We propose a novel factor model in the graph frequency domain for multivariate data residing on the vertices of a graph, referred to as a multivariate graph signal. By utilizing graph filters, our model extends the frequency-domain approach of the dynamic factor model from time series to graphs, enabling a graph-aware and multiscale interpretation of factors across graph frequencies. This latent modeling approach reduces the dimensionality of graph signals, thereby improving the understanding of their structure. It also allows the use of the extracted factors for subsequent analyses, such as clustering. We describe the estimation of factors and their loadings and investigate the consistency of the factor estimator. In addition, we propose two consistent estimators for determining the number of factors. The finite sample performance of the proposed method is demonstrated through simulation studies across various graph structures. We also compare it with classical factor analysis and examine how the choice of graph structure affects the results. The findings show that our model achieves lower reconstruction errors and successfully incorporates the graph structure. Furthermore, we illustrate the effectiveness of the proposed method by applying it to G20 economic data, water quality data from the Geum River, and passenger data from the Seoul Metropolitan subway.

With the advancements in technology and monitoring tools, we often encounter multivariate graph signals, which can be seen as the realizations of multivariate graph processes, and revealing the relationship between their constituent quantities is one of the important problems. To address this issue, we propose a cross-spectral analysis tool for bivariate graph signals. The main goal of this study is to extend the scope of spectral analysis of graph signals to bivariate graph signals. In this study, we define joint weak stationarity graph processes and introduce graph cross-spectral density and coherence for bivariate graph processes. We propose several estimators for the cross-spectral density and investigate the theoretical properties of the proposed estimators. Furthermore, we demonstrate the effectiveness of the proposed estimators through numerical experiments, including simulation studies and a real data application. Finally, as an interesting extension, we discuss robust spectral analysis of graph signals in the presence of outliers.