Solved – Are spectral decompositions of time-series useful for modeling/forecasting, or are they more of a tool for analysis

arimaspectral analysistime series

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!

Best Answer

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 $f = 0$.

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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Solved – Time series modeling with dynamic regressors in SAS vs R

While it is not built in, it is easy enough to do:

library(forecast)
fity <- auto.arima(y, xreg=x)
fitx <- auto.arima(x)
forecast(fity,h=10,xreg=forecast(fitx,h=10)$mean)

However, the prediction intervals obtained do not account for the variance in the forecasts of x. I doubt that SAS handles this either.

Solved – time-series analysis Vs statistical signal processing

It all boils down on how would you want to process a time series as to breakdown its components as to use these for later prediction or classification.

For one ARIMA is a parametric method (assumption of a fixed distribution) modeling a stationary time series based on static ARMA terms while with wavelets you model a wavelet function by selecting a list of characterictis you want the function to have as to best approximate a signal (wavelets can model a non stationary as well as stationary). In wavelets the length of the filter the number of vanishing moments and the symetry of the mother wavelet vs the signal will define how good the function is in providing representation of time resolution and frequency resolution. (existing a trade off between this two terms as you get shorter filters you get better in time resolution while decrease the frequency resolution.

In ARIMA you aproximate the signal components by selecting ARMA terms from acf and pacf while for wavelets you aproximate the signal by selecting a mother wavelet, filter length and number of vanishing moments. (the number of decomposition levels provides capturing certain frequencies per level). So an stationary ARMA model would have its pseudo equivalent for a non stationary signal by performing a wavelet transform only at one scale obtaining scaling and detail coefficients (sort of equivalent to having an ARMA with dynamic terms). The results for the two methods will be different, for as in Arima you have static ARMA terms across time, in wavelet decomposition you will have frequency ranges across time which will not correspond to the ARMA terms, although you could calculate what are the frequencies that the ARMA terms represent and approximate these by using wavelets, the dynamic approach of the wavelet will ensure that there is no one specific frequency across time where the ARMA terms will be represented).

In modeling a time series the first issue is to define the characteristics of the signal and the second is in knowing in what part of the signal we are most interested in. (modeling the entire signal or extracting a portion of it).

Which modeling method will provide a better representation of the signal or portions of it you are interested in, depends on the ability of the modeler, the properties of the signal (how easy is to model it) Sometimes an stockastic model such us ARMA or other modeling method model will be sufficient, other times using a built in transform by selecting carefully its terms will give better representation.

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Solved – Time series modeling with dynamic regressors in SAS vs R

Solved – time-series analysis Vs statistical signal processing

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