# Weighted \(z\)-Distance-Based Clustering and Its Application to Time-Series Data
## 4 Experimental Results
### Non Time-Series Datasets
For example, K-means outperforms SCC-I for Heart, Ionophere, and Seeds. Compared with K-means, SCC-I has advantages: (1) SCC-I considers deviation in the computation of distance; (2) SCC-I allows ellipsoidal shape of clusters. Note that SCC-I is less affected by the feeding order of instances, and thus can give a more stable and accurate clustering than SCC. However, given the number of clusters, the clusters obtained by K-means are not affected by the feeding order of instances. For the datasets with ellipsoidal shape of clusters, SCC-I is more likely to perform better on them. By contrast, for the datasets with spherical clusters, SCC-I may be inferior to K-means on such datasets. Table 2 also shows the execution time, in seconds, of each method on each dataset. The computer used for running the codes is equipped with Intel(R) Core(TM) i7-4770 CPU 3.40 GHz, 16 GB RAM, and Matlab R2011b. The times shown in the table only provide an idea of how efficiently these methods can run. Note that SCC-I runs slower than SCC. SCC only performs one training cycle on the instances, while SCC-I requires two or more training cycles. SCC-I takes more training time than other baselines due to some factors: (1) The codes of these baselines, e.g., K-means and FCM, were adopted from established websites, while SCC-I were written by graduate students; (2) SCC-I has to compute \(z\)-distances and Gaussian values, which is more computationally expensive; (3) In order to do re-assignment, the operation of removing instances from clusters is done during the clustering process. Comparisons of DBI, DI, and SI among different methods for each dataset are shown in Table 4. As can be seen from the table, SCC-I is better than other methods. SCC-I gets the lowest DBI for 9 out of 14 datasets, the highest DI for 9 out of 14 datasets, and the highest SI for 7 out of 14 datasets. Now we use the paired \(t\)-test [68] to test the real significant differences between SCC-I and other methods. Table 5 shows the t-values based on the values of Fscore, NMI, and DBI, respectively, under the 90% confidence level. Note that we have 14 datasets involved. So the degree of freedom is 13 and the corresponding threshold is 1.771. From Table 5, we can see that all the t-values are greater than 1.771, except for Gmm with NMI. Therefore, by statistical test we observe that SCC-I shows significantly better performance than other methods. A multiple comparison test may be used since there are multiple algorithms involved. Analysis of variance (ANOVA) [69] provides such tests. We have tried with ANOVA in two ways. Firstly, we used ANOVA to return a structure that can be used to determine which pairs of algorithms are significantly different. The results similar to those shown in Table 5 were obtained. Secondly, we used ANOVA to test the null hypothesis that all algorithms perform equally well against the alternative hypothesis that at least one algorithm is different from the others. A \(p\)-value that is smaller than 0.05 indicates that at least one of the algorithms is significantly different from the others. However, it is not certain which ones are significantly different from each other. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **K-Means** & **DSKmeans** & **FCM** & **Gmm** & **DBSCAN** & **SCC** & **SCC-I** \\ \hline Fscore & 4.2 & 3.8 & 4.4 & 4.4 & 4.7 & 4.4 & **2.1** \\ RI & **2.9** & 4.4 & 3.2 & 3.8 & 5.1 & 5.3 & 3.1 \\ NMI & 4.1 & 3.9 & 4.6 & 4.1 & 4.1 & 4.9 & **2.1** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Averaged ranking of 14 non-time-series datasets.