APPENDIX A 



BACKGROUND DISCUSSION OF MULTIPLE REGRESSION SCREENING (MRS) TECHNIQUE 



The MRS analysis fits, in the least squares sense, a sum of sinusoids with 

 preassigned frequencies to a given data record. 



It is desired to obtain an estimate of the time series in the form 



yg(nAt) = dg + d^X^(nAt) + d^X^CnAt) + ... + dpXp(nAt) p < N (A-1) 



where d^ are constants and X^^ the selected known functions of time. It is 

 assumed that the mean of the time series is zero so equation (A-1) can be 

 rewritten 



yg(nAt) = d^x^(nAt) + d2X2(nAt) + ... + dpXp(nAt) (A-2) 



where x- are selected known functions of time with mean removed. 



The constants d^ are selected to minimize the sum of the squared 

 differences between actual and estimated values in the time series 



N 



I [y(nAt) - ye(nAt)]2 (A-3) 



n=i 



By combining equations (A-2) and (A-3) , the sum of squared differences can be 

 written as 



N 



I [y(nAt) - d^x^(nAt) - d2X2(nAt) - ... - dpXp(nAt)]2 (A-4) 



n=i 



By using the method of least squares, the optimum values of d , d , ...» d 

 can be obtained as the solution to p simultaneous equations 



N N N N 



dl I x2 + d^ I x^X2+...+dp I x^Xp = I x^y 



n=i n=i n=i n=i 



N N N N 



d^ I x^x^ + d^ I y\ + ... + dp ); x^Xp = I x^y 

 n=i n=i n=i n=i 



N N N N 



d, y x„x, + d^ y x„x^ + ... + d„ y x^ = y x„y 



n=i n=i n=i n=i 



These equations are often called the normal equations. 



(A-5) 



79 



