Environmental Engineering Reference
In-Depth Information
8.4.3.1 Determination of S(x)
Preliminary OLS regression analysis utilizing all available variables yielded no sig-
ni
cant variables, very low adjusted R-squared, but a positive result for overall sig-
ni
cance. An investigation into possible violations of the Gaussian assumptions
reveals that multicollinearity, heteroscedasticity, and non-normality were all present.
A second regression was performed utilizing White
s matrix to eliminate het-
eroscedasticity. This regression revealed that the only signi
'
cant variables were age
and depth. In order to correct for non-normality, a Robust Least Absolute Error
(LAE) regression was performed. This
final regression con
rmed that age and depth
are signi
cant and resulted in Eq.
8.7
for cost of failure in City B:
S
ðÞ¼
1186
:
6
þ
258
35
4
:
707
:
Depth
i
þ
49
322
2
:
885
:
Age
i
ð
8
:
14
Þ
ð
Þ
ð
Þ
8.4.3.2 Determination of F(x)
The frequency of failure equation is determined by regressing frequency of failure
on age. The results of this regression were signi
cant. After correcting for heter-
oscedasticity and non-normality, a Robust LAE regression yielded Eq.
8.15
for
frequency of failure in City B:
F
ðÞ¼
10
:
846
þ
0
:
76923
2
:
688
Age
i
ð
8
:
15
Þ
ð
Þ
8.4.3.3 Solution of the DSS
The parameters of the DSS model for City B are as follows: Eqs.
8.14
and
8.15
,
I = 119,178 and d = 0.0423. The average depth of 3.24 is assumed for the network.
Substituting these parameters into Eq.
7.8
results in Eq.
8.16
:
R
t
1
1
349
:
546
þ
49
:
322x
ð
Þ
10
:
846
þ
0
:
76923x
ð
Þ
e
0
:
0423x
dx
þ
119178e
0
:
0423t
C
ðÞ¼
1
e
0
:
0423 t
þ
1
ð
Þ
ð
8
:
16
Þ
Continuous optimization of this model yields a solution of 26.66. Iterated
graphical minimization yields a consistent solution as depicted in Eq.
8.16
(Figs.
8.10
and
8.11
).
8.4.3.4 Nonlinearity
Nonparametric regressions were performed to determine whether the cost and
frequency functions in City B were nonlinear. The results have been shown in
Figs.
8.12
and
8.13
.
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