Environmental Engineering Reference
In-Depth Information
in the statistics. For example, a post-earthquake investigation at a given site often yields a
simple observation of whether or not a soil has liquefied, while the actual factor of safety
F
S
remains unknown. Hence, data censoring is commonly used in the development of liq-
uefaction potential assessment models. One advantage of the maximum likelihood method
over the method of moments in the model calibration is that it can deal with censored data
effectively. In the following, we will use a slope example to illustrate model calibration with
the censored data based on the maximum likelihood method.
eXaMPLe 4.4
For simplification, assumptions are often made in a slope reliability analysis and thus,
the calculated reliability index of a slope is not the actual reliability index (Zhang et al.
2007). When the performance of a large number of similar slopes in a region is known, it
is possible to calibrate the error in the calculated reliability index. Let β
c
and β
a
denote the
calculated and actual reliability indexes, respectively. Suppose the relationship between
β
c
and β
a
can be linked with a correction factor ε
β
as follows:
ββε
β
=+
a
(4.10)
Although other forms of relationships between β
a
and β
c
may also be assumed, the
simple additive relationship is considered here as an example for the illustration of model
calibration with censored data. As will be shown later, effectiveness of different assump-
tions can be ranked using the Bayesian information criterion (Zhang et al. 2007) as
needed. To calibrate the correction factor ε
β
, suppose we have collected data of
n
slopes
(including
n
F
failed slopes and
n
NF
non-failed slopes; thus,
n
=
n
F
+
n
NF
) and calculated the
reliability index for each of these
n
slopes. As an example,
Figure 4.5
shows the reliability
indexes of 25 slopes. Let
I
be an indicator variable with
I
= 1 denoting slope failure and
I
= 0 denoting no failure.
Let
I
i
denote the indicator variable of the
i
ith slope, that is,
d
i
= {
I
i
}, and
d
= {
d
1
,
d
2
, …,
d
n
} = {
I
1
,
I
2
, …,
I
n
} denote the calibration database. Assume the model parameter to be
estimated is θ = {ε
β
}. For the
i
ith slope, the probability to observe it as a failed slope if the
value of ε
β
is known can be written as
P
(
d
====−
1
|
θ
)
P I
(
1
|
θ
)
1
Φβ ε
β
(
+
)
i
i
ci
(4.11)
For the
j
th slope, the probability of observing it as a stable slope, if the value of ε
β
is
known, can be written as
P
(
d
==== +
0
|
θ
)
P I
(
0
|
θ βε
β
)
Φ
(
)
j
j
cj
(4.12)
Assuming the failure of slopes is statistically independent, the chance to observe the
n
F
failed slopes and
n
NF
stable slopes is then the product of the probability to observe the
performance of each slope, which can be calculated based on
Equations 4.11
and
4.12
for
a failed and nonfailed slope, respectively. Hence, the chance to observe
d
given θ = {ε
β
} or
the likelihood function can be written as follows:
n
n
F
NF
∏
∏
l
(
θ
|
d
=
)
1
−
Φ
(
β
+
ε
)
Φ
(
β
+
ε
)
(4.13)
ci
β
cj
β
i
=
1
j
=
1
where Φ is the cumulative density function of a standard normal variable. The log-
likelihood function can then be written as
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