## 3 Methodology
### Principal Component Analysis for Index Construction  
Weighting and aggregation play an important role in the construction of an index. Various techniques are currently available in the literature to weight and aggregate different indicators for different contexts [31]. However, there is no best one to use in all circum  
\begin{table}
\begin{tabular}{c c c} \hline
**Risk Domain** & **Selected SPIs** & **Abbreviation** \\ \hline Alcohol & I\({}_{1}\): The percentage of drivers below alcohol limit in roadside checks & _P\_alc_ \\ \hline Protective systems & I\({}_{2}\): Daytime seat belt wearing rates on front seats & _Belt\_front_ \\  & I\({}_{3}\): Daytime seat belt wearing rates on rear seats & _Belt\_rar_ \\ \hline Vehicles & I\({}_{4}\): Average score of occupant protection for new cars & _Prot\_occ_ \\  & I\({}_{5}\): Average score of pedestrian protection for new cars & _Prot\_pad_ \\  & I\({}_{6}\): Renewal rate of passenger cars & _Renewal_ \\  & I\({}_{7}\): The percentage of new passenger cars: less than 5 years & _Car\_age_ \\ \hline \end{tabular}
\end{table}
Table 2: Selected safety performance indicators for index construction.  
stances. In this study, we do not concentrate on introducing new techniques for index construction, but on the comparison of index scores over time via an easy-to-reproduce but effective approach, which is the principal component analysis (PCA).  
A PCA is a kind of multivariate data analysis method, and it is used frequently to reduce the dimensions of a problem. The main idea of this method is to generate a few so-called principal components (\(p\)) to replace a large number of original variables (\(l\)) without dropping important information. Several guidelines can be used to identify the optimal number of principal components. First, the reserved principal components should have associated eigenvalues larger than one; second, they should individually account for over 10% of the total variance; finally, they should cumulatively explain at least 60% of the overall variance [31].  
Prior to the application of PCA for index construction, we should first normalize the raw data so that the scale differences of the indicators and the effects of the measurement unit can be eliminated. In this study, the distance between the actual value and the best and the worst values is calculated, resulting in a normalized value between 0 and 1, in which a value of 1 indicates the highest safety performance, while a value of 0 means the lowest.  
After data normalization, we further obtain the matrix of component loading. Each indicator (\(i\)) can have a large loading score (\(a_{ij}\) (\(i=1,\ldots,l;j=1,\ldots,p\))) on only one of the principal components (\(j\)). The indicator weights can then be derived based on these loading scores by the following procedures [10].  
Define:  
\[u_{ij}=\frac{a_{ij}^{2}}{\sum\limits_{i=1}^{l}\sum\limits_{j=1}^{p}a_{ij}^{2}} \tag{3}\]  
Thus, the preliminary weight for each indicator \(i\) can be generated as follows:  
\[u_{i}=\max_{j}\bigl{(}u_{ij}\bigr{)} \tag{4}\]  
However, due to dimension reduction, the sum of \(u_{i}\) is less than one, i.e.,  
\[U=\sum\limits_{i}u_{i}<1 \tag{5}\]  
In order to enable the sum of weights to be one, the final weight of indicator \(i\) is calculated by:  
\[w_{i}=\frac{u_{i}}{U} \tag{6}\]  
Based on the derived weights, the index score of country \(k\) can be obtained as follows:  
\[Score_{k}=\sum\limits_{i=1}^{l}w_{i}v_{ik} \tag{7}\]  
where \(v_{ik}\) is the normalized value of indicator \(i\) for country \(k\).