Methods and guidelines for effective model calibration

APPENDIX A: THE MAXIMUM-LIKELIHOOD AND LEAST-SQUARES 
OBJECTIVE FUNCTIONS
The maximum-likelihood objective function is developed by considering the random na-
ture of y, the observations. This random nature results from conceptualizing measurement error as 
random. If Y is the vector of jointly distributed random variables of which y is a realization, the 
joint probability distribution function (pdf), f
Y
(y), depends on the true model and true parameter 
values. For the purpose of estimating parameters for a given assumed model, consider the joint pdf 
conditioned on a particular set of parameter values, f
Y
(y|b). This joint pdf can be thought of as the 
probability that different sets of possible observations would occur given the parameter values b.
In parameter estimation, the elements of y are known and we would like to estimate b. A reason-
able requirement of the estimates is that they maximize the probability of obtaining the observa-
tions, y. This requirement is imposed by defining the objective function using the likelihood 
function, l(b|y), which is defined as:
l(b|y) = f
Y
(y|b).
(A1)
If the true errors are from a joint, normal distribution, the likelihood function equals (Brockwell 
and Davis, 1987, p. 247):
l(b|y) = 
,
(A2)
where, as in equation 1 and 2, 
e = y - y’,
y’ is a function of b, and 
ND is the number of observations.
Replacing V(
ε
) using equation C21 (see below), taking the natural log, and multiplying by -2 pro-
duces the maximum-likelihood objective function:
S’(b) = -2 ln(l(b|y)) = ND ln2
π
- ln
.
(A3)
Because of the multiplication by a negative number, the maximization problem becomes a mini-
mization problem, and the objective is to determine the parameter estimates that minimize equation 
A3. To include prior estimates of the parameters, e and 
ω
are augmented as described in Appendix 
B, ND is replaced by ND+NPR, and the determinant of A3 is expanded so that A3 can be expressed 
as:
S’(b) = (ND+NPR) ln2
π
+ (ND+NPR) ln 
σ
2 
- ln |
ω
d
| - ln |
ω
p
| + 
, 
(A4)
1
2
π
------
 
 
ND 2
⁄
V
ε
( )
1 2
⁄
–
{
1
2
---
–
e
T
(V(
ε ) )
1
–
e
} 
exp
1
σ
2
------ 
ω
e
T
1
σ
2
------ 
ω




e
+
e
T
1
σ
2
------ 
ω




e

---

76
where 
ω
d
and 
ω
p
are the sections of the weight matrix applicable to dependent variable observa-
tions and prior estimates of the parameters, respectively.
For any assumed model, set of observations, and defined weight matrix used in the param-
eter-estimation procedure, 
ω
is approximated
and ND, 
σ
2
, and 
ω
are constant. Eliminating terms 
of equation A4 that do not depend on b and multiplying by 
σ
2
yields:
S(b) = e
T
ω
e.
(A5)
Thus, for the optimization process, the maximum-likelihood objective function equals the sum-of-
squares objective function (eq. 2).
The development of equation A5 from the maximum-likelihood objective function requires 
that the true errors be from a joint, normal distribution, a condition not required when the equation 
is derived in other ways.