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Graph-based consensus clustering for class discovery from gene expression data. Zhiwen Yum, Hau-San Wong and Hongqiang Wang Bioinformatics, 2007. Outline. Introduction Methods Experiment Conclusion. Introduction. Class discovery consists of two steps:

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## Graph-based consensus clustering for class discovery from gene expression data

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**Graph-based consensus clustering for class discovery from**gene expression data Zhiwen Yum, Hau-San Wong and Hongqiang Wang Bioinformatics, 2007**Outline**• Introduction • Methods • Experiment • Conclusion**Introduction**• Class discovery consists of two steps: • A clustering algorithm is adopted to partition the sample into K parts. • A cluster validity index is applied to determine the optimal K value. • For the class discovery problem, we focus on discovering the underlying classes from the samples.**Introduction**• Recently, researchers are paying more attention to class discovery based on the consensus clustering approaches. • They consist of two major steps: • Generating a cluster ensemble based on a clustering algorithm. • Finding a consensus partition based on this ensemble.**Introduction**• Consensus clustering have five types: • Using different clustering algorithms as the basic clustering algorithms to obtain different solutions. • Using random initializations of a single clustering algorithm. • Sub-sampling, re-sampling or adding noise to the original data. • Using selected subsets of features. • Using different K values to generate different clustering solutions**Methods**• In this paper, the approach belongs to type 4, in which the cluster ensemble is generated using different gene subsets. • Graph-based consensus clustering (GCC).**Methods**• Overview of the framework for GCC algorithm • Subspace generation • Subspace clustering • Cluster ensemble • Cluster discovery**The framework for GCC algorithm**• The framework:**The framework for GCC algorithm**• The framework:**Subspace generation**• A constant , which presents the number of genes in the subspace is generated by: where is a uniform random variable, and , for is the total number of genes.**Subspace generation**• Then, it selects the gene one by one until genes are obtained. • The index of each randomly selected gene is determined as: where denotes the hth gene, and is a uniform random variable.**Subspace generation**• Finally, the randomly selected genes are used to construct a subspace. one sample Randomly selection genes genes**The framework for GCC algorithm**• The framework:**Subspace clustering**• In the selected subspace, GCC performs two clustering approaches: • Correlation clustering • Correlation analysis • Graph partition • K-means**Correlation clustering**• Correlation analysis: calculate the correlation matrix (CM) whose entries , is the number of samples. where and denotes the ith and jth samples.**Correlation clustering**• Graph partition: use the normalized cut algorithm to partition the samples to K classes based on the CM. • A graph can be constructed, whose vertices correspond to samples , and edges are the correlation between the samples (i.e. CM).**Correlation clustering**• “Normalized cuts” is proposed by Shi and Malik in 1997, CVPR. • It’s an image segmentation method. • Pixels as vertices. • Similarity between pixels as weight edge.**Correlation clustering**• Like the normalized cuts method, we could find the label vector by solve the generalized eigenvalue problem: where is an diagonal matrix with as diagonal, is the correlation matrix. • The label vector is composed from the second smaller eigenvector .**K-means**• To minimize total intra-cluster variance, or the squared error function: where is the center of cluster .**Subspace clustering**• After obtaining the predicted labels, the adjacency matrix is constructed by the labels, whose elements are defined as: where and denote the predicted labels of the samples and .**The framework for GCC algorithm**• The framework:**Cluster ensemble**• For each , GCC repeats the above two steps B times, and obtains • B clustering solutions • B adjacency matrices • GCC constructs a consensus matrix by merging the adjacency matrix as: where represents the probability that two samples in the same class.**Cluster ensemble**• Then, GCC constructs a graph and applies the normalized cuts method. • It means the clustering result when the number of clusters is K.**The framework for GCC algorithm**• The framework:**Cluster discovery**• Define an aggregated consensus matrix : • Then, GCC converts it to a binary matrix : • By the same way, GCC converts to .**Cluster discovery**• We should compare clustering results with the aggregated matrix to decide the proper value of K. • Modified Rand Index: Penalty term for a large set of clusters. The degree of agreement between and**Cluster discovery**• The optimal number of classes is selected as • It considers the relationship between each clustering solution and the average clustering solution.**Experiment**• Experiment setting • Relationship between ARI and • Experiment results**Experiment setting**• Four combination algorithms comparison: • GCCcorr(GCC with correlation clustering) • GCCK-means(GCC with K-means) • CCHC(CC with hierarchical clustering with average linkage) • CCSOM(CC with Self-Organizing Maps) • Consensus Clustering (CC) is proposed by Monti et al. in 2003, a type 3(re-sampling) consensus clustering algorithm.**Experiment setting**• Parameters setting: • The datasets:**Experiment setting**• Adjusted Rand Index (ARI): Real index Expected index Maximum index The number of samples in the kth class in the true partition. The number of samples in the ith class in the predicted partition.**Relationship between ARI and**• The change of ARI with respect to different K:**Relationship between ARI and**• The change of with respect to different K:**Relationship between ARI and**• The correlation analysis of ARI and : The degree of dependence between ARI and is high.**Experiment results**• Estimated optimal K value by different approaches: Error terms ground truth**Experiment results**• The corresponding values of ARI: The GCC approaches outperform the CC approaches.**Experiment results**• The effect of the maximum K value: When Kmax increases, GCCcorr still correctly estimate the number of clusters in Synthetic2 dataset.**Experiment results**• The effect of the maximum K value: When Kmax increases, GCCcorr still correctly estimate the number of clusters in Leukemia dataset.**Experiment results**• The effect of the maximum K value: ζ decreases slightly when Kmax increases. ARI is not affected when Kmax increases.**Conclusion**• This paper proposes the design of a new framework, known as GCC, to discover the classes of the samples in gene expression data. • GCC can successfully estimate the true number of classes for the datasets in experiments.

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