Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 31

1.4.2 Multivariate normal distribution
The multivariate normal distribution can be easily understood as a generalization of the
bivariate normal distribution. Let us denote the d × 1 vector [Y 1 Y 2 … Y d ] T by y . It has a
mean vector μ = [μ 1 μ 2 … μ d ] T and a covariance matrix C :

σ

2

δ σσ δσσ

1 12 12 11 d d

σ

2

δ σσ

2 22 d d

C =

(1.53)

symmetric σ

2

d

The PDF for the multivariate normal distribution is

1 −− − ( y µ )

T

C

− 1

( y µ )

f () y = exp

(1.54)

d

2

2 π ⋅ C

If [X 1 X 2 … X d ] T are multivariate standard normal, the multivariate standard normal
distribution is simplified into

1 − xx

T

C

− 1

f () x = exp

(1.55)

d

2

2 π ⋅ C

where C is the correlation matrix:

1 δ …

…

δ
δ

12 1 d

1

2 d

C = (1.56)

sym. 1

It is important to note that C must satisfy a matrix property called positive definite-
ness. A positive-definite matrix is like a positive number. You can find the square root of
a positive number. In a similar way, you can find the Cholesky factor of a positive-definite
matrix. This requirement was not discussed in the bivariate case, because it is automati-
cally satisfied when δ 1 2 < 1. However, for the multivariate case, C may not be positive-
definite even if all δ ij lie between −1 and 1. This is a critical difference between the bivariate
and the multivariate model.

1.4.3 estimation of correlation matrix C
The correlation coefficients δ ij in the matrix C can be estimated using two methods:

a. Full multivariate manner based on a full multivariate dataset (X 1 , X 2 , …, X d ):