FCTGE: fractal-causal temporal graph embedding for unified time series representation learning

Abstract

Multivariate dynamical systems often generate high-dimensional, heterogeneous, and non-stationary time series, which pose significant challenges for learning expressive and robust representations for downstream tasks such as forecasting and anomaly detection. Existing time series embedding methods typically capture only limited aspects of temporal structure and therefore struggle to simultaneously characterize multiscale temporal dependencies, inter-variable causal relationships, and the nonlinear dynamics inherent in complex systems. To address this limitation, this paper proposes a novel fractal–causal temporal graph embedding (FCTGE) framework, which unifies multiscale fractal temporal dynamics, time-lagged causal interactions, and variable-level statistical characteristics within a unified graph-based representation learning paradigm. In the proposed framework, multivariate time series are represented as time-evolving graphs, where nodes correspond to system variables and edges represent temporally adaptive causal relationships inferred from the data. Extensive experiments on multiple benchmark multivariate time series datasets demonstrate that FCTGE consistently outperforms a wide range of state-of-the-art methods—including representative Transformer-based and graph-based architectures—in both long-term forecasting and unsupervised anomaly detection tasks. Specifically, FCTGE achieves a substantial reduction in prediction errors (MSE/MAE) and yields superior F1-scores compared to specialized anomaly detection frameworks. These results confirm that by explicitly coupling multifractal dynamics with adaptive causal priors, our framework effectively overcomes the inherent limitations of existing methods in characterizing non-stationary behaviors, providing more robust and physically interpretable representations for complex dynamical systems.