MATH-SYNC: Robust and Identifiable Affine-Clock Alignment of Heterogeneous Multirate Sensor Streams

Synchronizing heterogeneous multirate sensor streams is a prerequisite for valid event-related inference, yet practical multimodal systems combine independent clocks, unequal native sampling rates, missing synchronization anchors, timestamp jitter, and occasional gross timing errors. This study introduces MATH-SYNC, a robust affine-clock alignment framework that estimates a common temporal coordinate without forcing signals onto a shared sampling grid. An inverse affine-clock reparameterization yields a linear residual model after fixing a temporal gauge, and Huber M-estimation is implemented by iteratively reweighted least squares with a MAD-based robust scale update; conditional on a fixed robust scale, the resulting Huber objective is convex. We characterize formulation-specific structural properties, including fixed-scale convexity, exact rank-based identifiability, a sufficient anchor condition, conditional uniqueness, and fixed-weight conditioning. Validation combines controlled Monte Carlo experiments with semi-synthetic perturbation-and-replay on real public timing schedules while preserving native sampling rates. Within-participant replication on Harvard S0201 showed minimal session-to-session variation. In a frozen 10-participant Harvard cohort, Joint Huber reduced participant-level median macro-RMSE relative to Joint OLS by 73.7% under nominal contamination (4.052 to 1.064 ms) and 62.4% under stress contamination (15.190 to 5.713 ms), with improvements in all participants. In the external 15-participant WESAD dataset, the corresponding reductions were 82.1% and 67.5%, again with improvements in all participants. Under nearly clean timing, OLS and Huber were practically indistinguishable, with median differences below 1 microsecond. These findings support robust, identifiable, and native-rate-preserving affine-clock estimation for heterogeneous multimodal sensor alignment under controlled timestamp contamination in synthetic and semi-synthetic schedule replay.