GPS-X Technical Reference

409 Optimizer
APPENDIX C: NOMENCLATURE
= significance level for the parameter estimates
C = correlation matrix at the solution
^
= confidence coefficient
ei =
m x 1 residual vector that contains the differences between the measured
values of the response variables and the values predicted by the mathematical
process model
e
i , j = z
i, j
fi,
j
%Ei,j =
% error between the measured and predicted values at point for response j in
experiment i
fi,j = value of response variable j predicted by the process model in experiment i
F = objective function evaluated at k
k
F
k
= the partial derivative of the objective function with respect to parameter k
k
f i,
j
= partial derivative of respect to f i,j with respect to parameter k
2 F
= second partial derivative of the objective function with respect to k and l
k
j
l
= the heteroscedasticity parameter for response j
h
k =
the step or perturbation size used in the forward-difference derivative formula
for the kth parameter
H = Hessian matrix of objective function
Ĥ = Hessian matrix of objective function at the solution
m = the number of measured response variables
n
j = the number of experiments (i.e. observations) for response j
n =
the number of experiments (i.e. observations) assuming all responses have
the same number of observations
nr = number replicates for the vth value of the independent variable
v
= number of independent variable values for which there are replicates
j = standard deviation of the weighted residuals for response j
j
k
sample autocovariance function between the weighted residuals up to lag k
=
for response j
p = number of parameters being estimated
Q
j = Portmanteau statistic for response j
j
k
sample autocorrelation among the weighted residuals up to lag k for response
=
j
2
R =
% variation explained by regression expressed as a fraction (subscript used if
referring to a response variable)
GPS-X Technical Reference