PSI - Issue 5

Jing Zhang et al. / Procedia Structural Integrity 5 (2017) 1176–1183 Xia Yang et al. / Structural Integrity Procedia 00 (2017) 000 – 000

1181

6

360

0.0228 0.0220 0.0220

0.0209 0.0188 0.0184

0.0241 0.0161 0.0156

0.0311 0.0149 0.0129

0.0493 0.0184 0.0121

0.1060 0.0281 0.0137

3600

18000

Note: where ‘–‘ denotes that data is too few to be used, same denotation is adopted in the following table.

Table 4. Estimation results of 100-yearly EV (Theoretical value: E =68.44 ) Sampling time/d 50 52

54

56

58

( μ , σ )

(78.55,22.50) (67.22,130.81) (76.98,8.75) (73.26,80.55) (76.81,8.49) (74.32,79.65) (76.69,8.26) (75.79,77.51) (76.75,8.31) (76.37,77.08)

(82. 20,29.05) (62.86,162.02) (75.80,9.62) (70.39,83.47) (74.43,6.25) (71.43,77.40) (74.40,6.00) (73.22,75.52) (74.38,5.94) (73.90,74.78)

(75.02,20.11) (58.31,93.66) (74.71,8.28) (67.01,89.06) (73.89,7.48) (68.19,87.76) (72.69,4.31) (71.44,73.97) (72.62,4.19) (71.92,73.97)

- -

- - - -

30

95%CI

( μ , σ )

(72.98,7.74) (64.20,86.78) (71.94,5.95) (64.74,81.89) (71.37,3.18) (69.20,73.83) (71.25,2.85) (70.43,72.01)

180

95%CI ( , )  

(69.31,5.20) (60.42,79.85) (70.91,3.37) (67.83,76.59) (70.15,1.82) (69.06,71.50)

360

95%CI

( μ , σ )

3600

95%CI

( μ , σ )

18000

95%CI

Note: where μ, σ denote the mean value and standard deviation of characteristic values. From Table 4, it can be found that EV estimation depends on the threshold and sample size, and the threshold selection involves a trade-off between bias and variance. In order to evaluate the characteristic value and threshold, the MSE criterion is applied as

2 ) MSE bias E      2 (

(4)

where μ , E , σ 2 denote the mean value, the theoretical value and the variance of characteristic value. The threshold achieves optimal when MSE reach the minimum. Moreover, it is assumed that relative error less than 10% is within acceptance range. For the first parent distribution, the 95% CI of characteristic value in the interval [61.60, 75.28] is allowed. Based on Table 2 to 4, some conclusions are drawn as follows: generally, the estimated EV based on a GPD is larger than the theoretical value. It is not advisable to estimate the EV based on one day ’ s data. When the sampling time is less than one year, the estimates are not satisfied. In those cases, the threshold should be as large as possible. When the sampling time is ten years and the threshold is within the interval of [54, 56], the characteristic value can be well estimated. And it achieves the best when the threshold is 56. When the sampling time is fifty years, the estimate at the threshold of 58 is optimal. The variation of estimates with threshold of the other two parent distributions is similar to that of the first.

3.3. Threshold estimation

For each group of samples, the thresholds are estimated by the MRL plot, Hill plot and MSE method. The mean values of thresholds of the samples with the same sampling length are listed in Table 5.

Table 5. Estimated thresholds of the samples produced from the first parent distribution

Sampling time/d

Method

30

180

360

3600

18000