Methods and guidelines for effective model calibration

APPENDIX C: TWO IMPORTANT PROPERTIES OF REGRESSION
This appendix presents two basic properties of weighted linear regression, which are gen-
erally known as the Gauss-Markov theorem, in a manner which emphasizes the difficulties pro-
duced when the regression is nonlinear. More traditional derivations of the Gauss-Markov theorem 
can be found in Bard (1974) and Beck and Arnold (1977).
The two properties of concern are:
1. Parameters estimated by linear regression are unbiased.
2. The weight matrix needs to be defined a particular way for the parameter estimates to have the 
smallest variance, and for the parameter variance-covariance matrix to be calculated using 
equation 26.
Definitions and identities used in both proofs are presented first followed by the two proofs.
Identities
True linear model. The true model is unknown and correctly represents the system of con-
cern. A true linear model can be represented as:
y = 
β
0
+ 
β
1
X
1
+ 
β
2
X
2
+ . . .
β
j
X
j
. . . 
β
ν
X
ν
+ 
ε
Ε
(
ε
)=0
(C1)
where,
y is a measurement of the dependent variable (here, hydraulic heads, flows, and so on);
β
j
are the true (unknown) parameter values;
X
j
are the independent variables (here, location, depth, time, etc.);
ν
is the number of terms in the true model; and 
ε
is the true error, and needs to have a mean of zero, as shown, for the regression to be valid.
True nonlinear model. The true nonlinear model can not be represented as in C1, and re-
quires the more general form presented after equation 1 -- that is, using vector notation, y= F(
β
,
ζ
) 
+ 
ε
., where F repesents the form of the unknown nonlinear function, 
ζ
represents the independent 
variables, and the other symbols are as defined for equation C1.