Reasoning about Sets using Redescription Mining Mohammed J. Zaki - PowerPoint PPT Presentation

Reasoning about Sets using Redescription Mining Mohammed J. Zaki Naren Ramakrishnan zaki@cs.rpi.edu naren@cs.vt.edu

What are redescriptions? A shift-of-vocabulary, or a different way of communicating a given piece of information.

Input to Redescription Mining B R Cuba Canada Russia China Chile USA Brazil UK Argentina France G Y

Input to Redescription Mining (contd.) B R Cuba Canada Russia China Chile USA Brazil UK Argentina France G Y

Basic Problem Given • 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

A Redescription USA Canada Russia Chile Russia Russia China = EXCEPT AND China Brazil France China Cuba Canada UK USA USA Argentina ‘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’

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

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 • · · ·

How do redescriptions happen? RG RG RG RG BY B R Cuba BY Canada Russia China Chile BY USA Brazil UK BY Argentina France G Y

How do redescriptions happen? RG RG RG RG B R BY Cuba BY Canada Russia China Chile BY USA Brazil UK BY Argentina France G Y

A game on Karnaugh maps RG RG RG RG RG RG RG RG BY BY BY BY BY BY BY BY

Reading off a redescription RG RG RG RG RG RG RG RG BY BY BY BY BY BY BY BY

Reading off a redescription RG RG RG RG RG RG RG RG BY BY BY BY BY BY BY BY ( BY RG ∨ BY RG ∨ BY RG ∨ BY RG )

Reading off a redescription RG RG RG RG RG RG RG RG BY BY BY BY BY BY BY BY ( BY RG ∨ BY RG ∨ BY RG ∨ BY RG ) ⇔ ( BY RG ∨ BY RG ∨ BY RG ∨ BY RG )

Reading off a redescription RG RG RG RG RG RG RG RG BY BY BY BY BY BY BY BY ( BY ) ⇔ ( RG )

Redescriptions help reason about sets B R Cuba Canada Russia China Chile USA Brazil UK Argentina France G Y Q: How can B be made equal to R ? Ans: Subtract Y from B ; intersect G with R , yielding BY ⇔ RG .

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.

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.

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

Connections to Association Rule Mining RG RG RG RG BY BY Objects = Transactions Descriptors = Items BY BY

Connections to Association Rule Mining RG RG RG RG BY BY Objects = Transactions Descriptors = Items BY BY Colored cell = closed itemset (e.g., BY RG )

Connections to Association Rule Mining RG RG RG RG BY BY Objects = Transactions Descriptors = Items BY BY Reducible cluster of colored cells = closed itemset (e.g., BY R )

Connections to Association Rule Mining RG RG RG RG BY BY Objects = Transactions Descriptors = Items BY BY Reducible cluster of mixed cells = non-closed itemset (e.g., BY )

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

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 d4 d5 d6 dset: d1 d2 d3 d5 d6 d7 objset: o1 o3 o5 objset: o5 o6 objset: o4 o5 mingen: d1 d4, d4 d5, d4 d6 mingen: d3 d4 mingen: d1 d3, d1 d7, d3 d6, d6 d7 dset: d2 d3 d5 d7 dset: d1 d2 d5 d6 objset: o2 o4 o5 objset: o1 o3 o4 o5 mingen: d3 d5, d7 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

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 d4 d5 d6 dset: d1 d2 d3 d5 d6 d7 objset: o1 o3 o5 objset: o5 o6 objset: o4 o5 mingen: d1 d4, d4 d5, d4 d6 mingen: d3 d4 mingen: d1 d3, d1 d7, d3 d6, d6 d7 dset: d2 d3 d5 d7 dset: d1 d2 d5 d6 objset: o2 o4 o5 objset: o1 o3 o4 o5 mingen: d3 d5, d7 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

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 d4 d5 d6 dset: d1 d2 d3 d5 d6 d7 objset: o1 o3 o5 objset: o5 o6 objset: o4 o5 mingen: d1 d4, d4 d5, d4 d6 mingen: d3 d4 mingen: d1 d3, d1 d7, d3 d6, d6 d7 dset: d2 d3 d5 d7 dset: d1 d2 d5 d6 objset: o2 o4 o5 objset: o1 o3 o4 o5 mingen: d3 d5, d7 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

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

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

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 d4 d5 d6 dset: d1 d2 d3 d5 d6 d7 objset: o1 o3 o5 objset: o5 o6 objset: o4 o5 mingen: d1 d4, d4 d5, d4 d6 mingen: d3 d4 mingen: d1 d3, d1 d7, d3 d6, d6 d7 dset: d2 d3 d5 d7 dset: d1 d2 d5 d6 objset: o2 o4 o5 objset: o1 o3 o4 o5 mingen: d3 d5, d7 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

Reasoning about Sets using Redescription Mining Mohammed J. Zaki - PowerPoint PPT Presentation

Reasoning about Sets using Redescription Mining Mohammed J. Zaki Naren Ramakrishnan zaki@cs.rpi.edu naren@cs.vt.edu What are redescriptions? A shift-of-vocabulary, or a different way of communicating a given piece of information. Input to

Aligning Discourse and Argumentation Structures using Subtrees and Redescription Mining Laurine

Redescription Mining 10 July 2014 An Example In last season of Italys Serie A, the games in

Reasoning with Sets and Sums of Sets Markus Bender mbender@uni-koblenz.de Universitt

Redescription Mining Pauli Miettinen 17 November 2010 An Example VLDB ICDM SDM

M odels for Inexact Reasoning Fuzzy Logic Lesson 1 Crisp and Fuzzy Sets M aster in

Foundations Boolean Reasoning - George Boole, 1847, Brown 1990 Rough Sets - Zdzislaw

Association Rule Mining 1 What Is Association Rule Mining? Association rule mining is finding

Data Mining Learning from Large Data Sets Lecture 8

Mining for Contrasting Sets (STUCCO) Camilo Arango Department of Computing Science University of

Argument Mining: the Bottleneck of Knowledge and Reasoning Patrick Saint-Dizier IRIT - CNRS,

Automated Reasoning LTL Model Checking Alan Bundy Automated Reasoning LTL Model Checking

Intro to Mathematical Reasoning via Discrete Mathematics CMSC-37115 Instructor: Laszlo Babai

From Reasoning with Constraints to Mining Constraints: Multi-Objective Parameter Fitting in

Data Mining Learning from Large Data Sets Lecture 3

Data Mining Learning from Large Data Sets Lecture 2

Introduction Sequents and Multisets, Sets and Provability Logic Formalisation of cut-admissibility

Web Mining Web Mining Web Mining Web Mining Web mining is the use of data mining techniques

Discrete Mathematics & Mathematical Reasoning Basic Structures: Sets, Functions, Relations,

Discrete Mathematics & Mathematical Reasoning Basic Structures: Sets, Functions, Relations,

Discrete Mathematics & Mathematical Reasoning Basic Structures: Sets, Functions, Relations,

Mining and Pattern Analysis in Large Data Sets for Biological Information. David W. Mount

Mining and Pattern Analysis in Large Data Sets for Biological Information. David W. Mount

Data Mining Learning from Large Data Sets Lecture 1

Web Mining Web Mining Web mining is the use of data mining techniques to automatically

Reasoning about Sets using Redescription Mining Mohammed J. Zaki - PowerPoint PPT Presentation

Reasoning about Sets using Redescription Mining Mohammed J. Zaki Naren Ramakrishnan zaki@cs.rpi.edu naren@cs.vt.edu What are redescriptions? A shift-of-vocabulary, or a different way of communicating a given piece of information. Input to

Aligning Discourse and Argumentation Structures using Subtrees and Redescription Mining Laurine

Redescription Mining 10 July 2014 An Example In last season of Italys Serie A, the games in

Reasoning with Sets and Sums of Sets Markus Bender mbender@uni-koblenz.de Universitt

Redescription Mining Pauli Miettinen 17 November 2010 An Example VLDB ICDM SDM

M odels for Inexact Reasoning Fuzzy Logic Lesson 1 Crisp and Fuzzy Sets M aster in

Foundations Boolean Reasoning - George Boole, 1847, Brown 1990 Rough Sets - Zdzislaw

Association Rule Mining 1 What Is Association Rule Mining? Association rule mining is finding

Data Mining Learning from Large Data Sets Lecture 8

Mining for Contrasting Sets (STUCCO) Camilo Arango Department of Computing Science University of

Argument Mining: the Bottleneck of Knowledge and Reasoning Patrick Saint-Dizier IRIT - CNRS,

Automated Reasoning LTL Model Checking Alan Bundy Automated Reasoning LTL Model Checking

Intro to Mathematical Reasoning via Discrete Mathematics CMSC-37115 Instructor: Laszlo Babai

From Reasoning with Constraints to Mining Constraints: Multi-Objective Parameter Fitting in

Data Mining Learning from Large Data Sets Lecture 3

Data Mining Learning from Large Data Sets Lecture 2

Introduction Sequents and Multisets, Sets and Provability Logic Formalisation of cut-admissibility

Web Mining Web Mining Web Mining Web Mining Web mining is the use of data mining techniques

Discrete Mathematics &amp; Mathematical Reasoning Basic Structures: Sets, Functions, Relations,

Discrete Mathematics &amp; Mathematical Reasoning Basic Structures: Sets, Functions, Relations,

Discrete Mathematics &amp; Mathematical Reasoning Basic Structures: Sets, Functions, Relations,

Mining and Pattern Analysis in Large Data Sets for Biological Information. David W. Mount

Mining and Pattern Analysis in Large Data Sets for Biological Information. David W. Mount

Data Mining Learning from Large Data Sets Lecture 1

Web Mining Web Mining Web mining is the use of data mining techniques to automatically

Discrete Mathematics & Mathematical Reasoning Basic Structures: Sets, Functions, Relations,

Discrete Mathematics & Mathematical Reasoning Basic Structures: Sets, Functions, Relations,

Discrete Mathematics & Mathematical Reasoning Basic Structures: Sets, Functions, Relations,