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55 3>i + > This is expressed as a summation (similar to leastsquares) m which the values for and d misi be found. where f and d =y„, + j„+,. Summing the detrended values and pves 1 c = 6338.4 and Jcd = 9282 2, resulting in a 1 464. The value for a is substituted mto the intermediate equation and solved for the frequency tO: cos ¹ = .732 « = 42.9 The period is 360/i2.9 = 8 4 calendar quarters The last step in solviig the equatron for a single frequency is to wnie the normal equations; cosOK+1» cos CM sin CM = 2 J(,COS wf 2] * cos CM+* 2 sinffii = JV tot
and solve fur a and b. where ( = 1. 40. and ( = tZ 9 the other soluiions. a computer program is best for finding the sums (usmg detrended data) necessan to sohe the equations The sums arc a cosm sin / cos y, cos sin ,& Then, a and b can be found by substituting in the fullowing equations. SCOT sin cos cos sin V J,, cos OK  6 V cos OK sin OK The results « =  603 and b = 1.81 give the singlefrequency cune as >,=6(Hcosi2.9/+l 831 sin 42 1/ Taking f = 1 to be 1 67 and /  <i8 to be 19" \ and adding back the trend, the resulting periodic cune is shown m Figure 88 The singlefrequency curve shown in Figure 8b matches seven out ol the eight peaks in copper; howe\4r. it is not much more than could have been done using the Ehrlich FIGURE 88 Copper prices I963I979: smglefrequency copper cycle manually scaled to approximate amplitude. Cjcle Finder mentioned in a previous section. A singlefrequency curve can be created simply by identifjing the mosl dominant peaks, averaging the distance (period), and applying the singlefrequency formula TwoFrequency TrigonomeUic Regression The combination of more than one set of sine and cosine waves of varjing amplitudes and frequencies will create a better fit than a singlefrequency solution. This is analogous to the use of a secondorder (curvilinear) solution instead of the firstorder linear. The equation for the twofrequency cycle is
find the results of this complex wave, apply the same techniques used in the singlefrequency approach to the detrended copper data. The algebra for solving ttiis problem is an expanded form of the previous solution, and the use of a computer is a requirement. The programs necessary to solve this one appear in Appendix 4. The frequencies 0), and 0). are found by solving the quadratic equation: 2  , :(1 + /2)0 where x = cos co, using the standard formula. g, ± vai + 8(1 + /2) 4 The same leastsquares method as before cat! be used, derived from the getieral form: cc, (jb + +2) + «2 +1 = >„  1 + >„ + The leastsquares equations for Finding ai and a. are: a, + tt; v cd = 03 a, cd + «2 rf = where =y„ +y„ + z, = * i, andp =y„ , +>„ + These equations can be solved for a, and tt; using: Theti, oil and (oi are calculated from the two solutions :, anda:; of the quadratic equation. The next step is to solve the normal equations to find the amplitudes «i, fr„ « , and . a, cos (Alt + fri cos (oi( sin toi( + «2 cos tof cos tor +2 X *  Y.y
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