3.8 Sandia Challenge Problem
3.8.2 Problem 2
Table 3.8 reports the individual and overall sensitivity indices (to one decimal place) of the following quantities: (1) parameter uncertainty in b; (2) variability in b; (3) total uncertainty in b; and (4) uncertainty in a. It is seen that there is little interaction between variability and parameter uncertainty of b. The contribution of distribution parameter uncertainty is about 10%, and the variability in b contributes to about 60% of the overall variance; while the former can be reduced, the latter is irreducible uncertainty.
distribution for the variables a and b respectively. Then, 1−wa and 1−wb denote the weights assigned to the uniform distribution for the variablesaand brespectively.
For the sake of illustration, the prior distribution for all the distribution parameters is chosen to be uniform on the interval [0, 1].
The PDFs of all the above quantities are estimated using the available data, and the mean values, the standard deviations, maximum likelihood estimates, and 95%
bounds are shown in Table 3.9.
Table 3.9: Bayesian Model Averaging: Results
Variable Distribution Quantity Mean Standard 95% Bounds
Type Deviation
a
Normal
wa 0.36 0.23 [0.04,0.76]
µa 0.50 0.30 [0.03, 0.93]
σa 0.20 0.25 [0.03, 0.75]
Uniform
1−wa 0.64 0.23 [0.01,0.96]
La 0.73 0.17 [0.42, 0.91]
Ua 0.86 0.13 [0.61, 0.99]
b
Normal
wb 0.58 0.29 [0.05, 0.97]
µb 0.57 0.15 [0.23, 0.84]
σb 0.35 0.53 [0.01, 1.34]
Uniform
1−wb 0.42 0.29 [0.03, 0.95]
Lb 0.37 0.19 [0.04, 0.62]
Ub 0.67 0.25 [0.29, 0.95]
The results in Table 3.9 are difficult to interpret for a number of reasons, the primary reason being that all the estimates have very high degree of uncertainty (indicated by standard deviation). This happens because the method tries to estimate 5 parameters simultaneously using a small data set (3 for a and 4 forb). As a result, the 95% bounds are too large to be useful. The PDFs of the weights wa and wb are almost uniform, suggesting that even the maximum likelihood estimates may not be useful. Also, consider the uniform distribution estimated for a; the estimates of the lower and higher bounds are so close (but with high standard deviations) that it is difficult to derive any usefulness from such results.
Table 3.10: Bayesian Hypothesis Testing Results
Variable P(M|D) Quantity Mean Standard Deviation 95% Bounds a
Normal µa 0.57 0.16 [0.16,0.89]
0.32 σa 0.23 0.20 [0.01, 0.80]
Uniform La 0.41 0.16 [0.05, 0.65]
0.68 Ua 0.74 0.12 [0.54,0.97]
b
Normal µb 0.60 0.12 [0.30, 0.89]
0.28 σb 0.17 0.15 [0.01, 0.64]
Uniform Lb 0.43 0.15 [0.08, 0.60]
0.72 Ub 0.74 0.11 [0.60, 0.97]
Due to the large uncertainty in the input, further uncertainty propagation analysis is not useful. Instead, the Bayesian hypothesis testing approach is investigated next.
Note that the hypothesis testing approach does not estimate more than 2 parameters simultaneously, and hence is expected to produce results that have less uncertainty.
3.8.2.2 Bayesian Hypothesis Testing
Using the Bayesian hypothesis testing approach proposed in Section 3.6.2, the probabilities P(M1|D) and P(M2|D) can be directly calculated for both a and b.
Then, the PDFs of the distribution parameters (µa and σa for normal, and La and Ua for uniform) can also be calculated. The results of the distribution parameter estimation are shown in Table 3.10. Note that the estimation of the parameters of the normal distribution is totally independent of the estimation of the parameters of the uniform distribution, for both the variables a and b. However, this was not the case in the Bayesian model averaging approach.
Once the uncertainty in the model form and the distribution parameters are es- timated, then a and b are represented using a single, unconditional PDF each. The calculation of this unconditional PDF is using the simultaneous sampling approach
explained earlier in Section 3.6.3.2. This single PDF accounts for physical variabil- ity, distribution type uncertainty and distribution parameter uncertainty, and hence renders the uncertainty propagation analysis efficient.
The unconditional PDF of a and the unconditional PDF ofb is then propagated through Eq. 3.31 to calculate the PDF of Y. The PDFs of a and b are shown in Fig. 3.23.
−0.50 0 0.5 1 1.5
0.5 1 1.5 2 2.5
a
(a) PDF ofa
0 0.2 0.4 0.6 0.8 1
0 0.5 1 1.5 2 2.5 3 3.5
b
(b) PDF ofb
Figure 3.23: PDFs of Model Inputs
Using simple Monte Carlo simulation, the PDF of y is then calculated and shown in Fig 3.24. Since this PDF is calculated from multiple competing parametric PDFs, it is indicated as the parametric PDF. It accounts for all sources of uncertainty in the inputs - physical variability, distribution type uncertainty and distribution parameter uncertainty.
The computational method developed in Section 3.6.2.4 is then used to quantify the individual contributions of aleatory (physical variability) and epistemic uncer- tainty (distribution type and parameter uncertainty). Recall that it is not meaningful to calculate the effect of distribution parameter uncertainty alone, because the choice of distribution parameters is made only after selecting the distribution type. The individual and overall effects for all the three quantities a, b, and y are tabulated in Table 3.11.
0.5 1 1.5 2 0
1 2 3 4 5
Outputy
Parametric Non-Parametric
Figure 3.24: PDF of Model Output y
Table 3.11: Contributions of Physical Variability and Epistemic Uncertainty Quantity Effect Physical Epistemic Distribution
Variability Uncertainty Type
a Individual 31.5% 20.8% 0.0%
Overall 79.2% 68.5% 43.3%
b Individual 23.7% 36.5% 0.0%
Overall 63.5% 77.3% 9.9%
y Individual 4.2% 4.1% 1.0%
Overall 95.9% 95.8% 56.0%
It is seen from the results that the individual effect of distribution type uncertainty is almost zero; however it has a significant contribution together with distribution parameter uncertainty and physical variability. Also, there is significant interaction between the three sources of uncertainty - physical variability, distribution type and parameter uncertainties.
3.8.2.3 Non-parametric Approach
In addition to the parametric methods, the GP-based non-parametric PDFs are computed for both a and b. These PDFs are then used to compute PDF of y; this
is shown in Fig. 3.24. The non-parametric method does not assume any explicit dis- tribution type or distribution parameters. Thus, the use of a parametric distribution explicitly delineates variability, and epistemic uncertainty (distribution type and pa- rameters) whereas the non-parametric approach represents all the uncertainty in a single distribution.
Similar to the previous example problem, the proposed methodology produces a complete PDF for y, while some of the previously existing solution methodologies result in an interval value for the output y. Kozine and Utkin [94], De Cooman and Troffaes [95], and Ferson and Hajagos [96] produced [0.9, 1.5], [1.0, 1.2], and [0.8, 1.6]
respectively. Further, there is considerable agreement between the proposed PDF and the family of PDFs given by Zaman et al. [61].