Fourier Transform Continuous Signal for Stock Market
Harmonics In The Markets
Fourier Transforms As An Aid To Decision-Making
T raditionally, technical analysis has been used to detect and interpret patterns in past security prices to provide insight into future price movement. This, in turn, has lured researchers to try to beat the markets consistently, using a range of techniques varying from mathematics, physics, and economics to psychology. In this article, I will focus on the use of Fourier transforms together with technical analysis in making trading decisions.Is exiting a portion of your trade before your profit target is hit a viable strategy?
Fourier transforms
The application of Fourier transforms has been well established in fields like digital signal processing (DSP), medical diagnostics, image processing, the media, and so on. The Fourier transform breaks up a signal from the time domain to a frequency domain, characterizing signals and letting you see order where there appeared to be none. For this article I used the transform to convert raw data into useful information to aid in making trading decisions.
Many technical analysis indicators seem to work well for specific kinds of patterns. Unfortunately, there are no indicators that can take into account all of the possible characteristics that a price series can exhibit. I will show you how to incorporate stock price volatility so your indicators can adapt accordingly.
Apparently, stock price movement depends upon a number of known and unknown variables with a complex order of correlations existing among them. I will start by providing a brief overview of technical analysis and Fourier transforms, then state the problem context, followed by a brief description of the deficiencies of technical analysis, and finally describe the possibilities found by exploring Fourier transforms, frequency analysis, and inverse transformations.
Technical indicators
Although numerous indicators are available, I will focus on the effective usage of the two most significant indicator categories: averages and oscillators. Averages like the moving average and moving average convergence/divergence (Macd) are lagging indicators that take previous values into account to categorize the present. They do not tell us anything about the upcoming changes; hence, they work best when prices move in relatively long trends. Oscillators like stochastics and Williams' %R are leading indicators that assume price swings are bound to happen when a security becomes overbought or oversold. Thus, they work best in horizontal yet volatile price patterns.
Fourier transforms
Any continuous signal (that is, a signal that has only one value at any instant in time) can be represented by the sum of sine waves of varying frequency, amplitude, and phase. In the context of stock prices, the set of price values over a period of time becomes the signal. Two terms that commonly arise in discussions on frequency analysis are time domain and frequency domain. A time domain signal is a function of time, written as f(t), and a frequency domain signal, written as F(w).
Among waves of different frequencies, the lowest frequency depends upon the period taken for the input signal. That is, for a time period of size N, the lowest frequency would be 1/N. Any frequency lower than this cannot be fit, and goes into zero-frequency DC offset.
In order to accurately find the composite frequencies, the mean (DC offset) should be removed from the input signal. The subsequent higher frequencies would be higher-order harmonics (2/N, 3/N...up to 1/2) of the lowest frequency (fundamental wave).
The transformation from the time domain to frequency domain is called a forward transformation. The reverse is called an inverse or backward transformation. The forward Fourier transform of an array X of size n computes an array Y:
where the k output corresponds to frequency k/n.
Since there are no coefficients prior to summation, performing a forward followed by an inverse transformation would scale the output by a factor of n. The formulas mentioned here describe a nonnormalized transformation.
For the purposes of technical analysis, it does not matter whether the computation is done on normalized or nonnormalized input, since it focuses on the pattern and not the absolute values.
...Continued in the February issue of Technical Analysis of STOCKS & COMMODITIES
Excerpted from an article originally published in the February 2005 issue of Technical Analysis of STOCKS & COMMODITIES magazine. All rights reserved. © Copyright 2005, Technical Analysis, Inc.
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Source: http://traders.com/documentation/feedbk_docs/2005/02/Abstracts_new/Srivastava/Srivastava.html
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