Pattern-based Clustering

1. Pattern-based Clustering • How to cluster the five objects? • Hard to define a global similarity measure

2. What Is Pattern-based Clustering? • A cluster: a set of objects following the same pattern in a subset of dimensions (Wang et al, 2002)

3. 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

4. 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

5. Challenges of Mining GST Data • Most clustering algorithms were designed for 2D data, and cannot be directly extended for 3D data.

6. 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 )

7. Sample A Sample B Sample C Sample D Sample E Sample F Sample G Sample H 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)

8. Other Types of Coherent Clusters

9. 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.

10. 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 siand 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.

11. 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

12. 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.

13. 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.

14. s5 s4 s3 s4 s6 s3 s1 s5 s6 s2 Preprocessing Phase Suppose mins = 3 {s3,s4,s5,s6} is a coherent sample set of gene g The coherence matrix of gene g The coherence graph of gene g

15. 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

16. {} {a} {b} {c} {d} {a,b} {a,c} {a,d} {b,c} {b,d} {c,d} {a,b,c} {a,b,d} {a,c,d} {b,c,d} {a,b,c,d} Set Enumeration Tree The set enumeration tree for {a,b,c,d}

17. 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.

18. The Inverted List g2.b1 g2.b2 The table of maximal coherent sample sets for genes The table of inverted lists for samples

19. 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.

20. 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.

21. 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.

22. 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

23. 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.

24. { } {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}

25. 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

26. Gene-sample Search

27. 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)

28. A Coherent Gene Cluster from Real Data

29. Effect of Parameters Number of clusters vs. ming (mins=3,=0.8) Number of clusters vs. mins (ming=10, =0.8) Number of clusters vs.  (ming=10,mins=3)

30. 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

31. 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.

32. 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.