Frequency Analysis based on Monthly Comb
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Even a casual examination of historical price lines seems to reveal cycles or regular short term fluctuations that continue repeating. Through the years, technicians have expended much effort to measure and exploit these cycles. It would appear simple enough to do a frequency analysis on the time series. But the cycles or periods that seem so easy to see in the price data, have proven to be surprisingly difficult to measure by classical methods. The classical methods apply to stationary cycles of exactly regular period. We can easily remove the trends from historical price data to make the series stationary. However, the fluctuations seen in financial data are almost never exactly periodic, complicating most attempts to take advantage of these cycles. One factor in the behaviour of market fluctuations is "Anticipation". When speculators detect a cycle in prices, they will buy earlier before the peak, and sell earlier before the dip. This phenomena causes an increase in frequency, and eventually a breakup of the established cycle. Traditional frequency analysis methods such as the Fourier Transform, break down badly attempting to deal with these irregularities. Wilder estimated that there was an important period of about 28 trading sessions. The half cycle of this period (14 sessions) is a default factor in many of the technical indicators. We have found this to be a reasonable figure, even in today's markets, but there is significant variation between issues. Some companies just seem to have faster or slower cycles for one reason or another, and the indicators may be more accurate when fitted to the situation. New methods had to be invented to deal with the seeming mathematical absurdity in the notion of non-periodic cycles. |
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Frequency from Monthly Comb Filter
Representative Cycle (Trading Sessions) = 20.00
Sawtooth = 50.68
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 16.72 | 29.00 | 28.00 | 18.32 % | | Low | 17.18 | 29.00 | 17.00 | 15.36 % |
| Recent High | Price | Recent Low | Price | Cycle | | | Tuesday, January 26, 2010 | 14.56 | 30 | | Thursday, January 14, 2010 | 16.31 | | | 43 | | | Thursday, December 10, 2009 | 11.60 | 27 | | Wednesday, November 11, 2009 | 15.08 | | | 20 | | | Monday, November 02, 2009 | 12.56 | 21 | | Wednesday, October 14, 2009 | 15.11 | | | 20 | | | Friday, October 02, 2009 | 12.96 | 72 | | Wednesday, September 16, 2009 | 17.28 | | | 55 |
Fourier Transform after Digital Filter M | Cycle Length | Fourier Amplitude | | 14 | 0.52 | | 20 | 0.52 | | 23 | 0.31 | | 26 | 0.29 | | 29 | 0.23 | | 41 | 0.17 | | 33 | 0.16 | | 37 | 0.13 | | 49 | 0.13 | | 39 | 0.13 | | 45 | 0.11 |
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Frequency Analysis based on Quarterly Comb
Frequency from Quarterly Comb Filter
Representative Cycle (Trading Sessions) = 80.00
Sawtooth = 50.43
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 48.53 | 80.00 | 78.00 | 19.90 % | | Low | 47.41 | 81.00 | 47.00 | 20.57 % |
| Recent High | Price | Recent Low | Price | Cycle | | | Thursday, December 10, 2009 | 11.60 | 120 | | Wednesday, September 16, 2009 | 17.28 | | | 93 | | | Monday, June 22, 2009 | 7.36 | 84 | | Tuesday, May 05, 2009 | 10.42 | | | 80 | | | Friday, February 20, 2009 | 5.79 | 60 | | Thursday, January 08, 2009 | 11.21 | | | 76 | | | Friday, November 21, 2008 | 3.57 | 93 | | Friday, September 19, 2008 | 16.37 | | | 116 |
Fourier Transform after Digital Filter Q | Cycle Length | Fourier Amplitude | | 9 | 0.46 | | 14 | 0.28 | | 20 | 0.28 | | 27 | 0.17 | | 102 | 0.16 | | 24 | 0.16 | | 256 | 0.15 | | 87 | 0.13 | | 124 | 0.12 | | 29 | 0.12 | | 158 | 0.12 |
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Frequency Analysis based on Yearly Comb
Representative Cycle (Trading Sessions) = 0.00
Sawtooth = 0.00
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 0.00 | 0.00 | 0.00 | 0.00 % | | Low | 0.00 | 0.00 | 0.00 | 0.00 % |
| Recent High | Price | Recent Low | Price | Cycle | | | Friday, November 21, 2008 | 3.57 | 595 | | Friday, February 02, 2007 | 52.36 | | | 381 | | | Friday, July 14, 2006 | 37.03 | 497 | | Thursday, July 28, 2005 | 62.43 | | | 345 | | | Monday, July 26, 2004 | 37.07 | 346 | | Tuesday, March 16, 2004 | 50.52 | | | 430 | | | Tuesday, March 11, 2003 | 19.88 | 151 | | Monday, July 01, 2002 | 25.80 | | | 1056 |
Fourier Transform on Raw Closing Prices | Cycle Length | Fourier Amplitude | | 683 | 2.02 | | 372 | 2.01 | | 410 | 1.22 | | 455 | 1.20 | | 293 | 1.10 | | 14 | 1.07 | | 315 | 1.06 | | 512 | 1.04 | | 20 | 1.01 | | 241 | 0.90 | | 18 | 0.86 |
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Average Sawtooth based on Monthly Combo
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Average Sawtooth based on Bi-weekly Combo
A comparison of the average distance (in time) from each periodic high to to next periodic low, against the distance to the following high point, yields the Sawtooth Statistic. The mid-point, corresponding to a perfect sine wave, is 50 percent. Values less than this indicate the sell slope is steeper than the buy slope, while values over 50 percent show the opposite. The Sawtooth Statistic gives a clue to the general mood of the markets, functioning as a sort of sentiment indicator. The chart represents the sawtooth percentage of (approximately) monthly cycles over the past year, and of the (approximately) bi-weekly cycles, averaged over the past six months.
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