Hierarchical Regime-Conditioned Ornstein–Uhlenbeck Monitoring for Thermal Control in Memory Reliability Test Equipment

Burn-in fixtures for memory packages log the chamber temperature every second, but the logs contain no fault labels, the sensor resolution is comparable to the noise, and normal behaviour depends on both the operating regime and the individual fixture; a regime-blind reconstruction detector flagged 67–73% of its test period. This paper models the hold-phase tracking error as a three-level hierarchy (model, fixture, episode) driven by a regime-indexed Ornstein–Uhlenbeck (OU) process observed irregularly through a quantizer. A grid filter over level and state yields randomized probability-integral-transform (PIT) innovations that are independent standard normal for any quantizer when the hyperparameters are known, so cumulative-sum (CUSUM) charts run at theoretical thresholds; simulation shows the calibration survives parameter estimation. Five results are proved: an alarm-inflation formula for mismatched standardization, moment estimators for the variance components, a Fisher-information bound on the level information surviving quantization, a boundary-crossing lower bound for the cold-start false-alarm probability, and a sampling-interval law under which delay grows with coarse spacing and saturates with fine spacing. On six fixtures of two models the chart attains the nominal false-alarm rate where the hold level is stable, matches tuned baselines without calibration data, and detects a rack-level limit-cycle oscillation within 0.02 min.