## 4 Further Analysis with Discussions and Implications
### Angular Analysis and the Inferences of Fact
#### 4.2.1 Classifying the Technological Powers with the Angular Analysis Method  
The extractions of the two principal components in Section 4.1 (i.e., \(z_{1}\) and \(z_{2}\) by using Table 2 for the model parameters) is interesting. The influences of each technological power country on these two components are, respectively, estimated and identified as the weight priorities associated with the country's variable (e.g., 0.311 and 0.509 associated with \(x_{CAN}\) in Equations (10) and (11) for Canada). These values constitute useful information for the 'angular analysis'. For each country, these weights, when normalised, can be plotted as an arrow in the vector space weaved by the dimensions of the principal components, i.e., the (\(z_{1}\), \(z_{2}\)) coordinate system. When the arrows for all the countries areplotted, it forms a visualised angular-based diagram for further knowledge exploration (see the codes within the circle plotted in Figure 7a). Figure 7: The angular analysis results. Aside from the countries, the different types of technologies can also be the targets of interest in such an analysis. To explore the relations among them, another similar analysis that follows the correlation matrix analysis (Table 2 in Section 3.4) method and the PCA modelling (see Section 4.1) is also performed. However, this analysis is performed on a 'by-technology' (that is, by-IPC-code) basis, rather than on a 'by-country' basis. This additional PCA modelling yields another set of two principal components (\(z^{\prime}_{1}\) and \(z^{\prime}_{2}\)) that explains more than 85% of the total variance present among the variables of the IPC codes (i.e., \(\{x^{\prime}_{1},\ x^{\prime}_{2},\ldots,x^{\prime}_{q}\}\), where \(q\) is the number of IPC codes in Sections G and H). For space reasons, this equation system (that is, the PCA model) is omitted here because the RHS of these equations is too long. Instead, the weights that are associated with each \(x^{\prime}\) in the model compose a \((a^{\prime}_{1^{\prime}},a^{\prime}_{2^{\prime}})\) pair (where \(j=1,\ldots,q\) is an IPC code) that can be plotted as a point in the \((z^{\prime}_{1},\ z^{\prime}_{2})\) coordinate system. For simplicity in the presentation, in this study, each of the points, the locations of which show the features of a technological area, are separately plotted in Figure 7b, while the arrows that connote the patenting activities of the countries are plotted in Figure 7a. In both figures, the coordinate system is weaved by the first principal component (as the \(x\)-axis) and the second principal component (as the \(y\)-axis), but the meaning of an arrow and that of a point are different. Each point in Figure 7b, \((a^{\prime}_{1^{\prime}},a^{\prime}_{2^{\prime}})\), is for (the feature of) an IPC code, in terms of its weight in the equation for \(z^{\prime}_{1}\) and the weight in that for \(z^{\prime}_{2}\). Each arrow in Figure 7a, \((\overset{\rightharpoonup}{A}_{j}=(a_{1j},a_{2j}))\) is for (the feature of) a country, in terms of its weight in the equation for \(z_{1}\) and the weight in that for \(z_{2}\). This benefits the upcoming discussions. In Figure 7a, it can easily be seen that regardless of the angle and signs, each arrow vector, \(\overset{\rightharpoonup}{A}_{j}=(a_{1j},a_{2j})\), is defined by the contribution of country \(j\) to the first principal component and its contribution to the second. All of the arrows fall within the circle with a radius of 1. Since the length of each arrow is a Euclidean distance from the origin, it represents the amount of variance of the country and, thus, the importance of that country in the studied technological domains. In the figure, it can also be observed that among the 9 arrows plotted, except for the arrow for GER ('DE'), the amounts of variance indicated (that is, for the other 8 technological powers) are almost even. Other key observations, point by point, are as follows:  
* _Two groups of technological powers are classified naturally_.