2014). Although focusing on Gaussian regression here, a robust version of this framework assuming heavier tailed distributions on the random effects or residuals is available (Zhu et al, 2011) if robustness to outliers is desired, and can also be utilized with any other features or modeling components in the BayesFMM framework.…”

“…However, there is a lack of functional regression model selection methods for MCMC-based fully Bayesian models such as the semiparametric BayesFMM, which present special challenges. One could split the data into training and validation data sets, fit separate MCMC for each prospective model in the training data, and then compute ratios of predicted marginal densities, integrating over MCMC posterior samples, for the validation data, as done in Zhu et al (2014), for example. These predictive Bayes Factors would provide a rigorous model selection measure, or alternatively parallel MCMC could be run for each prospective model, and a multinomial random variable with Dirichlet prior used to select and perform Bayesian model averaging across models as the chains progress.…”

“…In this case, the random effect functions U hm (t) are iid mean zero Gaussian Processes with intrafunctional covariance cov{U hm (t 1 ), U hm (t 2 )} = Q h (t 1 , t 2 ) and the residual error functions E i (t) are iid mean zero Gaussian Processes with intrafunctional covariance cov{E i (t 1 ), E i (t 2 )} = S(t 1 , t 2 ). Other extensions of this framework allow the option of conditional autoregressive (CAR) (Zhang et al, 2014) or Matern spatial covariance or AR(p) temporal interfunctional correlation structures in the residual errors (Zhu et al, 2014).…”