Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 28

q

() =− −

F q 1 exp (1.47)

2

and

− ()

=

− 21 ln F qq (1.48)

If Q is indeed 2-DOF χ-square, it is clear from Equation 1.48 that a plot with q values
on the vertical axis and −2ln[1 − F( q )] on the horizontal axis will produce a 1:1 line. As a
result, if the actual q versus −2ln[1 − F n ( q )] relationship lies close to the 1:1 line, the bivari-
ate normal hypothesis is reasonable. F n ( q ) is the ECDF of Q, which can be estimated using
Equations 1.11 or 1.12 according to the Q data points.
Table 1.8 illustrates how the line test can be applied to the first 10 simulated bivari-
ate standard normal data in Figure 1.12a . The first column contains the simulated Q
data sorted in an ascending order. The second column contains the rank. The third col-
umn computes the ECDF F n ( q ) using Equation 1.12 . The fourth column is obtained by
applying the logarithm of 1 − F n ( q ) in the third column. Finally, the probability plot is
obtained by drawing the first column on the y -axis and the fourth column on the x -axis.
Figure 1.14 shows the resulting q versus −2ln[1 − F n ( q )] relationship. If it is indeed close
to the 1:1 line, we can conclude that there is no strong evidence to reject the bivariate
standard normal model.

1.3.4 Simulation of bivariate standard normal random variables
Given the Pearson correlation δ 12 , random samples of bivariate standard normal (X 1 , X 2 )
can be readily simulated by the following steps:

1. Simulate independent standard normal random variables (Z 1 , Z 2 ). The details have
been described in Section 1.2.4.

6

5

4

3

2

1

0 0

2 4 6

-2 ln[1-F n ( q )]

Figure 1.14 Line test: q versus − 2ln[1 − F n ( q )] plot. The dashed line is the 1:1 line.