L-moments-based uncertainty quantification for scarce samples including extremes

Abstract

Sampling-based uncertainty quantification demands large data. Hence, when the available sample is scarce, it is customary to assume a distribution and estimate its moments from scarce data, to characterize the uncertainties. Nonetheless, inaccurate assumption about the distribution leads to flawed decisions. In addition, extremes, if present in the scarce data, are prone to be classified as outliers and neglected which leads to wrong estimation of the moments. Therefore, it is desirable to develop a method that is (i) distribution independent or allows distribution identification with scarce samples and (ii) accounts for the extremes in data and yet be insensitive or less sensitive to moments estimation. We propose using L-moments to develop a distribution-independent, robust moment estimation approach to characterize the uncertainty and propagate it through the system model. L-moment ratio diagram that uses higher order L-moments is adopted to choose the appropriate distribution, for uncertainty quantification. This allows for better characterization of the output distribution and the probabilistic estimates obtained using L-moments are found to be less sensitive to the extremes in the data, compared to the results obtained from the conventional moments approach. The efficacy of the proposed approach is demonstrated on conventional distributions covering all types of tails and several engineering examples. Engineering examples include a sheet metal manufacturing process, 7 variable speed reducer, and probabilistic fatigue life estimation.

Appendices

Appendix 1: Distribution selection based on L-moment

Based on generalized distribution identification procedure mentioned in Fig. 1, the step-by-step distribution selection process based on L-moment ratio diagram is provided in Algorithm 1.

Appendix 2: Statistical distributions

1.1 2.1 Shape parameter and tail heaviness

Generalized pareto distribution (GPD) is used to approximate the tails of any parent distribution. ψ, the scale, and ξ, the shape, parameters characterize the GPD fit (Ramu et al. 2010). ξ plays a significant role in quantifying the weight of the tail or tail heaviness. Probability distributions can be classified as light (ξ < 0), medium (ξ = 0), and heavy tail (ξ > 0) distributions based on the tail heaviness. Let G be the performance measure which is random and u is threshold of G. The observations of G that surpass u are called, exceedance, Z in the following (12).

$$ Z = G-u $$

(12)

The conditional CDF of exceedance is modeled by GPD and denoted by $\hat {F}{_{\xi ,\psi }(Z)}$. The distribution function of (G − u) is mentioned in (13).

$$ \hat{F}{{~}_{\xi,\psi}(Z)}=\begin{cases}1-\left( 1+\frac{\xi}{\psi}Z\right)^{-\frac{1}{\xi}} & \text{if} \xi \not= 0\\1-\exp\left( -\frac{Z}{\psi}\right) & \text{if} \xi = 0\end{cases} $$

(13)

ξ for exponential distribution is approximately 0. Similarly, ξ is lesser than 0 for light tail and greater than 0 for heavy tail. Figure 19 shows different CDF of tails approximated by GPD of the selected distributions.

Fig. 19

CDF and shape parameter values of selected distribution

1.2 2.2 Results—One extreme

In Tables 4, 5, 6, and 7, the quartile (Q) values of the D_JS are presented which are obtained for different sample sizes based on the Pearson system and L-moment ratio diagram. In all cases, it can be observed that L-moment approach works better.

Table 4 Quartile values of JS divergence - light tail

Table 5 Quartile values of JS divergence - medium tail

Table 6 Quartile values of JS divergence - heavy tail I

Table 7 Quartile values of JS divergence - heavy tail II

1.3 2.3 Results—Two extremes

Two extremes are incorporated in the samples in this study. 10⁶ samples from the parent distribution are generated and the maximum value and the value at 9.9 × 10⁵ is considered as extremes in the ordered data. In Tables 8, 9, 10, and 11, and Figs. 20, 21, 22, and 23, the results are presented for two extremes case. For all the sample sizes, L-moment approach work better than the C-moment approach.

Fig. 20

JS divergence for light tail

Fig. 21

JS divergence for medium tail

Fig. 22

JS divergence for heavy tail I

Fig. 23

JS divergence for heavy tail II

Table 8 Quartile values of JS divergence - light tail

Table 9 Quartile values of JS divergence - medium tail

Table 10 Quartile values of JS divergence - heavy tail I

Table 11 Quartile values of JS divergence - heavy tail II

Appendix 3: Flat rolling metal sheet manufacturing process

1.1 3.1 Distribution selection process based on L-moment

As mentioned in Algorithm 1, the step by step distribution selection process is demonstrated for this example. In this example, coefficient of friction (μ_f) is considered as random variable and follows the distribution mentioned in Fig. 9. Sample size considered to demonstrate the distribution selection process is 100.

• 1. Samples are generated and the response, maximum change in thickness (ΔH) is computed using (10).

• 2. L-moment ratios L-skewness (t₃) and L-kurtosis (t₃) are estimated and the values are 0.3, 0.19. Estimated sample L-moment ratios are plotted in L-moment ratio diagram and shown in Fig. 24.

• 3. Shortest distance is computed for all possible distributions in the L-moment ratio diagram and presented in Table 12.

• 4. Generalized pareto distribution is considered as the best possible distribution which has less distance compared to all other distributions.

• 5. Parameters of generalized pareto distribution obtained from the sample and corresponding PDF is plotted with the actual PDF (sample size for population is 10000), presented in Fig. 25.

• 6. Distributions obtained from sample is compared with the population distribution using D_JS. The D_JS value is 0.002.

In Tables 13 and 14, the quartile (Q) values of the D_JS are presented, which are obtained for different sample sizes. L-moment approach works well in all cases compared to the C-moment approach.

Fig. 24

L-moment ratio diagram with sample L-moment ratio

Fig. 25

PDF obtained with sample size 100

Table 12 Shortest distance of all possible distributions in L-moment ratio diagram

Table 13 Quartile values of JS divergence - uncertainty quantification of ΔH of flat rolling sheet metal manufacturing process

Table 14 Quartile values of JS divergence - uncertainty propagation of ΔH of flat rolling sheet metal manufacturing process

1.2 3.2 Results—Two extremes

In this section, results of the two extremes in the scarce data are presented. The D_JS obtained for various sample size are presented in Figs. 26 and 27 and the corresponding Q values are provided in the Tables 15 and 16. Similar observations are obtained as mentioned in one extreme case.

Fig. 26

JS divergence - uncertainty quantification

Fig. 27

JS divergence - uncertainty propagation

Table 15 Quartile values of JS divergence - uncertainty quantification of ΔH of flat rolling sheet metal manufacturing process

Table 16 Quartile values of JS divergence - uncertainty propagation of ΔH of flat rolling sheet metal manufacturing process

Appendix 4: Design of speed reducer

1.1 4.1 Results—One extreme

In this section, the quartile (Q) values of the D_JS are presented for different sample sizes. The Q values clearly indicate that L-moment approach are less affected by the sampling variability and works better than C-moment approach.

1.2 4.2 Results—Two extremes

D_JS results are presented in Figs. 28, 29, 30, and 31 and the corresponding Q values are presented in Tables 21, 22, 23, and 24. The results clearly show that the superiority of the L-moment approach over C-moment approach.

Fig. 28

JS divergence - extreme at 1 variable at a time

Fig. 29

JS divergence - extreme at 3 variables at a time

Fig. 30

JS divergence - extreme at 5 variables at a time

Fig. 31

JS divergence - extreme at 7 variables at a time

Table 17 Quartile values of JS divergence - extreme at 1 variable at a time

Table 18 Quartile values of JS divergence - extreme at 3 variable at a time

Table 19 Quartile values of JS divergence - extreme at 5 variables at a time

Table 20 Quartile values of JS divergence - extreme at 7 variables at a time

Table 21 Quartile values of JS divergence - extreme at 1 variable at a time

Table 22 Quartile values of JS divergence - extreme at 3 variables at a time

Table 23 Quartile values of JS divergence - extreme at 5 variables at a time

Table 24 Quartile values of JS divergence - extreme at 7 variables at a time

Appendix 5: Probabilistic fatigue life assessment

In this section, probabilistic fatigue life assessment for different sample sizes are presented. μ + 0.5σ of fatigue life is considered the upper bound and the minimum value of ordered experimental data is considered as the lower bound for scarce sample selection. Samples are randomly drawn between the upper and lower bound of the ordered experimental data for all load cases. The survival % comparison results are presented in Fig. 32. The confidence bounds of 90th and 95th survival % are provided in Tables 25, 26, 27, and 28. In all cases, L-moments approach predicts the probability of fatigue life closer to the actual for different loading conditions.

Fig. 32

Comparison of 95th survival % of sample distribution to parent distribution with different sample sizes

Table 25 Survival % of fatigue life at different load cases and confidence bounds for sample size 10

Table 26 Survival % of fatigue life at different load cases and confidence bounds for sample size 15

Table 27 Survival % of fatigue life at different load cases and confidence bounds for sample size 20

Table 28 Survival % of fatigue life at different load cases and confidence bounds for sample size 25