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56 , sin 0),r cos «] + *i sin Wii +  sin Wjt cos mj + *2 sin o,r sin cojf = X a, cos mj cos Wir + *i 2 sin w,r + «2 cos mj + 62 cos cof sin mj = y, cos 102 a, sin mj cos w,r + b, sin Ojf sin + « X + *2 sin C02f  X 3 " Once the sums are obtained, the final step is to create a 4 x 5 matrix to solve the four normal equations for the coefficients „ 2, and fcj When plotting the answer it will be best to plot the original twofrequency equation in its component forms as well as in combination: yi = a, cos CO,/ + sin to,/ y, = 2 cos (OJ + bj sin t02f y, =y\ y, where «i = 3.635, *, = .317, «2 .930. and 6, 762. The solution to the twofrequency ftfoblem gives the follo\1ng values a, = .53 X, = .830 (X;= 133 X2=~J&i and finally the frequencies: to, = 33.9 and 102= 139.8 correspond to ltl.6 and 2.6 calendar quarters (Figure 89). Fourier Analysis: Complex Trigonometric Regression Developed by the French mathematician John Baptiste Joseph Fourier, tburier analysis is a method of complex trigonometnc regression, which expresses any data set as a senes of sine and cosine waves of the same type as discussed in the frevious section. Assuming that there is a cycle and that there are data points in each repetition of this cycle, the Fourier method of analysis shows that the N points lie on the regression curve: .,=i+v ( 72) ( 2)/ where the regression coefficients * and Vf, are given by 2 . . „ 1 . 2i[ki ev/2) .t"*" ( 72> 42 = 0 FIGURE 89 Twofrequency trigonometric afproximation.
It Is imponant to see that the mean of all the points on one cyde is equal to 1. The N values of v, will have the property Appljing the Fourier series to the seasonal component will help clarifj this method. Seasonal data form the most obvious cycle. Using average monthly prices, detrended to avoid letting the trend overwhelm the cycle, let N = 12. It is also known that seasonally adjusted prices will varj about the mean; hence, the weighting factors will have the same propertj as the above equation. With this information, the trigonometric curve, which approximates the seasonals, can be generated and compared with the results of other methods. Spectral Analysis Derived from the word apectrum, apecfral analjsis is a statiatical procedure that isolates and measures the cycles within a data series. The specific technique used is the Fourier series as previously discussed, although other series have also been used. When studjing the cycles that comprise a data series, it is important to refer to their phase with respect to each other. Phase is the relationship of the starting points of dififerent cycles. For example, if one cycle has the same period as another but its peaks and val 7 A continuation of this development can be found in Warren Gilchrist. Statistical Forecasting (,Jolm Wiley . Sons, London, 1976, pp. 139148); a more theoretical approach is to be found in C. Chatfidd, The Analjsis of a Time Series: Theory andPradice (Chapman andHall, London, 1975, Chapter 7). lejs are exactly opposite, it is 180 outofpbase. if the two cycles are identical in phase, they are coincident. Cjcles with the same period may lead or lag the other by being outohase to various degrees. A tool used in apecfral analjsis to visualize the relative significance of a series cyclic components is the periodogram. Weighting the cyclic components in the periodogram will give the more popular specfral density diagram, which will be used to illusfrate the results of the specfral analjsis. Density refers to the frequency of occurrence. Figures 8lOa and 8lOb show the apecfral density of a series composed of three simple waves (D is the
Fourier series made upofwavesA, B, and C).* The cycle length, shown at the bottom of the spectral density chart, corresponds exactly to the cycle length of the component waves A, B, and C. The spectral density, measured along the left side of Figure 8 lOb, varies with the amplitude squared of the cycle and the magnitude of the noise, or random price movements, which obscures the cycle. In Figure 8lOfr, the result is based on a series composed of only three pure waves. Had there been noise of the same magnitude as the underiying cycle amplitude, those cycles identified by the spectral analysis would have been completely obscured. Readers who have studied ARIMA will recognize the similarity between the spectral density and the correlogram. As in trigonometric regression analjsis, the other basic price components can distort the results. A noticeable trend in the data must be removed or it will be interpreted as the dominant cycle. The familiar methods of firsl differencing or linear regression can be used to accomplish this. The seasonal component is itself a cycle and does not need to be removed from the series. Because specfral analysis will identify both the seasonal and cyclic components, the success of the results will depend on the strength of these waves compared with the noise that remains. In appljing this technique to real data, it would not be surprising to see the results demonstrated in Figure 811. Three subcycles of length 10, 20, and 40 dajs are shown as part of a 250day (seasonal) cycle. Notice that, as the cycle lengthens, the width of the specfral density representation widens. This does not mean that the wider peaks are more important. The trader is most interested in those cycles with greater specfral density, corresponding to a larger price move. The minimum amount of data necessarj to find these cycles must include the full cycle that might be identified. For example, to see any seasonal pattem, a minimum of 12 months is needed. More data is better when using specfral analjsis to confirm the consistency of the cycle. A single year is not adequate to support any seasonal findings Weighting Factors The most important part of specfral analjsis is finding the proper estimators, or weighting factors, for the smglefrequency series of cosine waves. When looking for longterm cycles, it is worth being reminded that the frend and seasonal components must be removed, because the method of spectral analjsis will consider these the dominant characteristics, and other cycles may be obscured. As in the other trigonometric tormulas, ttie basic time scries notation is used, where >*„ t = 1,2, . . . , are the data points ana J>, wm be the restilting estimated poims on the spectral analysis. Then j),<co) = ~(co + 2 21 cos <akj ,4, WiUiamT ! . "Fourier Spectral Analysis, Yet:Im£caI Antaiysis of Stocks & Commodities av:/AugMal 19 4). FIGURE810 Specfral density (a) A compound wave D, formed from three primary waves. A, 8, and C. (b) Specfral densits of compound wave D.
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