Multivariate Transition Entropy–Fisher Information Plane Analysis of Electricity Load System

In this paper, we propose multivariate transition entropy–Fisher information plane (MTEFI), a framework designed to address the limitations of existing univariate complexity measures, which often fail to capture higher-order interactions and structural heterogeneity in multivariate systems. MTEFI integrates multivariate transition entropy (MTE) and multivariate transition Fisher information (MTFI), both computed from the transition probability matrix of a multivariate ordinal transition network (MOTN), thereby providing a unified characterization of global uncertainty and local variability in high-dimensional data. Through simulations on stochastic systems, chaotic systems and trivariate logistic systems, MTEFI demonstrates strong capability to distinguish stochastic from chaotic dynamics and to reveal causal structural differences in homogeneous systems. Finally, a comparative analysis of electricity load system across 14 European countries illustrates how geography, climate, energy structures and consumer behaviors jointly shape the complexity of real-world systems.