5. To find the optimal estimate, the partial derivative of R is taken 

 with respect to each parameter contained in the regression equation. The 

 equations thus obtained are set equal to zero to obtain an extremum (minimum) 

 for R. For the case of a linear regression equation involving m indepen- 

 dent variables, a linear set of m equations is obtained and may be solved 

 directly by matrix theory. If the system of equations has a nonlinear form, 

 the solution can be obtained numerically, usually by iteration. 



6. A nonlinear regression equation can sometimes be reduced to linear 

 form by an appropriate transformation of variables. For example, exponential 

 or power equations can be transformed to linear form by taking the logarithm. 

 However, this manipulation involves a modification of the original problem 

 since the minimization is carried out with respect to the logarithmic values 

 and not the original untransformed values. The difference is usually small 

 but can be significant if the measured values vary over a large range. 



7. A quantity expressing the ratio between the explained variation by 

 the regression model and the total variation in the data, denoted as the 

 coefficient of determination r^ , is defined as 



z 



(y? - y) 



r" = — :^^:: (A6) 



2 



y) 



,2 i=l 



^(yT- 



i=l 



The equation for the coefficient of determination may be rewritten in a 

 slightly different form to more easily allow interpretation: 



yT) 



^(yT- y)' - ^(yf- yT)' ^(y?- yT 



^^ i^i = 1 - i=i (A7) 



n n 



^(yT- y) ^(yT- y) 



i=l i=l 



A3 



