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, 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. The paradoxical condition of variable length periods forces the adoptation of some new forms of analysis. |
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Frequency from Monthly Comb Filter
Representative Cycle (Trading Sessions) = 29.00
Sawtooth = 50.24
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 16.63 | 30.00 | 22.00 | 21.89 % | | Low | 16.59 | 29.00 | 16.00 | 22.45 % |
| Recent High | Price | Recent Low | Price | Cycle | | Thursday, November 13, 2008 | 11.33 | | | 52 | | | Monday, October 27, 2008 | 7.87 | 64 | | Tuesday, September 02, 2008 | 13.72 | | | 15 | | | Monday, July 28, 2008 | 11.47 | 96 | | Monday, August 11, 2008 | 13.68 | | | 40 | | | Tuesday, March 11, 2008 | 9.89 | 34 | | Friday, June 13, 2008 | 15.00 | | | 29 | | | Tuesday, January 22, 2008 | 9.30 | 38 |
Fourier Transform after Digital Filter M | Cycle Length | Fourier Amplitude | | 10 | 0.69 | | 17 | 0.43 | | 22 | 0.34 | | 20 | 0.32 | | 25 | 0.30 | | 28 | 0.22 | | 39 | 0.21 | | 37 | 0.19 | | 41 | 0.19 | | 32 | 0.17 | | 34 | 0.16 |
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Frequency Analysis based on Quarterly Comb
Frequency from Quarterly Comb Filter
Representative Cycle (Trading Sessions) = 108.00
Sawtooth = 51.64
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 49.67 | 85.00 | 83.00 | 27.12 % | | Low | 53.29 | 98.00 | 108.00 | 29.20 % |
| Recent High | Price | Recent Low | Price | Cycle | | Tuesday, September 02, 2008 | 13.72 | | | 55 | | | Monday, July 28, 2008 | 11.47 | 130 | | Friday, June 13, 2008 | 15.00 | | | 159 | | | Tuesday, January 22, 2008 | 9.30 | 113 | | Thursday, October 25, 2007 | 13.15 | | | 88 | | | Thursday, August 09, 2007 | 11.90 | 108 | | Thursday, June 21, 2007 | 14.00 | | | 35 | | | Wednesday, March 07, 2007 | 12.23 | 160 |
Fourier Transform after Digital Filter Q | Cycle Length | Fourier Amplitude | | 10 | 0.37 | | 12 | 0.30 | | 16 | 0.26 | | 120 | 0.20 | | 25 | 0.18 | | 124 | 0.17 | | 21 | 0.16 | | 23 | 0.16 | | 51 | 0.15 | | 683 | 0.15 | | 41 | 0.15 |
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Frequency Analysis based on Yearly Comb
Representative Cycle (Trading Sessions) = 0.00
Sawtooth = 54.11
| 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, January 22, 2008 | 9.30 | 710 | | Thursday, June 21, 2007 | 14.00 | | | 290 | | Wednesday, April 26, 2006 | 14.26 | | | 556 | | | Monday, March 28, 2005 | 6.95 | 612 | | Tuesday, February 10, 2004 | 14.36 | | | 649 | | | Friday, October 18, 2002 | 5.80 | 241 | | | Thursday, November 01, 2001 | 9.10 | 406 | | Friday, July 06, 2001 | 19.74 | | | 738 |
Fourier Transform on Raw Closing Prices | Cycle Length | Fourier Amplitude | | 819 | 2.48 | | 683 | 2.22 | | 8 | 0.97 | | 585 | 0.93 | | 10 | 0.76 | | 455 | 0.65 | | 372 | 0.63 | | 228 | 0.55 | | 512 | 0.54 | | 256 | 0.50 | | 15 | 0.49 |
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Allied Waste Industries, Inc.
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Average Sawtooth based on Monthly Combo
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Allied Waste Industries, Inc.
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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.
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