# Incrementally Mining Column Constant Biclusters with FVSP Tree
## 1 Introduction  
Clustering is an important part of data mining [1]. Clustering can group samples into different clusters. Traditional one-way clustering methods such as K-means [2] take all features into consideration when calculating the similarity between samples. In many cases samples are similar only under partial features. Biclustering can cluster from both the row (sample) dimension and column (feature) dimension, extracting local coherent patterns. As shown in Figure 1, a bicluster can be categorized into constant bicluster, row constant bicluster, column constant bicluster, additive bicluster, multiplicative bicluster and coherent evolution bicluster [3]. Biclustering originates from gene expression data analysis [4], it has also been applied to other fields. In bicluster application, column constant biclusters (CCB) are most frequently used. In CCB, the values of each column are identical. As CCB can be transformed to be a horizontal discriminative vector through column averaging, to some extent, the horizontal discriminative vector has discriminating ability. CCB is expressed as follows: \(CCB=\{(f_{1},f_{2},\ldots,f_{i},\ldots,f_{n}):m\}\) where \(f_{i}\) denotes the \(i\)-th column value of the CCB containing \(m\) rows and \(n\) columns. In the literature, many studies based on CCB [5; 6; 7; 8; 9; 10; 11] have been proposed. In [5], detailed theoretical analysis of CCB based classification is given. In [6], CCB is used to mine breast cancer diagnosis rules, finally, with adaboost, the diagnosis rules can be combined to construct a strong diagnosis rule. In [7; 9], CCB is applied to mine stock price fluctuating rules, finally, with KNN or fuzzy influence, an optimal solution can be obtained. In [8], CCB is used to extract motor imagery classification rules that can construct a fuzzy rule base, subsequently, with fuzzy inference, an excellent classifier can be built. In [10], a newmethod for customer segmentation based on CCB is proposed. In [11], CCB is used to select features. In a review of the studies in the literature, existing CCB-based studies process data in a batch way. How to mine CCB in an incremental way has not been investigated yet. In real situations, since the computation sources are limited and fixed, if the dataset is very big, running will fail because the required computation resource is bigger than the maximal available computation resource. If the dataset is divided into several small parts, the required computation resource for incrementally processing each part is smaller than processing the whole dataset, therefore, processing the whole dataset can be successful. Therefore, designing a novel method for incrementally mining CCB [12; 13; 14; 15; 16] is critically necessary. Incrementally mining CCB from an original dataset and an incremental dataset can be divided into the following two steps: (1) Mine original constant column biclusters \(CCB_{o}\) from original dataset \(D_{o}\) and save the intermediates \(I_{o}\) that are used to generate \(CCB_{o}\). (2) Mine the whole column constant biclusters \(CCB_{a}\) from the new added dataset \(D_{a}\) and previously generated intermediates \(I_{o}\), \(CCB_{a}\) is the same as the CCB directly mined from the whole datasets (the combination of \(D_{o}\bigcup D_{a}\)). Incrementally mining CCB should have the following properties: (1) With the coming of a newly added dataset, the intermediates that are used to generate the previous mined biclusters can be updated to improve the quality of previously mined CCB and new CCB can be mined. (2) The original datasets that have been processed before cannot be scanned/processed again in the incremental mining stage. Nowadays, many biclustering methods have been proposed. Typical biclustering methods include the greedy search-based method [4], the evolutionary computation-based method [17], the exhaustive enumeration-based method [18], the statistical model-based method [19], etc.