Environmental Engineering Reference
In-Depth Information
8
7
6
5
4
3
2
1
0
0
2
4
6
8
F
-1
[F
n
(
q
d
)]
χ
d
Figure 1.15
Mahalanobis distance test:
q
d
versus F1F
χ
d
)] plot. The dashed line is the 1:1 line.
2
−
−
[
(
q
nd
2. Let
u
matrix be the Cholesky decomposition of
C:
(1.70)
uuC
T
×=
In MATLAB,
u
= chol(
C
).
3. Let
(1.71)
XZu
T
=×
It is noteworthy that the above steps break down if
C
is not positive-definite, because
the Cholesky decomposition
u
will contain complex numbers if
C
is not positive-definite.
This is analogous to producing a complex number if you take the square root of a negative
number.
1.4.5 Conditional normal and updating
Using a multivariate normal distribution, it is possible to update the marginal distribution
of any one parameter or the multivariate distribution of any group of parameters given
information from other parameters. This updated distribution is called the conditional dis-
tribution. To illustrate the conditioning, let us consider the following example with four soil
parameters:
s
u
(undrained shear strength),
σ
p
(preconsolidation stress), N (SPT-N value),
and
q
t
-σ
v
(net cone tip resistance). These four parameters are correlated.
To begin with, let us assume they are normally distributed with mean values and COVs
given in
Table 1.13
. As a result,
′
s
µσ
σµσ
µσ
σµσ
=+×
′ =+×
=+×
−=+×
X
X
u
1
1
1
p
2
2
2
(1.72)
N
X
3
3
3
q
X
t
v
4
4
4
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