Environmental Engineering Reference
In-Depth Information
200
4
p
-value = 0.97
3
150
2
1
100
0
-1
50
-2
-3
0
-4
-4
-2
0
x
2
4
-4
-2
0
Φ
-1
[F
n
(
x
)]
2
4
Figure 1.22
(a) Histogram for the X data converted from the simulated Y data; (b) x versus
Φ
−
1
[F(
x
)] plot
for the X data.
MATLAB kstest function assumes the target distribution is standard normal. The resulting
p
-value is shown in
Figure 1.22b
. It is clear that the X data converted from the Y data has a
p
-value larger than 0.05 (
p
-value = 0.97). Hence, the standard normal distribution hypoth-
esis cannot be rejected at a significant level of 0.05. Namely, the Johnson SU hypothesis for
the Y data cannot be rejected.
1.5.4 Simulation of the Johnson random variable
Given the family type (SU, SB, or SL) and the four parameters (
a
X
,
b
X
,
a
Y
,
b
Y
), random
samples of the Johnson random variable Y can be simulated by the following steps:
1. Simulate a standard normal random variable X.
2. Simulate the Johnson random variable Y using the inverse transform of
Equation 1.94
:
()
YF
=
−1
Φ
X
(1.96)
Y
With the family type chosen and the parameters (
a
X
,
b
X
,
a
Y
,
b
Y
) identified, this transfor-
mation has the following analytical form:
‡
+×
−
X
b
‰
‰
‰
‰
ba
sinh
X
SU
Y
Y
a
X
(
)
×
(
)
ba
++
b
exp
X
−
b
a
Y
Y
Y
X
X
Y
=
SB
(1.97)
ˆ
(
)
1
+
exp
X
−
ba
‰
‰
‰
‰
XX
X
−
*
b
X
b
+
exp
SL
Y
a
X
Š
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