Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 62

Table 1.17 Numbers of data points with bivar i ate information
Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 Y 9 Y 10 OCR
Y 1 3822 3822 3412 2084 1362 1835 1184 680 618 541 1475
Y 2 4265 3424 2169 1433 2173 1203 688 626 549 1745
Y 3 Index properties 3661 1999 1314 1709 1279 660 598 521 1388
Y 4 3370 1944 2419 853 965 862 668 1959
Y 5 2028 1423 554 780 691 543 1984
Y 6 3532 715 595 533 525 2120
Y 7 Stresses and strengths 1589 240 230 190 586
Y 8 1016 862 668 832
Y 9 Symmetry 862 590 692
Y 10 CPTU parameters 668 544
OCR 3531
Source: Ching, J. and Phoon, K.K. 2014b. Canadian Geotechnical Journal , 51(6), 686-704, reproduced with permission of the
NRC Research Press.
1.7.2 Construction of multivariate distribution
The multivariate non-normal distribution for (Y 1 , Y 2 , …, Y 10 ) is constructed using the
approach discussed in Section 1.6: (1) Fit a Johnson distribution to each component (Yi), i ), (2)
convert Y i into standard normal Xi i , and (3) compute the correlation matrix for (X 1 , X 2 , …,
X 10 ). The key challenge is to compute a valid positive-definite correlation matrix in step (3).
As discussed in Section 1.4.3, the correlation matrix is guaranteed to be at least semiposi-
tive definite if it is estimated from multivariate data ( Equation 1.57 ) . However, multivariate
data are rare in geotechnical engineering. The practical approach is to estimate each entry
in the correlation separately from bivariate data ( Equation 1.58 ) . The downside is that such
a piecemeal approach will not guarantee a positive-definite correlation matrix. This rather
abstract theoretical property cannot be dismissed without running the risk of producing
completely absurd answers such as a negative variance as shown in Section 1.4.3. This sec-
tion demonstrates how a correlation matrix can be estimated from actual bivariate data
while preserving the critical property of positive definiteness.
1.7.2.1 Fit a Johnson distribution to each component (Yi) i )
Each component (Y i ) can be fitted to a Johnson distribution using the procedures introduced
in Section 1.6.3. The distribution type and parameters are summarized in Table 1.18 .
1.7.2.2 Convert Y i into standard normal Xi i
It is easy to transform the soil parameters into standard normal random variables (X 1 , X 2 ,
…, X 10 ) using Equation 1.94 (or Equation 1.95 ) and the distribution type/parameters from
Table 1.18 . The normality of this transformed data can be checked using a probability plot
or the K-S test. Using MATLAB function kstest, the p -values associated with the K-S test
for (X 1 , X 2 , …, X 10 ) can be computed, as listed in Table 1.18 (the right-most column). There
are five p -values <0.05 (X 1 , X 2 , X 3 , X 6 , X 7 ), indicating that there is sufficient evidence to
reject the null hypothesis that Xi i is a standard normal random variable at a level of sig-
nificance of 5%. However, the Johnson distribution is still adopted in the ensuing analysis
because of its analytical elegance (discussed later).