Model-Based Depth with Applications to Functional Data

Statistical depth, a commonly used analytic tool in non-parametric statistics, has been extensively studied for multivariate and functional observations over the past few decades. Although various forms of depth were introduced, they are mainly procedure-based whose definitions are independent of the generative model for observations. To address this problem, we introduce a generative model-based approach to define statistical depth for both multivariate and functional data. The proposed model-based depth framework permits simple computation via Monte Carlo sampling and improves the depth estimation accuracy. When applied to functional data, the proposed depth can capture important features such as continuity, smoothness, or phase variability, depending on the defining criteria. Specifically, we view functional data as realizations from a second-order stochastic process, and define their depths through the eigensystem of the covariance operator. These new definitions are given through a proper metric related to the reproducing kernel Hilbert space of the covariance operator. We propose efficient algorithms to compute the proposed depths and establish estimation consistency. Through simulations and real data, we demonstrate that the proposed functional depths reveal important statistical information such as those captured by the median and quantiles, and detect outliers. Functional data appear in applications commonly, particularly in modern neuroscience such as electroencephalogram (EEG) and functional magnetic resonance imaging (fMRI), where phase variability plays an important role in the data. In this study, we introduce the Fisher-Rao registration to analysis neuronal signals and compare with Dynamic Time Warping (DTW) method from the aspects of basic framework, mathematical properties, and computational efficiency. We also introduce FRR to the research area of measures of synchronization and constructing brain network.