Real-world data, particularly from dynamic systems like industrial processes and weather phenomena, often exhibits complex, nonlinear behavior. Analyzing such data requires sophisticated mathematical tools capable of handling both temporal and spatial complexities. Multivariate entropy techniques, such as Multivariate Dispersion Entropy (mvDE), offer a robust approach to quantifying the complexity of multivariate time series. However, traditional methods often neglect the inherent topological structures present in many datasets. The increasing use of graph-based representations in data analysis motivates the development of methods that integrate both temporal and topological information. This paper addresses this need by introducing mvDEG, which builds upon existing graph-based entropy methods (Permutation Entropy and Dispersion Entropy for graph signals) by incorporating both temporal and topological dimensions for a more comprehensive analysis. This improved methodology aims to provide more accurate and efficient analysis of complex systems compared to existing techniques.

Existing multivariate entropy techniques, such as Multivariate Sample Entropy and mvDE, provide valuable tools for analyzing multivariate time series data. mvDE, in particular, shows better performance and stability than other methods, especially for shorter time series. However, these techniques often fail to consider the underlying topological structures in the data. Recent advancements have focused on extending entropy measures to graph signals, incorporating topological information. This includes the development of Permutation Entropy and Dispersion Entropy tailored for graph data, as well as the extension of Bubble Entropy to graph signals. While these methods advance the analysis of network data, they often do not fully integrate the temporal and topological dimensions, leading to incomplete insights, especially in systems where the interplay between spatial structure and temporal dynamics is crucial.

The proposed mvDEG algorithm enhances dispersion entropy by integrating both temporal and topological information. It leverages a coarse-graining process to analyze the data at multiple scales. The algorithm involves two main steps: 1) a coarse-graining process on the multivariate signals, and 2) the calculation of mvDEG at each scale. The coarse-graining process divides each channel of the multivariate signal into non-overlapping segments of length τ (scale factor), calculating the average of each segment to derive coarse-grained signals. An adjacency matrix, I_p, which can be predefined or learned from the data, encapsulates the connectivity between the channels. The mvDEG is computed using an embedding matrix constructed from the coarse-grained signals and the adjacency matrix. The embedding matrix is then mapped to classes, and the relative frequencies of each dispersion pattern are calculated to compute the entropy. The primary computational challenge lies in computing powers of large matrices. This is addressed by utilizing matrix properties and the Kronecker product to efficiently calculate the required matrix powers. This optimized implementation reduces the computational complexity of mvDEG from exponential to linear growth with the number of vertices, making it significantly more efficient than classical methods.

The performance of mvDEG was evaluated using synthetic and real-world datasets. Experiments with synthetic signals (multivariate 1/f noise and White Gaussian Noise) demonstrated mvDEG's ability to distinguish between different levels of complexity and correlation. In uncorrelated noise analysis, mvDEG produced results similar to mvDE, but with significantly improved computational efficiency. In correlated noise analysis, mvDEG demonstrated superior performance in distinguishing between varying degrees of correlation, unlike mvDE where entropy values overlapped across different correlation levels. A computational time comparison showed that mvDEG's computational time grows linearly with the number of sample points and channels, while the classical mvDE shows exponential growth. Real-world applications to weather data and two-phase flow data further validated mvDEG's effectiveness. In weather data analysis, mvDEG successfully differentiated between the complexity of rainfall, temperature, and wind data based on the respective entropy levels across different scales. In two-phase flow data analysis, mvDEG clearly distinguished six different flow regimes (bubbly, stratified, slug, plug, churn, and annular) based on their distinct entropy profiles across scales, demonstrating its ability to capture the nuanced dynamics of each flow regime. This is particularly notable at lower scales where traditional methods might struggle. In both real-world applications, mvDEG's computational efficiency allowed the analysis of larger datasets without memory errors encountered with the classical mvDE.

The results demonstrate that mvDEG offers a significant improvement over traditional multivariate entropy methods. Its ability to integrate both temporal and topological information provides a more comprehensive analysis of complex systems. The linear computational complexity of mvDEG addresses a major limitation of classical methods, making it suitable for large-scale and real-time applications. The application to real-world datasets showcases its versatility and robustness in handling complex, high-dimensional data. The improved accuracy and efficiency of mvDEG offer valuable advancements in fields such as weather forecasting, industrial process monitoring, and other applications involving multivariate time series data on networks. The findings highlight the importance of considering both temporal and topological dimensions in the analysis of complex systems.

This paper introduced mvDEG, a novel and computationally efficient method for analyzing multivariate time series data on graphs. Its ability to effectively combine temporal dynamics and topological information, coupled with its linear computational time complexity, positions it as a significant advancement in multivariate time series analysis. Future research could explore the application of mvDEG to other types of data and the development of more sophisticated methods for learning the adjacency matrix from data.

While mvDEG demonstrates significant advantages, several limitations should be considered. The performance of the method depends on the choice of parameters, such as the embedding dimension (m) and the number of classes (c). The selection of an appropriate adjacency matrix is also crucial, and further research is needed to develop more robust methods for learning the adjacency matrix from data, especially in noisy or incomplete datasets. The coarse-graining method employed is a straightforward approach; further investigation of other coarse-graining techniques may yield improved results in certain applications.