Frequency Analysis based on Monthly Comb
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Short term fluctuations that seem to repeat with a regular frequency are often (even generally) seen in stock price charts. 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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Subscribe More than 70 Hi-Rez charts daily for Boeing Aircraft: |
Frequency from Monthly Comb Filter
Representative Cycle (Trading Sessions) = 14.00
Sawtooth = 50.74
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 16.81 | 32.00 | 16.00 | 12.10 % | | Low | 17.22 | 30.00 | 20.00 | 10.36 % |
| Recent High | Price | Recent Low | Price | Cycle | | Thursday, January 28, 2010 | 62.14 | | | 14 | | | Tuesday, January 26, 2010 | 57.32 | 24 | | Thursday, January 07, 2010 | 61.78 | | | 16 | | | Friday, December 18, 2009 | 53.08 | 36 | | Monday, December 14, 2009 | 55.67 | | | 39 | | | Wednesday, October 28, 2009 | 46.49 | 18 | | Monday, October 19, 2009 | 52.62 | | | 14 | | | Friday, October 02, 2009 | 50.61 | 31 |
Fourier Transform after Digital Filter M | Cycle Length | Fourier Amplitude | | 12 | 0.54 | | 20 | 0.38 | | 23 | 0.35 | | 25 | 0.28 | | 28 | 0.24 | | 33 | 0.21 | | 48 | 0.20 | | 31 | 0.19 | | 46 | 0.19 | | 39 | 0.18 | | 41 | 0.15 |
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Frequency Analysis based on Quarterly Comb
Frequency from Quarterly Comb Filter
Representative Cycle (Trading Sessions) = 79.00
Sawtooth = 46.93
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 53.59 | 91.00 | 128.00 | 20.53 % | | Low | 50.65 | 95.00 | 95.00 | 21.39 % |
| Recent High | Price | Recent Low | Price | Cycle | | | Wednesday, October 28, 2009 | 46.49 | 80 | | Tuesday, September 29, 2009 | 53.78 | | | 79 | | | Tuesday, July 07, 2009 | 38.07 | 87 | | Monday, June 08, 2009 | 51.52 | | | 105 | | | Tuesday, March 03, 2009 | 28.35 | 68 | | Tuesday, January 06, 2009 | 44.27 | | | 160 | | | Thursday, November 20, 2008 | 35.48 | 79 | | Monday, May 19, 2008 | 82.12 | | | 159 |
Fourier Transform after Digital Filter Q | Cycle Length | Fourier Amplitude | | 12 | 0.33 | | 17 | 0.20 | | 124 | 0.19 | | 20 | 0.19 | | 23 | 0.16 | | 65 | 0.16 | | 25 | 0.15 | | 171 | 0.15 | | 195 | 0.14 | | 59 | 0.13 | | 48 | 0.13 |
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Frequency Analysis based on Yearly Comb
Representative Cycle (Trading Sessions) = 0.00
Sawtooth = 37.64
| 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 | | | Tuesday, March 03, 2009 | 28.35 | 623 | | Wednesday, July 25, 2007 | 99.47 | | | 303 | | | Friday, September 08, 2006 | 66.78 | 868 | | Wednesday, May 10, 2006 | 81.28 | | | 1050 | | | Monday, March 31, 2003 | 21.66 | 383 | | Monday, March 11, 2002 | 43.14 | | | 316 | | | Thursday, September 20, 2001 | 25.00 | 382 | | Wednesday, November 29, 2000 | 58.24 | | | 219 |
Fourier Transform on Raw Closing Prices | Cycle Length | Fourier Amplitude | | 819 | 3.93 | | 8 | 3.27 | | 455 | 3.19 | | 512 | 3.12 | | 372 | 3.03 | | 585 | 2.99 | | 683 | 2.68 | | 315 | 2.68 | | 12 | 2.40 | | 195 | 2.23 | | 410 | 1.39 |
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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 - 5 instances | | | | Future30SesAvgClose | 3.99 % minus 2.14 % = 1.85 % | 15.66 % | | Future5SesAvgClose | -0.09 % minus 0.44 % = -0.53 % | -26.75 % | | Future2SesAvgClose | -0.18 % minus 0.23 % = -0.40 % | -51.12 % | | Sell Signals - 2 instances | | Future30SesAvgClose | 4.26 % minus 2.14 % = 2.13 % | 18.00 % | | Future5SesAvgClose | -0.16 % minus 0.44 % = -0.59 % | -30.15 % | | Future2SesAvgClose | -0.03 % minus 0.23 % = -0.26 % | -32.76 % | | . | | | Semi-Annual Avg. of Bi-weekly Sawtooth Signals middle Crossing | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 66 instances | | | | Future30SesAvgClose | 1.38 % minus 2.14 % = -0.75 % | -6.39 % | | Future5SesAvgClose | 0.08 % minus 0.44 % = -0.36 % | -18.11 % | | Future2SesAvgClose | 0.12 % minus 0.23 % = -0.10 % | -12.99 % | | Sell Signals - 67 instances | | Future30SesAvgClose | 2.12 % minus 2.14 % = -0.01 % | -0.12 % | | Future5SesAvgClose | 0.30 % minus 0.44 % = -0.13 % | -6.75 % | | Future2SesAvgClose | -0.12 % minus 0.23 % = -0.35 % | -44.04 % | | . | | | Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30 | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 16 instances | | | | Future30SesAvgClose | 3.07 % minus 2.14 % = 0.93 % | 7.86 % | | Future5SesAvgClose | 1.72 % minus 0.44 % = 1.28 % | 65.17 % | | Future2SesAvgClose | 1.23 % minus 0.23 % = 1.00 % | 127.05 % | | Sell Signals - 7 instances | | Future30SesAvgClose | 6.95 % minus 2.14 % = 4.81 % | 40.75 % | | Future5SesAvgClose | -0.37 % minus 0.44 % = -0.80 % | -40.85 % | | Future2SesAvgClose | -0.18 % minus 0.23 % = -0.40 % | -51.13 % | | . | | | Projection of Representative Monthly Cycle | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 228 instances | | | | Future30SesAvgClose | 2.44 % minus 2.14 % = 0.31 % | 2.60 % | | Future5SesAvgClose | 0.42 % minus 0.44 % = -0.01 % | -0.74 % | | Future2SesAvgClose | 0.03 % minus 0.23 % = -0.19 % | -24.32 % | | Sell Signals - 232 instances | | Future30SesAvgClose | 1.89 % minus 2.14 % = -0.25 % | -2.08 % | | Future5SesAvgClose | 0.46 % minus 0.44 % = 0.02 % | 1.01 % | | Future2SesAvgClose | 0.11 % minus 0.23 % = -0.12 % | -14.90 % | | . | | | Projection of BiWeekly Modal Cycles | Signaled Appreciation minus non-signaled Appreciation | Annualized % | | Buy Signals - 496 instances | | | | Future30SesAvgClose | 1.95 % minus 2.14 % = -0.19 % | -1.58 % | | Future5SesAvgClose | 0.36 % minus 0.44 % = -0.07 % | -3.69 % | | Future2SesAvgClose | 0.09 % minus 0.23 % = -0.14 % | -17.54 % | | Sell Signals - 471 instances | | Future30SesAvgClose | 1.76 % minus 2.14 % = -0.38 % | -3.24 % | | Future5SesAvgClose | 0.27 % minus 0.44 % = -0.17 % | -8.52 % | | Future2SesAvgClose | 0.03 % minus 0.23 % = -0.19 % | -24.21 % | | . | | | | |
| The best buy signal for 30 future sessions was 1.85 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25,and the best sell signal for was -0.38 % from the Projection of BiWeekly Modal Cycles. | | For 5 future sessions the best buy signal was 1.28 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30,and the best sell signal was -0.80 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30. | | The best buy signal for 2 future sessions was 1.00 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30,and the best sell signal was -0.40 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30. |
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