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, attempts to exploit the market cycles are foiled by the deviation from strictly periodic behavior required by ordinary analysis. 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) = 29.00
Sawtooth = 49.96
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 16.24 | 29.00 | 18.00 | 7.33 % | | Low | 16.73 | 32.00 | 27.00 | 6.98 % |
| Recent High | Price | Recent Low | Price | Cycle | | | Friday, October 24, 2008 | 56.60 | 43 | | Wednesday, September 03, 2008 | 67.86 | | | 12 | | | Monday, August 25, 2008 | 67.26 | 84 | | Friday, August 15, 2008 | 67.80 | | | 89 | | | Friday, April 25, 2008 | 46.33 | 26 | | Thursday, April 10, 2008 | 47.66 | | | 36 | | | Wednesday, March 19, 2008 | 44.88 | 33 | | Tuesday, February 19, 2008 | 47.40 | | | 29 |
Fourier Transform after Digital Filter M | Cycle Length | Fourier Amplitude | | 13 | 0.51 | | 16 | 0.47 | | 21 | 0.35 | | 23 | 0.31 | | 28 | 0.27 | | 39 | 0.19 | | 34 | 0.18 | | 46 | 0.13 | | 63 | 0.12 | | 55 | 0.11 | | 42 | 0.11 |
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Frequency Analysis based on Quarterly Comb
Frequency from Quarterly Comb Filter
Representative Cycle (Trading Sessions) = 82.00
Sawtooth = 46.35
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 51.07 | 93.00 | 133.00 | 11.79 % | | Low | 46.38 | 88.00 | 71.00 | 14.07 % |
| Recent High | Price | Recent Low | Price | Cycle | | Wednesday, September 03, 2008 | 67.86 | | | 166 | | | Wednesday, March 19, 2008 | 44.88 | 82 | | Monday, January 07, 2008 | 52.83 | | | 58 | | | Monday, November 19, 2007 | 47.56 | 66 | | Friday, October 12, 2007 | 51.25 | | | 81 | | | Thursday, August 16, 2007 | 45.51 | 76 | | Tuesday, June 19, 2007 | 52.40 | | | 85 | | | Monday, April 30, 2007 | 47.10 | 39 |
Fourier Transform after Digital Filter Q | Cycle Length | Fourier Amplitude | | 9 | 0.48 | | 13 | 0.32 | | 16 | 0.24 | | 21 | 0.19 | | 23 | 0.16 | | 178 | 0.16 | | 39 | 0.15 | | 111 | 0.14 | | 186 | 0.14 | | 195 | 0.14 | | 27 | 0.13 |
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Frequency Analysis based on Yearly Comb
Representative Cycle (Trading Sessions) = 0.00
Sawtooth = 46.10
| 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 | | | Wednesday, March 19, 2008 | 44.88 | 600 | | Tuesday, June 19, 2007 | 52.40 | | | 737 | | | Thursday, October 27, 2005 | 37.42 | 669 | | Wednesday, July 14, 2004 | 48.80 | | | 271 | | Monday, June 16, 2003 | 47.43 | | | 163 | | | Tuesday, March 04, 2003 | 40.41 | 156 | | Monday, October 21, 2002 | 48.00 | | | 477 | | | Friday, July 19, 2002 | 38.27 | 588 |
Fourier Transform on Raw Closing Prices | Cycle Length | Fourier Amplitude | | 819 | 3.01 | | 9 | 2.56 | | 455 | 2.47 | | 683 | 2.36 | | 585 | 2.21 | | 372 | 2.14 | | 512 | 1.98 | | 410 | 1.89 | | 315 | 1.75 | | 12 | 1.66 | | 256 | 1.62 |
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Anheuser-Busch
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Average Sawtooth based on Monthly Combo
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Anheuser-Busch
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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. This is supposed to indicate the degree of fear or greed. 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 | 5.03 % minus 2.44 % = 2.59 % | 21.91 % | | Future5SesAvgClose | 1.13 % minus 0.49 % = 0.64 % | 32.46 % | | Future2SesAvgClose | 0.37 % minus 0.26 % = 0.12 % | 14.98 % | | Sell Signals - 1 instances | | Future30SesAvgClose | -3.61 % minus 2.44 % = -6.06 % | -51.28 % | | Future5SesAvgClose | -4.06 % minus 0.49 % = -4.55 % | -231.36 % | | Future2SesAvgClose | -3.59 % minus 0.26 % = -3.84 % | -487.87 % | | . | | | Semi-Annual Avg. of Bi-weekly Sawtooth Signals middle Crossing | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 33 instances | | | | Future30SesAvgClose | 1.69 % minus 2.44 % = -0.75 % | -6.37 % | | Future5SesAvgClose | 0.54 % minus 0.49 % = 0.05 % | 2.45 % | | Future2SesAvgClose | -0.05 % minus 0.26 % = -0.31 % | -39.03 % | | Sell Signals - 33 instances | | Future30SesAvgClose | 2.14 % minus 2.44 % = -0.30 % | -2.56 % | | Future5SesAvgClose | 0.59 % minus 0.49 % = 0.10 % | 5.08 % | | Future2SesAvgClose | 0.16 % minus 0.26 % = -0.10 % | -12.62 % | | . | | | Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30 | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 4 instances | | | | Future30SesAvgClose | 1.22 % minus 2.44 % = -1.22 % | -10.36 % | | Future5SesAvgClose | -0.88 % minus 0.49 % = -1.37 % | -69.55 % | | Future2SesAvgClose | -0.52 % minus 0.26 % = -0.77 % | -98.02 % | | Sell Signals - 8 instances | | Future30SesAvgClose | 2.36 % minus 2.44 % = -0.08 % | -0.68 % | | Future5SesAvgClose | -0.74 % minus 0.49 % = -1.23 % | -62.63 % | | Future2SesAvgClose | -1.00 % minus 0.26 % = -1.26 % | -159.90 % | | . | | | Projection of Representative Monthly Cycle | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 133 instances | | | | Future30SesAvgClose | 2.15 % minus 2.44 % = -0.29 % | -2.48 % | | Future5SesAvgClose | 0.65 % minus 0.49 % = 0.16 % | 7.89 % | | Future2SesAvgClose | 0.42 % minus 0.26 % = 0.16 % | 20.38 % | | Sell Signals - 154 instances | | Future30SesAvgClose | 2.24 % minus 2.44 % = -0.20 % | -1.72 % | | Future5SesAvgClose | 0.35 % minus 0.49 % = -0.14 % | -6.99 % | | Future2SesAvgClose | 0.17 % minus 0.26 % = -0.09 % | -11.34 % | | . | | | Projection of BiWeekly Modal Cycles | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 316 instances | | | | Future30SesAvgClose | 2.08 % minus 2.44 % = -0.37 % | -3.11 % | | Future5SesAvgClose | 0.62 % minus 0.49 % = 0.12 % | 6.34 % | | Future2SesAvgClose | 0.33 % minus 0.26 % = 0.07 % | 9.22 % | | Sell Signals - 341 instances | | Future30SesAvgClose | 2.06 % minus 2.44 % = -0.39 % | -3.29 % | | Future5SesAvgClose | 0.44 % minus 0.49 % = -0.05 % | -2.69 % | | Future2SesAvgClose | 0.19 % minus 0.26 % = -0.07 % | -8.46 % | | . | | | | |
| The best buy signal for 30 future sessions was 2.59 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25,and the best sell signal for was -6.06 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25. | | For 5 future sessions the best buy signal was 0.64 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25,and the best sell signal was -4.55 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25. | | The best buy signal for 2 future sessions was 0.16 % from the Projection of Representative Monthly Cycle,and the best sell signal was -3.84 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25. |
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