GPS-X Technical Reference

395 Optimizer
Hessian of the Objective Function
The Hessian is a matrix of second partial derivatives of the objective function with
respect to the optimization variables. For the case of two optimization variables, the
Hessian is defined as:
Equation 14.18
In GPS-X the elements of the Hessian are calculated using the Gauss-Newton
approximation. For the sum of squares objective function, each Hessian element is
defined as:
Equation 14.19
where:
m
n j
= number of target or response variables
= number of data points for target variable j
For the maximum likelihood objective function, Equation 14.5, each Hessian element is
defined as:
Equation 14.20
where γ j is the heteroscedasticity of the jth response variable.
GPS-X Technical Reference