Hello John

You could take the signal (of any dimensions) and break it down to say s = Trend + Noise and devise a forecast function f:

f(Trend) ---> next Trend

f(Noise) ---> next Noise

Or you could use

f (Trend, Noise) ---> next (Trend, Noise)

In other words use the Noise as if an important part of the data, which it is, or use it in tuple form (Trend, Noise) .

I feel in the discussion it Trend vs. Noise we have to choose one or use the raw signal, and there are other configurations as above indicates.

If you lowered a microphone into Beijing and recorded noises of the city you get Trend + Noise, Trend would give you important info about when people wake up and work and sleep basically their mass schedule and information about the days of the week and hollidays, Noise gives you what machinery other than humans are active in the environment. So you could use both to model or forecast next day activities of the city.