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) = 38.00
Sawtooth = 44.14
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 16.63 | 30.00 | 26.00 | 8.25 % | | Low | 17.45 | 35.00 | 20.00 | 5.76 % |
| Recent High | Price | Recent Low | Price | Cycle | | | Friday, February 05, 2010 | 7.68 | 19 | | Friday, January 22, 2010 | 7.88 | | | 34 | | | Friday, January 08, 2010 | 7.76 | 58 | | Wednesday, December 02, 2009 | 8.21 | | | 38 | | | Thursday, October 15, 2009 | 7.90 | 154 | | Thursday, October 08, 2009 | 8.14 | | | 38 | | Friday, August 14, 2009 | 7.80 | | | 52 | | | Monday, March 09, 2009 | 5.65 | 25 |
Fourier Transform after Digital Filter M | Cycle Length | Fourier Amplitude | | 12 | 0.54 | | 14 | 0.45 | | 16 | 0.43 | | 25 | 0.34 | | 18 | 0.34 | | 21 | 0.33 | | 27 | 0.27 | | 38 | 0.20 | | 29 | 0.20 | | 41 | 0.16 | | 35 | 0.15 |
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Frequency Analysis based on Quarterly Comb
Frequency from Quarterly Comb Filter
Representative Cycle (Trading Sessions) = 79.00
Sawtooth = 44.03
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 51.00 | 103.00 | 79.00 | 14.20 % | | Low | 49.27 | 96.00 | 116.00 | 11.53 % |
| Recent High | Price | Recent Low | Price | Cycle | | Wednesday, December 02, 2009 | 8.21 | | | 38 | | | Monday, March 09, 2009 | 5.65 | 72 | | Thursday, October 08, 2009 | 8.14 | | | 185 | | | Thursday, November 20, 2008 | 5.07 | 91 | | Wednesday, January 14, 2009 | 6.88 | | | 85 | | | Tuesday, July 15, 2008 | 6.96 | 102 | | Friday, September 12, 2008 | 7.36 | | | 79 | | | Tuesday, February 19, 2008 | 6.87 | 59 |
Fourier Transform after Digital Filter Q | Cycle Length | Fourier Amplitude | | 12 | 0.32 | | 14 | 0.25 | | 16 | 0.23 | | 18 | 0.20 | | 21 | 0.19 | | 228 | 0.17 | | 141 | 0.16 | | 128 | 0.16 | | 84 | 0.15 | | 24 | 0.14 | | 111 | 0.14 |
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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 | | | Thursday, November 20, 2008 | 5.07 | 320 | | Friday, September 12, 2008 | 7.36 | | | 640 | | | Thursday, August 16, 2007 | 5.89 | 286 | | Tuesday, February 28, 2006 | 6.22 | | | 481 | | | Tuesday, June 27, 2006 | 5.51 | 537 | | Wednesday, March 31, 2004 | 5.48 | | | 188 | | | Monday, May 10, 2004 | 4.54 | 194 | | Wednesday, July 02, 2003 | 5.51 | | | 415 |
Fourier Transform on Raw Closing Prices | Cycle Length | Fourier Amplitude | | 512 | 0.31 | | 683 | 0.31 | | 585 | 0.28 | | 410 | 0.25 | | 819 | 0.24 | | 372 | 0.22 | | 341 | 0.20 | | 455 | 0.19 | | 315 | 0.17 | | 15 | 0.15 | | 293 | 0.15 |
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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.
| . | | | Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25 | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 1 instances | | | | Future30SesAvgClose | -1.05 % minus 0.65 % = -1.70 % | -14.40 % | | Future5SesAvgClose | -0.13 % minus 0.12 % = -0.25 % | -12.78 % | | Future2SesAvgClose | -0.40 % minus 0.06 % = -0.46 % | -58.65 % | | Sell Signals - 1 instances | | Future30SesAvgClose | -4.22 % minus 0.65 % = -4.87 % | -41.23 % | | Future5SesAvgClose | 0.65 % minus 0.12 % = 0.53 % | 26.80 % | | Future2SesAvgClose | 0.36 % minus 0.06 % = 0.30 % | 37.95 % | | . | | | Semi-Annual Avg. of Bi-weekly Sawtooth Signals middle Crossing | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 33 instances | | | | Future30SesAvgClose | 0.95 % minus 0.65 % = 0.31 % | 2.60 % | | Future5SesAvgClose | 0.15 % minus 0.12 % = 0.02 % | 1.15 % | | Future2SesAvgClose | 0.24 % minus 0.06 % = 0.17 % | 21.92 % | | Sell Signals - 32 instances | | Future30SesAvgClose | 1.89 % minus 0.65 % = 1.24 % | 10.53 % | | Future5SesAvgClose | 0.18 % minus 0.12 % = 0.06 % | 2.84 % | | Future2SesAvgClose | -0.07 % minus 0.06 % = -0.13 % | -16.93 % | | . | | | Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30 | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 3 instances | | | | Future30SesAvgClose | -1.72 % minus 0.65 % = -2.37 % | -20.04 % | | Future5SesAvgClose | -1.35 % minus 0.12 % = -1.48 % | -75.05 % | | Future2SesAvgClose | -1.11 % minus 0.06 % = -1.17 % | -148.53 % | | Sell Signals - 7 instances | | Future30SesAvgClose | 0.89 % minus 0.65 % = 0.25 % | 2.09 % | | Future5SesAvgClose | -0.36 % minus 0.12 % = -0.49 % | -24.69 % | | Future2SesAvgClose | -0.75 % minus 0.06 % = -0.81 % | -102.69 % | | . | | | Projection of Representative Monthly Cycle | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 104 instances | | | | Future30SesAvgClose | 0.57 % minus 0.65 % = -0.08 % | -0.68 % | | Future5SesAvgClose | 0.02 % minus 0.12 % = -0.10 % | -5.22 % | | Future2SesAvgClose | -0.06 % minus 0.06 % = -0.12 % | -15.40 % | | Sell Signals - 120 instances | | Future30SesAvgClose | 0.46 % minus 0.65 % = -0.19 % | -1.59 % | | Future5SesAvgClose | 0.03 % minus 0.12 % = -0.10 % | -4.86 % | | Future2SesAvgClose | 0.06 % minus 0.06 % = 0.00 % | -0.10 % | | . | | | Projection of BiWeekly Modal Cycles | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 227 instances | | | | Future30SesAvgClose | 0.89 % minus 0.65 % = 0.24 % | 2.04 % | | Future5SesAvgClose | 0.31 % minus 0.12 % = 0.19 % | 9.60 % | | Future2SesAvgClose | 0.19 % minus 0.06 % = 0.13 % | 16.06 % | | Sell Signals - 237 instances | | Future30SesAvgClose | 0.67 % minus 0.65 % = 0.02 % | 0.16 % | | Future5SesAvgClose | 0.14 % minus 0.12 % = 0.01 % | 0.72 % | | Future2SesAvgClose | 0.05 % minus 0.06 % = -0.01 % | -1.36 % | | . | | | | |
| The best buy signal for 30 future sessions was 0.31 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals middle Crossing,and the best sell signal for was -4.87 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25. | | For 5 future sessions the best buy signal was 0.19 % from the Projection of BiWeekly Modal Cycles,and the best sell signal was -0.49 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30. | | The best buy signal for 2 future sessions was 0.17 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals middle Crossing,and the best sell signal was -0.81 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30. |
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