Environmental Engineering Reference
In-Depth Information
For SU,
a
=
2
z
cosh
−
1
[.(
0 5
mp
+
np
)]
a
>
0
X
X
{
}
ba
=
sinh
−
1
(
np
−
mp
) [(
21
D
−
)]
0 5
.
X
X
(1.91)
a
= − +− ++ >
=+ +
2
p D
(
1
)
05
.
[(
m pnp
2
)(
mpnp
2
)]
05
.
a
0
Y
Y
b
(
y
y
)
2
p np
(
−
mp
)[ (
2
mp
+−
np
2
)]
Y
c
b
For SB,
a
=
z
cosh
−
1
{.[(
05 1
+
pm
)(
1
+
pn
)]
0 5
.
}
a
>
0
X
X
ba
=
sinh
−
1
{(
pn
−
pm
)[(
1 +
pm
)(
1
+
pn
)
−
4
]
05
.
[(
2
D
−
1
−
1
)]}
X
X
(1.92)
a
=
p pm
{[(
1
+
)(
1
+
pn
)
−
24
]
2
−
}
0 5
.
(
D
−
1
−
1
)
a
>
0
Y
Y
b
=+ −
(
y
y
)
2
a
2
+
pp npmD
(
−
) [(
2
−
1
−
1
)]
Y
c
b
y
For SL,
a
=
= −
=+ −
2
z
ln(
mp
)
X
*
05
.
ba
ln{(
mp
1
205
)[ (
pm p
)
]}
(1.93)
X
X
by
(
y
)
.(
p mp
+
1
))(
mp
− 1
)
Y
c
b
in which
D
=
mn
/
p
2
.
As an example, consider the Johnson SU distribution with
a
X
= 1,
b
X
= −1,
a
Y
= 1, and
b
Y
= 0. By initiating the random at randn('state', 13), random samples of Y can be simulated
using the procedure introduced in Section 1.5.4. The sample size
n
= 1000. The histogram
of the simulated Y data is shown in
Figure 1.21a
, together with the underlying PDF. The
ECDF can be computed by
Equation 1.12
and is shown in
Figure 1.21b
. The percentiles
(
y
a
,
y
b
,
y
c
,
y
d
) can be readily identified using MATLAB command
y
i
= prctile(
y
, 100 *
p
i
).
Graphically,
y
a
is the location on the horizontal axis such that the ECDF is equal to
p
a
. The
resulting (
y
a
,
y
b
,
y
c
,
y
d
) are (−1.352, 0.285, 2.633, 10.773). As a result,
m
=
y
d
-
y
c
= 8.140,
n
=
y
b
-
y
a
= 1. 637,
p
=
y
c
-
y
b
= 2.347, and
mn
/
p
2
= 2.419. In this example, the SU family is
correctly identified from the data. In addition, the SU parameters can be identified using
sonably close to the actual values
a
X
= 1,
b
X
= −1,
a
Y
= 1, and
b
Y
= 0.
1.5.3.1 Probability plot and the goodness-of-fit test (K-S test)
1.5.3.1.1 Converting a Johnson random variable into standard normal
Given the simulated Y data, it is desirable to construct a probability plot similar to
Figure
to plot the “normal” probability plot for the X data converted from the Y data. In general,
this conversion has the following form:
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