Parallel Algorithms for general Galois lattices building

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Parallel Algorithms for general Galois lattices
building

• Fatma BAKLOUTI , Gérard LEVY
• CERIA
• fatma.baklouti_at_dauphine.fr,
  gerardlevy_at_dauphine.fr

Workshop WAS 2003
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Plan

• Knowledge Discovery in Databases (KDD)
• One tool for data mining Galois Lattices
• Problems and solutions
• Row-sharing
• Column-sharing
• Conclusion and perspectives

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Knowledge Discovery in Databases (KDD)

• Knowledge Discovery in Databases (KDD) or Data
  Mining (DM)
• Extraction of interesting (non-trivial,
  implicit, previously unknown and potentially
  useful) information (knowledge) or patterns from
  data in large databases or other information
  repositories
• Fayyad et al., 1996

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• DM emergence factors
• Wide Data bases volume from Gbyte to Tbyte
• Clientele report
• Example
• Analysis of a client basket in mass distribution
• Which group or set of products were frequently
  bought by a client during a passage
  in a shop?
• Disposition of product on shelves.
• Example Milk and bread
• when a client buys milk, does he buy bread too ?

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• Various applications
• Medecine, Finances, Distribution,
  telecommunication
• Fields of research

Data Base
Statistics
IHM
Learning
KDD
Etc
Information Science
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KDD General Process
Text Picture Sound Data
Data acquisition
Data Preparation
Selection,cleaning, integration
Transformations, editing construction of
attributes
Model Concept
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• Books
• Data Mining,
• Han Kamber (Morgan Kaufmann Pubs, 2001)
• Mastering Data Mining,
• Berry Linoff (Wiley Computer Publishing, 2000)
• Interesting sites
• http//www.kddnuggets.com
• http//www.crisp-dm.org CRoss-Industry
  Standard Process for Data Mining - effort de
  standardization

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Galois Lattices

• Using Galois Lattice (mathematical structure) for
  solving Data Mining problems.
• References
• Birkhoffs Lattice Theory 1940, 1973
• Barbut Monjardet 1970
• Wille 1982
• Chein, Norris, Ganter, Bordat,
• Diday, Duquenne,
• Emilion, Lévy, Diday, Lambert
• Basic Concepts
• Context, Galois connection, Concept.

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Galois Lattices - Definition

• Context (O, A, I)
• O finite set of examples
• A finite set of attributes
• I binary relation between O and A, (I ? O x
  A)
• Example

O
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Galois Lattices - Definition

• Galois connection
• Oi ? O and Ai ? A, we define f et g like this
• f P(O) ? P(A) f(Oi) a ? A / (o,a) ? I, ? o
  ? Oi intention
• g P(A) ? P(O) g(Ai) o ? O / (o,a) ? I, ? a ?
  Ai extension
• f et g are decreasing applications
• h g f and k f g, are
• Increasing O1 ? O2 ? h (O1) ? h (O2)
• Extensive O1 ? h (O1)
• Idempotent h (O1) h h (O1)
• h and k are closure operators.
• (f,g) Galois connection between P(O) and P(A)

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Galois Lattices - Definition

• Galois connexion Example
• O1 6,7 ? f(O1) a,c
  intention
• A1 a,c ? g(A1) 1,2,3,4,6,7 extension

Remark h(O1) g f(O1) g(A1) ? O1
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Galois Lattices - Definition

• Concept
• Oi ? O et Ai ? A,
• (Oi, Ai) is a concept iff Oi is the extension of
  Ai and Ai is the intention of Oi
• Oi g (Ai) and Ai f(Oi)
• L (Oi, Ai) ? P(O)?P(A) / Oi g(Ai) et Ai
  f(Oi) concepts set.
• L ordered set by the relationship
• (O1, A1) (O2, A2) iff O1 ? O2 (or A2 ? A1).
• Galois Lattice
• T(L, ) an ordered set of concepts.

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Galois Lattices - Definition

• Concept Example
• O1 6,7 ? f(O1) a,c
• A1 a,c ? g(A1) 1,2,3,4,6,7
• Remark h(O1) g f(O1) g(A1) ? O1
• (6,7 , a,c) ? L
• (1,2,3,4,6,7, a,c) ? L
• Because
• h(1,2,3,4,6,7) g f(1,2,3,4,6,7)
• g (a,c)
• 1,2,3,4,6,7

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Generalized Galois Lattices

• Context lt I, F, d gt
• T ltF, ?, ?, gt
• Tj ltFj, ?j, ?j, jgt for all j de J, J 1,n
• d I ? F
• di (di1,, dij,, din) description of the
  individual i relatively to the attributes j of J.

1 2 j n
1 i k

• x ? I
• f (x) ?d(i) i ? x Intention
• ? z ? F
• g (z) i ? I z d(i) Extension

Individuals I
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General Galois Lattice - Example
F F1 x F2 x F3 Size short, medium, high
1 lt 2 lt 3 Weight thin, fat
0 lt 1 Age child,
adolescent, adult 1 lt 2 lt 3
F1 F2 F3
f Cedric, Carine 1, 1, 2 g1, 1, 2
Cedric, Carine
Individuals I
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Ø, 313
4,312
3,203
34,202
24,112
134,201
234,102
1234,101
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Problems

• Large data volume
• Partition data on different server nodes
• Process in parallel locally
• Group results on one (client) node
• Post-process
• Our tool
• SDDS (Scalable Distributed Data Structures )

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Solutions
Column-sharing
Row-sharing
1 2 3
1 2 3
C
C
3
1 2
1 2 3
1 2 3
C2
C1
C3
C4
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Row-sharing
M2
M1
C1 T1TG(C1)
C2 T2TG(C2)
M
TTG(C)
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Example
C
C2
C1
T1GL(C1)
T2GL(C2)
T GL(C) is it egal to the horizontal product of
lattices T1 GL (C1) and T2 GL (C2) ?
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We apply an algorithm (here Bordats algorithm)
to context C1 and C2 to build respectively
lattice T1 GL(C1) and lattice T2 GL(C2).
Graph of lattice T1 GL(C1)
Graph of lattice T2 GL(C2)
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Total number of closed pairs ( X , z ) of
lattice T1 GL(C1) 12. pair(1) X,
z(2,3,3) pair(2) X1, z(1,0,2) pair(3)
X2, z(2,1,0) pair(4) X3,
z(0,3,1) pair(5) X4, z(1,1,1) pair(6)
X1,4, z(1,0,1) pair(7) X2,4,
z(1,1,0) pair(8) X3,4, z(0,1,1) pair(9)
X1,2,4, z(1,0,0) pair(10) X1,3,4,
z(0,0,1) pair(11) X2,3,4, z(0,1,0) pair(12)
X1,2,3,4, z(0,0,0). Total number of closed
pairs of T2 GL(C2) 5 pair (1) X,
z(2,3,3) pair(2) X5, z(0,1,3) pair(3)
X6, z(2,0,0) pair (4) X5,6, z(0,0,2)
pair(5) X5, 6, 7, z(0,0,0).
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X X1 ? X2 z z1 ? z2
Horizontal product of lattices T1 GL (C1) and
T2 GL (C2)
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We apply BORDATs algorithm to the full context
C.
Graph of lattice T GL(C)
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Total number of closed pairs (X, z) of T
GL(C) 15. pair(1) X, z(2,3,3) pair(2)
X1, z(1,0,2) pair(3) X2, z(2,1,0) pair
(4) X3, z(0,3,1) pair(5) X4,
z(1,1,1) pair(6) X5,, z(0,1,3) pair(7)
X1,4, z(1,0,1) pair(8) X1,5,6,
z(0,0,2) pair(9) X2,4, z(1,1,0) pair(10)
X2,7, z(2,0,0) pair(11) X3,4,5,
z(0,1,1) pair(12) X1,2,4,7, z(1,0,0)
pair(13) X1,3,4,5,6, z(0,0,1) pair(14)
X2,3,4,5, z(0,1,0) pair(15)
X1,2,3,4,5,6,7, z(0,0,0).
T GL(C) is the horizontal product of
lattices T1 GL(C1) and T2 GL(C2)
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Columnsharing
M2
M1
C1 T1TG(C2)
C2 T2TG(C2)
M
TTG(C)
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Example
C
C1
C2
T2GL(C2)
T1GL(C1)
T GL(C) is it egal to the vertical product of
lattices T1 GL (C1) and T2 GL (C2) ?
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Graph of lattice T1 GL(C1)
Graph of lattice T2 GL(C2)
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• Total number of closed pairs ( X , z ) of
  lattice T1 GL(C1) 8.
• pair(1)  X, z (2,3)
• pair(2)  X2, z (2,1)
• pair(3)  X 3, z (0,3),
• pair(4)  X 2,4, z (1,1)
• pair(5)  X 2,7, z (2,0)
• pair(6)  X 2,3,4,5, z(0,1),
• pair(7)  X 1,2,4,7, z(1,0)
• pair(8)  X 1,2 ,3,4,5,6,7, z(0,0).
• Total number of closed pairs ( X , z ) of
  lattice T1 GL(C1) 4.
• pair(1)  X 5, z(3)
• pair(2)  X 1,5,6, z(2)
• pair(3)  X 1,3,4,5,6, z(1)
• pair(4)  X 1,2 ,3,4,5,6,7, z (0).

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X X1 ? X2 z (z1 , z2 )
T GL(C) is the vertical product of lattices
T1 GL(C1) and T2 GL(C2)
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Conclusion and perspectives

• Generalized Galois Lattices.
• Problem of large data base can be perhaps
  resolved in our way.
• Sharing context into two subsets.
• Possibility of building different architectures
  for stations networks.

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Thank you for Your Attention Fatma
Baklouti Gérard LEVY fbaklouti_at_excite.com gerardl
evy_at_dauphine.fr