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. 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) = 32.00
Sawtooth = 50.47
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 16.38 | 29.00 | 28.00 | 10.83 % | | Low | 17.39 | 30.00 | 19.00 | 11.05 % |
| Recent High | Price | Recent Low | Price | Cycle | | | Thursday, October 09, 2008 | 15.21 | 61 | | Monday, June 30, 2008 | 33.41 | | | 49 | | | Tuesday, July 15, 2008 | 28.08 | 80 | | Monday, April 21, 2008 | 47.55 | | | 32 | | | Thursday, March 20, 2008 | 39.65 | 41 | | Wednesday, March 05, 2008 | 45.97 | | | 22 | | Friday, February 01, 2008 | 44.77 | | | 23 | | | Tuesday, January 22, 2008 | 39.55 | 40 |
Fourier Transform after Digital Filter M | Cycle Length | Fourier Amplitude | | 13 | 0.54 | | 18 | 0.43 | | 20 | 0.37 | | 22 | 0.28 | | 26 | 0.25 | | 29 | 0.20 | | 38 | 0.19 | | 46 | 0.17 | | 41 | 0.16 | | 31 | 0.15 | | 51 | 0.14 |
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Frequency Analysis based on Quarterly Comb
Frequency from Quarterly Comb Filter
Representative Cycle (Trading Sessions) = 61.00
Sawtooth = 48.12
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 50.52 | 94.00 | 61.00 | 23.29 % | | Low | 48.11 | 84.00 | 36.00 | 14.70 % |
| Recent High | Price | Recent Low | Price | Cycle | | | Thursday, October 09, 2008 | 15.21 | 141 | | Monday, April 21, 2008 | 47.55 | | | 32 | | | Thursday, March 20, 2008 | 39.65 | 141 | | Wednesday, March 05, 2008 | 45.97 | | | 45 | | Friday, December 28, 2007 | 46.33 | | | 111 | | | Tuesday, August 28, 2007 | 31.48 | 43 | | Monday, July 23, 2007 | 35.91 | | | 61 | | | Wednesday, June 27, 2007 | 31.55 | 111 |
Fourier Transform after Digital Filter Q | Cycle Length | Fourier Amplitude | | 13 | 0.28 | | 20 | 0.24 | | 22 | 0.19 | | 256 | 0.18 | | 171 | 0.17 | | 124 | 0.17 | | 51 | 0.14 | | 24 | 0.13 | | 164 | 0.12 | | 26 | 0.12 | | 117 | 0.12 |
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Frequency Analysis based on Yearly Comb
Representative Cycle (Trading Sessions) = 0.00
Sawtooth = 43.04
| 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, April 21, 2008 | 47.55 | | | 488 | | | Tuesday, August 28, 2007 | 31.48 | 154 | | | Thursday, January 18, 2007 | 29.82 | 621 | | Thursday, May 11, 2006 | 43.62 | | | 299 | | Friday, March 04, 2005 | 24.04 | | | 263 | | | Friday, July 30, 2004 | 14.48 | 349 | | Wednesday, February 18, 2004 | 16.43 | | | 553 | | | Wednesday, March 12, 2003 | 9.66 | 150 |
Fourier Transform on Raw Closing Prices | Cycle Length | Fourier Amplitude | | 819 | 2.08 | | 512 | 1.73 | | 455 | 1.62 | | 410 | 1.48 | | 585 | 1.47 | | 8 | 1.47 | | 372 | 1.20 | | 11 | 0.99 | | 683 | 0.96 | | 256 | 0.81 | | 15 | 0.79 |
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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. Bullish or bearish moods may be reflected in the steepness of the buying and selling slopes, particularly as they change through time. 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 - 0 instances | | | | Sell Signals - 0 instances | | . | | | Semi-Annual Avg. of Bi-weekly Sawtooth Signals middle Crossing | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 44 instances | | | | Future30SesAvgClose | 2.53 % minus 0.90 % = 1.64 % | 13.87 % | | Future5SesAvgClose | 0.62 % minus 0.18 % = 0.44 % | 22.28 % | | Future2SesAvgClose | 0.51 % minus 0.09 % = 0.41 % | 52.43 % | | Sell Signals - 45 instances | | Future30SesAvgClose | 1.07 % minus 0.90 % = 0.18 % | 1.52 % | | Future5SesAvgClose | -0.02 % minus 0.18 % = -0.20 % | -10.11 % | | Future2SesAvgClose | 0.04 % minus 0.09 % = -0.05 % | -6.02 % | | . | | | Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30 | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 5 instances | | | | Future30SesAvgClose | 1.46 % minus 0.90 % = 0.56 % | 4.78 % | | Future5SesAvgClose | 1.88 % minus 0.18 % = 1.70 % | 86.55 % | | Future2SesAvgClose | 1.19 % minus 0.09 % = 1.10 % | 139.39 % | | Sell Signals - 3 instances | | Future30SesAvgClose | 1.52 % minus 0.90 % = 0.63 % | 5.30 % | | Future5SesAvgClose | -0.26 % minus 0.18 % = -0.44 % | -22.16 % | | Future2SesAvgClose | -0.27 % minus 0.09 % = -0.36 % | -45.49 % | | . | | | Projection of Representative Monthly Cycle | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 146 instances | | | | Future30SesAvgClose | 0.72 % minus 0.90 % = -0.17 % | -1.45 % | | Future5SesAvgClose | 0.01 % minus 0.18 % = -0.17 % | -8.87 % | | Future2SesAvgClose | -0.01 % minus 0.09 % = -0.11 % | -13.46 % | | Sell Signals - 154 instances | | Future30SesAvgClose | 0.36 % minus 0.90 % = -0.54 % | -4.53 % | | Future5SesAvgClose | -0.02 % minus 0.18 % = -0.20 % | -10.26 % | | Future2SesAvgClose | -0.01 % minus 0.09 % = -0.10 % | -12.88 % | | . | | | Projection of BiWeekly Modal Cycles | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 340 instances | | | | Future30SesAvgClose | 0.74 % minus 0.90 % = -0.15 % | -1.31 % | | Future5SesAvgClose | 0.16 % minus 0.18 % = -0.02 % | -1.05 % | | Future2SesAvgClose | 0.15 % minus 0.09 % = 0.06 % | 7.72 % | | Sell Signals - 317 instances | | Future30SesAvgClose | 1.07 % minus 0.90 % = 0.18 % | 1.49 % | | Future5SesAvgClose | 0.22 % minus 0.18 % = 0.04 % | 1.96 % | | Future2SesAvgClose | 0.12 % minus 0.09 % = 0.02 % | 3.10 % | | . | | | | |
| The best buy signal for 30 future sessions was 1.64 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals middle Crossing,and the best sell signal for was -0.90 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25. | | For 5 future sessions the best buy signal was 1.70 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30,and the best sell signal was -0.44 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30. | | The best buy signal for 2 future sessions was 1.10 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30,and the best sell signal was -0.36 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30. |
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