Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 59

where δ Y1Y2 is the Pearson correlation between (Y 1 , Y 2 ); δ 12 is the Pearson correlation between
(X 1 , X 2 ); and f ( x 1 , x 2 ; δ 12 ) is the bivariate standard normal PDF defined in Equation 1.37 :

T − 1

x
x

1 δ x

x

( )

( ) =

1 12 1

1 2

f xx , ; δ exp. − 05 2 π ×− 1 δ

(1.109) 1212

δ 1

2 12 2

and μ Y i and σ Y i are the mean and standard deviation of Yi, i , respectively:

∞ ∞

∫

( ) × ()

2

∫

()

× ()

()

− 1 − 1

−

µ = F Φ x ϕ x d x σ = F Φ x µ ϕ xx d (1.110)

Y i Y i Y

i i i

−∞ − ∞

ϕ( x ) is the univariate standard normal PDF ( Equation 1.3 ) .
This relation between δ Y1Y2 and δ 12 is plotted on the left plot in Figure 1.32 . When evalu-
ating the CDF transform F

Φ x the parameters ( a X1 , b X1 , a Y1 , b Y1 ) in Table 1.15 are
used. The same approach is followed for evaluating F

1 − ()

[ ],

1

Φ x It is clear that δ Y1Y2 and δ 12
are not identical. Moreover, even though δ 12 spans the full range −1.0-1.0, δ Y1Y2 only spans
a nonfull range of −0.91-0.98. This gets even worse if we reduce a X1 and a X2 in Table 1.15
by a factor of 2 ( a X1 = 1.491/2 = 0.754; a X2 = 2.506/2 = 1.253). By doing so, the standard
deviations of (Y 1 , Y 2 ) will increase. The resulting relation between δ Y1Y2 and δ 12 is plotted on
the right plot in Figure 1.32 . It is seen that the lower bound for δ Y1Y2 is now only −0.4! The
inability of the CDF transform approach to reproduce strong negative correlations among
physical variables is arguably the most critical weakness of this approach.

2 − ()

[ ].

2

1.7 real eXaMPle

In this section, the construction of multivariate probability distributions of soil param-
eters will be demonstrated using the Clay/10/7490 database compiled by Ching and Phoon

(a) 1 (b) 1

a X1 and a X2
reduced by factor of 2

0.5 0.5

0 0

-0.5 -0.5

-1 -1

-1 -0.5 0

δ 12

0.5 1 -1 -0.5 0

δ 12

0.5 1

Figure 1.32 Relation between δ Y1Y2 and δ 12 (a) for parameters given in Table 1.15 ; and (b) a X1 and a X2 reduced
by a factor of 2. The dashed lines are the 1:1 lines.