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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!

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I'd like to informally try to approach a few of these.

1) Are spectral decompositions useful for modeling/forecasting, or are they typically used only for analysis purposes.

1A) When modelling, I use the spectrum to give information about the seasonal components of my data. Simplistically, I might consider a model of the form: $$ x_{t} = m_{t} + \sum_{i=1}^{S} s_{t}^{(i)} + Y_{t} $$

Where you would have a mean function ($m_{t}$), $S$ seasonal components (sinusoids) ($s_{t}^{(i)}$), and a zero-mean random process $Y_{t}$.

I use the spectrum to estimate the seasonal components amplitudes and phases and then an ARMA (ARIMA?) to model $Y_{t}$.

2) Are the forecast of spectral decompositions always some repeated periodic series?

2A) As far as I'm aware, yes. The motivation for the theory makes the assumption that the process of interest is a discrete parameter stochastic process of the form: $$ X_{t} = \sum_{l=1}^{L} D_{l}\cos(2\pi f_{l}t + \phi_{l}) $$ We let $L \rightarrow \infty$ in a "nice" way. I believe we would also say, plus noise?

This is on page 127 of Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques by Percival and Walden.

The only non-sinusoidal part is at $f = 0$.

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)

3A) My intuition is that I would be doubtful that the ARIMA would perform better than spectral decomposition, however without any concrete proof. The reasoning is that you should get a much better estimate of the frequencies of interest from a spectral decomposition. I'd like to reiterate: I'm not sure though.

I'm not too sure about 4), again my intuition would be that you would need to recalculate the spectrum using the new data as opposed to being able to update the existing spectrum.

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