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### Bi-Clustering (I): δ-Bi-Cluster

- For a submatrix
*I x J*

*T*he 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 ≥*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(
*e**ij*) =*e**ij**−**e**iJ**−**e**Ij**e**IJ*

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