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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
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c) scaling patterns
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
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3D gene-sample-time data
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 3D GST data set
The 2D representation
A coherent gene cluster
• 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
Sample A
Sample E
Sample B
Sample F
Sample C
Sample G
Sample D
Sample H
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 : |G|≥ming, |S|≥mins, 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 Person’s 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
s5
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
The coherence matrix
of gene g
s6
s4
s1
s3
s2
The coherence graph
of gene g
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s3
s4
s5
s6
{s3,s4,s5,s6} is a
coherent sample set of
gene g
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
{}
{a}
{a,b}
{b}
{a,c} {a,d} {b,c} {b,d}
{a,b,c} {a,b,d} {a,c,d}
{c}
{c,d}
{b,c,d}
{a,b,c,d}
The set enumeration tree for {a,b,c,d}
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{d}
Find the Maximal Coherent
Subset of Genes
 After the pre-processing phase:
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}
 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.
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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 S={si1,…,sik},
intersect the inverted lists of si1,…,sik.
 For example, given S={s1,s2,s3},
Ls1^Ls2^Ls3={g1.b1,g3.b1,g4.b1}, so Gs={g1,g3,g4}.
 Suppose the parent of S is S’={si1,…,sik-1}, then
LS=LS’ 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 S={si1,…,sik}, a
sample sjtails, if
 j > 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 S={si1,…,sik},
 if |S|+|tails|< 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
{}
{s2}
{s3,s4}
{s1}
{s2,s3,s4,s5}
{g4.b2, g5.b1}
{s1,s2}
{s3,s4}
{s1,s3}
{}
{s1,s4}
{}
{g1.b1, g2.b1, g3.b1, g4.b1}
{g1.b1, g3.b1, g4.b1}
{g1.b1, g2.b1, g3.b1}
{s1,s2,s3}
{}
{s1,s2,s4}
{}
{g1.b1,g3.b1,g4.b1}
{g1.b1,g2.b1,g3.b1}
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{s2,s3}
{}
{s3}
{}
{s4}
{}
{s2,s4}
{}
{g1.b1, g3.b1, g4.b1} {g1.b1, g2.b1, g3.b1}
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
Number of subjects to
enumerate
Coherent objects
Efficiency on GST data
samples
101~102
genes
103~104
Single set of maxmial
coherent genes
Single or multiple
sets of maxmial
coherent sample
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
(mins=3,=0.8)
Number of clusters vs. mins
(ming=10,  =0.8)
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Number of clusters vs. 
(ming=10,mins=3)
Scalability
Scalability of phase 1
Scalability w.r.t. number of
genes (number of samples: 30)
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Scalability w.r.t. number of
samples (number of genes:
3,000)
Conclusion
 We define the new problem of mining
coherent gene clusters from the novel genesample-time microarray data.
 We propose two approaches: the samplegene 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.
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