Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 33

Table 1.9 Simulated full multivariate X data with d = 3 and n = 10
k

X 1 X 2 X 3

1 1.20 0.35 − 0.63

2 0.67 − 0.69 0.40

3 1.74 0.27 0.94

4 -0.47 0.19 − 0.31

5 1.68 1.17

− 1.58

6 -1.28 0.17

− 1.08

7 -0.94 1.49 − 0.51

8 0.27 0.79 1.02

9 -1.43 1.41 − 1.92

10 -0.51 1.51 0.66

matrix is entered as C = [1 δ 12 ; δ 12 1] = [1 −0.570; −0.570 1]. The upper triangle Cholesky
matrix is computed as u = chol( C ) and two columns of correlated standard normal data of
(X 1 , X 2 ) are obtained from X T = Z T × u . The resulting dataset is called Bivariate dataset #1.
This procedure is subsequently repeated for n 13 = 11 and C = [1 δ 13 ; δ 13 1], and for n 23 = 9
and C = [1 δ 23 ; δ 23 1]. The resulting datasets are called Bivariate datasets #2 and #3. These
three bivariate datasets are shown in Table 1.10 . The data in Table 1.10 do not have full
multivariate information because (X 1 , X 2 , X 3 ) are not simultaneously known. For Bivariate
dataset #1, only (X 1 , X 2 ) are simultaneously known. This mimics the reality in geotechni-
cal literature when bivariate correlation datasets (e.g., simultaneously known OCR and s u )
are abundant, but full multivariate datasets are rare (e.g., simultaneously known OCR, s u ,
and S t ). The second method applies Equation 1.58 to estimate (1) δ 12 using Bivariate dataset
#1, (2) δ 13 using Bivariate dataset #2, and (3) δ 23 using Bivariate dataset #3. The resulting C
estimate is

1 − 0 713 . 0 706 .

C ≈ − 0 713 . 1 − 0 214 .

(1.61)

0 706 . − 0 214 . 1

1.4.3.1 Positive definiteness of the correlation matrix C
Let us consider a case with d = 3: there are three random variables X 1 , X 2 , and X 3 , and there
are three correlation coefficients δ 12 , δ 13 , and δ 23 . It is not possible for δ 12 , δ 13 , and δ 23 to take
arbitrary values between −1 and 1. Consider the following correlation matrix:

10108
01 110
08 10

. .

C = . .

(1.62)

. . 1

Note that δ 12 = 0.1 and δ 13 = 0.8. In other words, X 1 and X 2 are poorly correlated, whereas
X 1 and X 3 are highly correlated. It is obvious that δ 23 = 1.0 is absurd. If δ 23 were indeed 1.0,
X 2 and X 3 would have been perfectly correlated, then X 1 should have been highly correlated
to X 2 . This contradicts the fact that X 1 and X 2 are poorly correlated. In fact, δ 23 can only