# Incrementally Mining Column Constant Biclusters with FVSP Tree
## 2 Method
### Initial CCB Mining
#### 2.2.3. Initial CCB Mining  
Having built the initial FVSFP tree, the following step is mining CCB from the FVSFP tree. As mentioned above, one of the differences between the FVSFP tree and the FP tree is that the unfrequent nodes are deleted in the mining process instead of the FVSFP tree construction process. In the context of CCB-based applications, the main goal is to find the CCBs whose volume is as big as possible. Therefore, a novel bicluster mining method is proposed by mining one bicluster with the largest volume from each branch of the FVSFP tree. Sometimes, two CCBs may have identical volume; compared with feature number, row number is more important [5], therefore, the CCB with larger supports (row number) is outputted. Any CCBs containing less than \(NT\) rows or 3 columns are deemed to be meaningless [7] and should be deleted.  
\[NT=|D|*msr \tag{1}\]  
where \(|D|\)denotes the number of samples in the datasets \(D\), \(msr\) is the minimal support rate. The CCB mining strategy is as follows:  
* Find all leaf nodes;
* Find all branches by iteratively combining the whole nodes from the leaf node to the Root node;
* For each branch, delete the infrequent nodes whose count is less than threshold \(NT\). Then, find the whole candidate CCBs from each branch with the following steps. (1): Unique the counts of the nodes in the branch to obtain the unique counts array \([mcc_{1},\ldots,mcc_{i},\ldots,mcc_{n}]\)\((1\leq i\leq n)\) where \(n\) is the length of the array, \(mcc\) means the maximal common counts among the whole nodes in the CCB. For each \(mcc\), calculate the whole nodes whose count is greater than or equal to the \(mcc\), then the volume of the CCB can be obtained by Equation (2). Finally, the bicluster with the biggest volume is selected.  
Take the first branch (0:5 1:4 3:4 5:3) of the FVSFP tree generated by the original dataset in Figure 2, for instance, to illustrate mining CCB. Suppose the minimal support rate is 0.4, therefore, the minimal supports is 0.4 \(\times\) 5 = 2, the nodes whose count is less than 2 are unfrequent nodes, no node is unfrequent node, all the four nodes should be preserved. Then, the remaining effective branch becomes (0:5 1:4 3:4 5:3). Unique the counts [3 4 5], because the number of the nodes whose count is no less than 5 is less than 3, therefore remaining valid counts are [3 4]. Two candidate CCBs (shown in Table 3) can be generated, both CCBs have identical volume, but counts are more important than the node number, finally, the optimal CCB (0,1,3):4 is outputted.
* Delete the identical and subset CCBs. Because the optimal CCBs generated from different branches may be identical, identical CCBs should be deleted. Additionally, some CCBs may be the subsets of other CCBs, the subset CCBs also should be deleted. The maximal common count deleting subset pattern can guarantee that the obtained CCBs are inclusion-maximal.  
\[V_{i}=N_{i}*mcc_{i} \tag{2}\]  
where \(mcc\) denotes the least common counts of the nodes, \(N_{i}\) denotes the number of nodes whose count is no less than \(mcc_{i}\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**CCB No.** & **Nodes** &
\begin{tabular}{c} **Least Common** \\ **Counts (_lcc_)** \\ \end{tabular} & **Volume** \\ \hline
1 & (0, 1, 3, 5) & 3 & 3 \(\times\) 4 = 12 \\ \hline
2 & (0, 1, 3) & 4 & 4 \(\times\) 3 = 12 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Selecting the optimal CCB from the whole candidate patterns.  
As shown in the top part of Figure 2, \(D_{o}\) is composed of 5 samples, the feature values in sample are non-repetitive. The feature values in the original header table are sorted according to feature values' ascending order. With Algorithm 1, the original FVSFP tree can be constructed. The final step is to mine CCB from the branches of the tree. There are 3 branches, each of which is listed in the top subtable of the rightmost table. The middle subtable shows the optimal CCB mined from each branch. The subtable below is the final outputted CCB after deleting the identical and subset CCB.