94 C. Li and S. Mahadevan are the same and the lower level has a 100 % relevance from the viewpoint of uncertainty integration; if the two vectors are perpendicular, the physical structures are not similar at all and the relevance is zero. In addition, this definition of physical relevance gives a value on the interval [0, 1]; and its opponent, the square of sine value, indicates physical non-relevance; hence the sum of “relevance” and “non-relevance” is the unity. In the rollup methodology proposed in Sect. 9.4.2, this characteristic gives the possibility to treat the physical relevance similar to a probability and include the physical relevance in the uncertainty integration. A further question arises in the application of this physical relevance definition using sensitivity analysis. The global sensitivity analysis considers the entire distribution of variable, so we need to know the distributions of m and x, but the distribution of m is unknown. In order to solve this problem, a simple iterative algorithm to compute the physical relevance indexRis proposed: 1. Set initial values of Rbetween a lower level and the system level. 2. Obtain the integrated distribution of each model parameter using the current physical relevance and the proposed roll-up formula in Sect. 9.4.2. 3. Use the unconditional distributions from step 2 in the sensitivity analysis and compute the updated physical relevanceR. 4. Repeat step 2 and step 3 until the physical relevanceRconverges. 9.4.2 Roll-up Methodology The integrated distributions of model parameters need to be constructed before propagating them through the computational model at the system level to obtain the system output. Equation (9.1) expressed a roll-up methodology to integrate the results from model calibration and model validation; and this chapter modifies this methodology to include two additional concepts: 1. Stochastic model reliability: the event that the model prediction at Level i is correct is denoted by Gi, whose probability is P(Gi); and the stochastic reliability defines P(Gi) as a stochastic variable with PDFf (P(Gi)) which has been explained in Sect. 9.3. 2. Physical relevance: defined as the square of the cosine value of the angle between the sensitivity vectors. The corresponding non-relevance is the square of the sine value; since their sum is the unity, we can treat the physical relevance similar to probability or a weight in the roll-up equation. This chapter denotes the event that Level i is relevant to the system level bySi, and its probabilityP(Si) is the value of the physical relevance; andS 0 i denotes the event of non-relevance. Take the multi-level problem with two lower levels as an example. The integrated distribution of a model parameter conditioned on the calibration and validation data and a realization of model reliability P(Gi)(i D1,2) is: f ˇ ˇ ˇ DC;V 1 ;DC;V 2 ;P .G1/ ;P .G2/ DP .G1G2S1S2/f ˇ ˇ ˇ DC 1 ;DC 2 CP G1S1 \ G02 [S02 f ˇ ˇ ˇ DC 1 CP G2S2 \ G01 [S01 f ˇ ˇ ˇ DC 2 CP G01 [S01 \ G02 [S02 //f . / (9.11) From the view of generating samples, (9.11) indicates two criteria: (1) whether a level is relevant to the system level; (2) whether a level has the correct model prediction. A sample of is generated fromf ( jDC 1, DC 2 ) only when both levels satisfy both criteria; and a sample of is generated fromf ( jDCi ) if level i satisfies both criteria but the other level does not; a sample of is generated from the prior f ( ) if neither level satisfies both criteria. The so-called integrated distribution of , which is conditioned on the calibration and validation data, are computed as: f ˇ ˇ ˇ DC;V 1 ;DC;V 2 D ““ f ˇ ˇ ˇ DC;V 1 ;DC;V 2 ;P .G1/ ;P .G2/ f .P .G1//f .P .G2//dP .G1/dP .G2/ (9.12) Equations (9.11) and (9.12) express the proposed approach to integrate calibration, validation and relevance results. The analytical expression of f ( jD C;V 1 , D C;V 2 ) is difficult to derive since the results we collect in model calibration and validation are all numerical. A single loop sampling approach is proposed below to construct f ( jD C;V 1 , D C;V 2 ) numerically: 1. Generate a sample of each of P(G1) andP(G2) from their distributions.
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