Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 36

Table 1.11 Mahalanobis distances for the dataset in Table 1.9
k

X 1 − m 1 X 2 − m 2 X 3 − m 3 Q d

1 1.11 0.28 1.62 − 0.92

2 0.58 − 0.98 0.33 1.00

3 1.65 0.87 4.87

− 0.026

4 0.69

− 0.57 − 0.11 − 0.38

5 1.59 − 1.87 1.09 3.42

6 − 1.37 − 0.12 − 1.15 3.14

7 1.20 1.33

− 1.03 − 0.59

8 0.18 0.50 0.95 2.23

9 1.12 5.02

− 1.52 − 1.99

10 − 0.60 1.21 0.59 3.67

from Equation 1.69 that a plot with q d values on the vertical axis and F1 F

−

[ −

n q

( )] on the

2

χ d

− −[ ( )] rela-
tionship lies close to the 1:1 line, we can conclude that there is no strong evidence to reject
the multivariate normal model.
Table 1.12 illustrates how the line test can be applied to simulated multivariate standard
normal data in Table 1.9 . The first column contains the simulated Q d data sorted in ascend-
ing order. The second column contains the rank. The third column computes the ECDF
F n ( q d ) using Equation 1.12 . The fourth column is F1 F

horizontal axis will produce a 1:1 line. As a result, if the actual q d versus F1 F
χ d

n q

2

− −[ ( )], where F n ( q d ) is the data 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 1.15 shows the resulting q d versus
F1 F
χ d

n q

2

χ d

− −[ ( )], relationship. It is indeed close to the 1:1 line, indicating the multivariate
normal model is reasonable.

n q

2

1.4.4 Simulation of multivariate standard normal
random vector X
Given the correlation matrix C , random samples of multivariate standard normal (X 1 , X 2 ,
…, X d ) can be readily simulated by the following steps:

1. Simulate independent standard normal random vector Z = [Z 1 , Z 2 , …, Z d ] T .

Table 1.12 Sorted Q d data and the EDF
Sorted Q d data Rank k ECDF F n (q d ) from Equation 1.12 F[1F(q )] 2

− 1

−

nd

χ d

0.69 1 0.067 0.44

1.00 2 0.163 0.85

1.33 3 0.260 1.25

1.62 4 0.356 1.67

2.23 5 0.452 2.12

3.14 6 0.548 2.63

3.42 7 0.644 3.24

3.67 8 0.740 4.02

4.87 9 0.837 5.12

5.02 10 0.933 7.15