## 2 A Review of the Methods
### The K-means Technique  
In addition to the PCA method, another method, the K-means method [15], is also employed in this study. In the field of data mining, the K-means method has been, and still is, a widely accepted and popular method for data-clustering, even in recent years [16; 17; 18; 19]. It is a 'classifier' that is frequently applied by analysts in order to classify the data points in question into several 'clusters'. A point in a cluster is said to be identical or similar to other points in the same cluster when their similarity can be justified based on some properties (i.e., 'attributes', 'factors', or even 'features'; the usual term is subject to the field or domain), but points in the same cluster are all quite different from the points in any other cluster. Figure 2 illustrates such a clustering process, and such a process can be performed by using the K-means method. Figure 2: The fundamental logic of the K-means classification method (uniform colour: non-clustered data points; red, green and blue: data points in clusters 1, 2 and 3, respectively). Mathematically speaking, the mechanism behind the K-means method can be written as the following optimisation problem, wherein the objective is to minimize the aggregated / statistic in the objective function of the model:  
\[Min(J) \tag{5}\]  
\[\text{s.t.}\ J=\sum_{i=1}^{k}\left(\sum_{x_{j}\in S_{i}}\left(x_{j}-\mu_{S_{i}} \right)^{2}\right) \tag{6}\]  
\[\underset{i=1}{\overset{k}{\cup}}S_{i}=S \tag{7}\]  
\[\underset{i=1}{\overset{\ell}{\cap}}S_{i}=\Phi \tag{8}\]  
where \(x_{j}\) is a data point that belongs to cluster \(i\), as represented by a set \(S_{i}\) which has one or more data points, among all, as its elements; \(\mu_{S_{i}}\) is the mean value of all the data points in the set: \(S_{i}\); \(k\) is a predefined number of groups to be clustered as desired by the analyst; \(J\), therefore, is the summation of all the 'within-group variances' that is to be minimised; and \(S\) is the set of all source data points. This K-means model seeks to cluster the data points into a number of \(k\) clusters. The model reaches optimality when, in each cluster, the sum of the Euclidean distances between each point and the virtual mean point of the cluster is sufficiently minimised so that, overall, the sum of the 'within-group variances' (for all groups) is minimised, given the condition that every data point in \(S\) has been classified into one of the \(k\) clusters. If such optimality is reached, the model solution (i.e., the solution indicating which source data point should belong to which cluster) is regarded as an optimal clustering scenario of the source data, while the centre of each cluster can be located. Normally, the constant \(k\) can be pre-determined for the above process, but in real-world applications, this value is usually uncertain and not easily fixed [20]. For this reason, some studies have suggested that trying any value within the [1,10] interval for \(k\) would be sufficient in most cases [21]. However, given the computational efficiency available nowadays, limiting the number of trials is perhaps unreasonable. Instead, incrementing the value for \(k\) above 2 and then watching the objective function value of Equation (5) with regard to the optimal solution obtained every time seems to be a better approach. As will be shown in Section 4, in this study, it was found that \(k=7\) yields the optimal solution for clustering the relevant patents. In addition, the results for the identified set of the 10'most important technologies' (non-ordered) exactly match those for the 10 'hottest technologies' identified previously using the descriptive methods detailed in Section 3. This match confirms the main empirical finding of this study, despite the fact that such a reconfirmation is rarely seen in the patent study literature. Therefore, this study may provide supplements to the field through its interdisciplinary use of methods that are reflexive to the theories of big data analytics.