Environmental Engineering Reference
In-Depth Information
The correlation matrix
C
can be partitioned as
1090507
09 10406
05 04 104
07 06 04
.
.
.
.
.
.
C
C
CC
[11]
[12]
(1.76)
C
=
=
.
.
.
[21]
[22]
.
.
.
1
It can be proved that the conditional distribution of
X
[1]
given
X
[2]
is a multivariate normal
distribution with the following updated mean vector and covariance matrix:
−
1
=
05 07
04 06
.
.
×
104
04
.
−
−
1 111
0 667
.
.
=
−
−
069
056
.
.
×
(
)
−1
[1]
µ
update
=
CC
[12]
[22]
×
X
[2]
×
.
.
.
1
×
(
)
−1
[11]
[11]
[12]
[22]
[21]
(1.77)
C
update
=
CCC C
−
×
C
−
104
04
1
109
09
.
05 07
04 06
.
.
.
05 04
06 07
.
.
045044
044061
.
.
=
−
×
×
=
.
1
.
.
.
1
.
.
.
.
The conditional mean value for X
1
is −0.69 and the conditional variance is 0.45. The con-
ditional mean and variance of
s
u
can be calculated using the relationship s
u
= 200 + 40X
1
: the
conditional mean value for
s
u
= 200 + 40 × (- 0.69) = 172.49 kPa, and the conditional variance
for
s
u
= 40
2
× 0.45 = 723.81 kPa
2
. The conditional COV is therefore 723.81
0.5
/172.49 = 0.16.
agreement with the histogram of the conditional
s
u
samples. The conditional mean, vari-
ance, and COV for
σ
p
can also be calculated in a similar way.
The conditional bivariate distribution for (, )
′
s
u
σ
p
′
can also be obtained. Recall that
(, )
s
u
σ
are related to (X
1
, X
2
) through the following equation:
s
200
2500
40
0
X
X
=
+
×
u
′
1
(1.78)
σ
0
750
2
The conditional mean vector for
(, )
s
u
σ
is therefore
µ
µ
σ
200
2500
40
0
172 48
710 26
.
.
=
+
µ=
s
u
,
update
[1]
µ
update
=
×
(1.79)
update
0
750
,
update
′
p
The conditional covariance matrix for
(, )
s
u
′σ
is therefore
40
0
40
0
723 81
.
2803
.
81
=
×
×
=
[11]
C
C
updat
(1.80)
update
e
0
750
0
750
2803 81
.
15603 81
.
This conditional bivariate distribution can be evaluated using
Equation 1.36
. Its contours
are plotted in
Figure 1.18
. The left plot shows the contours for the unconditional PDF for
(, )
σ , whereas the right plot shows the contours for the conditional PDF for (, )
σ . Shown
s
u
s
u
together with the contours are the unconditional samples of (, )
σ (left plot) and the condi-
s
u
tional samples of (, )
σ (right plot).
s
u
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