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I've been reading up on seasonal adjustment (removing "seasonal" periodic components from a time series) recently and although I see a lot of fancy work around ARIMA models and fancy ways to detect the seasonality, I see comparatively little work on moving to the frequency domain and looking at series with Fourier transforms or wavelets or anything along those lines. There are some older papers on the topic but it looks like the mainstream approaches don't do anything directly in that space.

Does anyone know why? Does naively applying the Fourier transform, removing undesired periodic components, and inverting it lead to bad results? Is there a survey of seasonal adjustment from the perspective of "you might think to use XYZ, but this is why that doesn't work and how these mainstream techniques improve on that"?

Mysterious Dan
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There is quite a bit on this, but I think that you may not be using the terminology of that field, so nothing much is coming up. The technique you are referring to is called "spectral analysis" of time series and its a well established technique. This is a nice primer on the basic concepts.

With regard to deseasonalizing, this SAS white paper discusses spectal analysis to identify seasonality of various frequencies. If a time series is represented as a sum of harmonics, then a stong seasonality will show up as a peak at a particular periodicity. Removing this component will effectively deseasonalize the time series. There is really nothing special about taking the Fourier approach, as traditional deseasonalizing methods (e.g., seasonal adjustment factors) are essentially just the same thing, but represent the Fourier harmonic as a reciprocal multiplier instead of an additive adjustment.