Stationary Graph Processes and Spectral Estimation (1603.04667v2)

Abstract: Stationarity is a cornerstone property that facilitates the analysis and processing of random signals in the time domain. Although time-varying signals are abundant in nature, in many practical scenarios the information of interest resides in more irregular graph domains. This lack of regularity hampers the generalization of the classical notion of stationarity to graph signals. The contribution in this paper is twofold. Firstly, we propose a definition of weak stationarity for random graph signals that takes into account the structure of the graph where the random process takes place, while inheriting many of the meaningful properties of the classical definition in the time domain. Our definition requires that stationary graph processes can be modeled as the output of a linear graph filter applied to a white input. We will show that this is equivalent to requiring the correlation matrix to be diagonalized by the graph Fourier transform. Secondly, we analyze the properties of the power spectral density and propose a number of methods to estimate it. We start with nonparametric approaches, including periodograms, window-based average periodograms, and filter banks. We then shift the focus to parametric approaches, discussing the estimation of moving-average (MA), autoregressive (AR) and ARMA processes. Finally, we illustrate the power spectral density estimation in synthetic and real-world graphs.

• The paper introduces weak stationarity for graph signals, defining them as outputs of linear graph filters applied to white noise.
• It adapts traditional spectral estimation techniques like periodograms and window methods to accurately estimate power spectral density on graphs.
• Parametric models such as MA, AR, and ARMA are extended to graph signals, offering improved PSD estimation and insights into graph connectivity.

Stationary Graph Processes and Spectral Estimation

The paper entitled "Stationary Graph Processes and Spectral Estimation" extends the notion of stationarity, a fundamental property in time domain signal processing, to graph signals, facilitating the analysis and processing of random processes on graph domains. The authors propose a definition of weak stationarity for random graph signals that considers the graph's inherent structure through the application of a normal matrix representing the graph shift operator (GSO).

Weak stationarity for graph processes is defined by the ability to model these processes as the output of a linear graph filter applied to a white noise input, entailing that such processes have a covariance matrix diagonalizable by the graph Fourier transform (GFT). This is analogous to stationarity in the time domain, where power spectral density (PSD) plays a crucial role by summarizing the signal’s frequency behavior. The paper highlights different methods for estimating the PSD of graph signals, including extensions of traditional spectral estimation techniques like periodograms and windowing, adapted to handle the complexity and irregularity of graph domains.

Key Numerical Results and Analysis

Key numerical analyses demonstrate that the proposed methods extend the conventional periodogram and window-based estimation techniques to graph signal processing, providing unbiased estimates and reducing MSE when applied intelligently. For instance, the paper shows that utilizing a set of local windows or a filter bank that respects inherent graph connectivity properties can significantly improve PSD estimation accuracy. The simulation results illustrate how various window and filter bank configurations affect estimation performance, particularly when exploiting graph locality and connectivity.

Parametric methods, including moving-average (MA), autoregressive (AR), and autoregressive moving average (ARMA) models, are also adapted for graph signals, allowing for efficient modeling and PSD estimation. These methods illustrate promising performance, especially when the graph structure induces specific spectral characteristics. For example, autoregressive processes on graphs correctly model local dependencies and their implications on the PSD.

Implications and Future Directions

This research opens new avenues in graph signal processing by extending classical spectral analysis techniques to graph domains, providing tools for applications where data reside on irregular supports, such as social networks, brain connectivity networks, and infrastructure grids. Future work could explore more sophisticated graph topologies and dynamics, develop adaptive methods for estimating stationarity properties, and integrate machine learning to enhance model robustness. The exploration of how these theoretical tools apply to practical problems indicates substantial potential for further interdisciplinary research in network science, control theory, and complex systems analytics.

The work introduces a comprehensive foundation for the analysis of stationary processes on graphs, enhancing the signal processing toolkit for modern data science applications. As computational capabilities evolve, the employment of these methods in dynamic and large-scale networks promises practical utility in domains as diverse as urban planning, biological network analysis, and cyber-physical system design.