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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. The paradoxical condition of variable length periods forces the adoptation of some new forms of analysis. |
Frequency Analysis based on Monthly Comb
| Independent Analysis of AstroPower: |
Frequency from Monthly Comb Filter
Representative Cycle (Trading Sessions) = 29.00
Sawtooth = 48.72
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 17.58 | 31.00 | 14.00 | 41.82 % | | Low | 16.62 | 29.00 | 16.00 | 31.64 % |
| Recent High | Price | Recent Low | Price | Cycle | | | Thursday, October 02, 2003 | 1.44 | 23 | | Thursday, September 18, 2003 | 1.95 | | | 21 | | | Friday, August 29, 2003 | 1.31 | 25 | | Tuesday, August 19, 2003 | 1.85 | | | 35 | | | Friday, July 25, 2003 | 1.15 | 29 | | Monday, June 30, 2003 | 3.38 | | | 30 | | | Friday, June 13, 2003 | 2.26 | 12 | | Friday, May 16, 2003 | 3.99 | | | 40 |
Fourier Transform after Digital Filter M | Cycle Length | Fourier Amplitude | | 7 | 0.54 | | 13 | 0.29 | | 18 | 0.24 | | 20 | 0.24 | | 22 | 0.21 | | 16 | 0.18 | | 35 | 0.18 | | 29 | 0.17 | | 54 | 0.17 | | 26 | 0.16 | | 24 | 0.15 |
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Frequency Analysis based on Quarterly Comb
Frequency from Quarterly Comb Filter
Representative Cycle (Trading Sessions) = 63.00
Sawtooth = 58.77
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 48.67 | 63.00 | 39.00 | 40.13 % | | Low | 44.40 | 80.00 | 136.00 | 43.58 % |
| Recent High | Price | Recent Low | Price | Cycle | | Thursday, September 18, 2003 | 1.95 | | | 86 | | | Friday, July 25, 2003 | 1.15 | 63 | | Friday, May 16, 2003 | 3.99 | | | 116 | | | Friday, April 25, 2003 | 1.84 | 136 | | Wednesday, November 27, 2002 | 8.92 | | | 168 | | | Wednesday, October 09, 2002 | 6.52 | 159 | | Monday, April 01, 2002 | 27.94 | | | 61 | | | Thursday, February 21, 2002 | 22.03 | 104 |
Fourier Transform after Digital Filter Q | Cycle Length | Fourier Amplitude | | 7 | 0.32 | | 205 | 0.24 | | 171 | 0.21 | | 73 | 0.21 | | 128 | 0.19 | | 10 | 0.19 | | 79 | 0.18 | | 114 | 0.18 | | 14 | 0.18 | | 341 | 0.17 | | 57 | 0.13 |
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Frequency Analysis based on Yearly Comb
Frequency from Yearly Comb Filter
Representative Cycle (Trading Sessions) = 0.00
Sawtooth = 56.50
| Peak Type | Avg. Cycle | Median Cycle | Cycle Mode | Cycle Amplitude | | High | 0.00 | 147.00 | 0.00 | 0.00 % | | Low | 0.00 | 0.00 | 0.00 | 0.00 % |
| Recent High | Price | Recent Low | Price | Cycle | | Monday, April 01, 2002 | 27.94 | | | 213 | | | Thursday, November 11, 1999 | 7.67 | 276 | | Thursday, May 17, 2001 | 36.62 | | | 147 | | | Thursday, October 08, 1998 | 4.04 | 165 | | Monday, October 16, 2000 | 40.42 | | | 150 | | | Monday, January 01, 0001 | 0.00 | 0 | | Tuesday, March 07, 2000 | 31.46 | | | 177 | | | Monday, January 01, 0001 | 0.00 | 0 |
Fourier Transform on Raw Closing Prices | Cycle Length | Fourier Amplitude | | 171 | 2.79 | | 512 | 2.06 | | 7 | 2.04 | | 256 | 1.94 | | 146 | 1.93 | | 73 | 1.60 | | 205 | 1.54 | | 10 | 1.47 | | 12 | 1.26 | | 79 | 1.22 | | 68 | 1.21 |
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| Independent Analysis of AstroPower: |
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Average Sawtooth based on Monthly Combo
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Average Sawtooth based on Bi-weekly Combo
The Sawtooth statistic measures the time from each periodic high to the next periodic low, in comparison to the time from that low to the next high point . 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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