Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 34

Table 1.10 Simulated bivariate X data with n 12 = 10, n 13 = 11, and n 23 = 9
Bivariate dataset #1

Bivariate dataset #2 Bivariate dataset #3

k X 1 X 2 X 1 X 3 X 2 X 3

1 1.20 1.72

− 0.63 − 0.59 − 1.30 − 0.16

2 0.67 − 0.69 0.28 1.22 − 0.53 − 0.35

3 1.74 0.27 − 1.22 − 1.49 0.30 − 1.94

4 0.19 0.023

− 0.47 − 0.83 − 0.11 − 0.0038

5 1.68 0.81 2.08 0.020 − 1.58 − 1.87

6 − 1.28 0.17 − 0.0093 − 1.43 0.64 − 0.94

7 1.49 0.60

− 0.94 − 0.77 − 0.87 − 2.99

8 0.27 0.79 0.49 0.87

− 0.39 − 1.38

9 − 1.43 1.41 − 2.16 − 2.12 1.34 0.59

10 − 0.51 1.51 0.41 − 1.07

11 0.12

− 0.78

take values <0.677. This restriction is related to the concept of matrix positive definiteness.
The eigenspectrum of C contains only positive values if and only if C is positive-definite.
Namely, the C matrix in Equation 1.62 is not positive-definite. Indeed, it has a negative
eigenvalue of −0.2089. Positive definiteness can be guaranteed only if the correlation matrix
C is estimated from a full multivariate dataset (X 1 , X 2 , …, X d ) as shown in Table 1.9 (i.e.,
using Equation 1.57 ) and if n ≫ d . The C matrix estimated using the entry-by-entry bivari-
ate method in Equation 1.58 is not guaranteed to be positive-definite. Examples of produc-
ing nonpositive definite C based on actual data are shown in Section 1.7.3.
To illustrate the absurdity of the C matrix in Equation 1.62 , consider a random variable
Y = X 1 + X 2 - X 3 . It is common practice to encounter this linear sum, usually in the context
of a first-order Taylor series expansion of a nonlinear function. The variance of Y is equal
to

Var()= +++ − σσσ δσσδ σσ δσσ

2 2 2

2 2 − 2 =+ − 3 2 δ 2 δ − 2

2 δ=− .

0 4

1 2 3 12 12 13 13 23 23 12 13

(1.63)

where σ i = 1 is the standard deviation of Xi. i . Note that the variance of any random variable
is positive by definition. The nonpositive definite C matrix in Equation 1.62 can produce a
negative variance as shown in Equation 1.63 . Hence, positive definiteness is not an academic
concept that we can safely ignore in practice, notwithstanding the rather abstract nature of
this concept.

1.4.3.2 Goodness-of-fit test
Multivariate normality requires separate checks. For example, if the scatter plot of Xi i versus
X j shows a distinct nonlinear trend, then the multivariate normal distribution assumption
is not suitable. There are numerous formal tests for multivariate normality in the statis-
tics literature, but the state of practice is less established than formal tests for univariate
normality (e.g., K-S test). The first method is the generalization of the line test in Section
1.3.3. This method is applicable to nonstandard multivariate normal distribution with an
arbitrary dimension ( d ) and is based on the fact that the Mahalanobis distance Q d between