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Bi-Clustering (I): δ-Bi-Cluster

  • For a submatrix I x J
    • The mean of the i-th row:

\[e_{iJ}=\frac{1}{|J|} \sum_{j\epsilon J}e_{ij}\]

    • The mean of the j-th column:

\[e_{Ij}=\frac{1}{|I|} \sum_{i\epsilon I}e_{ij}\]

    • The mean of all elements in the submatrix:

\[e_{IJ}=\frac{1}{|I||J|} \sum_{i\epsilon I,j\epsilon J}e_{ij}=\frac{1}{|I|} \sum_{i\epsilon I}e_{iJ}=\frac{1}{|J|} \sum_{j\epsilon J}e_{Ij}\]

    • The quality of the submatrix as a bi-cluster can be measured by the mean squared residue value

\[H(I\times J)=\frac{1}{|I||J|} \sum_{i\epsilon I,j\epsilon J}(e_{ij}-e_{iJ}-e_{Ij}+e_{IJ})^{2}\]

  • A submatrix I x J is δ-bi-cluster if H(I x J) ≤ δ where δ ≥ 0 is a threshold. When δ = 0, I x J is a perfect bi-cluster with coherent values. By setting δ > 0, a user can specify the tolerance of average noise per element against a perfect bi-cluster
    • residue(eij) = eij eiJ eIj + eIJ

Speaker notes:

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