Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 56

200 200 200

p -value = 0.98 p -value = 0.79 p -value = 0.40

150 150 150

100 100 100

50 50 50

0 0 0

-4 -2 0

X 1

2 4 -4 -2 0

X 2

2 4 -4 -2 0

X 3

2 4

Figure 1.27 Histograms of the converted ( X 1 , X 2 , X 3 ) data. Solid lines are the standard normal PDF.

1 − 0 599 . 0 645 .

ˆ

C = − 0 599 . 1 − 0 049 . (1.104)

0 645 . − 0 049 . 1

As discussed earlier, the Pearson product-moment correlations among (X 1 , X 2 , X 3 ) are
nearly the same as the Spearman rank correlations among (Y 1 , Y 2 , Y 3 ). Hence, the C matrix
can be also estimated as the Spearman rank correlations among (Y 1 , Y 2 , Y 3 ). This can be
easily done in MATLAB using C = corr( y , 'type', 'Spearman'). The resulting C matrix is

1 − 0 571 . 0 630 .

ˆ

C = − 0 571 . 1 − 0 045 . (1.105)

0 630 . − 0 045 . 1

When only bivariate data are available, Equation 1.58 should be used in an entry-by-
entry manner. This entry-by-entry approach will be demonstrated in Section 1.7 for the
Clay/10/7490 database.

4 4 4

δ 12 = -0.599 δ 13 = -0.645 δ 23 = -0.049

2 2 2

0 0 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

X 1 X 1 X 2

Figure 1.28 Transformed ( X 1 , X 2 , X 3 ) data points.