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    This study introduces cross-spectral analysis for bivariate graph signals, enabling the examination of relationships within multivariate graph processes. The new methods provide effective estimators for graph cross-spectral density and coherence.

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    Area of Science:

    • Signal Processing
    • Graph Signal Processing
    • Time Series Analysis

    Background:

    • Multivariate graph signals are increasingly common due to technological advancements.
    • Understanding relationships between quantities in these signals is crucial.
    • Existing spectral analysis tools are limited for bivariate graph signals.

    Purpose of the Study:

    • To extend spectral analysis to bivariate graph signals.
    • To introduce methods for analyzing relationships in multivariate graph processes.
    • To develop estimators for graph cross-spectral density and coherence.

    Main Methods:

    • Definition of joint weak stationarity graph processes.
    • Introduction of graph cross-spectral density and coherence for bivariate processes.
    • Development and theoretical investigation of estimators for cross-spectral density.

    Main Results:

    • Proposed estimators for graph cross-spectral density were developed.
    • Theoretical properties of the estimators were analyzed.
    • Effectiveness demonstrated through simulations and a real-world data application.

    Conclusions:

    • The proposed cross-spectral analysis tool effectively analyzes bivariate graph signals.
    • The developed estimators provide reliable insights into signal relationships.
    • Future work includes robust spectral analysis for outlier detection.