Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 29

Table 1.8 Sorted Q data and the ECDF
Sorted Q data Rank k ECDF F n (q) from Equation 1.12

0.62 1 0.067 0.14

0.72 2 0.164 0.36

0.96 3 0.260 0.60

0.97 4 0.356 0.88

1.33 5 0.452 1.20

1.57 6 0.548 1.59

1.69 7 0.644 2.07

3.12 8 0.740 2.70

3.23 9 0.837 3.62

3.78 10 0.933 5.40

2. Let

XZ
X

=
=⋅ +−⋅

1 1

δ Z 1 δ

1 2

Z (1.49)

2 12 1 2

It is easy to verify that both X 1 and X 2 are still standard normal. Unlike Z 1 and Z 2
being independent, X 1 and X 2 are correlated with Pearson correlation = δ 12 .

Another way of writing Equation 1.49 is

XZu

T =×

(1.50)

where

1
01

δ

X Z

=

12

1 1

X = Z u = (1.51)

X Z

− δ

1 2

2 2

Note that

1 δ

T ×==

12

uuC (1.52)

δ 1

12

The u matrix is called the upper Cholesky factor of C . The Cholesky factor can
be viewed as the “square-root” of a matrix. In MATLAB, the function 'chol' can be
used to compute u : u = chol( C ). Figure 1.12 shows the simulated samples of (X 1 , X 2 )
with mean values = 0, standard deviations = 1, and with the Pearson correlation = δ 12
(δ 12 = 0.9 and 0.0).

1.4 MultIVarIate norMal VeCtor

1.4.1 Multivariate data
Multivariate data can be understood as a generalization of bivariate data to higher dimen-
sions. We recall that bivariate data occupy two columns in EXCEL. Multivariate data