By Gratuito

CYCLES TUTORIAL

by

John Ehlers

INTRODUCTION

The use of cycles is perhaps the most widely misunderstood aspect of technical analysis of the markets. This is due, in part, to a wide variety of disparate approaches ranging from astrology to wavelets being lumped into a cycles category. The purpose of this tutorial is to present a logical and consistent perspective on what cycles are and how they can be used to enhance technical analysis. I was originally attracted to the use of cycles because it is one parameter on the charts that can be scientifically measured. These measurements can be used to dynamically modify conventional indicators such as RSI, Stochastics, and Moving Averages. Better yet, our research has provided superior indicators derived directly from cycle theory. The successful application of cycles to technical analysis is proven by mechanical trading systems which we offer for both intraday and position trading are ranked #1 in their respective categories.

The following sections are more or less independent, but weave together to establish a basis for a scientific approach to trading. Some sections should be an easy read. Other sections might become too technical for many traders. If you feel uncomfortable in a section, just skip it for the time being and plan to return to it later. The punch line of this tutorial is in the final section, where we show how to correlate the indicators for a consistent analytical approach.

HISTORICAL PERSPECTIVE

Cyclic recurring processes observed in natural phenomena by humans since the earliest times have embedded the basic concepts used in modern spectral estimation. Ancient civilizations were able to design calendars and time measures from their observations of the periodicities in the length of the day, the length of the year, the seasonal changes, the phases of the moon, and the motion of the planets and stars. Pythagoras developed a relationship between the periodicity of musical notes produced by a fixed tension string and a number representing the length of the string in the sixth century BC. He believed that the essence of harmony was inherent in the numbers. Pythagoras extended the relationship to describe the harmonic motion of heavenly bodies, describing the motion as the “music of the spheres”.

Sir Isaac Newton provided the mathematical basis for modern spectral analysis. In the seventeenth century, he discovered that sunlight passing through a glass prism expanded into a band of many colors. He determined that each color represented a particular wavelength of light and that the white light of the sun contained all wavelengths. He invented the word spectrum as a scientific term to describe the band of light colors.

Daniel Bournoulli developed the solution to the wave equation for the vibrating musical string in 1738. Later, in 1822, the French engineer Jean Baptiste Joseph Fourier extend the wave equation results by asserting that any function could be represented as an infinite summation of sine and cosine terms. The mathematics of such representation has become known as harmonic analysis due to the harmonic relationship between the sine and cosine terms. Fourier transforms, the frequency description of time domain events (and vice versa) have been named in his honor.

Norbert Wiener provided the major turning point for the theory of spectral analysis in 1930, when he published his classic paper “Generalized Harmonic Analysis.” Among his contributions were precise statistical definitions of autocorrelation and power spectral density for stationary random processes. The use of Fourier transforms, rather than the Fourier series of traditional harmonic analysis, enabled Wiener to define spectra in terms of a continuum of frequencies rather than as discrete harmonic frequencies.

John Tukey is the pioneer of modern empirical spectral analysis. In 1949 he provided the foundation for spectral estimation using correlation estimates produced from finite time sequences. Many of the terms of modern spectral estimation (such as aliasing, windowing, prewhitening, tapering, smoothing, and decimation) are attributed to Tukey. In 1965 he collaborated with Jim Cooley to describe an efficient algorithm for digital computation of the Fourier transform. This Fast Fourier Transform (FFT) unfortunately is not suitable for analysis of market data.

The work of John Burg was the prime impetus for the current interest in high-resolution spectral estimation from limited time sequences. He described his high-resolution spectral estimate in terms of a maximum entropy formalism in his 1975 doctoral thesis and has been instrumental in the development of modeling approaches to high-resolution spectral estimation. Burg’s approach was initially applied to the geophysical exploration for oil and gas through the analysis of seismic waves. The approach is also applicable for technical market analysis because it produces high-resolution spectral estimates using minimal data. This is important because the short-term market cycles are always shifting. Another benefit of the approach is that it is maximally responsive to the selected data length and is not subject to distortions due to end effects at the ends of the data sample. The trading program, MESA, is an acronym for Maximum Entropy Spectral Analysis.

PHILOSOPHICAL FOUNDATION FOR MARKET CYCLES

It has been written that the market is truly efficient and follows the random walk principle. The fact that Paul Tudor Jones, Larry Williams, and a host of other notable traders consistently pull money from the market disproves the categorical assertion. However, a more detailed analysis of the random walk theory could yield some interesting results.

Brownian motion is a random walk, where for example, it describes the path of a molecule of oxygen in a cubic foot of air. That molecule is free to move in three-dimensional space. The market is more constrained. Prices can only move up and down. Time can only go forward. There is a more constrained version of random walk, called the Drunkards Walk. In this version, the “Drunk” staggers from point A to point B. We want to examine two formulations of the problem.

In the first formulation, the “Drunk” flips a coin, and depending on whether the coin turns up heads or tails takes a step to the right or left with each step forward. That is, the random variable is direction. The solution to this formulation is a rather famous differential equation called