Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 26

1.3.3.4 Statistical uncertainties in the δ 12 estimate
The bootstrapping can be used to assess the sampling distribution of δ 12 . The steps for boot-
strapping are as follows:

1. Resampling the sample index with replacement. For the data in Equation 1.43 , there
are in total five data points ( n = 5). As a result, the sample index includes 1, 2, 3,
4, 5. Resampling these indices with replacement can be done by implementing the
MATLAB function randsample( n , n , 1). Note that after the resampling, there may be
repetitive indices.

2. Suppose the resampled indices are (5, 3, 1, 1, 2). The resampled samples are therefore

′ =

() 5 () 3 () 1 () 1 () 2

XXXXX
XXXXX

1 1 1 1 1

X T

() 5 () 3 () 1 () 1 () 2

2 2 2 2 2

(1.45)

168174 . . 120120 . . 067 .

=

− 051027 . . − 128128 . − . − 094 .

3. Estimate the sample Pearson correlation between the resampled (X 1 , X 2 ) using one of
the methods presented above. This is a resampled δ 12 value.

4. Repeat steps 1 and 2 to obtain B resampled δ 12 value. Note that B is not the same as n .

The B resampled δ 12 values can be viewed as approximate realizations of the sampling
distribution of δ 12 .
Now, consider the (X 1 , X 2 ) data in the left plot in Figure 1.12 . The actual value of δ 12
is 0.9. The sample value of δ 12 can be obtained from the methods of moments ( Equation
1.40 ). For n = 50, the estimated value of δ 12 is 0.937. For n = 200, the estimated value of
δ 12 is 0.912. However, these are point estimates of δ 12 , and it is not clear how large the
statistical uncertainties are. Figure 1.13 shows the histograms of 10,000 resampled δ 12 val-
ues for n = 50 and n = 200 based on the bootstrapping procedure (bootstrap sample size
B = 10,000). The 95% bootstrap confidence interval of δ 12 can be estimated as the interval
bounded by the 0.025 and 0.975 sample percentiles of the resampled δ 12 values. This is
called the percentile method (Efron 1981). It is clear that the confidence interval is narrower
when the number of data points gets larger.

2000 2000

Point estimate
= 0.912

n = 50

Point estimate
= 0.937

n = 200

1500 1500

1000 1000

500 500

0 0

0.8 0.85 0.9

Bootstrap samples of δ 12

0.95 1 0.8 0.85 0.9

Bootstrap samples of δ 12

0.95 1

Figure 1.13 Bootstrap samples of δ 12 . The arrows indicate the point estimates.