NOW COMPUTE THE CORRESTONDING AMPLFTUDES

The reason for separating your original price series into two new sequences was to allow computation of the amplitudes of not only a sine-wave series, but an associated cosine series as well. A cosine wave is shaped just like a sine wave but is 90 degrees out of phase (one-fourth cycle) with the reference sine wave. The sum of two such wave forms is simply another wave form with the same frequency, but with differing amplitude and phase. We will now calculate the amplitudes of the Fourier sine and

Let us assume that you have selected "m" price data points for analysis. Form the quantity Then divide this into the constant, ir = 3,14159. .. Call the resulting

quantity: =(-l)/2

Now form a new page for data tabulation. Head the first column "w," representing the angular frequencies about which your analysis is going to provide you with information. Calculate these frequencies as follows:

• The first one is zero (zero frequency represents infinitely long periods, or in other words a value without oscillation).

• The second one is just the value of "Z" that you found earlier, divided by the digital data spacing you have chosen.

• The third one is two times "Z," divided by digital data spacing.

• The fourth one is three times "Z," divided by digital data spacing,

• Continue calculating frequencies and tabulating them in this manner until the number you are multiplying "Z" by is the number, that you computed earlier.

At this point you can see cleariy just how many samples your analysis will provide of the frequency spectrum of the stock you are analyzing. You may convert any of these angular frequencies (" ") to the corresponding sinusoidal period by the relationship:

A word here about units. If you have selected daily data to analyze, the angular frequencies calculated above will be in units of radians per day. If yoiu" selection was of weekly data, the frequency will be in units of radians per week, etc. It is a good idea always to convert your computed frequencies to some common unit of measure to avoid confusion. Radians per year is a good choice. Thus, frequency in radians per week must be multiplied by 52 to get radians per year, and so forth. Similarly, when you convert angular frequency in radians per year to period by using the relationship l-liE. the period "7 will come out in years. If you had used frequency in radians

per week in this conversion, the period would have come out in weeks.

cosine waves associated with each of the frequencies youve found that the analysis considers-for the particular stock price sequence youre using. Afterwards, you will be shown how to combine these into a single spectrum.

We wfll start with sequence No. 1, which provides the amplitudes of the cosinusoidal components.

First, find the amphtude of the cosinusoidal component associated wdth the first angular frequency ( == 0) in your tabulation. To do this, add up all the numbers in sequence No. 1, and divide by the number, Lz J,that you found earUer. The result is the ampUtude you seek. 2

The ampUtude of the cosinusoidal component associated wdth the second angular frequency in your tabulation is found as foUows:

1. Note the value of the first number in sequence No. 1.

2. Now fmd the cosine of the number "Z" from a set of trigonometric tables. Multiply this by the second number in sequence No. I. Add the result to Item 1 above.

3. Now find the cosine of the number 2 X Z. Multiply this by the third number in sequence No. 1. Add the resuK to Item 2 above.

4. Find the cosine of the number 3 X Z. Multiply this by the fourth number in sequence No. 1, and add the result to Item 3 above.

5. Proceed in this manner until you run out of numbers in sequence No. 1.

6. Divide the final result (sum of results of all intervening steps) by the number,--.

Enter this as the amplitude of the cosinusoidal component associated with the second angular frequency in your list.

The ampUtude of the third cosinusoidal component is found as follows:

1. Proceed exactly as for component No. 2-except before looking up any of the required cosines in the trigonometric tables, multiply the quantity obtained before by two, then look up the cosine of the result.

For the fourth component, proceed as for the third, but multiply by three before finding the cosine in the tables.

Proceed in this manner untU you have found a corresponding cosinusoidal ampUtude for each angular frequency in your tabulation.

Now compute the ampUtudes of the sinusoidal components associated wdth the same frequencies. The process is siraUar to the one you have just completed with three small differences:

1. You wfll use the numbers in sequence No. 2 now instead of No. 1.

2. Instead of looking up cosines in the tables, you will look up sines.

3. The amphtude of the first sinusoidal component is zero.

With these exceptions, proceed exactly as you did when finding the cosinusoidal amplitudes.

HOW TO GET COMPOSITE AMPLITUDES

Now go back to your frequency tabulation. For each frequency you now have an associated amplitude for a co lusoidal and a sinusoidal component. Form the square root of the sum of the squares of these amplitude pairs for each frequency. This is the composite amplitude of the oscillatory component of the frequency associated with it in your table, which exists in the original stock price data. Plot the pairs of amplitudes and frequencies as shown in Figure A I-l. You may either plot amplitude vs. angular frequency, or you can first convert angular frequency to period, plotting this against amplitude. With this step your Fourier analysis is complete!

THE KIND OF RESULTS YOU CAN EXPECT

This kind of analysis has been completed usmg 2300 weekly close data points of the DJ 30 Industrial Average. The results are plotted m the Appendix, and the interpretation of them is given along with correlations to spectral analysis using other methods.

You must always remember that such a Fourier analysis does not necessarily imply that the resulting oscillatory components are m the original data because of having been generated separately and summed to the total. For example, you may draw a sloping line on a piece of paper, then tabulate the pairs of pomts making up this line. A Fourier analysis of the resulting data will produce a set of oscillating components which can be made to approximate the original line (when summed) as closely as you desire. The analysis has found the spectrum of the line for you, but this does not imply that the line was originally formed by adding up this series of components. As a matter of fact, you formed it by simply drawing a ruled line!

A Fourier analysis is always a good starting point when you suspect hidden periodicities, but other techniques must be used if you wish to learn more about whether the originating process put all or part of the periodic components into the spectrum to begin with. If they were not brought about in such a maimer, the resulting spectral analysis may tell you many things, but it will not permit prediction by extrapolation of periodicities.

This is the really fascinating point about which this whole book revolves: the fact that about 23% of all price motion is not part of an artificial frequency spectrum, but instead represents a basic, intrinsic process whereby that part of price motion is formed. The case for this is further developed in the Appendix.

HOW NUMERICAL FILTERS CAN HELP YOU

The most familiar example of a filter is a sieve. Such a device allows particles under a certain size to pass, while holding back those that are larger. A sieve thus separates granular material into two parts. One of these contains particles smaller than a certain size while the other contains particles larger than that size. The basic function of a filter is to separate something into parts according to some specified characteristic of that which is being separated.