An Irregular Patch is a Local Function: Rethinking Patch Representation for Irregular Time Series Forecasting

Patch-based forecasters for irregular multivariate time series have mainly improved where patches are placed; we study what a patch should represent. For a geometryweighted mean, a locally linear signal incurs a level error equal to its local slope times the weighted offset of observations from the patch center. We therefore represent each patch as a weighted local-linear estimate of its level and change. The closed-form estimator is exact for locally linear signals and retains secondorder bias under non-degenerate asymmetric sampling, where the weighted mean becomes first-order. It uses only timestamps and response-independent support weights, so it applies to fixed hard windows and adaptive soft supports. Replacing the patch interior lowers MSE and MAE in all eight dataset–metric pairs for both a fixed-grid PatchTST-style backbone and APN; the fixed-grid comparison shares hard weights, whereas APN variants share the support parameterization but are trained separately. Our full model, Adaptive Kernel Patching (AKP), adds sampleconditioned supports, support-mass-gated aggregation, and cross-variable mixing. Against 17 baselines, each tuned with 100 Optuna trials and evaluated over five final runs, AKP has the best mean on five of eight pairs; on USHCN it improves MSE/MAE by 4.0%/17.4% over the strongest competitor.