This is a bit of a theoretical question. I'm also new to time-series analysis, and trying to learn fast. Sorry if some of my terminology is off.
You can loosely categorize methods to analyze and model time-series into time-domain and frequency-domain approaches. In the time domain, models like ARIMA forecast based on the recent measurements. A prediction of some time X in the future will get better as you get closer to it (with the one step prediction being the best).
Instead of linear combination of recent measurements, the signal can be decomposed into a sum of sines and cosines. This seems particularly apt when the signal has strong periodic/seasonal components. However, won't the forecast of this be an infinitely repeating signal of some set period? So that the prediction of some future value X would not change as new information came in, unless you simply redo the decomposition.
Let me lay out some exact questions.
1) Are spectral decompositions useful for modeling/forecasting, or are they typically used only for analysis purposes.
2) Are the forecast of spectral decompositions always some repeated periodic series?
3) Would using a seasonal ARIMA likely outperform (in terms of forecasting) a spectral decomposition, even with a ARIMA model on the residuals of the spectral model? (assuming data with strong seasonal/periodic trends)
4) Is there anyway to online or iteratively update the spectral decomposition of a time series?
No need to answer all of these in detail. I imagine they give you an idea of what I'm looking for. If you know of a method or model that seems relevant, a name is a good enough lead for me to investigate. Likewise, if frequency decompositions are a dead-end in terms of modeling and forecasting, that would be great to know.
I appreciate the help!