Environmental Engineering Reference
In-Depth Information
The SL member (which is the shifted lognormal member) is bounded from
below
by
b
Y
and is defined by
(1.89)
κ() ln()
Y
=
Y
n
n
Clearly,
Equations 1.84
and
1.89
are identical if
bbaa
*
=−
X
ln
( .
Their PDFs are as
X
X
Y
follows:
(
)
‡
2
(
)
(
)
aa
⋅
exp.
−
05
ba
+
sinh
−
1
y
21
π
+
y
2
for SU
‰
‰
‰
XY
XX
n
n
(
)
2
(
)
=
(
(
)
)
fy
aa
exp.
−
05
ba y
+
ln
1
−
y
2
π
⋅
y
1
−
y
for SB
ˆ
XY
XX
n
n
n
n
‰
‰
‰
2
ba yb
(
)
(
)
a
exp.
−
05
*
+
ln
−
2
π
⋅ −
yb
for SL
X
XX
y
y
Š
(1.90)
Figure 1.20
shows some distributions in the Johnson system—the Johnson system can
generate distributions with a wide range of shapes. For each of the SU, SB, and SL distribu-
tions, a baseline case is plotted. Then, the effect of each of the parameters (
a
X
,
b
X
,
a
Y
,
b
Y
) is
shown in the figure.
1.5.3 Selection and parameter estimation for the Johnson distribution
Slifker and Shapiro (1980) proposed an elegant selection and parameter estimation approach
for the Johnson distribution using percentiles:
1. Choose a number
z
> 0. We assume
z
= 0.7 in this chapter, as recommended in Slifker
and Shapiro (1980).
2. Compute the percentiles corresponding to −3
z
, −
z
,
z
, 3
z
using the standard CDF.
Hence, for
z
= 0.7, the four percentiles are
p
a
= Φ(−2.1) = 0.018,
p
b
= Φ(- 0.7) = 0.242,
p
c
= Φ(0.7) = 0.758, and
p
d
= Φ(2.1) = 0.982. A useful rule of thumb is to ensure that
the sample size is larger than 10/
p
a
. If this rule is not satisfied, the value of
z
should
be reduced. For
z
= 0.7, 10 /
p
a
= 556, and hence, a sample size of about 550 is required
for this choice of
z
. The typical sample size for geotechnical engineering data is in the
order of 100 or less. Further research is needed to look into fitting data from more
realistic sample sizes to the Johnson system.
3. Compute the values of Y corresponding to these four percentiles. Formally,
y
a
= F
−1
(
p
a
),
y
b
= F
−1
(
p
b
),
y
c
= F
−1
(
p
c
), and
y
d
= F
−1
(
p
d
). However, this is not practical because the CDF
of Y, F(
y
), is unknown. Fortunately,
y
a
,
y
b
,
y
c
, and
y
d
can be obtained directly from
data without the knowledge of F(
y
) using sample percentiles:
y
i
=
p
i
sample percentile.
In MATLAB command,
y
i
= prctile(
y
, 100 *
p
i
), where the
y
vector contains all Y data
points.
4. Three parameters are computed from
y
a
,
y
b
,
y
c
, and
y
d
:
m
=
y
d
-
y
c
,
n
= y
b
-y
a
, and
p
=
y
c
-
y
b
.
5. Finally, identify the Johnson member as SU if
mn
/
p
2
> 1, SB if
mn
/
p
2
< 1, and SL if
mn
/
p
2
= 1.
Once the distribution type has been identified (SU, SB, or SL), the distribution parameters
can be computed as follows:
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