Numerion: A Multi-Hypercomplex Model for Time Series Forecasting
Hanzhong Cao, Wenbo Yan, Ying Tan
TL;DR
Numerion addresses long-horizon time series forecasting by embedding sequences into multiple hypercomplex spaces of power-of-two dimensions and modeling them with a Real-Hypercomplex-Real domain MLP (RHR-MLP). Each hypercomplex space captures distinct frequency components, and an adaptive fusion mechanism combines their predictions, yielding state-of-the-art MAE results across nine public datasets. The method provides empirical evidence that higher-dimensional hypercomplex spaces tend to model low-frequency features, while lower-dimensional spaces handle high-frequency content, offering a natural multi-frequency decomposition without hand-crafted architectures. This approach offers a principled inductive bias, interpretability in frequency-domain decomposition, and practical forecasting benefits, with future work targeting hardware-accelerated hypercomplex computations and expanded theoretical analysis.
Abstract
Many methods aim to enhance time series forecasting by decomposing the series through intricate model structures and prior knowledge, yet they are inevitably limited by computational complexity and the robustness of the assumptions. Our research uncovers that in the complex domain and higher-order hypercomplex spaces, the characteristic frequencies of time series naturally decrease. Leveraging this insight, we propose Numerion, a time series forecasting model based on multiple hypercomplex spaces. Specifically, grounded in theoretical support, we generalize linear layers and activation functions to hypercomplex spaces of arbitrary power-of-two dimensions and introduce a novel Real-Hypercomplex-Real Domain Multi-Layer Perceptron (RHR-MLP) architecture. Numerion utilizes multiple RHR-MLPs to map time series into hypercomplex spaces of varying dimensions, naturally decomposing and independently modeling the series, and adaptively fuses the latent patterns exhibited in different spaces through a dynamic fusion mechanism. Experiments validate the model`s performance, achieving state-of-the-art results on multiple public datasets. Visualizations and quantitative analyses comprehensively demonstrate the ability of multi-dimensional RHR-MLPs to naturally decompose time series and reveal the tendency of higher dimensional hypercomplex spaces to capture lower frequency features.