Someone mentioned accelerometer based sleep monitors and there so happens to be a popular one for my Android (Motorola Android Razr Maxx) called "Sleep like Android" so I downloaded, installed, read through the instructions. You plug your phone into the power (so it does not fully discharge) and put your phone on the mattress and "Sleep like Android" uses the phone's accelerometer to detect movement to try to classify your sleep in simplistic terms, on a continuum from Deep to Light. The idea is if you are moving, you are awake or in light sleep, or maybe REM, or something. Below is an image that compares my ResMed S9 AutoSet respiration (low and stead during Deep, volatile during REM) and my Zeo (big 'eyes' during Deep, volatile during REM) with the Zeo classifications from their website at the bottom. The black chart is the "Sleep like Android" output.
I may try some additional nights to double check, but in my initial view... "Sleep like Android" does not work and is not useful, at least for me. This does not suggest other accelerometer based sleep monitors do not work, just that the "phone on a mattress" look to be a weak solution.
Major advantage: My phone was fully charged in the morning, ready for a busy day. :)