  Stationarity is a cornerstone in classical signal processing (CSP) for
modeling and characterizing various stochastic signals for the ensuing
analysis. However, in many complex real world scenarios, where the stochastic
process lies over an irregular graph structure, CSP discards the underlying
structure in analyzing such structured data. Then it is essential to establish
a new framework to analyze the high-dimensional graph structured stochastic
signals by taking the underlying structure into account. To this end, looking
through the lens of operator theory, we first propose a new bivariate isometric
joint translation operator (JTO) consistent with the structural characteristic
of translation operators in other signal domains. Moreover, we characterize
time-vertex filtering based on the proposed JTO. Thereupon, we put forth a new
definition of joint wide-sense stationary (JWSS) signals in time-vertex domain
using the proposed isometric JTO together with its spectral characterization.
Then a new joint power spectral density (JPSD) estimator, called generalized
Welch method (GWM), is presented. Simulation results are provided to show the
efficacy of this JPSD estimator. Furthermore, to show the usefulness of JWSS
modeling, we focus on the classification of time-series on graph. To that end,
by modeling the brain Electroencephalography (EEG) signals as JWSS processes,
we use JPSD as the feature for the Emotion and Alzheimer's disease (AD)
recognition. Experimental results demonstrate that JPSD yields superior Emotion
and AD recognition accuracy in comparison with the classical power spectral
density (PSD) and graph PSD (GPSD) as the feature set for both applications.
Eventually, we provide some concluding remarks.
