GPS-X Technical Reference

389 Optimizer
In GPS-X the log-likelihood function is used instead of the likelihood function for
mathematical convenience. Maximizing the log-likelihood function is the same as
maximizing the likelihood function. The log-likelihood function is derived by taking the
natural logarithm of Equation 14.6, as shown below:
Equation 14.7
where:
= variance of response j in experiment i
In this derivation it is assumed that the variance of a response variable is not constant
across all the observations. To take this into account, GPS-X uses the following
expression for the estimated variance (Reilly et al., 1977):
Equation 14.8
where:
= estimate of the variance,
j
f i,j
j
= proportionality constant that will be called the standard deviation of the
weighted residuals for response j
= value of response variable j predicted by the process model in
experiment i
= heteroscedasticity parameter for response j
This expression relates the variability of response variable j to the magnitude of the
predicted value for response j.
The heteroscedasticity parameter controls how the variance depends on the predicted
values. This parameter is bounded between 0 and 2 and is continuous within this range. A
value of 0 indicates constant absolute variability across all of the observations for
response variable j. A value of 2 indicates constant relative variability across all of the
observations for response variable j.
GPS-X Technical Reference