Reasoning about Sets using Redescription Mining
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1 Reasoning about Sets using Redescription Mining Mohammed J. Zaki Naren Ramakrishnan
2 What are redescriptions? A shift-of-vocabulary, or a different way of communicating a given piece of information.
3 Input to Redescription Mining B Canada Chile USA Brazil Cuba Russia China UK R Y Argentina France G
4 Input to Redescription Mining (contd.) B Canada Chile USA Brazil Cuba Russia China UK R Y Argentina France G
5 Input to Redescription Mining (contd.) B Canada Chile USA Brazil Cuba Russia China UK R Y Argentina France G
6 Input to Redescription Mining (contd.) B Canada Chile USA Brazil Cuba Russia China UK R Y Argentina France G
7 Input to Redescription Mining (contd.) B Canada Chile USA Brazil Cuba Russia China UK R Y Argentina France G
8 Input to Redescription Mining (contd.) B Canada Chile USA Brazil Cuba Russia China UK R Y Argentina France G
9 Given Basic Problem a set O of objects (e.g., countries) a collection of subsets (descriptors) of O Find subsets of O that can be defined in at least two ways
10 A Redescription Canada Russia China USA EXCEPT USA Chile Brazil Canada Argentina = Russia China France UK USA AND Russia China Cuba Countries with land area > 3,000,000 sq. miles Tourist Destinations in the Americas Permanent members of U.N. Security Council Countries with history of communism
11 Redescription is sort of like... association rule mining generalize from implications to equivalences conceptual clustering find clusters with dual characterizations constructive induction build features that mutually reinforce each other
12 Applications in Bioinformatics (Gene) subsets galore! Genes localized in the mitochondrion Genes up-expressed two-fold or more in heat stress Genes encoding for proteins forming the immunoglobin complex Genes involved in glucose biosynthesis Genes handpicked by Prof. Genie for further study Genes clustered together by your favorite algorithm
13 How do redescriptions happen? RG RG RG RG B Cuba R Canada Chile USA Brazil Russia China UK Y Argentina France G
14 How do redescriptions happen? RG RG RG RG B Cuba R Canada Chile USA Brazil Russia China UK Y Argentina France G
15 A game on Karnaugh maps RG RG RG RG RG RG RG RG
16 A game on Karnaugh maps RG RG RG RG RG RG RG RG
17 A game on Karnaugh maps RG RG RG RG RG RG RG RG
18 A game on Karnaugh maps RG RG RG RG RG RG RG RG
19 A game on Karnaugh maps RG RG RG RG RG RG RG RG
20 Reading off a redescription RG RG RG RG RG RG RG RG
21 Reading off a redescription RG RG RG RG RG RG RG RG ( RG RG RG RG)
22 Reading off a redescription RG RG RG RG RG RG RG RG ( RG RG RG RG) ( RG RG RG RG)
23 Reading off a redescription RG RG RG RG RG RG RG RG ( ) (RG)
24 Redescriptions help reason about sets B Canada Chile USA Brazil Cuba Russia China UK R Y Argentina France G Q: How can B be made equal to R? Ans: Subtract Y from B; intersect G with R, yielding RG.
25 Some Definitions Given a collection of objects O and descriptors D: A redescription X Y (X, Y D) holds when X Y = and X and Y induce the same set of objects.
26 Some Definitions Given a collection of objects O and descriptors D: A redescription X Y (X, Y D) holds when X Y = and X and Y induce the same set of objects. A conditional redescription X Y Z (Z D) holds when X Y = X Z = Y Z = and X Z and Y Z induce the same set of objects.
27 Some Definitions Given a collection of objects O and descriptors D: A redescription X Y (X, Y D) holds when X Y = and X and Y induce the same set of objects. A conditional redescription X Y Z (Z D) holds when X Y = X Z = Y Z = and X Z and Y Z induce the same set of objects. A redescription X Y is a non-redundant redescription iff there does not exist another redescription X Y for the same set of objects, such that X X and Y Y
28 Connections to Association Rule Mining RG RG RG RG Objects = Transactions Descriptors = Items
29 Connections to Association Rule Mining RG RG RG RG Objects = Transactions Descriptors = Items Colored cell = closed itemset (e.g., RG)
30 Connections to Association Rule Mining RG RG RG RG Objects = Transactions Descriptors = Items Reducible cluster of colored cells = closed itemset (e.g., R)
31 Connections to Association Rule Mining RG RG RG RG Objects = Transactions Descriptors = Items Reducible cluster of mixed cells = non-closed itemset (e.g., )
32 Adapting association mining algorithms Mining redescriptions reduces to: mining closed itemsets (descriptor sets) obtain submatrices reducible to these closed sets (generators) Object Descriptors o 1 d 1 d 2 d 4 d 5 d 6 o 2 d 2 d 3 d 5 d 7 o 3 d 1 d 2 d 4 d 5 d 6 o 4 d 1 d 2 d 3 d 5 d 6 d 7 o 5 d 1 d 2 d 3 d 4 d 5 d 6 d 7 o 6 d 2 d 3 d 4
33 Lattice of Closed Sets dset: d1 d2 d3 d4 d5 d6 d7 objset: o5 mingen: d1 d3 d4, d3 d4 d5, d3 d4 d6, d4 d7 dset: d2 d3 d4 dset: d1 d2 d3 d5 d6 d7 dset: d1 d2 d4 d5 d6 objset: o5 o6 objset: o4 o5 objset: o1 o3 o5 mingen: d3 d4 mingen: d1 d3, d1 d7, d3 d6, d6 d7 mingen: d1 d4, d4 d5, d4 d6 dset: d2 d3 d5 d7 objset: o2 o4 o5 mingen: d3 d5, d7 dset: d1 d2 d5 d6 objset: o1 o3 o4 o5 mingen: d1, d6 dset: d2 d3 dset: d2 d5 dset: d2 d4 objset: o2 o4 o5 o6 objset: o1 o2 o3 o4 o5 objset: o1 o3 o5 o6 mingen: d3 mingen: d5 mingen: d4 dset: d2 objset: o1 o2 o3 o4 o5 o6 mingen: d2
34 Lattice of Closed Sets dset: d1 d2 d3 d4 d5 d6 d7 objset: o5 mingen: d1 d3 d4, d3 d4 d5, d3 d4 d6, d4 d7 dset: d2 d3 d4 dset: d1 d2 d3 d5 d6 d7 dset: d1 d2 d4 d5 d6 objset: o5 o6 objset: o4 o5 objset: o1 o3 o5 mingen: d3 d4 mingen: d1 d3, d1 d7, d3 d6, d6 d7 mingen: d1 d4, d4 d5, d4 d6 dset: d2 d3 d5 d7 objset: o2 o4 o5 mingen: d3 d5, d7 dset: d1 d2 d5 d6 objset: o1 o3 o4 o5 mingen: d1, d6 d1 => d5; d6 => d5 dset: d2 d3 dset: d2 d5 dset: d2 d4 objset: o2 o4 o5 o6 objset: o1 o2 o3 o4 o5 objset: o1 o3 o5 o6 mingen: d3 mingen: d5 mingen: d4 dset: d2 objset: o1 o2 o3 o4 o5 o6 mingen: d2
35 Lattice of Closed Sets dset: d1 d2 d3 d4 d5 d6 d7 objset: o5 mingen: d1 d3 d4, d3 d4 d5, d3 d4 d6, d4 d7 dset: d2 d3 d4 dset: d1 d2 d3 d5 d6 d7 dset: d1 d2 d4 d5 d6 objset: o5 o6 objset: o4 o5 objset: o1 o3 o5 mingen: d3 d4 mingen: d1 d3, d1 d7, d3 d6, d6 d7 mingen: d1 d4, d4 d5, d4 d6 dset: d2 d3 d5 d7 objset: o2 o4 o5 mingen: d3 d5, d7 dset: d1 d2 d5 d6 objset: o1 o3 o4 o5 mingen: d1, d6 dset: d2 d3 dset: d2 d5 dset: d2 d4 objset: o2 o4 o5 o6 objset: o1 o2 o3 o4 o5 objset: o1 o3 o5 o6 mingen: d3 mingen: d5 mingen: d4 dset: d2 objset: o1 o2 o3 o4 o5 o6 mingen: d2
36 Up Closed and Personal d1 d2 d5 d6 d1 d2 d5 d1 d5 d6 d1 d2 d6 d2 d5 d6 d1 d2 d1 d5 d1 d6 d2 d6 d5 d6 d1 d6
37 Up Closed and Personal d1 d2 d5 d6 d1 d2 d5 d1 d5 d6 d1 d2 d6 d2 d5 d6 d1 d2 d1 d5 d1 d6 d2 d6 d5 d6 d1 d6 d1 <=> d6
38 Finding Minimal Generators dset: d1 d2 d3 d4 d5 d6 d7 objset: o5 mingen: d1 d3 d4, d3 d4 d5, d3 d4 d6, d4 d7 dset: d2 d3 d4 dset: d1 d2 d3 d5 d6 d7 dset: d1 d2 d4 d5 d6 objset: o5 o6 objset: o4 o5 objset: o1 o3 o5 mingen: d3 d4 mingen: d1 d3, d1 d7, d3 d6, d6 d7 mingen: d1 d4, d4 d5, d4 d6 dset: d2 d3 d5 d7 objset: o2 o4 o5 mingen: d3 d5, d7 dset: d1 d2 d5 d6 objset: o1 o3 o4 o5 mingen: d1, d6 dset: d2 d3 dset: d2 d5 dset: d2 d4 objset: o2 o4 o5 o6 objset: o1 o2 o3 o4 o5 objset: o1 o3 o5 o6 mingen: d3 mingen: d5 mingen: d4 dset: d2 objset: o1 o2 o3 o4 o5 o6 mingen: d2
39 Finding Minimal Generators dset: d1 d2 d3 d4 d5 d6 d7 objset: o5 mingen: d1 d3 d4, d3 d4 d5, d3 d4 d6, d4 d7 dset: d2 d3 d4 dset: d1 d2 d3 d5 d6 d7 dset: d1 d2 d4 d5 d6 objset: o5 o6 objset: o4 o5 objset: o1 o3 o5 mingen: d3 d4 mingen: d1 d3, d1 d7, d3 d6, d6 d7 mingen: d1 d4, d4 d5, d4 d6 dset: d2 d3 d5 d7 objset: o2 o4 o5 mingen: d3 d5, d7 dset: d1 d2 d5 d6 objset: o1 o3 o4 o5 mingen: d1, d6 dset: d2 d3 dset: d2 d5 dset: d2 d4 objset: o2 o4 o5 o6 objset: o1 o2 o3 o4 o5 objset: o1 o3 o5 o6 mingen: d3 mingen: d5 mingen: d4 dset: d2 objset: o1 o2 o3 o4 o5 o6 mingen: d2
40 Finding Minimal Generators dset: d1 d2 d3 d4 d5 d6 d7 objset: o5 mingen: d1 d3 d4, d3 d4 d5, d3 d4 d6, d4 d7 dset: d2 d3 d4 dset: d1 d2 d3 d5 d6 d7 dset: d1 d2 d4 d5 d6 objset: o5 o6 objset: o4 o5 objset: o1 o3 o5 mingen: d3 d4 mingen: d1 d3, d1 d7, d3 d6, d6 d7 mingen: d1 d4, d4 d5, d4 d6 dset: d2 d3 d5 d7 objset: o2 o4 o5 mingen: d3 d5, d7 dset: d1 d2 d5 d6 objset: o1 o3 o4 o5 mingen: d1, d6 Diff = {d1,d6} dset: d2 d3 dset: d2 d5 dset: d2 d4 objset: o2 o4 o5 o6 objset: o1 o2 o3 o4 o5 objset: o1 o3 o5 o6 mingen: d3 mingen: d5 mingen: d4 dset: d2 objset: o1 o2 o3 o4 o5 o6 mingen: d2
41 Not so fast... Datasets are 50% dense! cannot rely on pruning to help handle large datasets Solution approach CHARM-L: Mining with constraints Only expand lattice around objects/descriptors of interest
42 Exploring Gene Sets in Bioinformatics Vocabularies GO functional categories (BIO, CEL, and MOL) Expression range buckets in specific microarray experiments Gene clusters
43 Interactive Exploration w/ CHARM-L What is the relationship between... d183 (ORFs 5 expressed in 15 minutes of heat shock) d184 (ORFs 5 expressed in 20 minutes of heat shock) Answer: d183 d388 d460 d515 d184 d309 d388: (GO MOL mannose transporter) d460: (GO CEL external protective structure) d515: (GO BIO fructose metabolism) d309: (GO MOL molecular function unknown)
44 Another example What is the relationship between... d141 (ORFs 2 expressed in 10 minutes of heat shock) d184 (ORFs 5 expressed in 20 minutes of heat shock) Answer: d141 d515 d608 d184 d183 d515: (GO BIO fructose metabolism) d608: (ORFS 4 expressed in histone depletion) d183: (ORFs 5 expressed in 15 minutes of heat shock)
45 Performance Results Time (s) Total Lattice Mingen Rules G1 Dset Length G Minimum Support (%) Minimum Support (%) Total Lattice Mingen Rules G G3 Time (s) Dset Length e Minimum Support (%) Minimum Support (%)
46 Recap Redescriptions help reason about set collections Conjunctive forms handle set intersections and negations Empowers biologist to create and work with vocabularies Algorithmic innovations Lattice mining, finding minimal generators, constraint propagation Established connections to boolean formula manipulation
47 Future Work Story telling Find a sequence of redescriptions connecting disjoint sets X and Y Schema matching X O 1, Y O 2, O 1 and O 2 are related by relation R Generalized boolean expressions Mine redescriptions in more expressive forms
48 Acknowledgements Collaborators Deept Kumar (Virginia Tech) Laxmi Parida (IBM TJ Watson) Funding NSF CAREER IIS , DOE Career DE-FG02-02ER25538, NSF grants EIA and EMT (Zaki) NSF grants IBN and EIA (Ramakrishnan)
49 Questions? For related work, see: N. Ramakrishnan et al., Turning CARTwheels: An Alternating Algorithm for Mining Redescriptions, in Proceedings of KDD 04, pages , L. Parida and N. Ramakrishnan, Redescription Mining: Structure Theory and Algorithms, in Proceedings of AAAI 05, pages , July Contact: Naren Ramakrishnan Department of Computer Science Virginia Tech, Blacksburg, VA naren
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