Environmental Engineering Reference
In-Depth Information
AB
C
D
E
F
G
H
I
J
2
3
4
5
Spreadsheet Template for Calibrating Probability Model
Parameter to
be estimated
Q
p
Q
s
w
F
w
NF
ε
β
6
0.05
0.4
0.125
1.58333333
0.27528444
7
8
Observed data
Eq. (4.10)
Case No
Failed?
β
c
β
a
p
f
w
i
ln
P
(
d
i
|
ε
β
)
L
(
ε
β
|
D
)
9
1
0
3.25
3.
5
2528444
3.52528444
0.00021151
-0.000334933
-0.01300741
0
10
2
0
3.48
3.
7
5528444
3.75528444
8.6572E-05
-0.000137079
0
11
3
0
3.24
3.
5
1528444
3.51528444
0.00021964
-0.000347804
0
12
4
0
3.56
3.
8
3528444
6.271E-05-9.92932E-05
0
3.83528444
Log-likelihood
13
5
0
2.79
3.
0
6528444
0.00108732
-0.001722519
0
3.06528444
14
6
0
3.7
3.
9
7528444
3.97528444
3.5148E-05
-5.56513E-05
0
15
7
0
3.68
3.
9
5528444
3.95528444
3.8222E-05
-6.0519E-05
0
16
8
0
3.62
3.
8
9528444
3.89528444
4.9042E-05
-7.76514E-05
0
17
9
0
4.52
4.
7
9528444
4.79528444
8.1222E-07
-1.28602E-06
0
18
10
0
3.97
4.
2
4528444
4.24528444
1.0916E-05
-1.72835E-05
0
19
11
0
3.3
3.
5
7528444
3.57528444
0.00017492
-0.000276987
0
20
12
0
3.37
3.
6
4528444
0.00013355
-0.000211465
0
3.64528444
21
13
0
3.91
4.
1
8528444
4.18528444
1.424E-05-2.25476E-05
0
22
14
0
2.87
3.
1
4528444
0.00082963
-0.001314121
0
3.14528444
23
15
0
3.02
3.
2
9528444
3.29528444
0.00049161
-0.000778575
0
24
16
1
-3.35
-3.07471556
-3.07471556
0.99894648
-0.000131759
1
25
17
1
-2.25
-1.97471556
0.97584978
-1.97471556
-0.003055827
1
26
18
1
-4.23
-3.95471556
-3.95471556
0.99996169
-4.7892E-06
1
27
19
1
-3.65
-3.37471556
0.99963054
-3.37471556
-4.6191E-05
1
28
20
1
-3.34
-3.06471556
0.99891061
-3.06471556
-0.000136247
1
29
21
1
-3.36
-3.08471556
0.99898127
-3.08471556
-0.000127407
1
30
22
1
-2.89
-2.61471556
-2.61471556
0.99553491
-0.000559386
1
31
23
1
-2.67
-2.39471556
0.99168337
-2.39471556
-0.001043926
1
32
24
1
-2.36
-2.08471556
-2.08471556
0.98145244
-0.002340216
1
33
25
1
-3.42
-
3.
1
4471556
-3.14471556
0.99916876
-0.000103948
1
34
35
36
37
Notes:
(1) The setting in Solver is "Maximize the value in Cell I9 by changing the value in Cell H6."
Figure 4.5
Spreadsheet template for model calibration using 25 censored data inputs.
n
n
F
NF
∑
∑
L
(
θ
|
d
=
)
ln
1
−
Φ
(
β
+
ε
)
+
ln
Φ
(
β
+
ε
)
(4.14)
ci
β
cj
β
i
=
1
j
=
1
value of ε
β
is −0.181. A negative value of ε
β
indicates that the slope reliability is overesti-
mated. Thus, the procedure for calculating the slope reliability index is, on average, on
the nonconservative side.
Adhering to the rule of random sampling (i.e., randomly selecting samples from the pop-
ulation) is important when applying statistical methods. For the slope example illustrated
above, the random sampling criterion implies that the ratio of failed slopes to stable slopes
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