Methods and guidelines for effective model calibration

The importance of 
X
 and X. The different symbols, 
X
and X, are used in C1 and C2 be-
cause they may be different. For linear problems they often are the same, but differences occur 
when the approximate model includes more or fewer terms than the true model (n does not equal 
ν
). In addition, errors in measuring the independent variables could affect 
X
j 
and X
j
, but this prob-
lem is not addressed in this report.
For a nonlinear model, an equation of the form C2 is used to represent the linearized ap-
proximate nonlinear model, as discussed above. For nonlinear problems, the X
j 
also vary depend-
ing of the set of parameter values about which model is linearized, and the difference between 
X
j 
and X
j
, becomes greater as the optimized parameter values differ more from the true parameter val-
ues.
Functional form of observations. The ith observation used in the regression can be ex-
pressed in terms of the true linear model as:
y
i
= 
β
0
+ 
β
1
X
1i
+ 
β
2
X
2i
+ . . .
β
j
X
ji
. . . 
β
ν
X
ν
i
+ 
ε
i
(C3)
and in terms of the approximate linear or linearized model as:
y
i
= b
0
+ b
1
X
1i
+ b
2
X
2i
+ . . .
β
j
X
j
. . . 
β
n
X
ni
+ e
i
= y
’
i
+ e
i
(C4)
All observations used in the regression together can be expressed in terms of the true linear 
model using matrix notation (vectors are underlined lower case or greek letter, matrices are under-
lined capital letters) as:
y = 
X
β
+ 
ε
(C5)
and in terms of the approximate linear or linearized model as:
y = X b + e
(C6)
For the linearized approximate nonlinear model, each element of the X array is one of the 
derivatives, or sensitivities, discussed above. An expanded form was shown in appendix B.