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10

Time Period Duration (weeks) A-B 23

B-C 14

C-D 9

D-E 21

F-F 12

F-G 22

G-H 24

H-l 17

1-J 20

J-K 23

average of the seven samples, which is +2.572 and-4.428 weeks. Average these limits to obtain ±3.5 weeks. In view of the relative inaccuracy of the method, round off decimals-and your expectation is for 21.4 ± 3.5-week cycles to continue to make an appearance in the near future in this indicator. These results must, of course, be subjected to continuous up-dating since the principle of variation is in constant operation. One final bit of very useful information can be obtained from this exercise by forming a center Une mid-way between envelope bounds. The significance of this line is that it represents the sum of all cyclic components of duration longer than 21.4 weeks-added to any fundamental trends existing at the time.

"NESTING" ENVELOPES UPWARD

Using the same data, let us now apply the same technique to extract information regarding the next longer duration cycle of the model.

Turning to Figure II-4, we see the same chart as before, but now a second envelope has been added. The characteristics of this one are exactly as before, except that they apply to the first envelope, rather than to the original data. It is immediately seen that the first envelope oscillates back and forth between the bounds of the second, establishing contact at the lettered intervals. Proceeding as before, count the weeks between lows. Two samples are available with durations of 67 and 75 weeks. Averaging these produces 71 ± 4 weeks as the nominal expected duration and extent of variation in the near future. The cyclic model calls for the presence of an 18-month cycle. Eighteen months is 78 weeks-and the 71 ± 4-week fluctuation noted is the counterpart of the 18-month component of the model for this period of time in the DJIA.

We now have a set of "nested" envelopes, and the process of constructing a second envelope about the first one we wUl call "nesting" upward. The mechanics of envelope construction does not permit the existence of envelopes containing fluctuations of duration between 21.4 and 71 weeks in this particular sample of DJIA data. Noting this, and referring back to the cyclic model, you will see that a 9- and 12-month component appear to be missing. They are, in truth, present-but of such small magnitude at this particular time as not to be "observationally significant," This is an example of the "dominance" characteristic of the principle of variation in action.

During this same period of time, the "missing" components can be found dominant in other indicators and specific issues. Once again, a center line can be drawn



FIGURE S-4

Nesting Envelopes

between the bounds of the second envelope. The significance, as before, is the representation of the sum of fundamental trends and all longer duration components (in this case longer than 71 weeks). This type of information will be put to good use in later chapters.

"NESTING" DOWN

After nesting upwards to the point where you run out of cyclic samples (as in this case), the same data can be made to produce still more information by nesting down. However, you will find that envelope smoothness deteriorates steadily as enclosed cyclic durations decrease. Part of this is due to the use of weekly data, and can be overcome by shifting to daily plots. But the bulk of the problem is intrinsic to the expression of the principle of variation.

Accordingly, another technique is introduced that you will often find useful. Lets go back and re-examine Figure 11-3. Observing the region of this figure between letters F and G (lows of the 21.4-week cycle), we note that prices did not move smoothly up and then down again, but did so in now famiHar gallops. In fact, between F and G, three such gallops are seen. What about G and H? Again three. Between H and I the same occurs, except that the imagination is stretched a bit on gallops two and three. But, between I and J, once again three movements are clearly noted.

Now the average duration of the F-G, G-H, H-I, etc. type movements was previously established as 21.428 weeks. One third of this is 7.142 weeks. This is the expression at this time in the DJIA of the (nominal) 6.5-week component of the cyclic model.



Anofter Envelope Technique

The result is seen in Figure -5 with all extraneous information removed-and the fluctuations of interest are clearly seen. Now run a center line through this envelope as in Figure -6 (where the lows of interest are numbered). Referring these back to a time scale, weeks between each can be counted and averaged. The outcome-15 samples with an average duration of 6.766 weeks. This is within 5.3% of the original estimate based on one-third of the 21.4-week cycle-and is considered more accurate. The average of the extremes of the deviation from nominality is ±.8 weeks.

USING EXPANDED OR CONTRACTED DATA

All components of tlie cyclic model which are present in this sample of the DJIA between the nominal ones of 6.5 weeks and 18 months have now been identified. To proceed further, the price action must be presented in a different form.

To extract still shorter duration fluctuations, daily data must be used. Any time the price chart permits less than six or seven data points in the time span covered by the duration of one cycle of a given component, the display must be expanded by going to closer data spacing. To extract longer duration fluctuations, a longer time period

A third technique can be employed to provide additional estimate accuracy.

• Form a new envelope between the letters E and J of Figure II-5 by carefully connecting each weekly low on the plot to its nebor with short, straight lines to obtain the bottom of the envelope.

• Similarly, connect the weekly higlis to form the envelope upper bound.



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