All this despite the fact that phase is only explicitly defined for a pure sinusoid, or a narrowband oscillation. If the data is anything else, we are constructing one potential phase estimate of many.

Classically, I've seen the filter-Hilbert transform-analytic signal approach used to define phase. This will work fine in many high rhythm power scenarios because the phase that comes out using this approach will strongly correlate with that from other approaches.

But in moments when there's less rhythm power the filter-Hilbert approach has to create a cycle since the instantaneous frequency is bounded by the filter. But this may not be what is desired for the phase estimate. Alternatives will suggest different estimates for the phase.

We consider two alternatives: a state space model of rhythms and using zero-crossings to define phase. Each of these suggests a different phase estimate during times when the rhythm power decreases or if the waveform of the rhythm is non-sinusoidal.

We show how we can use confidence limits on the phase coming from a Hilbert transform and credible intervals from the state space model to define moments of time ("high confidence") when the phase will be correlated across these different methods. Here's an example for AR(2) sim data.

The big message we wanted to convey is that depending on what is intended for the phase to track (do you want it to act like a clock or do you want it to tell you when the peak/trough is reached?) you might want to consider alternative methods and use uncertainty metrics.

I focused in this #tootprint on one situation when the phase can become quite ambiguous - amplitude modulation, but in the paper we consider other cases as well including non-sinusoidal oscillations which lead to other considerations - do check it out! - https://doi.org/10.1101/2023.01.05.522914