Frequency Analysis based on Monthly Comb
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It is obvious from inspection of price lines, that many seem to contain cycles that may persist in repeating for a long time. 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) = 32.00
Sawtooth = 52.25
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 16.63 | 32.00 | 34.00 | 11.22 % | | Low | 17.19 | 32.00 | 33.00 | 12.21 % |
| Recent High | Price | Recent Low | Price | Cycle | | | Monday, February 08, 2010 | 61.06 | 40 | | Tuesday, December 29, 2009 | 65.95 | | | 29 | | | Wednesday, December 09, 2009 | 61.70 | 27 | | Monday, November 16, 2009 | 65.56 | | | 20 | | | Friday, October 30, 2009 | 61.68 | 41 | | Monday, October 19, 2009 | 65.55 | | | 50 | | | Wednesday, September 02, 2009 | 52.63 | 47 | | Friday, August 07, 2009 | 57.33 | | | 48 |
Fourier Transform after Digital Filter M | Cycle Length | Fourier Amplitude | | 12 | 0.54 | | 15 | 0.49 | | 20 | 0.39 | | 26 | 0.32 | | 24 | 0.30 | | 29 | 0.23 | | 34 | 0.21 | | 39 | 0.17 | | 31 | 0.16 | | 37 | 0.15 | | 43 | 0.14 |
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Frequency Analysis based on Quarterly Comb
Frequency from Quarterly Comb Filter
Representative Cycle (Trading Sessions) = 46.00
Sawtooth = 47.45
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 47.14 | 89.00 | 46.00 | 21.01 % | | Low | 48.50 | 88.00 | 56.00 | 30.84 % |
| Recent High | Price | Recent Low | Price | Cycle | | Tuesday, December 29, 2009 | 65.95 | | | 147 | | | Monday, March 09, 2009 | 37.91 | 72 | | Monday, June 01, 2009 | 58.25 | | | 169 | | | Thursday, November 20, 2008 | 42.41 | 94 | | Friday, September 26, 2008 | 65.93 | | | 79 | | | Thursday, July 10, 2008 | 53.94 | 120 | | Thursday, June 05, 2008 | 67.89 | | | 46 | | | Thursday, January 17, 2008 | 53.85 | 106 |
Fourier Transform after Digital Filter Q | Cycle Length | Fourier Amplitude | | 12 | 0.34 | | 14 | 0.26 | | 18 | 0.22 | | 26 | 0.20 | | 20 | 0.19 | | 24 | 0.17 | | 241 | 0.15 | | 22 | 0.14 | | 87 | 0.14 | | 77 | 0.13 | | 111 | 0.13 |
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Frequency Analysis based on Yearly Comb
Representative Cycle (Trading Sessions) = 0.00
Sawtooth = 36.64
| 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 | | | Monday, March 09, 2009 | 37.91 | 647 | | Thursday, June 05, 2008 | 67.89 | | | 621 | | | Thursday, August 10, 2006 | 35.88 | 333 | | Wednesday, December 14, 2005 | 42.73 | | | 250 | | | Friday, April 15, 2005 | 35.11 | 633 | | Friday, December 17, 2004 | 42.31 | | | 693 | | | Wednesday, October 09, 2002 | 17.69 | 652 | | Wednesday, March 20, 2002 | 28.64 | | | 295 |
Fourier Transform on Raw Closing Prices | Cycle Length | Fourier Amplitude | | 512 | 3.31 | | 683 | 3.08 | | 372 | 3.01 | | 455 | 2.72 | | 315 | 2.49 | | 11 | 2.22 | | 410 | 2.18 | | 819 | 2.11 | | 341 | 1.80 | | 241 | 1.70 | | 293 | 1.62 |
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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.
| . | | | Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25 | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 2 instances | | | | Future30SesAvgClose | 2.76 % minus 2.05 % = 0.71 % | 6.00 % | | Future5SesAvgClose | 4.29 % minus 0.38 % = 3.91 % | 198.48 % | | Future2SesAvgClose | 1.22 % minus 0.19 % = 1.03 % | 131.43 % | | Sell Signals - 1 instances | | Future30SesAvgClose | -4.36 % minus 2.05 % = -6.42 % | -54.33 % | | Future5SesAvgClose | -5.38 % minus 0.38 % = -5.76 % | -292.39 % | | Future2SesAvgClose | -3.51 % minus 0.19 % = -3.70 % | -470.13 % | | . | | | Semi-Annual Avg. of Bi-weekly Sawtooth Signals middle Crossing | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 28 instances | | | | Future30SesAvgClose | 4.49 % minus 2.05 % = 2.43 % | 20.58 % | | Future5SesAvgClose | 0.66 % minus 0.38 % = 0.28 % | 14.43 % | | Future2SesAvgClose | 0.51 % minus 0.19 % = 0.32 % | 40.29 % | | Sell Signals - 27 instances | | Future30SesAvgClose | 3.16 % minus 2.05 % = 1.11 % | 9.39 % | | Future5SesAvgClose | 0.04 % minus 0.38 % = -0.34 % | -17.51 % | | Future2SesAvgClose | -0.14 % minus 0.19 % = -0.33 % | -41.57 % | | . | | | Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30 | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 5 instances | | | | Future30SesAvgClose | 13.57 % minus 2.05 % = 11.51 % | 97.46 % | | Future5SesAvgClose | 3.28 % minus 0.38 % = 2.89 % | 147.04 % | | Future2SesAvgClose | 1.06 % minus 0.19 % = 0.87 % | 110.33 % | | Sell Signals - 2 instances | | Future30SesAvgClose | 3.90 % minus 2.05 % = 1.84 % | 15.61 % | | Future5SesAvgClose | 0.64 % minus 0.38 % = 0.26 % | 13.38 % | | Future2SesAvgClose | 0.25 % minus 0.19 % = 0.06 % | 7.92 % | | . | | | Projection of Representative Monthly Cycle | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 121 instances | | | | Future30SesAvgClose | 2.54 % minus 2.05 % = 0.49 % | 4.14 % | | Future5SesAvgClose | 0.69 % minus 0.38 % = 0.31 % | 15.81 % | | Future2SesAvgClose | 0.32 % minus 0.19 % = 0.13 % | 16.12 % | | Sell Signals - 126 instances | | Future30SesAvgClose | 2.22 % minus 2.05 % = 0.17 % | 1.43 % | | Future5SesAvgClose | 0.52 % minus 0.38 % = 0.14 % | 7.33 % | | Future2SesAvgClose | 0.06 % minus 0.19 % = -0.13 % | -16.76 % | | . | | | Projection of BiWeekly Modal Cycles | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 259 instances | | | | Future30SesAvgClose | 2.38 % minus 2.05 % = 0.33 % | 2.79 % | | Future5SesAvgClose | 0.54 % minus 0.38 % = 0.16 % | 7.93 % | | Future2SesAvgClose | 0.33 % minus 0.19 % = 0.14 % | 17.92 % | | Sell Signals - 246 instances | | Future30SesAvgClose | 2.65 % minus 2.05 % = 0.59 % | 5.04 % | | Future5SesAvgClose | 0.81 % minus 0.38 % = 0.43 % | 21.90 % | | Future2SesAvgClose | 0.53 % minus 0.19 % = 0.34 % | 42.67 % | | . | | | | |
| The best buy signal for 30 future sessions was 11.51 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30,and the best sell signal for was -6.42 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25. | | For 5 future sessions the best buy signal was 3.91 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25,and the best sell signal was -5.76 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25. | | The best buy signal for 2 future sessions was 1.03 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25,and the best sell signal was -3.70 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25. |
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