interval pattern structures: an introdution

Building Concept Lattices fromNumerical Data with Pattern Structures

A tutorial

Mehdi Kaytoue and Amedeo Napoli

[email protected]

November, 9th 2010

Context

Formal Concept AnalysisWorks on binary relationsClassification of objects w.r.t. the attributes they have incommon within formal concepts (extent, intent)Ordering concepts gives a mathematical structure

Ganter & Wille, Springer mathematical foundations 99

Concept lattice : useful for many tasksSimultaneous classification of objects and their attributesInformation organizationKnowledge discovery in databases (closed itemsets,associations rules)Information retrieval...

Valtchev & al., ICFCA 04 – Wille, JETAI 02

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Problem and proposition

When facing numerical data ?

Transform data into binary, a general problemConceptual scaling (binarization)Important choices to be madeLoss of information, of links between objects

Avoiding binarization ? Considering a similarity relationbetween values ?

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Outline

1 Formal Concept Analysis

2 Pattern structures and intervals

3 Introducing a similarity relation

4 Conclusion

Formal context

Given by (G,M, I) withG a set of objectsM a set of attributesI a binary relation between objects and attributes :(g,m) ∈ I means that “object g owns attribute m”

Represented by a binary table

m1 m2 m3

g1 × ×g2 × ×g3 × ×g4 × ×g5 × × ×

G = {g1, . . . , g5}M = {m1,m2,m3}(g1,m3) ∈ I

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Galois connection

Two derivation operators forming a Galois connectionGives the set of common attributes owned by a set ofobjects A ⊆ G

A′ = {m ∈ M | ∀g ∈ A ⊆ G : (g,m) ∈ I}

Gives the set of objects owning all attributes in B ⊆ M

B′ = {g ∈ G | ∀m ∈ B ⊆ M : (g,m) ∈ I}

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Formal concepts

Given by (A,B), withwith A′ = B and B′ = AA is the concept extentB is the concept intent

Illustration

{g1}′ = {m1,m3}

{m1,m3}′ = {g1,g5}

{g1,g5}′ = {m1,m3}

m1 m2 m3

g1 × ×g2 × ×g3 × ×g4 × ×g5 × × ×

({g1,g5}, {m1,m3}) is a formal concept

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Concept lattice

Ordering relation on concepts

(A1,B1) ≤ (A2,B2)⇔ A1 ⊆ A2 (⇔ B2 ⊆ B1)

({g1,g5}, {m1,m3}) ≤ ({g1,g2,g5}, {m1})

Concept lattices have interesting propertiesMaximalitySpecialization/generalisation hierarchySynthetic representation of the data without loss ofinformation

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Problem

How to build a lattice from numerical data ?

How to consider “similar” objects in concepts ?

m1 m2 m3

g1 5 7 6g2 6 8 4g3 4 8 5g4 4 9 8g5 5 8 5

while avoiding discretization and associated problems(thresholds, size of binary table, information loss,calculability, interpretation)

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Outline




4 Conclusion

First elements

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How to order object descriptions

Classical caseLattice of attributes (2M ,⊆). With N,O ∈ 2M , one has

N ⊆ O ⇐⇒ N ∩O = N

For example, with M = {a,b}

{a} ⊆ {a,b} ⇐⇒ {a} ∩ {a,b} = {a}

Pattern case∩ has the properties of a meet u in a semi lattice

A “similarity operator” that gives a description handlingthe similarity of its arguments

{a,b} ∩ {a,d} = {a}

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Pattern structures

Given by (G, (D,u), δ)G a set of objectsD a meet semi-lattice of object descriptions called patternsδ a mapping associating to each object g ∈ G itsdescription δ(g) ∈ D

Patterns from (D,u) are ordered by

c v d ⇐⇒ c u d = c ∀c,d ∈ D

A Galois connection between (2G,⊆) and (D,v) gives rise toa (pattern) concept lattice

Existing algorithms of FCA (based on closure computation) caneasily be adapted

Ganter & Kuznetsov, ICCS01

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Intervals are patterns

Let be [a1,b1] and [a2,b2] two intervals

Their meet is[a1,b1] u [a2,b2] = [min(a1,a2),max(b1,b2)]

[4,4] u [5,5] = [4,5]

Their order is given by

[a1,b1] v [a2,b2] ⇐⇒ [a1,b1] u [a2,b2] = [a1,b1][4,5] v [5,5] ⇐⇒ [4,5] u [5,5] = [4,5]

Semi lattice (D,u), or (D,v)

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Interval vectors are patterns

Given the two following interval vectors

e = 〈[ai ,bi ]〉i∈[1,p] et f = 〈[ci ,di ]〉i∈[1,p]

Their meet is

e u f = 〈[ai ,bi ] u [ci ,di ]〉i∈[1,p]

〈[4,4], [3,4]〉 u 〈[2,3], [2,6]〉 = 〈[2,4], [2,6]〉

Their order is given by

e v f ⇔ [ai ,bi ] v [ci ,di ], ∀i ∈ [1,p]

〈[2,4], [2,6]〉 v 〈[4,4], [3,4]〉 car [2,4] v [4,4] et [2,6] v [3,4]

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Galois connection

Two operators

Gives the description representing similarity of a set ofobjects

A� =l

g∈A

δ(g) pour A ⊆ G

Gives the maximal set of objects sharing a givendescription

d� = {g ∈ G|d v δ(g)} pour d ∈ (D,u)

Ganter & Kuznetsov, ICCS01

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Numerical data are pattern structures

m1 m2 m3

g1 5 7 6g2 6 8 4g3 4 8 5g4 4 9 8g5 5 8 5

{g1,g2}� =l

g∈{g1,g2}

δ(g) = δ(g1) u δ(g2)

= 〈[5,5], [7,7], [6,6]〉 u 〈[6,6], [8,8], [4,4]〉= 〈[5,5] u [6,6], [7,7] u [8,8], [6,6] u [4,4]〉= 〈[5,6], [7,8], [4,6]〉

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Numerical data are pattern structures

m1 m2 m3

g1 5 7 6g2 6 8 4g3 4 8 5g4 4 9 8g5 5 8 5

〈[5,6], [7,8], [4,6]〉� = {g ∈ G|〈[5,6], [7,8], [4,6]〉 v δ(g)}= {g1,g2,g5}

({g1,g2,g5}, 〈[5,6], [7,8], [4,6]〉) is a concept

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Semantics in R|M|

Patterns are |M|-hyperrectanglesOrdering of patterns corresponds to rectangle inclusion

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General problem

Lowest concepts : few objects, small intervalsHighest concepts : many objects, large intervalsOverwhelming : a single concept for each interval of value

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4 Conclusion

Introducing a similarity relation between objects

How to group within the same concept objects havingsimilar values ?

A simple similarity relation

a 'θ b ⇔ |a− b| ≤ θ

Examples

2 '2 4, 2 6'3 7

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The meet : a similarity operator

Given two objects g and htheir descriptions are respectively δ(g) et δ(h)the similarity between g et h is represented byδ({g,h}) = δ(g) u δ(h)

For any arbitrary set of objectsevery objects are similar (since we can compute u)their level of similarity depends of the level of the meet oftheir description in the semi-lattice

How to consider a similarity relation w.r.t. a distance ?23 / 32

Towards a similarity between objects

Introduce an element ∗ ∈ (D,u) denoting dissimilarity

c u d 6= ∗ ⇐⇒ c and d are similar

c u d = ∗ ⇐⇒ c are d are not similar

For intervals, the meet is constrained by a threshold θ

[a,b]uθ[c,d ] = [min(a, c),max(b,d)] if max(b,d)−min(a, c) ≤ θ[a,b] uθ [c,d ] = ∗ otherwise

with θ = 0.2

Actually, we just “cut” the semi-lattice24 / 32

Going further ?

'θ is not a transitive relation, i.e. a tolerance relation

Projecting each pattern d ∈ D, i.e. ψ(d) v dFor each dimension, replacing each value with a largerintervalFrom a value d and its attribute domain

“Dilatation” : ball of patterns of radius θ (similarity)“Erosion” : delete pairs of values violating the similarity(maximality)Computing the meet of remaining values

Projection can be computing as preprocessing

Each projected pattern determines an equivalence class ofsimilar values : it reduces the number of concepts.

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Projecting for changing lattice granularity

FIGURE : Classical case (no projection)

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FIGURE : With θ = 0

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4 Conclusion

Conclusion

In a few wordsPattern structures for numerical dataIntroducing a similarity relation

Other worksLinks between binarization and projection of patternsAlgorithms for interval pattern structuresInterval dataMining closed interval patterns and their generatorsEnhancing information fusion with pattern structuresMining bi-sets in numerical data

ApplicationsGene expression data analysisFarmer practices evaluationRecommendation systems (movielens)

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Some references

M. Kaytoue, S. Duplessis, S. O. Kuznetsov, and A. Napoli. Mining Gene Expression Data with PatternStructures in Formal Concept Analysis. In Information Sciences. Spec.Iss. : Lattices, 2010.

M. Kaytoue, Z. Assaghir, A. Napoli, and S. O. Kuznetsov. Embedding tolerance relations in Formal ConceptAnalysis for classifying numerical data In 19th Conference on Information and Knowledge Management(CIKM), 2010.

Z. Assaghir, M. Kaytoue, A. Napoli, and H. Prade. Managing Information Fusion with Formal ConceptAnalysis. In Modeling Decisions for Artificial Intelligence, 6th International Conference (MDAI), 2010.

B. Ganter et S. O. Kuznetsov. Pattern Structures and Their Projections. In International Conference onConceptual Structures, LNCS (2120), Springer, 2001

M. Kaytoue, S. Duplessis, S. O. Kuznetsov, et A. Napoli. Two FCA-Based Methods for Mining GeneExpression Data. In Formal Concept Analysis, LNCS (5548), Springer, pages 251–266, 2009.

M. Kaytoue, Z. Assaghir, N. Messai, et A. Napoli. Two Complementary Classification Methods for Designinga Concept Lattice from Interval Data. In Foundations of Information and Knowledge Systems, LNCS (5956),Springer, pages 345–362, 2010.

M. Kaytoue, S. Duplessis, and A. Napoli. Toward the Discovery of Itemsets with Significant Variations inGene Expression Matrices. In Studies in Classification, Data Analysis, and Knowledge Organization,Springer, 2010.

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interval pattern structures: an introdution

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