Multivariate Time Series Forecasting needs Cross Variable Loss

2026-08-06

AI Summary

This work addresses a critical yet overlooked issue in multivariate time series modeling: existing direct prediction methods often ignore the co-evolutionary dynamics and lagged dependencies among variables, leading to a mismatch between the learning objective and the true underlying temporal structure. To resolve this objective misalignment, the paper introduces CvLoss, a plug-and-play cross-variable loss regularizer that enforces consistency between synchronous and asynchronous variable interactions through constraints on the residual graph of predictions. By integrating graph-structured residual regularization, cross-variable interaction modeling, and a multi-step direct forecasting framework, CvLoss consistently enhances the performance of diverse state-of-the-art models. Extensive experiments across multiple benchmark datasets demonstrate its generality and effectiveness.

Abstract

Multivariate time series forecasting presents unique challenges because future variables often co-evolve under shared system dynamics. While existing studies mainly focus on cross-variable dependencies in historical observations, dependencies among future values are much less explored. Specifically, modern forecasting models largely follow the Direct Forecasting (DF) paradigm, generating multi-step forecasts with point-wise objectives that do not explicitly constrain cross-variable structure. In this work, we show that the DF objective is mismatched in the presence of cross-variable and lagged dependencies, revealing an objective gap. To address this issue, we propose \textbf{C}ross-\textbf{V}ariable \textbf{Loss} (CvLoss), a plug-in structural regularizer that constrains forecast residuals on a cross-variable graph. CvLoss penalizes inconsistent edge-wise residual differences over forecast patches, encouraging consistency across both synchronous and asynchronous interactions. Our experiments show that CvLoss consistently improves competitive forecasting models, outperforms representative learning objectives, and is compatible with a variety of forecasting backbones.

Problem

• Multivariate Time Series Forecasting
• Cross-variable Dependencies
• Direct Forecasting
• Forecasting Objective
• Structural Consistency

Innovation

• Multivariate Time Series Forecasting
• Cross-Variable Dependencies
• Structural Regularization
• Forecasting Objective
• CvLoss