# A Novel K-Means Clustering Algorithm with a Noise Algorithm for Capturing Urban Hotspots
## 5 Statistical Analysis of Wilcoxon  
A Wilcoxon rank sum test is a non-parametric null hypothesis statistical testing method [50] that is often used to test the significant difference and distribution state of a clustering training process. When statistical validation is performed, it is usually composed of \(p\), \(h\) and \(stats\), in which \(p\) represents the results \(u\) and \(v\) of a clustering evaluation. \(p\) is the continuous distribution of data samples, which is used to test \(u\) and \(v\) under the non-null hypothesis (noise-based K-means vs. FCM, noise-based K-means vs. K-means, noise-based K-means vs. K-means plus). As \(p\)\(\rightarrow\)0, the difference between \(u\) and \(v\) becomes more obvious. \(h\) is the logical value for testing 0 or 1, \(h\) = 1 means reject the null hypothesis, and \(h\) = 0 means reject the null hypothesis at \(\alpha\) (for example, \(\alpha\) = 0.05, \(\alpha\) is the significance level parameter, value range: 0 < \(\alpha\) < 1). That is, \(h\) = 1 means that the difference between \(u\) and \(v\) is significant, while \(h\) = 0 means that the difference between \(u\) and \(v\) is not significant. \(stats\) consists of two statistics, \(zval\) and \(ranksum\). \(zval\) represents a normal distribution estimate of \(p\), while \(ranksum\) represents a statistic [51]. The statistical analysis results of the Wilcoxon rank sum testing are shown in Tables 8-10. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Noise-Based K-Means Versus FCM**} \\ \cline{3-6}
**Taxi GPS Dataset** & **Clustering Result** & \multirow{2}{*}{\(k\)} & \multirow{2}{*}{\(p\)} & \multirow{2}{*}{\(h\)} & \multicolumn{2}{c}{_Stats_} \\ \cline{4-6}  & & & & & \(Zval\) & _Ranksum_ \\ \hline \multirow{6}{*}{
\begin{tabular}{c} Aracaju \\ (Brazil) \\ \end{tabular} } & **SC** & \multirow{6}{*}{120} & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 610 \\  & SSE & & 6.79561 \(\times\)\(10^{-8}\) & 1 & \(-\)5.3965 & 210 \\  & DBI & & 6.79561 \(\times\)\(10^{-8}\) & 1 & \(-\)5.3965 & 210 \\  & PBM & & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 610 \\  & **SC** & \multirow{6}{*}{140} & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 610 \\ San Francisco & SSE & & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 610 \\  & SSE & & 6.79561 \(\times\)\(10^{-8}\) & 1 & \(-\)5.3965 & 210 \\  & DBI & & 6.79561 \(\times\)\(10^{-8}\) & 1 & \(-\)5.3965 & 210 \\  & PBM & & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 610 \\  & SC & & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 610 \\ Roma & SSE & & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 210 \\ (Italy) & DBI & & 6.79561 \(\times\)\(10^{-8}\) & 1 & \(-\)5.3965 & 210 \\  & PBM & & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 227 \\  & PBM & & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 610 \\  & SC & & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 610 \\ Chongqing & SSE & & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 610 \\ (China) & DBI & & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 210 \\  & PBM & & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 210 \\  & **SC** & \multirow{6}{*}{125} & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 610 \\ Beijing & SSE & & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 610 \\ (China) & DBI & & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 610 \\  & PBM & & 6.79561 \(\times\)\(10^{-8}\) & 1 & 5.3965 & 610 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Statistical analysis results of Wilcoxon testing for noise-based K-means vs. FCM.