Graph-Based Multivariate Multiscale Dispersion Entropy: Efficient Implementation and Applications to Real-World Network Data

TL;DR

We address the challenge of analyzing multivariate time series on graphs by introducing $\mathop{\mathrm{mvDE_G}}$, a graph-aware, multiscale dispersion entropy. The method combines a coarse-graining step with a graph-structured embedding to compute normalized entropy over dispersion patterns, and it exploits Kronecker-product-based decompositions to achieve linear-time scaling in the number of nodes. Through synthetic tests (uncorrelated and correlated noise) and real-world datasets (weather-ground stations and two-phase flow), $\mathop{\mathrm{mvDE_G}}$ demonstrates superior discriminative ability and substantial computational gains over classical mvDE. The approach enables robust, real-time-capable analysis of large-scale multivariate time series on graphs, opening avenues for distributed sensing, climate science, and industrial monitoring.

Abstract

We introduce Multivariate Multiscale Graph-based Dispersion Entropy (mvDEG), a novel, computationally efficient method for analyzing multivariate time series data in graph and complex network frameworks, and demonstrate its application in real-world data. mvDEG effectively combines temporal dynamics with topological relationships, offering enhanced analysis compared to traditional nonlinear entropy methods. Its efficacy is established through testing on synthetic signals, such as uncorrelated and correlated noise, showcasing its adeptness in discerning various levels of dependency and complexity. The robustness of mvDEG is further validated with real-world datasets, effectively differentiating various two-phase flow regimes and capturing distinct dynamics in weather data analysis. An important advancement of mvDEG is its computational efficiency. Our optimized algorithm displays a computational time that grows linearly with the number of vertices or nodes, in contrast to the exponential growth observed in classical methods. This efficiency is achieved through refined matrix power calculations that exploit matrix and Kronecker product properties, making our method faster than the state of the art. The significant acceleration in computational time positions mvDEG as a transformative tool for extensive and real-time applications, setting a new benchmark in the analysis of time series recorded at distributed locations and opening avenues for innovative applications.

Figures (10)

• Figure 1: Mean value and standard deviation (SD) for the classical multivariate Dispersion Entropy $\mathop{\mathrm{mvDE}}\nolimits$.
• Figure 2: Entropy value comparison across scale factors for $\mathop{\mathrm{mvDE_G}}\nolimits$, the mean and SD with (a) identity adjacency graph, indicating no prior inter-channel correlation, and (b) correlation matrix graph, reflecting inherent inter-channel dependencies.
• Figure 3: Correlated noise entropy across various structures, ranging from uncorrelated to fully connected across scale factors.
• Figure 4: Entropy for varying degrees of correlated noise.
• Figure 5: Computational times comparison for $\mathop{\mathrm{mvDE_G}}\nolimits$ against classical $\mathop{\mathrm{mvDE}}\nolimits$, for different data lengths.