7 T

where/ = frequency in cycles per unit time; and T = period in the same units of time.

The relationship between these two quantities is most easily seen by means of an example.

Given: a sine wave with a period of six months or a half-year. The corresponding frequency is: /= = 2 cycles per year-or, leaving time in units of months: f~~\ cycle per month.

There is another form of the frequency quantity which is still more meaningful in some applications. This is called angular frequency. It comes about because the amplitude of a sine wave can be related to the angular measure of a circle, 360 degrees (or 27 radians) of a circle corresponding to one period of a sine wave. It is simply derived as follows:

- - 2 w = 2iTf- Y

where: = angular frequency in radians per unit time / = frequency in cycles per unit time T ~ period in units of time JT = constant = 3.14159 ...

Now, just how does all of this relate to the market?

It is a true fact that any given time history of any event (including the price history of a stock) can always be considered as reproducible to any desired degree of accuracy by the process of algebraically summing a particular series of sine waves. This is intuitively evident if you start with a number of sine waves of differing frequencies, amplitudes, and phases, and then sum them up to get a new and more complex wave form. In such a case you already know the nature of the sine wave content of the final result. As you recall, several very simple examples of this process were shown in Chapter Three, except that a sloping straight line was added in as well. But if the straight line had not been included, the frequency spectrum of the resultant would have been described by a plot similar to that of Figure A I-l, but with only one or two frequency "lines" present.

In the references to Fourier Analysis in the bibliography you can find proof of the fact that even the straight Une of the last paragraoh can be accurately represented by the sum of a series of properly chosen sine waves. If the spectral plot were expanded to include these components as well, we would say that we then have a complete spectral analysis of the wave forms generated as examples in Chapter Three.

In similar manner, the frequency, ampUtude, and phase of a sine wave series which adds up to any given wave form, no matter how complex, can always be found to any desired degree of accuracy by using the techniques of Fourier Analysis (subject to certain mathematical and conceptual restrictions which do not apply to market data).

HOW TO DO FOURIER ANALYSIS

As you look into the references on this subject in the bibliography, you will find that there are several varieties of the Fourier methods. The one chosen for presentation here is described by Lanczos, m a simple and straightforward maimer, and will suffice as your introduction to the subject.

ASSEMBLING YOUR DATA

Your stock price data must be equi-spaced in time-that is, daily, weekly, monthly, etc.

You must choose one value that you consider representative of the price over the chosen increment of time. In the case of daily data this can be the closing price, the mean between high and low for the day, or any other price you feel typifies the day. Once you have chosen it, however, you should use the same criterion for each days data. The price most often used is the closing one (for the day or for the week, etc.).

Choose an odd number of such price datums, in the proper time sequence, for your analysis. The larger the number of such that you use the better your resulting frequency spectrum will be resolved.

Have your data tabulated before you in chronological order.

The results of such an analysis are called the frequency spectrum of the wave form, and the process is called spectral (sometimes harmonic) analysis.

It should be clearly understood that whDe nearly all wave forms have a frequency spectrum, this jloes not necessarily imply that the wave form was initially generated by a process which sums sine waves. In our case, the fact that stock price histories have frequency spectrums does not guarantee that they were generated by a process which starts out with sine waves and adds them up. However, methods arc available whereby such processes can be detected. In the case of stocks, the random and fundamentally motivated parts of price motion are not generated in this manner, whfle the "X" motivated portion of price motion is generated in this way. This is a vital consideration since if this were not true, knowledge of spectral components would not necessarily imply predictability. In any case, the starting pomt in looking for such useful results as "X motivation" cyclicaUty is the process of spectral or harmonic analysis.

At this point we can state the following in support of the need for numerical analysis in any comprehensive study of the market.

1. Stock price histories consist of sequences of numbers.

2. We need to be able to accomplish spectral analysis of price motion using these numbers.

3. Spectra! analysis may be accomplished in several ways but the one which permits direct application to stock price numbers is numerical analysis.

SEPARATING YOUR DATA INTO TWO SEQUENCES

Note and mark in your tabulation the one data point in the exact center of the series. You are going to form two new data sequences now, and this central number is going to be the starting place.

To form the first new sequence, proceed as follows:

1. The first number in the sequence is just the middle number of your stock price series.

2. Now add together the two numbers on either side of the mid-point number of your original price series. This is the second number in the new sequence.

3. Next, add together the two numbers one removed from, and to either side of, the mid-point number of your original price series. This is the third number in the new sequence.

4. The fourth number is obtained by adding together the two numbers of the original series that are two removed from the central number.

5. Proceed in this manner until you get the last element of your new series by adding together the first and last prices in your original stock price series of numbers.

6. Now take the last number only in your new series, and change it by dividing by 2.0. This operation completes the first new sequence you must make.

To form the second sequence, proceed as foUows:

1. Do just the same as for sequence No. 1, except for the following:

• Enter zero as the first number of this sequence.

• Instead of adding the appropriate numbers to get new ones, subtract. Always subtract the earliest number in your original series of stock prices from the later number.

• Do not divide the final number by two as you did before. Instead, change the last number to zero.

2. identify this whole new tabulation as sequence No. 2.

This completes your two new sequences of numbers. All further operations will be conducted on these number series rather than on the original stock prices.

DETERMINING THE FREQUENQES IN YOUR ANALYSIS

Remember what it is you are accomplishing in doing a Fourier analysis. You are trying to determine the angular frequency, amphtude, and phase of a predetermined number of "slices" through the frequency spectrum of the stock price history being analyzed. The thing that determines how finely your analysis wUl "slice," and therefore how weU you are able to separate any frequency peaks or valleys that may exist, is the amount of data you assemble to analyze and the length of the time period involved. So the very next thing you must do is to determine the precise angular frequencies associated with your particular analysis.