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experiments on the data, which consists of samples from patients diagnosed with
acute lymphoblastic leukemia (ALL) or acute myeloid leukemia (AML) diseases
(see [1, 2, 7, 12, 13]), confirm that the algorithm outperforms the previous results
in the quality of solution as well as computation time. Consistent biclustering is
also used to analyze scalp EEG data obtained from epileptic patients undergoing
treatment with a vagus nerve stimulator (VNS) (see [4]).
The rest of the chapter is organized as follows: We first introduce mathemat-
ical formulations for consistent biclustering and introduce the application of fea-
ture selection for consistent biclustering. Next, the complexity results are shown
and the chapter is concluded with closing remarks.
13.2. Consistent Biclustering
Given a classification of the samples, S r ,let S =( s jr ) n×k denote a 0-1 matrix
where s jr =1if sample j is classified as a member of the class r (i.e., a j
S r ),
and s jr =0otherwise. Similarly, given a classification of the features, F r ,let
F =( f ir ) m×k denote a 0-1 matrix where f ir =1if feature i belongs to class r
(i.e., a i
F r ), and f ir =0otherwise. Construct corresponding centroids for the
samples and features using these matrices as follows
C S = AS ( S T S ) 1 =( c ) m×r
(13.1)
C F = A T F ( F T F ) 1 =( c ) n×r
(13.2)
The elements of the matrices, c and c , represent the average expression of the
corresponding sample and feature in class ξ , respectively. In particular,
c = j =1 a ij s
= j|a j ∈S ξ a ij
|
j =1 s
,
S ξ |
and
= i|a i ∈F ξ a ij
|
c = i =1 a ij f
i =1 f
.
F ξ |
Using the elements of matrix C s , one can assign a feature to a class where it
is over-expressed. Therefore feature i is assigned to class r if c ir =max ξ {
c }
,
i.e.,
F r =
c ir >c ,
a i
ξ,ξ
= r.
(13.3)
Note that the constructed classification of the features, F r , is not necessarily
the same as classification F r . Similarly, one can use the elements of matrix C F to
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