The circadian clock is an internal timer that coordinates the daily rhythms of behavior and physiology, including sleep and hormone secretion. Accurately tracking the state of the circadian clock, or circadian phase, holds immense potential for precision medicine. Wearable devices present an opportunity to estimate the circadian phase in the real world, as they can non-invasively monitor various physiological outputs influenced by the circadian clock. However, accurately estimating circadian phase from wearable data remains challenging, primarily due to the lack of methods that integrate minute-by-minute wearable data with prior knowledge of the circadian phase. To address this issue, we propose a framework that integrates multi-time scale physiological data to estimate the circadian phase, along with an efficient implementation algorithm based on Bayesian inference and a new state space estimation method called the level set Kalman filter. Our numerical experiments indicate that our approach outperforms previous methods for circadian phase estimation consistently. Furthermore, our method enables us to examine the contribution of noise from different sources to the estimation, which was not feasible with prior methods. We found that internal noise unrelated to external stimuli is a crucial factor in determining estimation results. Lastly, we developed a user-friendly computational package and applied it to real-world data to demonstrate the potential value of our approach. Our results provide a foundation for systematically understanding the real-world dynamics of the circadian clock.
