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57 I1IIIq
Methods of spectral analysis vary due to the choice of weighting functions tha compensa for th fact that th accuracy of decreases as k increases The two most popular techniques for adjusting (or this problem introduce an estimator called a / § window and a truncation point M /V so that the values of c* for / < Jb < /V are no longer used and the values of c for Jb M are we ghted by >*. The spectral analysis approidmarion is then written m) = (Vo + 2 £ V. cos u*) where tan be cither of the following; (a) lUkeyumdow (b) window Using a Fast Fourier Transform Program There are computer programs that apply a Fast Fourier Transform to perform a spectral analjsis and create a Fourier power spectrum such as the one in Figure 8lOb. Anthony Warrens approach can be found in Appendix 5 written in BASIC program code. The pro Anthony Warren,A Mini Guide to Fourier Spectrum Analysis," Technical Analjsis ofStocks . Commodities (January 1983). A very useful series of articles on spectral analjsis has been published in Technical Analjsis beginning in Januarj 1983, authored by both Anthony W Warren and jad; K. Hutson. Much of the infonnation in this section was drawn from that material. gram deUends the data and reduces endpoint discontinuity, which can produce large unwanted cycles. This is accomplished by multiplying the data by a bellshaped window and extending the endpoints to give a more definitive structure to the deUended data, without affecting the results (as discussed in the previous section). A second filter is applied using selected moving averages. The moving average will reduce or eliminate the importance of those cycles, which are equal to or shorter than two times the length of the moving average, letting the more dominant cycles appear. For example, the use of a 10d moving average will eliminate cycles of length less than 20 ds (frequencies greater than 125 per year). Figure 812showstheoutput of the computer program. Subsequent woits by Warren and Hutson present a computer program to calculate moving average weighted filters using linear, friangular, and Hanning weights. Interpreting the Results of the Fourier Power Spectrum Both Figures 8lOb and 812 show a power spectrum resulting from a Fourier fransform. Figure 8lOb is an ideal representation, in which the cycles stand out with no ambiguity; Figure 812 is more realistic, showing both the dominant cycles and a certain amount of variance around those values. In the power spectrum, the cycle power shown along the jaxis is the cycle amplitude squared. In Figure 8lOa, cycle D peaks at a price of about 425, which yields a specfral density, or specfral power, of 180,625 when squared, corresponding roughly to the 40day cycle in Figure 81 Ob. Using the infonnation fran the beginning of this chapter, the frequency is the inverse of the cycle length: therefore, if the cycle length is 40 dajs. the frequencyF = 360/40 = 9. Anthony Warren and jackK. Hutson, "Finite Impulse Response Filter," Technical Analjsis of Stocks & Commodities
(May 1983) FIGURE 812 Output of spectral analysis program. TITLE T BILLS S03/B3 01/07/83 FOURIER ANALYSIS DF 7OfiY MOVING ftUG. 9.88 Fourier N = ZSE . Source., jad; . Hutson, "Using Fourier," Technical Analjsis of Stocks & Commodities, 1, no. 2 January & February 1983). m 1983 Technical Analjsis, Inc. Used with permission. The sine wave changes phase at the rate of 9 degrees per d, completing one fun cycle every 40 ds. A fast method for observing the possible results is to use weekly rather than daily data. This will be a close approximation for lowfrequency waves but will he less representative for the high frequencies. Averaging the data points can yield results very similar to the daily analjsis. MAXIMUM ENTROPY Maximum Enfropy Specfral Analjsis (MESA) is a technique that filters noise (entropy) from a time series and exposes the useful cycles:" it provides a very practical altemative to Fourier analjsis that makes it possible to find cycles usmg a verv small amount of data. The use of Fourier fransforms requires at least 256 da points and a minimum of 16 consistent cycles of 16 bars. That would eliminate the possibility of uncovering cycles for the shortterm trader. Jolm Ehiers describes the existence of shortterm cycles as a natural phenomenon. it is part of the process thai causes rivers to meander back and forth as water seeks to fiow in a sfraight line, or a drankard who walks through an alley bumping against the walls but moving steadily forward. From these pattems, useful cycles can be found aboul 20% of the time. Using the Phase Angle In an ideal situation, in which the market cycle can be shown as a pure sine wave, the phase angle constantlj increases throughout the cycle, beginning at 0 and ending at 360. The phase angle then drops to 0 when the new cycle begins and increases again at a constant rate until it ends at 360. This repeated pattem forms a sawtooth chart, as shown in Figure 813. Although the cycle goes fran peak to value, the phase angle moves constanUy in one direction in the practical analjsis of shortterm cycles, Ehiers compresses tick data into bars of equal numbers of ticks, then examines the phase for uniformity. Once found, the uniform phase, which appears as a sawtooth chart, will become erratic as the shortterm cycle begins to break down, maricing the end of a current maitet event. Ehlerss Lateral Shift in Thinking At first glance, the use of only a small amount of data needed by MESA seems to confradict the basic rules of statistics, which demand that results be based on as much data as possible to be reliable. But Ehiers, who has been the dominant influence in cycles since about 1990, is too knowledgeable to have made a mistake so simple. His book MESA and Trading Maitet Cjcles focuses on the use of shortterm cycles based on short sample time periods. Instead, he has used this very attribute to apply cycles inside out.
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