GPS-X Technical Reference

Optimizer 396
The variable e i,j is the residual and is defined as:
Equation 14.21
The variable z i,j is the measured value of response j at the ith data point.
The expressions in Equation 14.19 and Equation 14.20 were derived assuming that the
response variables are equivalent to the state variables. Even if this is not the case, this is
not a limitation because the sensitivity coefficients are being calculated using finitedifferences.
Therefore any dependencies in the responses are being taken into account.
The Hessian matrices for the other objective functions are not calculated because they
cannot be approximated using the Gauss-Newton approximation. As a result, they
require the calculation of second order sensitivities.
If any of the optimization variables are at their bounds, the elements of the Hessian
involving these variables are zero.
The Hessian matrix is only reported when the report objective function gradient and
Hessian switch in the Derivative Information sub-section in the Optimizer form is set
to ON.
Confidence Limits
GPS-X calculates linear-approximation confidence limits for the parameter estimates
using the variance-covariance matrix. For the sum of squares objective function, the
variance-covariance matrix of the parameter estimates is defined as (Bard, 1974):
Equation 14.22
The matrix Ĥ -1 1 is the inverse of the Gauss-Newton Hessian approximation to the sum of
squares objective function at the solution. The variable s 2 is the variance estimate for the
measurement errors and is defined as:
Equation 14.23
SS ˆ
is the value of the sum of squares objective function at the solution, n is the
where
number of data points (total number considering all target variables), and p is the number
of parameters (i.e. optimization variables).
GPS-X Technical Reference