GPS-X Technical Reference

Optimizer 388
In GPS-X it is assumed that the experimental errors are normally distributed random
variables with a mean of zero. As a result, the likelihood function used in GPS-X is:
Equation 14.6
where:
n
m
V i
= number of experiments (i.e. observations)
= number of measured response variables
= variance-covariance matrix for the ith experiment
| V i | = represents the determinant of V i
ei
= m x 1 residual vector that contains the differences between the
measured values of the response variables and the values predicted by
our mathematical process model.
= the vector of parameters to be estimated in our mathematical process
model.
This expression is derived by substituting the residual vector, ei for the error vector in the
multivariate normal probability density function (pdf). The error vector in the
multivariate normal pdf contains the differences between the measured values of the
response variables (the variables that we are fitting) and the true values. See Bard (1974)
for details.
GPS-X also assumes that the measurement errors are independent from observation to
observation and from response variable to response variable. This means that the
variance-covariance matrices found in Equation 14.6 are diagonal but not necessarily
equal.
GPS-X Technical Reference