Interval Pattern Structures: An introdution

Building Concept Lattices from
Numerical Data with Pattern Structures
A tutorial
Mehdi Kaytoue and Amedeo Napoli
mehdi.kaytoue@insa-lyon.fr
November, 9th 2010

Context
Formal Concept Analysis
Works on binary relations
Classiﬁcation of objects w.r.t. the attributes they have in
common within formal concepts (extent, intent)
Ordering concepts gives a mathematical structure
Ganter & Wille, Springer mathematical foundations 99
Concept lattice : useful for many tasks
Simultaneous classiﬁcation of objects and their attributes
Information organization
Knowledge 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 problem
Conceptual scaling (binarization)
Important choices to be made
Loss of information, of links between objects
Avoiding binarization ? Considering a similarity relation
between 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) with
G a set of objects
M a set of attributes
I 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 connection
Gives the set of common attributes owned by a set of
objects 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), with
with A = B and B = A
A is the concept extent
B 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 properties
Maximality
Specialization/generalisation hierarchy
Synthetic representation of the data without loss of
information
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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 6
g2 6 8 4
g3 4 8 5
g4 4 9 8
g5 5 8 5
while avoiding discretization and associated problems
(thresholds, size of binary table, information loss,
calculability, interpretation)
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How to order object descriptions
Classical case
Lattice 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 in a semi lattice
A “similarity operator” that gives a description handling
the similarity of its arguments
{a, b} ∩ {a, d} = {a}
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Pattern structures
Given by (G, (D, ), δ)
G a set of objects
D a meet semi-lattice of object descriptions called patterns
δ a mapping associating to each object g ∈ G its
description δ(g) ∈ D
Patterns from (D, ) are ordered by
c d ⇐⇒ c d = c ∀c, d ∈ D
A Galois connection between (2G, ⊆) and (D, ) gives rise to
a (pattern) concept lattice
Existing algorithms of FCA (based on closure computation) can
easily 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] [a2, b2] = [min(a1, a2), max(b1, b2)]
[4, 4] [5, 5] = [4, 5]
Their order is given by
[a1, b1] [a2, b2] ⇐⇒ [a1, b1] [a2, b2] = [a1, b1]
[4, 5] [5, 5] ⇐⇒ [4, 5] [5, 5] = [4, 5]
Semi lattice (D, ), or (D, )
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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 f = [ai, bi] [ci, di] i∈[1,p]
[4, 4], [3, 4] [2, 3], [2, 6] = [2, 4], [2, 6]
Their order is given by
e f ⇔ [ai, bi] [ci, di], ∀i ∈ [1, p]
[2, 4], [2, 6] [4, 4], [3, 4] car [2, 4] [4, 4] et [2, 6] [3, 4]
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Galois connection
Two operators
Gives the description representing similarity of a set of
objects
A =
g∈A
δ(g) pour A ⊆ G
Gives the maximal set of objects sharing a given
description
d = {g ∈ G|d δ(g)} pour d ∈ (D, )
Ganter & Kuznetsov, ICCS01
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Numerical data are pattern structures
m1 m2 m3
g1 5 7 6
g2 6 8 4
g3 4 8 5
g4 4 9 8
g5 5 8 5
{g1, g2} =
g∈{g1,g2}
δ(g) = δ(g1) δ(g2)
= [5, 5], [7, 7], [6, 6] [6, 6], [8, 8], [4, 4]
= [5, 5] [6, 6], [7, 7] [8, 8], [6, 6] [4, 4]
= [5, 6], [7, 8], [4, 6]
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Numerical data are pattern structures
m1 m2 m3
g1 5 7 6
g2 6 8 4
g3 4 8 5
g4 4 9 8
g5 5 8 5
[5, 6], [7, 8], [4, 6] = {g ∈ G| [5, 6], [7, 8], [4, 6] 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|-hyperrectangles
Ordering of patterns corresponds to rectangle inclusion
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General problem
Lowest concepts : few objects, small intervals
Highest concepts : many objects, large intervals
Overwhelming : a single concept for each interval of value
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1 Formal Concept Analysis
2 Pattern structures and intervals
3 Introducing a similarity relation
4 Conclusion

Introducing a similarity relation between objects
How to group within the same concept objects having
similar values ?
A simple similarity relation
a θ b ⇔ |a − b| ≤ θ
Examples
2 2 4, 2 3 7
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The meet : a similarity operator
Given two objects g and h
their descriptions are respectively δ(g) et δ(h)
the similarity between g et h is represented by
δ({g, h}) = δ(g) δ(h)
For any arbitrary set of objects
every objects are similar (since we can compute )
their level of similarity depends of the level of the meet of
their description in the semi-lattice
How to consider a similarity relation w.r.t. a distance ?
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Towards a similarity between objects
Introduce an element ∗ ∈ (D, ) denoting dissimilarity
c d = ∗ ⇐⇒ c and d are similar
c d = ∗ ⇐⇒ c are d are not similar
For intervals, the meet is constrained by a threshold θ
[a, b] θ[c, d] = [min(a, c), max(b, d)] if max(b, d)−min(a, c) ≤ θ
[a, b] θ [c, d] = ∗ otherwise
with θ = 0.2
Actually, we just “cut” the semi-lattice
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Going further ?
θ is not a transitive relation, i.e. a tolerance relation
Projecting each pattern d ∈ D, i.e. ψ(d) d
For each dimension, replacing each value with a larger
interval
From 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 of
similar 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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Conclusion
In a few words
Pattern structures for numerical data
Introducing a similarity relation
Other works
Links between binarization and projection of patterns
Algorithms for interval pattern structures
Interval data
Mining closed interval patterns and their generators
Enhancing information fusion with pattern structures
Mining bi-sets in numerical data
Applications
Gene expression data analysis
Farmer practices evaluation
Recommendation systems (movielens)
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Some references
M. Kaytoue, S. Duplessis, S. O. Kuznetsov, and A. Napoli. Mining Gene Expression Data with Pattern
Structures 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 Concept
Analysis 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 Concept
Analysis. 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 on
Conceptual Structures, LNCS (2120), Springer, 2001
M. Kaytoue, S. Duplessis, S. O. Kuznetsov, et A. Napoli. Two FCA-Based Methods for Mining Gene
Expression 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 Designing
a 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 in
Gene 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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