ZCN: Adaptive Multi-Scale Complex-Domain Representation for Multivariate Time-Series Forecasting

Abstract

Multivariate time series forecasting requires capturing long-term trends, periodic patterns, local transients, and cross-variable dependencies within a single framework. This paper proposes ZCN, an adaptive complex-domain model that addresses this challenge through a learnable numerical Z-transform. Specifically, after reversible instance normalization, the input sequence is evaluated on multiple trainable radii, which extend the standard unit-circle Fourier analysis by introducing complementary radial weightings. The resulting complex-valued responses are projected onto a bounded Riemann sphere and processed by a multi-scale complex U-Net. A channel-mixing module then recalibrates cross-variable features, followed by an MLP prediction head that outputs the future trajectory. We evaluate ZCN on five public datasets spanning energy, transportation, and finance, across four prediction horizons, comparing against PatchTST, FITS, TimesNet, and DLinear. Among 40 dataset-horizon-metric combinations, ZCN achieves the best or tied-best performance in 35 cases and ranks second in the remaining five. Ablation studies confirm the contributions of the multi-scale architecture, spherical mapping, and channel-mixing components, though the benefit of spherical mapping on one traffic dataset requires further validation. Overall, these results demonstrate that learnable radial complex-domain features are an effective representation for long-horizon forecasting. Importantly, these features should be interpreted as forecasting-oriented representations rather than direct identification of physical system poles.