平穩性是經典信號處理（CSP）中用於對各種隨機信號進行建模和表徵以進行後續分析的基石。但是，在許多複雜的真實世界場景，其中隨機過程處於不規則的圖結構上，CSP在分析此類結構化數據時會拋棄基礎結構。建立新的框架用以分析高維圖隨機信號並考慮底層結構是至關重要的。為此，從運算子理論的角度來看，我們首先提出一個新的雙變量等距聯合平移算子（JTO），此算子與其他信號域中之算子有一致的抽象形式。此外，基於所提出的JTO，我們表徵時間-頂點濾波的概念。隨即，我們使用提出的等距JTO及其頻譜特徵，提出了時域中的聯合廣義平穩（JWSS）信號的新定義。然後提出一個新的聯合功率譜密度（JPSD）估算器，稱為廣義韋爾奇方法（GWM）。也提供仿真結果以驗証該JPSD估計器的有效性。為了展示這種建模的有效性，我們將目標專注於圖上時間序列的分類。然後，通過將腦電圖（EEG）信號建模為隨機時變圖信號，我們使用JPSD作為特徵以進行十分有挑戰性的情緒識別工作。實驗結果顯示，JPSD特徵與經典功率譜密度（PSD）和圖形PSD（GPSD）作為兩種應用的特徵集相比，在情緒識別上產生較高的準確性。

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. 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.

1 Introduction 9
2 Mathematical Background 13
2.1 Vertex Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Discrete-Time Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Joint Time-Vertex Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . 17
3 Bivariate Transition Operator in Time-Vertex Domain 31
3.1 Graph Transition/Translation Operator . . . . . . . . . . . . . . . . . . . . 23
3.2 Isometric Bivariate Joint Transition Operator . . . . . . . . . . . . . . . . . 25
3.3 Joint Filtering via Joint Transition Operator . . . . . . . . . . . . . . . . . 30
4 Stationarity of Time-Series on Graph 35
4.1 Joint Wide-Sense Stationarity via Transition Invariance . . . . . . . . . . . 35
4.2 Separable JWSS Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Joint Power Spectral Density Estimation . . . . . . . . . . . . . . . . . . . . 40
4.4 Simulation and Experimental Results . . . . . . . . . . . . . . . . . . . . . . 43
5 Concluding Remarks 51

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