Environmental Engineering Reference
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350
90%
confidence
interval
300
90%
confidence
interval
250
200
Median
150
5%
percentile
95%
percentile
100
Histogram of δ
12
50
Histogram of δ
79
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
δ
ij
Figure 1.34
Histograms of the bootstrap
δ
ij
samples. (From Ching, J. and Phoon, K.K. 2014b.
Canadian
Geotechnical Journal
, 51(6), 686-704, reproduced with permission of the NRC Research Press.)
interval for δ
12
is significantly narrower than that for δ
79
probably because there are more
data points in the X
1
− X
2
pair (
n
12
= 3822) than in the X
7
− X
9
pair (
n
79
= 230). On the
basis of δ
ij
samples, the 90% confidence intervals (5% and 95% percentiles) and the median
the 5% and 95% percentiles) and the median of the δ
ij
estimates (median values are in the
parentheses).
The correlation matrix
C
presented in
Table 1.19
,
which is formed from the median
values of δ
ij
, may not be positive-definite, because δ
ij
is estimated
independently
of other
entries in
Table 1.19
.
There is one negative eigenvalue in the
C
matrix based on the median
values of δ
ij
.
The choice of using the median value as a point estimate of each entry in
C
is a matter
of convenience. The median value is not the only possible value of δ
ij
, given the statistical
uncertainty shown in
Figure 1.34
. Ching and Phoon (2014b) formed a positive-definite
C
matrix using the following steps:
1. For each bivariate correlation, a bootstrap sample of δ
ij
is obtained. There are
d
(
d
− 1)/2 = 45 possible bivariate correlations; so, this step is conducted for 45 times.
2. Determine whether the resulting
C
matrix is positive-definite by checking its eigenval-
ues. If
C
is positive-definite, it is accepted (and the δ
ij
values are accepted). Otherwise,
C
is rejected.
3. The above steps are repeated until 1000
C
matrices are accepted. The ratio of the
acceptance is about 16% in this example.
4. The final matrix is obtained by averaging the 1000 accepted
C
matrices. This matrix
is shown in
Table 1.20
.
Table 1.20
happens to be similar to
Table 1.19
in this example.
However,
Table 1.20
is positive-definite, because it can be mathematically proved that
the average of positive-definite matrices is also positive-definite.
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