Phase Estimation from Noisy Data with Gaps

Determining the phase of a rhythm embedded in a time series is a key step in understanding many oscillatory systems. While existing approaches such as harmonic regression and cross-correlation are effective even when some data are missing, we show that they can produce biased estimates of phase when missing data are consecutive (i.e., there is a gap). We propose a simple modification of the least-squares approach, Gap Orthogonalized Accelerated Least Squares (GOALS), which addresses this issue with a negligible increase in computational expense. We test GOALS against other approaches on a synthetic dataset and on a real-world dataset of activity recorded by an Apple Watch, showing in both cases that GOALS is effective at recovering phase estimates from noisy data with gaps.