Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 48

(a) (b)

500 1

n = 1000

P d = 0.982

Johnson SU
a X = 1, b X = -1, a Y = 1, b Y = 0

0.8

400

P c = 0.758

300 0.6

0.4

200

P b = 0.242

100 0.2

y c = 2.633

P a = 0.018

0

0

-2 0 2 4 6 8 10 0 5 10

y d = 10.773

y a = -1.352 y a = 0.285

y y

Figure 1.21 (a) Histograms for the simulated Y; (b) ECDF of Y, and the four percentiles.

()

X =

Φ 1

F Y (1.94)

Y

where F Y is the CDF of Y. With the family type chosen and the parameters ( a X , b X , a Y , b Y )
identified, this conversion has the following convenient analytical form:

‡

2

+× − ++ −

Y b Y b

‰
‰
‰
‰
‰

Y Y

ba ln 1 SU

XX

a a

Y Y

( )

Y − ba

YY

X = ba +× ln SB (1.95)

ˆ

( )

XX

1−− Y ba

‰
‰
‰
‰
‰

Y Y

( )

ba

*

+× − ln Y b SL

XX Y

Š

Figure 1.22a shows the histogram of the X data that was converted from the aforemen-
tioned simulated Y data using Equation 1.95 . The conversion is based on the model parame-
ters: a X = 1.027, b X = −1.019, a Y = 1.040, and b Y = −0.042. The standard normal PDF is also
plotted for comparison. Visually, the X converted from the Y data seems to fit a standard
normal model reasonably well.

1.5.3.1.2 Normal probability plot and K-S test
The normal probability plot for the converted X data can be obtained using the proce-
dure shown in Figure 1.9 . Figure 1.22b shows the normal probability plots for the con-
verted X data. The p-value for the K-S test can be computed by using MATLAB function
[ h , p ] = kstest( X ). Note that X must be taken as the input to this function (not y ), because the