It's worth noting that this doesn't work for any multi-dimensional time series, but only some. For example, the lat-long-elevation plot of an airplane's flight

It's worth noting that this doesn't work for any multi-dimensional time series, but only some. For example, the lat-long-elevation plot of an airplane's flight path could not be recovered from just the latitude plot, or one channel could simply be noise. Although, it's incredibly interesting to find cases where this does work, as it indicates there's some special underlying manifold structure.