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64 exactly the observed nature of the periodic fluctuation of price with time. Special purpose filters can now be designed to isolate and study individual side bands. This has been done and the results confirm the detailed nature of the spectral model as described above. It is even possible to assemble a modulation model which links the elements of the spectral model in such a way as to explain the relationship a, = - noted in the Fourier analysis! Yet once again the question is posed: whence comes all this order in the spectral signature of a time series that is so widely believed to be random? It is clearly impossible for such relationships to exist in the frequency domain without counterpart order being present in the time domain. The spectral model of Figure A 1-8 indicates a lower limit to the cycUc portion of price motion in the vicinity of 18 years duration. This is not necessarily the case. This limit is imposed simply because resolution of the analytical methods used to date show .3676 radians per year to be a possible minimum spacing. Other cychc components may be crowded between .3676 radians per year and zero frequency, but their existence is academic to the apphcations. This is true not only because fluctuations of such long periods can be approximated over any reasonable transaction interval by a linear function, but also because of the constant maximum time rate of change relationship derived previously. This relationship essentially gives the vast number of high frequency components (whose periods grow ever more closely packed together as they decrease) dominance in the description of price fluctuations over the sparsely spaced, longer period components. In effect, each modulated component of the model has exactly the same maximum impact on price motion as that of any other, no matter what the disparity in amplitude may be. Regarding the high-frequency end of the spectral model, no limit has been found to the order and precision of the spectral signature-right up to frequencies so high that they can be resolved only by using trade-by-trade data. While the above results would seem to be strictly true only for the DJIA, extension of each finding to individual issues is possible. The results shown used samplings from many specific issues as well as the Averages, and are substantiated by some thousands of cases of graphical analysis and inferential testing. While it is impractical to repeat such extensive analysis on all of the listed issues, a good feeling for the coramonaUty of the phenomenon is provided in Appendix Two.
appendix two Extension of "Average" Results to Individual Issues A Basis for the Principle of Commonality Spectral Signatures, Fundamentals, and Time Synchronization A BASIS FOR THE PRINCIPLE OF COMMONALITY The problem of demonstrating tliat the unusual and useful traits of the spectral signature of the Dow Jones Industrial Average apply to indhfidual issues as well seems an overwhelming one. This would be most convincingly accompHshed if a repetition of the studies summarized for the DJIA could be completed for every stock for which there is sufficient recorded data. This, of course, is a problem common to all generalizations. For example, Newtons famous law concerning the force of attraction between masses could scarcely be verified for every mass in existence. The procedure in such cases is to note effects from small samples, draw conclusions, extend the range of applicability to as large a sample as practicable, and then test the resulting hypothesis or "model" by forcing it to produce predictions which can be verified by real world action. In this situation, the DJIA constitutes the smaD sample, what follows is the extension to a large sample, and the predictive results of previous chapters are a sampling of real world tests of the hypothesis. SPECTRAL SIGNATURES, FUNDAMENTALS, AND TIME SYNCHRONIZATION Figure A Il-l is a plot, on a logarithmic scale, of the weekly closing values of the DJIA and the S&P "500" average from 1949 into 1961. The S&P "500" has each weekly value modified by a constant scale factor so that the total range of value over this time period matches that of the DJIA. This was done to provide direct comparison of the timing of value fluctuations without regard for relative volatility. The implications are unmistakable. The long, sweeping undulations of each average are, of course, the nominal 4.5-year modulated cycle of the spectral model. This clearly exists in both.
FIGURE -1  if COMPARISON JAL > STOCK AVE. DJ 30 INDUSTRIAL AVERAGE S.AND P.SOCSTC 1954 1955 1956 1957 1958 1959 The Law Of Commonality; Generalized
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