Stockhausen and Fogarty: Removing observational noise from time series data using ARIMA models 



101 



The largest degree of F on the righthand side of this 

 equation is Q. By matching coefficients of F' on both sides 

 of the equation, Box et al. (1978) found, for _; = 0,1,...Q, 

 that 



i]X = C<p, 



(A.8) 



where X, (p are Q+1 column vectors X^=(X,,, X,,..., Xq), 

 qP'=(l, -q),,..., -<Pq) and »], C are (Q+l).t(Q+l) matrices 

 such that 



C = 



1 c, 



Cr. 



1 



(A.9) 



Thus, Equations A.8 and A.9 define a set of Q+1 equa- 

 tions in Q+1 unknowns iXg, X^,..., Xq) which can be 

 solved for by matrix inversion such that 



X = fj-'Cfl>. 



(A.IO) 



For />Q, the X^'s are computed recursively by using the 

 relation 



^.=1^.-1 a- (A.ll) 



1=1 



Finally, the coefficients of w(B} are given by 



«o = i — |-^o; «j= — |-^,'i = i'- 



(A.12) 



1 -'?! 



-1q 



-1q 



-"Iq 



-Iq 





 1 



 ll 







-nq-i 



-n, 1 



