I very keen to learn about successful application of Fourier analysis in systematic trading .. anyone ?

No one. What are you all guys keen of Fourier — what are you going to do with phase in a non-stationary process?

ah .. this is what I thought ) it's a fancy method but unfortunately I have never seen anyone actually using it in trading ...

I wonder how all (!) newcomers to the market with a background in signal processing rush to apply FFT, LPC, and other transforms up to Kalman filters without understanding of two simple things:

1. Half of them are suitable only for (pseudo)periodical processes;

1. Success in application of others depends only on knowledge of what actually is being sought.

As you can see, market data as time series doesn't conform to either condition.

Alex: you are very quick to personally insult people and I bet noone appreciates that. Jon is a seasoned professioanl, I have a proven multi year track record and we were simply discussing an objective method to a very subjective task - namely that of selecting a timeframe. Notice noone mentioned strategies as per se.

Kirill, where in my latest post did I reference you or Jonathan personally? And what are the relationships between your experience (which I haven't ever questioned btw), strategies, and my comment?

Alex, have you considered that a non-stationary process may be the product (multiplicative or additive) of several different processes, some of which are stationary and periodic?

Jonathan, unfortunately not. Even though sometimes visually the sum of periodic signals may resemble a non-stationary signal, especially when their phases are also being changed over time, but in reality we simply have periodic signals with very long periods and shifted phase. A true non-stationary signal (process) is not the one which is aperiodic, but the one in which amplitude may change by virtually any value in any amount of time, and the distribution of these changes in generally considered as normal.

However even if you decide that you can afford to confine your model to only some first harmonics, chances are that some of them will be too close to each other. In this case all the aforementioned spectral analysis methods will give you a "general" picture, but as always the devil is in the details: you won't be able to decide whether this particular series was caused by a harmonic in this frequency or in the adjacent one. A typical example: if you run an FFT over EURUSD hourly bars you will find easily three most dominant periods: 4 hours, 6 hours and 24 hours. Now the question is what to do with 4 and 6 as they are too close to each other and spectrums overlap.

It's hard to explain it without a picture or a formula, but I hope you understand.

Application of FFT in financial time series is generally a problem because the TS are non-stationary. Therefore typically a modified Discret Fourier Transformation (DFT), or even better Maximum Entropy Spectral Analysis (MESA) , the Goertzel algorithm, band pass filter banks or autocorrelation periodograms are used for spectral analysis of financial TS. John Ehlers has done here a lot of pioneering work. Empirical Mode Decomposition (EMD) is a new method which can be used for a first detrending and TS decomposition of data to receive pretty good spectral analysis results.

Boris, could you please elaborate a bit on the criteria used to judge whether the spectral analysis results applied to financial TS are "pretty good"? Thank you.

Hi Alex, with „pretty good“ I mean the determination of dominant cycle frequencies or periods with peaks in the amplitude or even better in my eyes peaks in cycle strength (=amplitude/cycle period). Nevertheless this determination of peak frequencies is not enough because the significance of cycle periodicity has to be still verified. One method for doing this is applying the old Bartels test for cycle significance. Cycles in financial TS appear often in different time frames over a certain time and then disappear again. If they are significant there is a high probability that they will repeat periodically.