Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 63

Table 1.18 Distribution type and distribution parameters for (Y 1 , Y 2 , …, Y 10 )

Soil parameter
or its log
transform

Distribution parameters

Random
variable a X b X a Y b Y

Y 1 ln(LL) SU 1.636 − 1.166 0.616 3.479 5.7e-07
Y 2 ln(PI) SU 1.433 − 0.265 0.918 3.178 3.0e-05
Y 3 LI SU 1.434 − 1.068 0.629 0.358 1.2e-07
Y 4 ln( σ′ v / P a ) SB 3.150 0.256 14.458 − 7.010 0.40
Y 5 ln( σ′ p / P a ) SB 4.600 21.548 576.785 − 4.793 0.16
Y 6 ln( s u / σ′ v ) SU 2.039 − 0.517 1.427 − 1.461 2.9e-09
Y 7 ln( S t ) SU 2.393 − 2.080 1.885 0.461 7.1e-14
Y 8 B q SU 2.676 0.161 0.513 0.615 0.31
Y 9 ln[( q t − σ v )/ σ′ v ] SU 1.340 − 0.572 0.659 1.476 0.53
Y 10 ln[( q t - u 2 )/ σ′ v ] SU 2.134 − 1.102 1.154 0.657 0.57
Source: Ching, J. and Phoon, K.K. 2014b. Canadian Geotechnical Journal , 51(6), 686-704, reproduced with permission of the
NRC Research Press.

Distribution type p-Value

1.7.2.3 Compute the correlation matrix for (X 1 , X 2 , …, X 10 )
Figure 1.33 presents the bivariate correlation structure underlying the 10 soil parameters
after they have been transformed into standard normal random variables using Equation
1.95 . As mentioned previously, there are 45 possible bivariate correlations for a database
containing d = 10 parameters. The simplest method to quantify the bivariate correlation
between X i and X j is to compute the Pearson correlation coefficients δ ij using Equation 1.58 .
Here, a simplified version is used:

n

ij

1

∑

() k () k

(1.111)

δ ij ≈ XX ⋅

n

i j

ij

k = 1

where n ij is the total number of bivariate (Xi, i , X j ) data points. This simplified version is based
on the fact that the mean and standard deviation of Xi i are equal to 0 and 1, respectively,
because X i is standard normal. Therefore,

COVX X (, ) ( XX µµ

σσ

) −

i j i j i j

(1.112)

δ = = = EXX ( )

ij

σσ ⋅ ⋅

i j

i j i j

It is useful to recollect that this additional step of converting a non-normal variable Y into
a standard normal variable X is necessary if one were to exploit the multivariate normal
distribution to couple individual components together in a consistent way. The bivariate cor-
relation structure presented in Figure 1.33 is sufficient to fully characterize a multivariate
probability distribution only if the multivariate normal hypothesis is true.

1.7.2.4 Problem of nonpositive definiteness
The bootstrapping technique (Efron and Tibshirani 1993) introduced in Section 1.3.3 is
applied to obtain 1000 bootstrap samples of δ ij . Figure 1.34 shows the histograms of the
1000 δ ij estimates for the X 1 − X 2 and X 7 − X 9 pairs, namely δ 12 and δ 79 . The 90% confidence