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60

appendix one

The Notto-Be-Expected "Order" of Spectral Relationships in Stock Price Data

• The Implications of Fourier Analysis of Stock Prices

• Coarse Frequency Structure

• Fine Frequency Structure

• Amplitude-Frequency Relationship

• The Use of Comb Filters

• The Variables Involved

• Best Estimate of Spectral Line Spacing

• The Line Spectral Model

In Chapter Two, a price-motion model was formulated that implied a surprising degree of order in the frequency spectra of stock prices. The basic tenets of the model were demonstrated primarily by observational methods, and were then shown to be sound by use of a number of price-predictive techniques which could work only if the theory behind them was reasonably correct.

Then, in Chapter Three, it was noted that the otherwise incomprehensible formation and repetition of specific chart patterns was fully explained by the assumed model, lending still further credence to the theory upon which the model is based.

In this Appendfac, more powerful tools will be apphed to the problem which demonstrate not only the remarkable breadth and consistency of the phenomena involved, but also bring to light new aspects, understanding of which can help extricate one from the occasional difficult situation.

THE IMPLICATIONS OF FOURIER ANALYSIS OF STOCK PRICES

A hint of the unusual spectral order involved can be obtained from a high resolution harmonic analysis. Figure AI-I is a plot of the results of such a project.



70 60

FOURIER ANALYSIS DJIA

3 4 5 6 7 ANGULAR FREQUENCY RAOYR.

44 Years Of Maiket Cyclicality



This analysis was conducted on the weelcly closing values of the Dow Jones Industrial Average over the time period 29 April 1921 through 25 June 1965. The resulting 2229 data points provided a frequency resolution of .568 radians per year.

Three major elements oi spectral order are to be noted from the plot:

1. The resolution of spectral amplitudes into several broad segments with minimums approximately located at .95, 1.65, 2.8,4.75,7.0, and 9.8 radians per year.

2. The regularity of the fine frequency structure between the above major separation points.

3. The shape and smoothness of the upper envelope bound of the broad segments as drawn from peak to peak of each.

Lets discuss the significance of these unusual symptoms in order.

COARSE FREQUENCY STRUCTURE

The coarse structure is seen to be divided into sections of increasing range of frequency as frequency increases. One is immediately forced to ask why this bit of order should exist in the spectra of a time series that is widely held to be randomly generated. Of even greater interest is the fact that the frequency peaks central to these large segments correspond to periodic price motion in the time domain that is readily observable (as demonstrated in several of the examples of Chapter Two). The sinusoidal periods associated with the central frequencies of these lobes is indicated (in years) in the figure, and each of these can be visually noted in the various samples of the DJIA presented in the chapters.

However, in the observational samples, fairly short periods of time are involved. In this case, we see evidence that the observationally isolated periodicities have persisted for at least 44 years. Any significant deviation from such a consistency would otherwise have averaged out both the broad peaks noted and the associated valleys.

FINE FREQUENCY STRUCTURE

The regularity of the fine lobular structure between the coarse structure valleys is very apparent. In fact, these peaks average close to .8 radians per year in separation. It is significant that the broader lobes increase in width as frequency increases while the fine structure maintains equal spacing regardless of frequency. Although not shown in the figure, this statement applies all the way up to frequencies that can be resolved only by the use of trade-by-trade data.

The important observation that can be made is the fact that any degree of regularity is present at all. We arrive at the tentative conclusion that we may be dealing with a so-called "Ime" spectrum in the analysis of stock prices, and once again we are forced to ask why this element of spectral order should exist, let alone persist over long periods of time as we have seen demonstrated. Such order implies possible predictability in the time domain, and caimot be present if price fluctuations are randomly generated.



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