Environmental Engineering Reference
In-Depth Information
n
n
1
2
(
d
−
µ
)
2
∏∏
1
i
l
(
θ
|
d
)
=
f
(
d
|
θ
)
=
exp
−
(4.3)
i
2
σ
2
πσ
i
=
i
=
1
Substituting
Equation 4.3
into
Equation 4.2
,
its log-likelihood can be written as follows:
n
−
1
2
(
d
−
µ
)
2
∑
L
(
θ
|
d
=
)
ln
i
(4.4)
2
σ
2
πσ
i
=
1
For this particular problem, the optimal values of μ and σ can be determined analytically
by equating the gradients of
Equation 4.4
with respect to μ and σ, respectively, equal to
zero (e.g., Ang and Tang 2007), which yields the following analytical solutions:
n
1
∑
µ
*
=
d
(4.5)
n
i
i
=
1
n
1
∑
σ
*
=
(
d
−
µ
*
)
2
(4.6)
i
n
i
=
1
where μ
∗
and σ
∗
are the maximum likelihood estimates for μ and σ, respectively. Based
on
Equations 4.5
and
4.6
, the analytical solution given data listed in
Table 4.1
yields
μ
∗
= 18.05° and σ
∗
= 5.01°.
In many cases, the analytical solution for maximizing the likelihood function is unat-
tainable, which necessitates a numerical implementation of the maximum likelihood
method. Many existing software packages such as MATLAB
®
and Microsoft Excel™ can
be used to perform this numerical optimization. In the examples detailed in this chapter,
Excel was used to maximize the likelihood function whenever possible. For example,
Figure 4.1
shows an Excel spreadsheet that is configured to maximize the likelihood
function using Solver, a built-in optimization tool in Excel. Here, the optimal values of μ
and σ are found to be 18.05° and 5.01°, respectively, which are exactly the same as that
obtained from the analytical solution.
eXaMPLe 4.2
Using the same data listed i
n
Table 4.1,
we now want to calibrate the parameters for the
joint distribution of
c
and φ, assuming it is a bivariate normal distribution.
Let μ
1
and μ
2
denote the mean values of
c
and φ, respectively, and σ
1
and σ
2
denote
the standard deviations of
c
and φ, respectively. Let ρ denote the correlation coefficient
between
c
and φ. In the bivariate normal distribution, the uncertain parameters to be cal-
ibrated can be denoted as θ = {μ
1
, σ
1
, μ
2
, σ
2
, ρ}. Let
d
i
= {
d
i
1
,
d
i
2
} denote the
i
ith observation
of {
c
, φ} and
d
= {
d
1
,
d
2
, …,
d
n
}. Based on the probability density function of a bivariate
normal distribution (e.g., Ang and Tang 2007), the chance to observe the
i
ith data if the
values of θ are known can be written as follows:
1
f
(
d
|θ
)
=
i
2
πσ σ
1
−
ρ
2
12
‡
‰
‰
Ž
‰
‰
2
2
1
d
−
µ
d
−
µ
d
−
µ
d
−
µ
+
i
1
1
i
1
1
i
2
2
i
2
2
×
exp
−
−
2
ρ
(4.7)
σ
σ
σ
σ
21
(
−
ρ
2
)
1
1
2
2
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