Sampling the input parameter space

A Latin Hypercube (LHC) sampling scheme is used to create the Nu sets of in-

put parameters used in the PAWN SA experiment. LHC sampling is a stratified random sampling technique which ensures all portions of each input parameter is represented in the sample. As in the LHC sampling method explained by McKay et al. (2000), the range of each input parameter is divided into Nu strata of equal

marginal probability, 1/Nu, and one value is sampled from each stratum. The Nu

sampled values of each input parameter are then matched at random. The advan- tage of using the LHC method is that each input parameter range is represented in a fully stratified manner, giving more reliable SA results. An example of a LHC sample from a two-dimensional input parameter space, where Nu = 3000, is shown

in Figure 6.4. 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 Parameter 1 Pa ra me te r 2

Figure 6.4. An example of a Latin Hypercube (LHC) sample for a two-dimensional input parameter space where Nu = 3000.

For this investigation, the input parameter space will be sampled by generating values that represent a relative change from the best fit parameters estimated in Chapter 5, denoted pµ, pσ, pξ, pθ, pφ1 and pφ2. Relative changes are used because

they have a straightforward interpretation and are directly comparable between in- put parameters e.g. at different locations. The left truncated GEV parameters are different for each grid cell, however the same relative change will be applied at all grid cells, for example if pµ = 0.9 then all left truncated GEV location parame-

Table 6.1.. Input parameter ranges used to explore the sensitivity of input parameter ranking to this specification.

Input Parameter pµ pσ pξ pθ pφ1 pφ2

Parameter range 1 (0.2, 2.2) (0.2, 2.2) (0.2, 2.2) (0.2, 2.2) (0.2, 2.2) (0.2, 2.2)

Parameter range 2 (0.5,1.5) (0.5,1.5) (0.5,1.5) (0.5,1.5) (0.5,1.5) (0.5,1.5)

Parameter range 3 (0.7, 1.3) (0.7, 1.3) (0.7, 1.3) (0.7, 1.3) (0.7, 1.3) (0.7, 1.3)

Parameter range 4 (0.5, 2) (0.5, 2) (-2.2, 2.2) (-10, 10) (0.2, 2.2) (0.2, 2.2)

ters are decreased by 10%, throughout the domain. This means that only 6 input parameter values need to be generated for each of the Nu samples from the input

parameter space.

The results of the SA experiments will depend on the range of the input parameters used because the PAWN sensitivity index, T , is calculated as a statistic over the range of the input parameter, the median (see Figure 6.2). For this reason, four sep- arate PAWN SA experiments will be carried out, each with a different set of input parameter ranges, to explore the sensitivity of the results to this parameter range choice. Each experiment requires a new sample of Nu sets of input parameters and

therefore Nu model evaluations to calculate the output AAL and M AL. The input

parameter ranges used to test this sensitivity are shown in Table 6.1. Parameter range 1 represents relative ranges from an 80% decrease and 120% increase. This large interval is used to ensure that a wide range of possible model behaviour is captured. The variation in the model parameters based on the relative changes in Parameter range 1 is presented in Figure 6.5, showing that this range in the input parameters results in a large variation in both the local wind gust distribution and the footprint spatial dependence structure.

Parameter ranges 2 and 3 (Table 6.1) explore how the SA results change when the input parameter ranges are reduced to a 50% and 30% increase or decrease respectively. Parameter range 4 increases the range of input factors pξ and pθ. The

best fit values of ξ and θ are relatively small compared to the other parameters of the model, therefore this set of relative change parameter ranges is used to explore whether these two parameters have a greater relative influence on the model output when varied more.

The variation in ξ and θ based on Parameter range 4 is presented in Figure 6.6. The GEV shape parameter ξ varies between high positive values which will produce very heavy tailed GEV distributions with no upper limit, to low negative values which

Figure 6.5. Variation in footprint model parameters when input factors (a) pµ, (b)

pσ, (c) pξand (d) pθ, pφ1 and pφ2 apply a decrease of 80% (top row), apply no change

(middle row) and apply an increase of 120% (bottom row). The colour scale in (d) represents the correlation between locations.

Figure 6.6. Variation in footprint model parameters when input parameters (a) pξ

applies a change of −120% (top row), applies no change (middle row) and applies an increase of 90% (bottom row), (b) pφ1 and pφ2 apply a decrease of 120% (top

row), apply no change (middle row) and apply an increase of 120% (bottom row), and (c) pθ applies a decrease of 1000% (top row), applies no change (middle row)

will produce a GEV distribution with a low upper limit. The spatial dependence rotation angle, θ, varies from the longitude axis pointing north-west to south-east to pointing south-west to north-east, therefore encompassing a much larger range in spatial dependence angle.

PAWN SA requires the specification of n, Nu and Nb, which were selected by trial

and error in Pianosi and Wagener (2015). The values used in this investigation are based on those used by Pianosi and Wagener (2015). They used n = 10 conditioning intervals, Nb = 200 bootstrap samples and, for an example application, showed that

the sensitivity indices start to converge when Nu = 3000.