Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 67

Table 1.20 A verage of 1000 positive-de i nite matrices obtained from bootstrap samples
X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 10
X 1 1.00 0.91 − 0.25 − 0.24 − 0.30 0.10 − 0.21 0.09 0.09 0.07
X 2 1.00 − 0.32 − 0.21 − 0.27 0.04 − 0.25 0.11 0.00 − 0.01
X 3 Index properties 1.00 − 0.49 − 0.57 0.01 0.59 − 0.05 0.06 − 0.05
X 4 1.00 0.72 − 0.50 0.00 0.20 − 0.38 − 0.32
C = X 5 1.00 0.01 0.06 − 0.03 0.11 0.04
X 6 1.00 0.18 − 0.24 0.73 0.63
X 7 Stresses and strengths 1.00 0.18 0.15 -0.08
X 8 1.00 − 0.45 -0.63
X 9 Symmetry 1.00 0.74
X 10 CPTU parameters 1.00
Source: Ching, J. and Phoon, K.K. 2014b. Canadian Geotechnical Journal , 51(6), 686-704, reproduced with permission of the
NRC Research Press.

is a realistic example as Y 8 and Y 10 are routinely measured simultaneously in piezocone
surroundings.
The solution requires the distribution of a six-dimensional random vector. In standard
normal space, this random vector can be partitioned into two subvectors:

X
X

5

6

X

=

X []
[]

1

3

X = (1.113)

X
X
X

X

2

4

8

10

The correlation matrix for the six-dimensional random vector (which is simply a subma-
trix extracted from Table 1.20 ) is partitioned as shown below to make explicit the correlation
matrix for each subvector (C [11] and C [22] ) and the cross-correlation matrix (C [12] = transpose
of C [21] ):

XX XXXX

5 6 3 4 8 10

X
X
CX
X
X
X

1 . 01 − 057072 . − . − 003004 . .

5

00110 . . 01 − 050024 . − . 063 .

6

= − 057001 . . 1 − 0 49 . − 005005 . − .

=

CC
CC

[] 11 [] 12

(1.114) 3

− 072050 . − . − 049 . 11 . 20 − 032 .

[] 21 [] 22

4

− 003024 . − . − 005020 . . 1 − 0 63 .

8

004063 . . − 005032 . − . − 06 . 331

10

The conditional distribution of X [1] given X [2] is a bivariate normal distribution with the
following mean vector and covariance matrix:

− 0 228 . 0 691 . − 0 035 . 0 230 .

( )

− 1

[] 1

µ update = CC

[] 12 [] 22

X

[] 2

= X

[] 2

− 0 152 . − 0 419 . 0 233 . 0 634 .

(1.115)

0 359 . 0 204 .

=

( ) −

1

[] 11

C = C

[] 11

− C

[] 12

C

[] 22

C []

21

update

0 204 . 0 451 .