Environmental Engineering Reference
In-Depth Information
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
of δ ij can be identified. Table 1.19 shows the 90% confidence intervals (range bounded by
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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