Global Sensitivity Analysis

Global sensitivity analysis is a method of decomposing output variance into contributions from the individual model parameters and the interactions between parameters. Unlike LSA, it is conducted only for those inputs that can be expressed as probabilistic distributions. The process of variance apportionment in GSA is carried out with the Sobol’ method, which uses MCS to calculate the Main-effect Sensitivity Index (MSI) and the Total-effect Sensitivity Index (TSI) for each parameter [Homma and Saltelli (1996), Sobol’ (2003)]. The MSI of a parameter signifies the contribution to output variance due to that parameter alone, whereas the TSI denotes the contribution to output variance due to that parameter and its interactions with other model parameters. The TSI calculations are performed using the mean-subtracted alternative GSA approach, which enhances computational stability [Sobol’ (2001)]. The MSI and TSI values can be used to rank the model inputs based on their contribution to output variability. The sum of all MSI for the model should be roughly equal to one, whereas the sum of the TSI should be greater than or equal to one, depending on the magnitude of the interaction effects. The Sobol’ method has been employed extensively for GSA of various modules within APMT-Impacts

[Mahashabde (2009), Brunelle-Yeung (2009), Kish (2008), Jun (2007)].

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Global sensitivity analysis was conducted only for the distributional model parameters – namely, the background noise level, contour uncertainty, and the three regression parameters. Since GSA inputs must be independent distributions, the three regression parameters were considered

collectively, as together they represent one independent regression distribution obtained from the bootstrapping procedure described in Section 5.1.2.6. Therefore, a total of three inputs were examined, and for each parameter a MCS with 2000 runs was performed where the distribution of the given parameter was fixed at its base sample values, while all other parameters were resampled from their respective distributions. A total of five runs were required for each GSA scenario – one resampled case for each of the three model parameters, one base case without resampling, and one case where all parameters were resampled. The MSI and TSI for the model parameters were calculated based on the NPV distributions obtained from the five evaluations.

For the deterministic parameters, an inner loop/outer loop procedure may be employed in order to investigate what interaction effects, if any, they have on the MSI and TSI of the distributional parameters. In the outer loop, a deterministic input, such as income growth rate, is set to its extreme value (as defined in the LSA) while holding all other parameters at their nominal values.

The inner loop consists of conducting the GSA for the distributional inputs at their nominal values, as described above. In this thesis, three outer loop settings were considered,

corresponding to the nominal case for all model parameters, a significance level of 65 dB DNL, and an income growth rate of 3%. The discount rate was not varied from 3.5% in the inner loop/outer loop procedure because it solely affects the post-processing of the NPV results, and therefore has no impact on the MSI and TSI of the other inputs.

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Figure 20: Global sensitivity indices – outer loop: nominal case

Figure 20 shows the MSI and TSI for the nominal baseline and policy minus baseline scenarios.

In both scenarios, the regression parameters had by far the largest MSI and TSI, followed by the background noise level, with the contour uncertainty having the smallest indices. This suggests that the majority of the output variability is attributable to scientific and modeling uncertainties associated with the WTP versus income regression relationship implemented in the model.

Figure 21: Global sensitivity indices – outer loop: significance level = 65 dB

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Figure 21 shows the GSA results for the outer loop setting where the significance level is set to 65 dB. With this change, the background noise level becomes a negligible contributor to output variability because no longer is each grid point in the noise contours considered to have

significant monetary impact. Therefore, only a subset of the noise grid points – those exceeding 65 dB DNL – are included in the NPV calculation; those points all have a ΔdB of at least 10 dB, hence reducing the relative influence of the variability in the background noise level in the monetary impact calculation. Correspondingly, the contour uncertainty has a larger relative contribution to output variability, as it plays a key role in determining whether or not certain grid points are included in the impact calculation.²⁷ The decrease in the MSI and TSI for the

regression parameters is also explained by the larger ΔdB values in this particular outer loop setting, which downplays the relative influence of the WTP (and therefore the regression parameters) in the computation of monetary noise impacts.

Figure 22: Global sensitivity indices – outer loop: income growth rate = 3%

Figure 22 shows the GSA results for the outer loop setting where the income growth rate is set to 3% per year. There are only minor changes in the MSI and TSI between Figure 20 and Figure 22, and no shifts in relative ranking of the three inputs the based on their indices. This result is

27 For example, if the noise contour level at a grid point is 65 dB DNL, it will be judged to have significance noise impact if the contour uncertainty associated with that point is greater than or equal to zero. Otherwise, if the contour uncertainty is negative, the noise level at that point will not meet the 65 dB significance threshold, and is therefore excluded from the monetary impact calculation.

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not surprising, given that the income growth rate should have no effect on the noise level (ΔdB), but rather scale the WTP by a constant value for each airport in each year. For this setting, the magnitude of the NPV is much larger than for the nominal case, but the breakdown of model parameters by contribution to output variability remains relatively unchanged.