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62

of the frequency domain. Likewise, we can hope to learn whether such lines (if they exist) tend to shift slowly with time or not. If sharp over short segments of time as indicated by filter output, the blurring of such lines indicated by the Fourier analysis would indicate slow drift with time.

Figure A 1-3 is typical of the output generated from such combs. Each sine curve in time was frequency analyzed to permit study of frequency variations as a function of time.

Results for the 23 filters of this example are shown in Figure A 1-4. Here, it is noted that the output of filters 1,2, and 3 are clustered in a narrow frequency band, then there is a "leap" in frequency to the output of filters 4, 5, 6, and 7. Outputs from several filters in the comb (such as 8, 12, and 16), whose response curves straddled the frequency gaps shown, fell completely outside the possible response pass-bands of the respective filter and were discarded as meaningless. All other filters provided outputs well within their response range.

It is seen that over the frequency range of 7.5 to 12 radians per year, filter outputs clustered in frequency bands. If the bands of these clusters are shaded in and

FREQUENCY VERSUS TIME

FIGURE 4

2S 50 75 K»ieS60r;S200 225 2SO?7S WEEKS

The Incredible Frequency-Separation Effect!



center lines are drawn an arbitrary .8 radians per year apart through them (the spacing of the fine structure noted in the Fourier analysis), the results appear as in Figure A 1-5. Returning to Figure A I-l, it is seen how well these spectral "clumps" correspond to the fine frequency structure of the last coarse-frequency structure lobe of the Fourier analysis.

Furthermore, band definition is very sharp, lending credence to the suspicion that a slow drift of discrete spectral lines takes place with time. This effect can be tracked with special purpose filters and proven to exist.

Such comb filter analysis has been conducted over varying time periods for the DJIA, a number of individual issues, and over the entire frequency range from approximately .4 radians per year to the highest frequencies resolvable by daily data-with unvarying results.

The rate of time variance is a negligible factor in applications, as was to be suspected from the fact that the line character of the spectrum was not masked by a Fourier analysis over a 44-year interval. It nevertheless exists and in some situations can be profitably taken into account.

MODULATION SIDEBANDS

ti-RAa/YR.

11.8

T-WKS4

28.7

32.0

FIGURE A1-5

The "Line" Frequency Phenomena



THE VARIABLES INVOLVED

The potential conclusion, of course, is that the spectrum of at least the DJIA consists of discrete frequency lines. (This conclusion is extended to a large segment of individual issues in Appendix Two.) This in turn leads to potential for predictability in the time domain, and to considerable amazement that the phenomena should exist at all!

However, before drawing such a conclusion, test of the invariance of the spectral signature is required over a large range of variables.

For example, in order to be sure that the noted effects are not a product of the analytical method or of the process of sampHng a possible continuous function, results must be verified using random filter pass-bands in the combs, random variation of equal digital data spacing, variations in the degree of filter overlap, and non-equal digital data spacing. This was completed for the entire usefiil frequency range, and a sample of the results is shown in Figure A 1-6.

Here, each filter output was frequency analyzed as a function of time. Additional smoothing of the results was obtained by passing a least-square-error straight line through the frequency versus time data. Each line on this figure represents such an output from one filter-and these outputs are taken from filters designed to sample the variables mentioned above.

The results show the same spectral grouping effects in the same frequency regions as for the Fourier analysis and for the equi-spaced filters working on equal digital data spacing. The conclusion reached is that the spectral signature effects noted are real and not due to vagaries of method of analysis.

BEST ESTIMATE OF SPECTRAL LINE SPACING

If we now make the assertion that the hypothesis of a line spectrum is a valid one, we may obtain a highly smoothed best estimate of the minimum spacing between such lines by ranking the means of the least-square-error lines through frequency versus time data in order of increasing frequency.

This is done in Figure A 1-7, where the resulting mean frequencies are plotted against the index number "N." The scatter along this is very small and the slope of the least-square-error straight hne through the data points is the desired line spacing estimate. This does not mean, of course, that still more closely spaced lines do not exist, beyond the resolution of the filters used. The implication is certainly strong, however, not only that stock prices (or at least the fluctuations in the DJIA) exhibit a line frequency spectrum, but also that the lines are equally spaced.

The minimum Hne spacing is seen to be .3676 radians per year-just about one-half the spacing estimated from Fourier analysis. Going back to Figure A 1-1, traces of such lines are seen to be visible between the peaks of the fine structure. Presence is, of course, largely inhibited by the limiting resolution of the analysis of approximately .5 radians per year.



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