QUANTITATIVE ANALYSIS
Dominant Cycle = Fog Lights
Fourier Transform For Traders
by John F. Ehlers
When market conditions are variable, adapting to them becomes a challenge.
Here's how you can use a dominant cycle to tune the relevant indicators
so you don't have to drive through the fog.
IT is intrinsically wrong to use a 14-bar
relative strength index (RSI), a nine-bar stochastic, a 5/25 double moving
average crossover, or any other fixed-length indicator when the market
conditions are variable. It's like driving on a curving mountain road in
a fog bank with your cruise control locked — and you've desperately got
to clean your eyeglasses.
That market conditions are continuously changing is not even a subject
of debate. There have been a number of attempts to adapt to changing market
conditions. Volatility-based nonlinear moving averages are just one example
of adapting to market changes. As I come from an information theory background,
my answer to the question of how to adapt to changing conditions is to
first measure the dominant market cycle and then tune the various indicators
to that cycle period, or at least a fraction of it. Theoretically, an RSI
performs best when the computation period is just half of a cycle period
— that is, when all the movement is in one direction and then reverses
so all the movement is in the other direction over the period of one cycle
— and you get a full amplitude swing from the RSI.
FOURIER TRANSFORMS
Make no mistake: Measuring market cycles is difficult. Not only is there
the problem of simultaneously solving for frequency, amplitude, and phase
to arrive at an accurate estimate, but we must also realize the measurement
is being made in a low signal-to-noise environment. Further, we must be
concerned with the responsiveness of the measuring technique to capture
the cycle periods that are continuously changing without introducing transient
artifacts into the measurement. A variety of spectrum estimation techniques
is available, ranging from the Fourier periodogram to modern high-resolution
spectral analysis approaches.
I have long railed against the use of Fourier transforms in estimating
market cycles because of their lack of resolution. Figure 1 represents
a typical spectrum measurement. The horizontal axis is the frequency (or
its reciprocal, cycle period) scale. The vertical axis is the amplitude
scale. The frequency with the highest amplitude identifies the measured
cycle.
If the width of the spectral line is narrow, just a spike like the solid
line, then the cycle is identified with high resolution. If we have a high-resolution
technique we could, in fact, identify two closely spaced cycle periods
if they are present in the data. On the other hand, if we have a low-resolution
measurement technique and the width of the spectral line is broad, two
closely spaced cycles could be averaged together and you would not be able
to identify them, as demonstrated by the dotted line (Figure 1).
