GPS-X Technical Reference

Optimizer 390
To determine the optimal value of j , given the other adjustable parameters in the loglikelihood
function, the estimated variance given by Equation 14.8 is substituted into
Equation 14.7. The resulting equation is then differentiated with respect to j and the
derivative is set to zero, leading to the following expression:
Equation 14.9
where:
= measured value of response j in experiment i
Substituting this expression into Equation 14.8 yields the estimate for the variance that
accounts for non-homogeneous measurement errors:
Equation 14.10
If Equation 14.10 is substituted into Equation 14.7, the log-likelihood function
becomes:
Equation 14.11
To allow for the possibility that the number of observations is different for each response
variable, Equation 14.11 is re-written as (Steiner et al., 1990):
Equation 14.12
This is the function that is maximized by GPS-X to fit process models to measured data
and obtain optimal parameter estimates. Equation 14.12 is a measure of the probability
that the measurements were generated by the process model.
GPS-X Technical Reference