The principles of multiple regression analysis are used to estimate Y from 

 X^, X^, and X^. 



The following analysis is simplified by transforming X's and Y's to deviations 

 from the mean : 



X = X - X 



and 



y = Y - Y. 



Normal set of equations for Y = f^X^jX^jX^) + e is 



and 



lih - L^ii^2i^2 " E^ii^3i^3 = i;^li>^i 

 L^2i^li^ " E^^i^2 " E^2i^3i^3 = S^2i>^i 

 E^3i^li^ " E^3i^2i^2 " E^^ib3 = E^3i>^i 



Vy. - y;(X. -b. + X_.b_ + X_.bJ = nb . 

 ^ 1 ^ li 1 2i 2 3i 3 o 



Note that the variables in the fourth equation are uncorrected for deviations from 

 the mean. 



The serial correlation coefficient obtained from the bulk depth transect is 

 related to the product mean and has this relation: 



r, = corfx.jX. ), = 

 ab ^ J j+a-^b 



covfx. ,x. ) 



.V var (x. ) var (x . )^ 



where a is the lag and b is the orientation and have values a = 1, 2; b 

 The above expression can be written; 



1, 2, 3. 



ab 



X . X . 



2 2 

 X . X. J 



J J+a 



Thus 



x.x. 

 J J + aJ 



2 2 

 X. x . 



ab L j j + a 



31 



