Pattern-based Clustering

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Pattern-based Clustering

• How to cluster the five objects?
• Hard to define a global similarity measure

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What Is Pattern-based Clustering?

• A cluster a set of objects following the same
  pattern in a subset of dimensions (Wang et al,
  2002)

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Challenges

• Most clustering approaches do not address the
  temporal variations in time series gene
  expression data, which is an important feature
  and affect the performance.
• Previous approaches try to find coherent patterns
  and clusters w.r.t. the entire set of attributes
  
• Patterns may be embedded in sub attribute spaces
• Only a subset of genes participate in any
  cellular processes of interest
• Any cellular process may take place only in a
  subset of experiment conditions.

a) raw data b) shifting
patterns c) scaling patterns
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Gene-Sample-Time (GST) Microarray Data
A collection of samples
2D time-series data

• The GST microarray data consist of three
  dimensions
• The samples often exhibit various phenotypes,
  e.g., cancer vs. control

3D gene-sample-time data
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Challenges of Mining GST Data

• Most clustering algorithms were designed for 2D
  data, and cannot be directly extended for 3D data.

Challenges 2D data 3D data
Mining Process Partition genes Partition genes and samples simultaneously
Cluster model Two types of variables Three types of variables
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Coherent Gene Cluster
A coherent gene cluster
The 2D representation
A 3D GST data set

• The group of samples (sj1, sj2, sj3 ) may
  exhibit the same phenotype
• The group of genes (gi1,gi2,gi3) may be strongly
  correlated to the phenotype shared by (sj1, sj2,
  sj3 )

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Results from a Real Data Set

• The Multiple Sclerosis (MS) data consist of
• 4324 genes
• 13 MS patients
• 10 time points before and after IFN-? treatment
• 25 coherent gene clusters were reported

An example of coherent gene clusters (107 genes,
8 samples)
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Other Types of Coherent Clusters
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Problem Definition

• Given a GST microarray data matrix M, a maximal
  coherent gene cluster C(G?S) is a combination of
  a subset of genes G and a subset of samples S
  such that
• Coherent the subset of genes G are coherent
  across the subset of samples S
• Significant Gming, Smins, where ming and
  mins are user-specified parameters
• Maximal any insertion of g?G or s?S will make C
  not coherent.
• The problem of mining coherent gene clusters is
  to find the complete set of maximal coherent gene
  clusters in M.

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Coherence Measure

• Various coherence measures exist.
• Measure selection is application dependent.
• A general coherence model
• Given a coherence measure sim() and a
  user-specified threshold ?,
• A gene ga is coherent on samples si and sj, if
  sim(pai,paj) ?.
• Coherent gene matrix (G1,S1) if every gene gi ?
  G1 is coherent across samples in S1.
• Trivial coherent gene matrix (gi, sj), (G,
  sj)
• We choose the Persons correlation coefficient.
• Other coherence measures are also applicable.

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Related Work

• Clustering algorithms on Gene-Sample or
  Gene-Time microarray data
• The cluster model is completely different
• Subspace clustering
• Find subsets of objects coherent with subsets of
  attributes
• Frequent pattern mining
• Find subsets of items frequently appearing in
  transaction databases

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Algorithm Outline

• Phase 1 (Pre-processing) For each gene g, find
  the complete set of maximal coherent sample sets
  of gene g.
• Phase 2 Compute the complete set of maximal
  coherent gene clusters based on pre-processing
  results.

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Coherent Sample Sets

• Given a gene g, a maximal coherent sample set of
  g is a subset of samples Si such that
• coherent g is coherent across Si
• significant Si ? mins
• maximal there exists no superset S?Si such
  that g is also coherent with S.
• (g? Si ) is a building block for coherent gene
  clusters including g.

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Preprocessing Phase
Suppose mins 3
s1 s2 s3 s4 s5 s6
s1 1 1 0 1 0 0
s2 1 1 0 0 0 0
s3 0 0 1 1 1 1
s4 1 0 1 1 1 1
s5 0 0 1 1 1 1
s6 0 0 1 1 1 1
s3,s4,s5,s6 is a coherent sample set of gene g
The coherence matrix of gene g
The coherence graph of gene g
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Sample-gene Search

• Set enumeration tree
• Enumerate all subsets of samples systematically.
• Each node on the tree corresponds to a subset of
  samples.
• For each node S
• Find the maximal set of genes Gs which is
  coherent with S

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Set Enumeration Tree
The set enumeration tree for a,b,c,d
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Find the Maximal Coherent Subset of Genes

• After the pre-processing phase
• Given a subset of samples S, how to find the
  maximal coherent set of genes GS?
• Expensive approach scan the table once
• For each S, Gs can be derived by a single
  scan of the maximal coherent samples of all
  genes. If S ? Sj, g ? Gs.
• Efficient approach use the inverted list.

g1 s1, s2, s3, s4, s5
g2 s1,s2,s4, s1,s5
g3 s1,s2,s3,s4,s5
g4 s1,s2,s3,s5,s6
g5 s1,s5,s6
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The Inverted List
Gene Maximal Coherent sample sets
g1 s1, s2, s3, s4, s5
g2 s1, s2, s4, s1, s5
g3 s1, s2, s3, s4, s5
g4 s1, s2, s3, s5, s6
g5 s1, s5, s6
g2.b1
g2.b2
The table of maximal coherent sample sets for
genes
Sample The inverted list
s1 g1.b1, g2.b1, g2.b2, g3.b1, g4.b1, g5.b1
s2 g1.b1, g2.b1, g3.b1, g4.b1
s3 g1.b1, g3.b1, g4.b1
s4 g1.b1, g2.b1, g3.b1
s5 g1.b1, g2.b2, g3.b1, g4.b2, g5.b1
s6 g4.b2, g5.b1
The table of inverted lists for samples
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Intersection Instead of Scanning

• Given a subset of samples Ssi1,,sik,
  intersect the inverted lists of si1,,sik.
• For example, given Ss1,s2,s3,
  Ls1Ls2Ls3g1.b1,g3.b1,g4.b1, so
  Gsg1,g3,g4.
• Suppose the parent of S is Ssi1,,sik-1,
  then LSLS ?Lsik.

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Anti-monotonic Property

• Given a combination (G?S),
• if G is not coherent on S,
• then for any superset S?S, G cannot be coherent
  on S.
• For any descendant S of S on the tree
• let GS be the maximal coherent gene set of S,
• let GS be the maximal coherent gene sets of S,
• since S?S, we have GS? GS.

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Pruning Irrelevant Samples

• Given a subset of samples Ssi1,,sik, a
  sample sj?tails, if
• j gt ik
• there exists at least ming genes g such that g
  is coherent with S?sj
• Samples sl?tails(irrelevant samples) cannot be
  used to extend S.

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Pruning Unpromising Nodes

• Given a subset of samples Ssi1,,sik,
• if Stailslt mins, then prune the subtree of
  S.
• let the maximal coherent subset of genes of S
  be Gs,
• if there exists (G?S) such that
• (S?tails) ? S
• Gs?G,
• the prune the subtree of S

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Determination of Maximal Coherent Gene Clusters

• The depth-first search strategy
• For any superset S of S, S is
• visited before S
• or a child of S.
• To determine whether a coherent gene cluster
  (Gs?S) is maximal,
• check (Gs?S) after visiting all its children,
• report (Gs?S) if it is not subsumed.

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Sample The inverted list
s1 g1.b1, g2.b1, g2.b2, g3.b1, g4.b1, g5.b1
s2 g1.b1, g2.b1, g3.b1, g4.b1
s3 g1.b1, g3.b1, g4.b1
s4 g1.b1, g2.b1, g3.b1
s5 g1.b1, g2.b2, g3.b1, g4.b2, g5.b1
s6 g4.b2, g5.b1

s2 s3,s4
s1 s2,s3,s4,s5
s3
s4
s1,s4 g1.b1, g2.b1, g3.b1
s2,s3 g1.b1, g3.b1, g4.b1
s2,s4 g1.b1, g2.b1, g3.b1
s1,s2 s3,s4 g1.b1, g2.b1, g3.b1, g4.b1
s1,s3 g1.b1, g3.b1, g4.b1
s1,s2,s3 g1.b1,g3.b1,g4.b1
s1,s2,s4 g1.b1,g2.b1,g3.b1
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Mining Coherent Gene Clusters

• Systematic enumeration of genes and samples
• Sample-Gene Search
• Gene-Sample Search
• Pruning rules
• Determination of whether a coherent gene cluster
  (G?S) is maximal

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Gene-sample Search
Sample-Gene Search Gene-Sample Search
Subjects to enumerate samples genes
Number of subjects to enumerate 101102 103104
Coherent objects Single set of maxmial coherent genes Single or multiple sets of maxmial coherent sample
Efficiency on GST data High Low
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Experiment Data Sets

• Real-world gene expression data
• 4324 genes
• 13 multiple sclerosis (MS) patients
• before and at 1,2,4,8,24,48,120 and 168 hours
  after IFN-? treatment
• Synthetic data
• Given the number of genes NG, samples NS and
  coherent gene clusters NC
• Simulate the pre-processing results
• Embed NC maximal coherent gene clusters (G?S)

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A Coherent Gene Cluster from Real Data
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Effect of Parameters
Number of clusters vs. ming (mins3,?0.8)
Number of clusters vs. mins (ming10, ? 0.8)
Number of clusters vs. ? (ming10,mins3)
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Scalability
Scalability w.r.t. number of genes (number of
samples 30)
Scalability w.r.t. number of samples (number of
genes 3,000)
Scalability of phase 1
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Conclusion

• We define the new problem of mining coherent
  gene clusters from the novel gene-sample-time
  microarray data.
• We propose two approaches the sample-gene
  search and the gene-sample search.
• We conduct an extensive empirical evaluation on
  both real and synthetic data sets.

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Future Work

• New problems from the gene-sample-time
  microarray data
• Coherent sample clusters (G?S)
• for each s?S, any pair of genes gi, gj?G has
  coherent patterns.
• Coherent gene-sample clusters (G?S),
• both a coherent gene cluster and a coherent
  sample cluster.