Rethinking Patch Aggregation for Irregular Time Series Forecasting with Voronoi Quadrature

Patching has become the efficient backbone of irregular multivariate time series (IMTS) forecasting, yet the step that turns scattered records into a patch token has remained largely unexamined. We revisit that step by formalizing a patch operator as a pair: a support kernel delimiting its temporal scope, and an aggregation measure assigning integration mass to the observations within that support. Prior work has increasingly refined the support kernel while leaving the aggregation measure at one unit of base mass per observation, so patch content estimates an average under the observation-induced measure rather than over physical time. We formalize this discrepancy and prove that it equals the patch-local covariance between temporal content and observation intensity, divided by the mean intensity. Being deterministic, the discrepancy resists averaging: it vanishes on a regular grid, grows whenever recording density correlates with content, and is fixed before any learnable module reads the token, beyond the reach of added capacity. Guided by this identity, we propose VQPatch, which treats every record as the representative of its Voronoi temporal cell, converts cell widths into quadrature weights, and retains the observation pattern as a bounded prediction condition. The resulting token provably approximates the physical-time patch integral at a rate governed by the cell mesh and independent of the sampling intensity, at no asymptotic cost beyond sorting. Across four IMTS benchmarks and seventeen baselines, VQPatch attains the lowest MAE on every dataset, improves MSE over the strongest patch baseline by up to 9.36%, and cuts peak training memory by 61.4%.