Wasserstein Boxplots for the Analysis of EEG Power Spectral Densities With Applications to Autism.

Functional boxplots are an informative exploratory tool for visualizing functional data and providing a concise summary of a dataset's central tendency, spread, and potential outliers. Motivated by within- and across-sample visualization of electroencephalogram (EEG) power spectral densities in autism studies, we propose Wasserstein boxplots for summarizing a sample of densities. Densities take on nonnegative values and integrate to 1. In order to take into account the constraints of the signal and to avoid metric distortions associated with transformations, the proposed Wasserstein boxplots utilize the 2-Wasserstein metric for quantifying distances between densities. In addition, the proposed boxplots extend the traditional approach of summarizing central tendency within a single dataset to comparative settings where central tendency of data within a target sample is quantified with respect to a reference group. Cross-sample Wasserstein boxplots are motivated by quantification of deviations in power spectra of autistic children (target group) from neurotypical development (reference group). Finally, covariate-adjusted boxplots are proposed for quantifying deviations in the target group from the Fréchet mean in the reference group, conditional on covariates. A unique feature of EEG power spectra is the peak alpha frequency (PAF), which shifts to higher frequencies as children age. Hence, covariate-adjusted boxplots are used to quantify deviations in the autistic sample from neurotypical spectra, conditional on age. The proposed exploratory tools, especially the comparative analyses are applicable more broadly, beyond the motivating autism context, to studies involving both a target and a reference group.