Biclustering, co-clustering, or two-mode clustering is a data mining technique which allows simultaneous clustering of the rows and columns of a matrix. The term was first introduced by Mirkin (recently by Cheng and Church in gene expression analysis), although the technique was originally introduced much earlier (i.e., by J.A. Hartigan).
Given a set of m rows in n columns (i.e., an matrix), the biclustering algorithm generates biclusters - a subset of rows which exhibit similar behavior across a subset of columns, or vice versa.
The complexity of the biclustering problem depends on the exact problem formulation, and particularly on the merit function used to evaluate the quality of a given bicluster. However most interesting variants of this problem are NP-complete requiring either large computational effort or the use of lossy heuristics to short-circuit the calculation.
Type of Bicluster
Different biclustering algorithms have different definitions of bicluster. 
- Bicluster with constant values (a),
- Bicluster with constant values on rows (b) or columns (c),
- Bicluster with coherent values (d, e).
a) Bicluster with constant values 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 b) Bicluster with constant values on rows 1.0 1.0 1.0 1.0 1.0 2.0 2.0 2.0 2.0 2.0 3.0 3.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 c) Bicluster with constant values on columns 1.0 2.0 3.0 4.0 5.0 1.0 2.0 3.0 4.0 5.0 1.0 2.0 3.0 4.0 5.0 1.0 2.0 3.0 4.0 5.0 1.0 2.0 3.0 4.0 5.0 d) Bicluster with coherent values (additive) 1.0 4.0 5.0 0.0 1.5 4.0 7.0 8.0 3.0 4.5 3.0 6.0 7.0 2.0 3.5 5.0 8.0 9.0 4.0 5.5 2.0 5.0 6.0 1.0 2.5 e) Bicluster with coherent values (multiplicative) 1.0 0.5 2.0 0.2 0.8 2.0 1.0 4.0 0.4 1.6 3.0 1.5 6.0 0.6 2.4 4.0 2.0 8.0 0.8 3.2 5.0 2.5 10.0 1.0 4.0
There are many biclustering algorithms developed for bioinformatics, including: block clustering, CTWC (Coupled Two-Way Clustering), ITWC (Interrelated Two-Way Clustering), δ-bicluster, δ-pCluster, δ-pattern, FLOC, OPC, Plaid Model, OPSMs (Order-preserving submatrixes), Gibbs, SAMBA (Statistical-Algorithmic Method for Bicluster Analysis), , Robust Biclustering Algorithm (RoBA), Crossing Minimization  , cMonkey, PRMs, DCC, LEB (Localize and Extract Biclusters), QUBIC (QUalitative BIClustering), BCCA (Bi-Correlation Clustering Algorithm) and FABIA (Factor Analysis for Bicluster Acquisition). Biclustering algorithms have also been proposed and used in other application fields under the names coclustering, bidimentional clustering, and subspace clustering.
Given the known importance of discovering local patterns in time series data, recent proposals have addressed the biclustering problem in the specific case of time series gene expression data. In this case, the interesting biclusters can be restricted to those with contiguous columns. This restriction leads to a tractable problem and enables the development of efficient exaustive enumeration algorithms such as CCC-Biclustering  and e-CCC-Biclustering . These algorithms ﬁnd and report all maximal biclusters with coherent and contiguous columns with perfect/approximate expression patterns, in time linear/polynomial in the size of the time series gene expression matrix using eﬃcient string processing techniques based on suffix trees.
Some recent algorithms have attempted to include additional support for biclustering rectangular matrices in the form of other datatypes, including cMonkey.
There is an ongoing debate about how to judge the results of these methods, as biclustering allows overlap between clusters and some algorithms allow the exclusion of hard-to-reconcile columns/conditions. Not all of the available algorithms are deterministic and the analyst must pay attention to the degree to which results represent stable minima. Because this is an unsupervised-classification problem, the lack of a gold standard makes it difficult to spot errors in the results. One approach is to utilize multiple biclustering algorithms, with majority or super-majority voting amongst them deciding the best result. Another way is to analyse the quality of shifting and scaling patterns in biclusters. Biclustering has been used in the domain of text mining (or classification) where it is popularly known as co-clustering . Text corpora are represented in a vectorial form as a matrix D whose rows denote the documents and whose columns denote the words in the dictionary. Matrix elements Dij denote occurrence of word j in document i. Co-clustering algorithms are then applied to discover blocks in D that correspond to a group of documents (rows) characterized by a group of words(columns).
Several approaches have been proposed based on the information contents of the resulting blocks: matrix-based approaches such as SVD and BVD, and graph-based approaches. Information-theoretic algorithms iteratively assign each row to a cluster of documents and each column to a cluster of words such that the mutual information is maximized. Matrix-based methods focus on the decomposition of matrices into blocks such that the error between the original matrix and the regenerated matrices from the decomposition is minimized. Graph-based methods tend to minimize the cuts between the clusters. Given two groups of documents d1 and d2, the number of cuts can be measured as the number of words that occur in documents of groups d1 and d2.
More recently (Bisson and Hussain) have proposed a new approach of using the similarity between words and the similarity between documents to co-cluster the matrix. Their method (known as χ-Sim, for cross similarity) is based on finding document-document similarity and word-word similarity, and then using classical clustering methods such as hierarchical clustering. Instead of explicitly clustering rows and columns alternately, they consider higher-order occurrences of words, inherently taking into account the documents in which they occur. Thus, the similarity between two words is calculated based on the documents in which they occur and also the documents in which "similar" words occur. The idea here is that two documents about the same topic do not necessarily use the same set of words to describe it but a subset of the words and other similar words that are characteristic of that topic. This approach of taking higher-order similarities takes the latent semantic structure of the whole corpus into consideration with the result of generating a better clustering of the documents and words.
In contrast to other approaches, FABIA is a multiplicative model that assumes realistic non-Gaussian signal distributions with heavy tails. FABIA utilizes well understood model selection techniques like variational approaches and applies the Bayesian framework. The generative framework allows FABIA to determine the information content of each bicluster to separate spurious biclusters from true biclusters.
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- ^ a b Mirkin, Boris (1996). Mathematical Classification and Clustering. Kluwer Academic Publishers. ISBN 0792341597.
- ^ Cheng Y, Church GM (2000). "Biclustering of expression data". Proceedings of the 8th International Conference on Intelligent Systems for Molecular Biology: 93–103.
- ^ Hartigan JA (1972). "Direct clustering of a data matrix". Journal of the American Statistical Association (American Statistical Association) 67 (337): 123–9. doi:10.2307/2284710. JSTOR 2284710.
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