TRADING SYSTEMS
Instantaneous Investing
The Instantaneous Trendline
by John F. Ehlers, Ph.D.
Is it possible to have an instantaneous trendline?
To say that a trendline is "instantaneous"
may be presumptuous, but because you can compute a continuous trendline
with modern technology and thus assess the action in the markets, the term
is somewhat appropriate.
A market can be in two modes - a trend mode and a cycle mode. That means
you can describe the general market as a combination of the two. If you
apply a simple moving average (SMA) over the period of a dominant cycle,
the dominant cycle component can be completely eliminated.
Identifying a continuously varying dominant cycle and applying a simple
average over the period of the dominant cycle on a bar-by-bar basis results
in a variable-length moving average. That moving average is significant
because it notches out the dominant cycle component. If the composite analytic
waveform consists of a trend component and a cycle component, what remains
after removing the cycle component is the trend. This is not precisely
true, because there will always be components other than the dominant cycle
present. But these secondary cycles usually have small amplitudes, providing
a workable solution for trading purposes.
When you use an SMA with the length of the measured dominant cycle,
the lag produced by the SMA is (dominant cycle - 1)/2. This suggests that
if you have a 21-bar measured dominant cycle, the instantaneous trendline
would be lagging the price by 10 bars. This is one of the limiting factors
of such a technique, because it would be more advantageous to be as close
to zero lag as possible.
This lag can be minimized, however, by using a smoothing filter specifically
designed for minimum lag and then using a frequency notch filter to remove
undesired frequency components. This strategy not only removes the dominant
cycle, but also smooths the price waveform to form a superior instantaneous
trendline.
The elliptic lowpass filters are known to produce the minimum amount
of lag for a given reduction in strength, or attenuation. I selected a
three-pole elliptic filter with a 0.8 decibel (dB) in-band ripple and a
30 dB bandstop attenuation to filter out the high-frequency components.
By setting the 0.8 dB passband at a normalized frequency of 0.22 (a nine-bar
cycle period), the filter has a notch exactly at a five-bar cycle and 30
dB or more attenuation for cycle periods shorter than five bars. The frequency
response of this filter can be seen in Figure 1. The equation for this
filter in EasyLanguage code is:
Filt1 = 0.0542*Price + 0.021*Price[1] + 0.021*Price[2]
+ 0.0542*Price[3] + 1.9733*Filt1[1] - 1.6067*Filt1[2] + 0.4831*Filt1[3]
Figure 1: amplitude response of a three-bar elliptic filter. The filter has a notch at a five-bar cycle and 30 dB or more reduction for cycle periods shorter than five bars.
...Continued in the February 2002 issue of Technical Analysis of
STOCKS & COMMODITIES
Excerpted from an article originally published in the February 2002
issue of Technical Analysis of STOCKS & COMMODITIES magazine. All rights
reserved. © Copyright 2002, Technical Analysis, Inc.
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