# Weighted \(z\)-Distance-Based Clustering and Its Application to Time-Series Data
## Appendix A  
A simple example is given here to illustrate how SCC-I works. Suppose **X** has the following 12 training instances:  
\[ \begin{array}{llllll}\textbf{x}_{1}&=&[0.30&0.60]^{T};\textbf{x}_{2}&=&[0.70 &0.35]^{T};\textbf{x}_{3}&=&[0.50&0.52]^{T};\\ \textbf{x}_{4}&=&[0.35&0.38]^{T};\textbf{x}_{5}&=&[0.19&0.89]^{T};\textbf{x}_{6}& =&[0.78&0.20]^{T};\\ \textbf{x}_{7}&=&[0.62&0.25]^{T};\textbf{x}_{8}&=&[0.24&0.81]^{T};\textbf{x}_{9 }&=&[0.29&0.89]^{T};\\ \textbf{x}_{10}&=&[0.40&0.65]^{T};\textbf{x}_{11}&=&[0.28&0.48]^{T};\textbf{x}_{ 12}&=&[0.24&0.89]^{T}.\end{array} \]  
Note that \(N=12\) and \(n=2\). These instances are shown in Figure 11, marked as circles. Let \(\rho=0.55\) and \(\sigma_{0}=0.2\). Below, we consider two feeding orders of instances. \(\bullet\) The feeding order of \(\textbf{x}_{1}\),..., \(\textbf{x}_{12}\). After performing SCC in the first iteration, there are six clusters: \(C_{1}\), \(C_{2}\), \(C_{3}\), \(C_{4}\), \(C_{5}\), and \(C_{6}\), as shown in Figure 11, with  
\[ \begin{array}{llllll}\textbf{c}_{1}&=&[0.350&0.625]^{T},&\sigma_{1}&=&[0.271 &0.235]^{T};\\ \textbf{c}_{2}&=&[0.660&0.300]^{T},&\sigma_{2}&=&[0.257&0.271]^{T};\\ \textbf{c}_{3}&=&[0.500&0.520]^{T},&\sigma_{3}&=&[0.200&0.200]^{T};\\ \textbf{c}_{4}&=&[0.315&0.430]^{T},&\sigma_{4}&=&[0.250&0.271]^{T};\\ \textbf{c}_{5}&=&[0.240&0.870]^{T},&\sigma_{5}&=&[0.241&0.240]^{T};\\ \textbf{c}_{6}&=&[0.780&0.200]^{T},&\sigma_{6}&=&[0.200&0.200]^{T}.\end{array} \]  
The clusters are numbered and wrapped in dashed contours, with their centers marked with crosses. Instances \(\textbf{x}_{1}\) and \(\textbf{x}_{10}\) are assigned to \(C_{1}\), \(\textbf{x}_{2}\) and \(\textbf{x}_{7}\) are assigned to \(C_{2}\), \(\textbf{x}_{3}\) is assigned to \(C_{3}\), \(\textbf{x}_{4}\) and \(\textbf{x}_{11}\) are assigned to \(C_{4}\), \(\textbf{x}_{5}\), \(\textbf{x}_{8}\), \(\textbf{x}_{9}\), and \(\textbf{x}_{12}\) are assigned to \(C_{\[\mathbf{c}_{1} =[0.366\ \ 0.526]^{T},\ \sigma_{1} =[0.288\ \ 0.305]^{T};\] \[\mathbf{c}_{2} =[0.700\ \ 0.267]^{T},\ \sigma_{2} =[0.280\ \ 0.276]^{T};\] \[\mathbf{c}_{3} =[0.240\ \ 0.870]^{T},\ \sigma_{3} =[0.241\ \ 0.240]^{T}.\]  
Instances \(\mathbf{x}_{1}\,\mathbf{x}_{3}\,\mathbf{x}_{4}\,\mathbf{x}_{10}\), and \(\mathbf{x}_{11}\) are assigned to \(C_{1}\,\mathbf{x}_{2}\,\mathbf{x}_{6}\), and \(\mathbf{x}_{7}\) are assigned to \(C_{2}\), and \(\mathbf{x}_{5}\), \(\mathbf{x}_{8}\,\mathbf{x}_{9}\), and \(\mathbf{x}_{12}\) are assigned to \(C_{3}\). In the fourth iteration, no assignment has been changed. Therefore, SCC-I stops with three clusters \(C_{1}\), \(C_{2}\), and \(C_{3}\) as shown above, with 5, 3, and 4 instances assigned to them, respectively. \(\bullet\) The feeding order of \(\mathbf{x}_{9}\,\mathbf{x}_{10}\,\mathbf{x}_{4}\,\mathbf{x}_{7}\,\mathbf{x}_{5}\, \mathbf{x}_{11}\,\mathbf{x}_{2}\,\mathbf{x}_{12}\,\mathbf{x}_{3}\,\mathbf{x}_{1}\,\mathbf{x}_{6}\, \mathbf{x}_{8}\). After performing SCC in the first iteration, there are 5 clusters: \(C_{1}\,C_{2}\,C_{3}\,C_{4}\), and \(C_{5}\), as shown in Figure 1(a), with:  
Figure 1: Clusters produced with the first feeding order. Figure 2: Clusters produced with the second feeding order. \[ \begin{array}{llllll}\mathbf{c}_{1}&=&[0.240&0.870]^{T},&\sigma_{1}&=&[0.240&0.240]^{T} ;\\ \\ \mathbf{c}_{2}&=&[0.350&0.625]^{T},&\sigma_{2}&=&[0.271&0.235]^{T}\;;\\ \\ \mathbf{c}_{3}&=&[0.315&0.430]^{T},&\sigma_{3}&=&[0.250&0.271]^{T}\;;\\ \\ \mathbf{c}_{4}&=&[0.700&0.267]^{T},&\sigma_{4}&=&[0.280&0.276]^{T}.\\ \\ \mathbf{c}_{5}&=&[0.500&0.520]^{T},&\sigma_{5}&=&[0.200&0.200]^{T}.\\ \end{array} \]  
Instances \(\mathbf{x}_{1}\), \(\mathbf{x}_{5}\), \(\mathbf{x}_{8}\), and \(\mathbf{x}_{12}\) are assigned to \(C_{1}\), \(\mathbf{x}_{2}\) and \(\mathbf{x}_{10}\) are assigned to \(C_{2}\), \(\mathbf{x}_{3}\) and \(\mathbf{x}_{6}\) are assigned to \(C_{3}\), \(\mathbf{x}_{4}\), \(\mathbf{x}_{7}\), and \(\mathbf{x}_{11}\) are assigned to \(C_{4}\), and \(\mathbf{x}_{9}\) is assigned to \(C_{5}\). Iterations 2 and 3 are performed subsequently.