BiClustering II

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Bi-Clustering II

• COMP 790-90 Seminar
• Spring 2009

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Coherent Cluster
Want to accommodate noises but not outliers
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Coherent Cluster

• Coherent cluster
• Subspace clustering
• pair-wise disparity
• For a 2?2 (sub)matrix consisting of objects x,
  y and attributes a, b

mutual bias of attribute a
mutual bias of attribute b
attribute
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Coherent Cluster

• A 2?2 (sub)matrix is a ?-coherent cluster if its
  D value is less than or equal to ?.
• An m?n matrix X is a ?-coherent cluster if every
  2?2 submatrix of X is ?-coherent cluster.
• A ?-coherent cluster is a maximum ?-coherent
  cluster if it is not a submatrix of any other
  ?-coherent cluster.
• Objective given a data matrix and a threshold ?,
  find all maximum ?-coherent clusters.

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Coherent Cluster

• Challenges
• Finding subspace clustering based on distance
  itself is already a difficult task due to the
  curse of dimensionality.
• The (sub)set of objects and the (sub)set of
  attributes that form a cluster are unknown in
  advance and may not be adjacent to each other in
  the data matrix.
• The actual values of the objects in a coherent
  cluster may be far apart from each other.
• Each object or attribute in a coherent cluster
  may bear some relative bias (that are unknown in
  advance) and such bias may be local to the
  coherent cluster.

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Coherent Cluster
Compute the maximum coherent attribute sets for
each pair of objects
Two-way Pruning
Construct the lexicographical tree
Post-order traverse the tree to find maximum
coherent clusters
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Coherent Cluster

• Observation Given a pair of objects o1, o2 and
  a (sub)set of attributes a1, a2, , ak, the 2?k
  submatrix is a ?-coherent cluster iff, for every
  attribute ai, the mutual bias (do1ai do2ai)
  does not differ from each other by more than ?.

If ? 1.5, then a1,a2,a3,a4,a5 is a coherent
attribute set (CAS) of (o1,o2).
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Coherent Cluster

• Observation given a subset of objects o1, o2,
  , ol and a subset of attributes a1, a2, ,
  ak, the l?k submatrix is a ?-coherent cluster
  iff a1, a2, , ak is a coherent attribute set
  for every pair of objects (oi,oj) where 1 ? i, j
  ? l.

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Coherent Cluster

• Strategy find the maximum coherent attribute
  sets for each pair of objects with respect to the
  given threshold ?.

? 1
The maximum coherent attribute sets define the
search space for maximum coherent clusters.
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Two Way Pruning
(o0,o2) ?(a0,a1,a2) (o1,o2) ?(a0,a1,a2)
(a0,a1) ?(o0,o1,o2) (a0,a2) ?(o1,o2,o3) (a1,a2)
?(o1,o2,o4) (a1,a2) ?(o0,o2,o4)
(o0,o2) ?(a0,a1,a2) (o1,o2) ?(a0,a1,a2)
(a0,a1) ?(o0,o1,o2) (a0,a2) ?(o1,o2,o3) (a1,a2)
?(o1,o2,o4) (a1,a2) ?(o0,o2,o4)
delta1 nc 3 nr 3
MCAS
MCOS
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Coherent Cluster

• Strategy grouping object pairs by their CAS and,
  for each group, find the maximum clique(s).
• Implementation using a lexicographical tree to
  organize the object pairs and to generate all
  maximum coherent clusters with a single
  post-order traversal of the tree.

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a0,a1 (o0,o1)
(o1,o2)
(o0,o2)
a0,a2 (o1,o3),(o2,o3)
(o1,o2)
(o0,o2)
a1,a2 (o0,o4),(o1,o4),(o2,o4)
(o1,o2)
(o0,o2)
a2,a3 (o0,o1),(o1,o2)
(o0,o2)
a0,a1,a2 (o1,o2)
(o0,o2)
a0,a1,a2,a3 (o0,o2)
a0
a2
a1
assume ? 1
a1
a2
a2
a3
(o0,o1)
(o1,o3)
(o0,o4)
(o0,o1)
(o2,o3)
(o1,o4)
(o1,o2)
a2
(o2,o4)
(o1,o2)
a3
(o0,o2)
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o0,o2 ? a0,a1,a2,a3
o1,o2 ? a0,a1,a2
o0,o1,o2 ? a0,a1
o1,o2,o3 ? a0,a2
o0,o2,o4 ? a1,a2
o1,o2,o4 ? a1,a2
o0,o1,o2 ? a2,a3
(o0,o2)
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Coherent Cluster

• High expressive power
• The coherent cluster can capture many interesting
  and meaningful patterns overlooked by previous
  clustering methods.
• Efficient and highly scalable
• Wide applications
• Gene expression analysis
• Collaborative filtering

subspace cluster
coherent cluster
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Remark

• Comparing to Bicluster
• Can well separate noises and outliers
• No random data insertion and replacement
• Produce optimal solution

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Definition of OP-Cluster

• Let I be a subset of genes in the database.
  Let J be a subset of conditions. We say ltI, Jgt
  forms an Order Preserving Cluster
  (OP-Cluster), if one
  of the following relationships exists for any
  pair of conditions.

Expression Levels
A1 A2 A3 A4

when
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Problem Statement

• Given a gene expression matrix, our goal is to
  find all the statistically significant
  OP-Clusters. The significance is ensured by the
  minimal size threshold nc and nr.

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Conversion to Sequence Mining Problem
Sequence
Expression Levels
A1 A2 A3 A4
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Ming OP-Clusters A naïve approach

• A naïve approach
• Enumerate all possible subsequences in a prefix
  tree.
• For each subsequences, collect all genes that
  contain the subsequences.
• Challenge
• The total number of distinct subsequences are

root
a
b
c
d
a
b
c
d
a
c
d

b
c
d
b
d
b
c
c
d
a
d

d
d
c
b
d
b
c
d
c
a
d

A Complete Prefix Tree with 4 items a,b,c,d
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Mining OP-Clusters Prefix Tree

• Goal
• Build a compact prefix tree that includes all
  sub-sequences only occurring in the original
  database.
• Strategies
• Depth-First Traversal
• Suffix concatenation Visit subsequences that
  only exist in the input sequences.
• Apriori Property Visit subsequences that are
  sufficiently supported in order to derive longer
  subsequences.

Root
a1,2
b3
a1,2
a1,2,3
a1,2,3
d1
b2
a3
d1,3
d1,3
d1,2,3
d1,2,3
d2
b1
c1,3
d3
c1,2,3
c1
c2
c3
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References

• J. Yang, W. Wang, H. Wang, P. Yu, Delta-cluster
  capturing subspace correlation in a large data
  set, Proceedings of the 18th IEEE International
  Conference on Data Engineering (ICDE), pp.
  517-528, 2002.
• H. Wang, W. Wang, J. Yang, P. Yu, Clustering by
  pattern similarity in large data sets, to appear
  in Proceedings of the ACM SIGMOD International
  Conference on Management of Data (SIGMOD), 2002.
• Y. Sungroh,  C. Nardini, L. Benini, G. De
  Micheli, Enhanced pClustering and its
  applications to gene expression data
  Bioinformatics and Bioengineering, 2004.
• J. Liu and W. Wang, OP-Cluster clustering by
  tendency in high dimensional space, ICDM03.