# Evaluation of the Urban Weather Generator on the City of Toulouse (France)
## 2 Materials and Methods
### UWG Evaluation
#### 2.3.2 UWG Sensitivity Analysis Methodology
After this process, we have a set of \(R\) trajectories where each trajectory \(\Theta^{(i)}\) corresponds to \(N+1\) specific variations of parameters (the difference between two successive sets \(\Theta^{(i)}(j)\) and \(\Theta^{(i)}(j+1)\) is only on parameter \(\theta_{j+1}\). Simulations are then performed on all \(R\) trajectories. The Morris method relies on several concepts to analyze the sensitivity of a given parameter \(\theta_{j}\). They are described below. **The mean of the absolute elementary effects**:  
For each parameter \(\theta_{j}\), its effect on the variation of \(\hat{T}\) among all the \(R\) trajectories is quantified, for any time step \(t\), by \(\mu_{j}(t)\) computed as:  
\[ \begin{split}&\mu_{j}(t)=\frac{1}{R}\sum\limits_{i=1}^{R}|d_{i,j}(T,t )|\text{ with }\\ & d_{i,j}(T,t)=\frac{UWG(\Theta^{(i)}(j),t)-UWG(\Theta^{(i)}(j-1), t)}{\Delta_{j}},\text{ the elementary effect of parameter }\theta_{j}\text{ at the trajectory}\end{split}  \tag{3}\]  
Roughly, \(d_{i,j}\) quantifies the effect on the output of variations of parameters \(\theta_{j}\), since only \(\theta_{j}\) varies between \(\Theta^{(i)}(j+1)\) and \(\Theta^{(i)}(j)\). In practice, to evaluate the relative importance of all variables among them, we calculate the normalized mean of absolute elementary effects \(\mu_{j}^{*}\) defined as:  
\[\mu_{j}^{*}(t)=\frac{\mu_{j}(t)}{\sum\limits_{k\in\Theta}\mu_{k}(t)}. \tag{4}\]  
This normalized quantity enables us to evaluate the relative importance of one variable \(\theta_{j}\) with respect to all others by providing a value \(\mu_{j}^{*}(t)\) in the interval \([0,1]\) with \(\sum\limits_{k\in\Theta}\mu_{k}^{*}(t)=1\). **The standard deviation \(\sigma_{j}\) of the elementary effects**:  
For each input parameter \(\theta_{j}\), the standard deviation of elementary effects is defined as:  
\[\sigma_{j}(t)=\sqrt{\frac{1}{R-1}\sum\limits_{i=1}^{R}\bigl{(}d_{i,j}(T,t)-\mu _{j}(t)\bigr{)}^{2}}. \tag{5}\]  
This value accounts either for non-linearities in the influence of \(\theta_{j}\) (the higher the value, the more non-linear the influence of the associated parameter) or for interactions between input variables (\(\theta_{j}\) interacts with at least another variable). On the contrary, a low standard deviation means that the influence of \(\theta_{j}\) output is linear on the temperature's estimations and has no interaction with other inputs. As a matter of fact, the standard deviation is computed for elementary effects associated with a single input parameter. More precisely, if the studied parameter \(\theta_{j}\) varies by a given quantity \(\Delta_{j}\), the associated elementary effect for \(d\) is the variation of the output divided by \(\Delta_{j}\). Computing the standard deviation of the elementary effect for all possible variations of \(\Delta_{j}\) is therefore informative. Indeed, if this standard deviation is equal to zero, it indicates that regardless of the variation in \(\Delta_{j}\), the elementary effect is homogeneous (in other words, the output varies linearly with respect to \(\theta_{j}\)). In contrast, if this standard deviation is not equal to zero, it indicates that the output varies either non-linearly with respect to \(\theta_{j}\) or that the studied parameter is correlated with another one. It has been shown in [57] that values of \(\mu_{j}\) and \(\sigma_{j}^{2}\) computed on the set of trajectories are unbiased estimators of the actual mean and variance of the true distribution of input parameters. This process is then used to analyze the sensitivity of UWG with respect to the set of input parameters. In practice, atmospheric and surface parameters have been set based on plausible (and acceptable for UWG) extreme values. The complete list of input parameters used, their extreme values, and associated discretization steps in the sensitivity analysis process are visible in Table 5.