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MESA COMPARED TO FOURIER TRANSFORMS: FFT is the acronym for Fast Fourier Transform. FFT is a computer algorithm to perform the Fourier Transforms rapidly. The correct use of Fourier Transforms is subject to several constraints. First, the data must be stationary (non-shifting) over the observation period. Secondly, an integer number of cycles must be used in the analysis. The Nyquist theory of sampled data systems states that there must be at least two samples per cycle. These constraints pose a dilemma for analysis of price data. For example, if we have data consisting of 64 points (64 days in the market), the longest cycle we can measure is 64 days. The next longest cycle is 64/2=32 days. The next longest available cycle is 64/3=21.3 days. The next cycle is 64/4=16 days, etc. The 64 day data simply does not provide good resolution to identify the cycles because there is a 5 day gap between measured results right in the most active cycle region. The only way to increase resolution is to increase the data length. However, if the data length is increased there is a significantly lower probability that the cycle has not shifted over the entire data length. In fact, it is downright unlikely. Thus the use of FFTs for trading is not advisable . MESA accurately extracts short term cycles using an adaptive short data length, and therefore MESA should be your cycle measurement method. Measured spectra are usually plotted as amplitude versus cycle period as shown in the figure below. The amplitude is often measured in decibels (dB) because this logarithmic scale can display a wider dynamic range. For example, a 20 db amplitude range is a power ratio of 100:1. The display usually appears as a bell-shaped curve. The sharper the bell the more the energy is focused at a single cycle period. A highly focused spectrum is a spike in the spectrum display. To enable plotting the spectral estimate in time synchronization with the barcharts, we convert the amplitude of the spectrum to colors as shown in the figure below. Using these colors, we can plot the spectrum continuously as a contour plot below the barchart. A highly focused measurement would show as a yellow line in this contour plot. A poorly focused or low resolution measurement is a fat bell curve, and would show up as having the yellow color splattered over a range of cycle periods.
The charts below contain a theoretical 20 bar Sinewave and compare the FFT measurements on the left to the MESA measurements on the right. Clearly, the FFT measurement has poor resolution while the MESA measurement is sharply focused at the theoretical period. Further, the FFT measurements contains artifacts relating to the phase angle of the waveform.
The chart below shows one year of the S&P futures continuous contract for the year prior to November 2005. The first subgraph shows the high resolution spectra as measured by MESA8 and the second subgraph shows the fuzzy, inexact spectral measurements produced by the Fourier Transform. This chart demonstrates that if cycle measurements are to be used to adapt your indicators and trading systems to current market conditions, then the only logical technique selection is MESA8. Clearly, the FFT measurement has poor resolution while the MESA measurement is sharply focused at the theoretical period. Further, the FFT measurements contains artifacts relating to the phase angle of the waveform.
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