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, a fact that has played havoc with most schemes to exploit the market 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. In order to determine the actual frequences in a particular case, we must conduct an new form of analysis that is better suited to deal with the oxymoronic condition of variable length cycles. |
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Frequency from Monthly Comb Filter
Representative Cycle (Trading Sessions) = 31.00
Sawtooth = 49.32
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 16.77 | 31.00 | 15.00 | 21.77 % | | Low | 16.91 | 31.00 | 19.00 | 15.32 % |
| Recent High | Price | Recent Low | Price | Cycle | | Monday, March 01, 2010 | 57.83 | | | 17 | | | Monday, December 21, 2009 | 47.95 | 25 | | Wednesday, February 03, 2010 | 57.00 | | | 55 | | | Friday, November 13, 2009 | 49.28 | 31 | | Thursday, November 12, 2009 | 53.17 | | | 40 | | | Thursday, October 01, 2009 | 55.22 | 38 | | Thursday, September 17, 2009 | 58.24 | | | 34 | | | Friday, August 07, 2009 | 48.19 | 37 |
Fourier Transform after Digital Filter M | Cycle Length | Fourier Amplitude | | 14 | 0.42 | | 19 | 0.39 | | 21 | 0.34 | | 24 | 0.29 | | 29 | 0.27 | | 26 | 0.24 | | 34 | 0.19 | | 41 | 0.16 | | 45 | 0.14 | | 36 | 0.14 | | 76 | 0.14 |
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Frequency Analysis based on Quarterly Comb
Frequency from Quarterly Comb Filter
Representative Cycle (Trading Sessions) = 76.00
Sawtooth = 47.16
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 50.31 | 81.00 | 56.00 | 24.20 % | | Low | 47.68 | 82.00 | 38.00 | 27.94 % |
| Recent High | Price | Recent Low | Price | Cycle | | | Monday, December 21, 2009 | 47.95 | 94 | | Thursday, September 17, 2009 | 58.24 | | | 74 | | | Friday, August 07, 2009 | 48.19 | 75 | | Wednesday, June 03, 2009 | 62.44 | | | 76 | | | Wednesday, April 22, 2009 | 51.34 | 35 | | Thursday, February 12, 2009 | 73.21 | | | 125 | | | Tuesday, March 03, 2009 | 52.48 | 62 | | Thursday, August 14, 2008 | 83.25 | | | 95 |
Fourier Transform after Digital Filter Q | Cycle Length | Fourier Amplitude | | 16 | 0.26 | | 19 | 0.20 | | 76 | 0.17 | | 114 | 0.16 | | 23 | 0.16 | | 26 | 0.14 | | 29 | 0.14 | | 85 | 0.14 | | 41 | 0.13 | | 164 | 0.12 | | 102 | 0.12 |
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Frequency Analysis based on Yearly Comb
Representative Cycle (Trading Sessions) = 0.00
Sawtooth = 41.39
| 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 | | Thursday, August 14, 2008 | 83.25 | | | 145 | | | Tuesday, August 28, 2007 | 58.97 | 323 | | Thursday, January 17, 2008 | 81.13 | | | 313 | | | Tuesday, May 16, 2006 | 55.00 | 500 | | Wednesday, October 18, 2006 | 70.07 | | | 231 | | | Thursday, May 20, 2004 | 41.00 | 470 | | Wednesday, November 16, 2005 | 77.53 | | | 452 | | | Wednesday, July 10, 2002 | 16.88 | 291 |
Fourier Transform on Raw Closing Prices | Cycle Length | Fourier Amplitude | | 512 | 4.32 | | 683 | 4.15 | | 819 | 3.54 | | 585 | 2.22 | | 341 | 2.09 | | 164 | 1.91 | | 256 | 1.87 | | 410 | 1.82 | | 19 | 1.69 | | 16 | 1.52 | | 372 | 1.51 |
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Average Sawtooth based on Monthly Combo
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Average Sawtooth based on Bi-weekly Combo
Comparing the length of time from high points to successive lowpoints, against the corresponding time span from the lows to the following periodic highs, gives us the Sawtooth Indicator. 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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