---

 SYSTEM DESIGN

---

The Endpoint Fast Fourier Transform System

---

by Dennis Meyers, Ph.D.

---

Last time, we explored the Fourier transform, a mathematical technique for analyzing data to determine cyclical component. This time, we use the fast Fourier transform as a trend determinant for a model for trading the Standard & Poor's 500.

---

In my previous article, "The Discrete Fourier Transform Illusion," we demonstrated the misuses of the Fourier transform mathematical technique as applied to the Standard & Poor's 500 index. We showed how fitting the Fourier transform to the S&P 500 index data series produced a perfect curve-fit on past data, giving the illusion that this technique would predict the major turning points of the S&P 500. However, when we examined the Fourier transform on a day-by-day walk-forward basis, this seemingly wondrous predictive capability disappeared.

FIGURE 1: NOISE-FILTERED FFT. Here's the noise-filtered Fft from January 16, 1998, to January 22, 1999. The SP closing high was on July 20, 1998. As can be seen from Figure 1, the FFT curve clearly leads the July 20, 1998, top and is pointing down on that day. This gives the illusion that the FFT curve is a predictive indicator that leads the price series, enabling the market participant to escape the ensuing bear market drop. However, this FFT curve was generated on data from January 16, 1998, to January 22, 1999. What would the FFT curve look like if the curve were generated on July 20, 1998?

This time, we will demonstrate how to use Fourier transform using the computational algorithm called the fast Fourier transform (FFT) on a walk-forward basis on the S&P 500 continuous futures contract (SP).

THE FFT ILLUSION REVIEW

Figure 1 presents the noise-filtered FFT on the SP from January 16, 1998, to January 22, 1999. The SP closing high was on July 20, 1998. As can be seen from Figure 1, the FFT curve clearly leads the July 20, 1998, top and is pointing down on that day. This gives the illusion that the FFT curve is a predictive indicator that leads the price series, enabling the market participant to escape the ensuing bear market drop. However, this FFT curve was generated on data from January 16, 1998, to January 22, 1999. What would our FFT curve look like if we generated our curve on July 20, 1998?

Figure 2 presents the noise-filtered FFT on the SP from July 15, 1997, to July 20, 1998. As can be seen from the noise-filtered Fft curve generated on July 20, the curve is pointing straight up, giving no indication whatsoever of the coming market drop. Why does this happen? When the FFT went to fit the data, it already knew where all the tops and bottoms were. The FFT mathematics minimize the error between the curve it generates and the real datapoints. This error minimization process forces the generated curve to fit the past data like a glove. As a matter of fact, it’s almost impossible not to get an excellent fit.

THE ENDPOINT FFT

To avoid the past-data curve-fit illusion, we will create an indicator that walks forward one day at a time. This indicator will calculate the noise-filtered FFT curve but only save the last point, or endpoint, of the curve on the day it is calculated. We will then connect the generated endpoints to produce a curve that matches what we would have seen if we performed the noise-filtered FFT on the endpoint dates.

Excerpted from an article originally published in the May 1999 issue of Technical Analysis of STOCKS & COMMODITIES magazine. All rights reserved. © Copyright 1999, Technical Analysis, Inc.