returns the largest integer less than or equal to $z$.

The computational complexities of histogram construction and Otsu’s method are $\mathcal{O}(N_i)$ ($N_i$: number of points in the cluster) and $\mathcal{O}(L)$, respectively. $L$ is constant in our experiments and therefore the proposed modification does not alter the linear time complexity of Var-Part and PCA-Part.

Figure \[fig\_histo\] shows a histogram where using the mean point as a threshold leads to poor results. This histogram is constructed during the first iteration of PCA-Part from the projections of the points in the Shuttle data set (see Table \[tab\_data\_set\]). As marked on the figure, the mean point of this histogram is $61$, whereas Otsu’s method gives a threshold of $105$. The SSE of the initial cluster is $1,836$. When the mean point of the histogram is used a threshold, the resulting two subclusters have SSE’s of $408$ and $809$. This means that splitting the initial cluster with a hyperplane orthogonal to the principal eigenvector of the cluster covariance matrix at the mean point results in approximately $34$% reduction in the SSE. On the other hand, when Otsu’s threshold is used, the subclusters have SSE’s of $943$ and $101$, which translates to about $43$% reduction in 