Methods and guidelines for effective model calibration

Normal equations. To calculate parameter values that produce the closest match to the ob-

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servations, the weighted least-squares objective function is minimized with respect to the parame-
ter values. Using the approximate linear model, this produces what are called the normal equations, 
expressed in matrix notation as:
b = (X
T
ω
X)
-1
X
T
ω
y
(C7)
Despite some variation, the similarity between equations C7 and 4a is apparent, with the 
major difference being that C7 produces the actual optimal parameter values after being evaluated 
just once, while equation 4a produces a vector that is used to update the parameter values, and op-
timal parameter values are obtained only after a number of parameter-estimation iterations. Be-
cause, as noted above, the interative nature of the equations are not central to the issue addressed 
here, equation C7 is used. 
Random variables. The primary random variable in the above equations are the true errors 
ε
. Then, noting that functions of random variable are random, y is random from C1, e is random 
from C2, and b is random from C7. Because for any step of the analysis teh nonlinear model is lin-
earized and X is evaluated for a defined set of parameters, X it is not thought of as being random. 
Expected value. The expected value can be taken of any term, and is represented as E(.) or 
E[.], where the term appears within the parentheses or brackets. As noted above, 
ε
has a mean of 
zero, so E(
ε
)=0.
Variance-covariance matrix of a vector. Proof 2 requires the evaluation of the variance-
covariance matrix of the vector of estimated parameters and the true errors. The variance-covari-
ance matrix of any vector v is calculated as E[ (v-E(v)) (v-E(v))
T 
].