## 2 A Review of the Methods
### The PCA Model  
This study uses the principal component analysis (PCA) method to extract the principal factors behind the correlations among the technological powers, in terms of their patenting activities. The influences of each country on the two major factors are thus identified. The PCA method was proposed by [PERSON] in 1901 [PERSON] (1901) and further developed by [PERSON] (1901). The purpose of PCA is to identify a set of unobservable variables that are the 'principal components' behind a set of observable facts. It is a conversion process wherein the identified 'first principal component' has the largest variance (i.e., it accounts for the main part of variability in the given data set). Furthermore, while the'second principal component' is orthogonal to the first one in the vector space, it has the highest variance among all the other factors except for the first one. So, in fact, this process orthogonally and linearly transforms the data into a new coordinate system; this can be mathematically written as follows:  
\[\left\{ \begin{array}{l}z_{1}=\sum\limits_{j=1}^{p}a_{1j}x_{j}\\ z_{2}=\sum\limits_{j=1}^{p}a_{2j}x_{j}\\ \vdots\\ z_{p}=\sum\limits_{j=1}^{p}a_{pj}x_{j}\end{array} \right. \tag{2}\]  
Or alternatively, it can be written as:  
\[z_{i}=\sum\limits_{j=1}^{p}a_{ij}x_{j},\forall i\in\{1,2,\ldots,p\} \tag{3}\]  
where \(i\) is the identifier for the principal components, which identifies \(z_{i}\); \(p\) is the number of original variables; \(j=1\),..., \(p\) and \(x_{j}\) are the original variables; and \(a_{ij}\) is the 'factor loading' of the \(j\)-th original variable for the \(i\)-th principal component. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Indicator** & **Definition** & **Meaning** \\ \hline
**Patent Activity** & The number of patent applications & The level of R\&D activities \\ \hline
**Share of Granted Patents** & Granted patents/patent activity & Higher technological capacity \\ \hline
**Technological Scope** & Diversity and number of IPC classes a  & Technological quality of patent applications \\ \hline
**International Scope** & Size of patent family and share of triad (US, JP and EFO) patents of patent activities.[ENDFOOTNOTE] & Economic quality of patent applications \\  & patent activities & \\ \hline
**Technology Share** & The ratio of patent activity to patent & Competitive technological position in \\  & applications of all competitors in & technological field \\  & technological field & \\ \hline
**Relative Technology Share** & Patent strength/maximum patent & Distance of firm \(i\) from the technological \\  & strength in technological field & leader in technological field \\ \hline
**Citation Ratio** & The number of patent citations against & The number of citations received by a \\  & the number of patent applications & patent in subsequent patent documents can \\  & the number of patent applications & also be viewed as a sign for an economically \\  & important invention & \\ \hline
**Patent Growth Rate** & Average growth of patent applications & Measures the average growth of patent \\  & per technological field/average growth & applications in a technological field relative \\  & of patent applications in all & to the average growth of total \\  & technological fields & patent applications \\  & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The main indicators in the systematic patent-indicator framework. With PCA, one can often use a lower number of \(m\) variables to represent the original system of \(p\) variables (\(m<p\)). Though after the linear transformation we still have a number of \(p\) components (\(z_{1},\ \ldots,\ z_{p}\)) (strictly speaking), a threshold between 70% and 90% can be set, such that if some components (\(z^{\prime},\ \ldots,\ z^{\prime}_{m}\)) are sufficient to resolve enough of the variance (against the total variance) so that the amount resolved is above the threshold, then these components can be used as the 'principal factors' to resolve a large part of the total variance.