## 4 Further Analysis with Discussions and Implications
### The Most Important Technologies and the Associated Implications  
This Section provides supplementary support to the findings regarding the 'world's hottest technologies' that are presented in Figure 6 in Section 3.3. By introducing the K-means method (see Section 2.4) to classify the technologies in terms of IPC, another set of'most important technologies'(i.e., most influential areas of patent investment) are identified. It is surprising to find that the results from this additional experiment can cross-validate the previous results, such that further confidence can be gained regarding the relevant findings. In our analysis, not only is the 'best clustering scenario' for the source data points (of the IPC codes, as shown in Figure 7b) identified; rather, the suitable value for \(k\), which is the optimal number that is to be set for the number of clusters, is also obtained. The results of this clustering process are also visualised in Figure 8, with the most appropriate \(k\) value set at \(k=7\), because the minimal value of the objective function in Equation (5) is obtained (that is, the model reaches optimality) when \(k=7\). In Figure 8, each point is plotted for an IPC. The source data points of the IPC codes are classified into 7 clusters, each of which is displayed with a different colour. The texts of the IPC codes are intentionally unmarked, making the figure clearer for illustration purposes, with only the locations of them being displayed. Further scrutiny yields several implications, which are as follows:  
* _The most important (influencing) technologies are identified._ As can be imagined, the farther a point is from the origin, the more important the technology should be (this is again indicated by the magnitude of the vector). So, the top 10'most important technologies' during the study period, at the'subclass' level, are (ranked by vector length): G06F, H01L, H04L, H04N, G01N, H04W, G06Q, G02B, G06Q, H01M and H04B. As a supplementary material, the coordinate and the  
Figure 8: The clusters of technologies. magnitude (Euclidean distance toward the origin) of the vectors of all IPCs are also listed and ranked in Table 3, while the abovementioned most important technologies are highlighted. * The set of most important technologies identified above is identical to the set of most popular (hottest) technologies, when the top 10 in either case are considered. When the order in the'most important technologies list' above is ignored, it becomes a set of elements, i.e., the'most important technologies set' (see the highlighted entries for the domains in Table 3). As the patterns in Table 4 are extensively analysed (and also mapped to Figure 8), these 10 technologies fall in (and perfectly match) the four clusters (i.e., CID = 1, 3, 4 and 7) that not only have the centres that are the farthest from the origin O (see the 'Dist. to O' column in Table 4; 505,921 for CID = 3, 330,414 for CID = 4, 163,899 for CID = 4 and 78,659 for CID = 7) but also the highest average within-group variances for the IPC codes (i.e., 51468.61 for CID = 7 and 22483.96 for CID=1, except that 0 was observed for both CID = 3 and CID = 4 since for these clusters, each contains only one technological domain). These observations of the patterns rendered by the four clusters (which consists of a few very diversified points and at the same time are very far away from the origin) may help ascertain the importance of the 10 technologies in the set. Going back to Figure 6 in Section 3.3, the 'hottest technologies list' also implies another 'hottest technologies set' regardless of the rank order. Comparing these two non-ordered sets, it is found that the'most important technologies set' obtained by the K-means machine learning technique is exactly identical to the 'hottest technologies set' identified before by the descriptive statistical method. This is perhaps the most surprising finding of this study, and one which leads to further implications. * _The popularity of a technology may connote its importance, and vice versa_.