Mining bi-sets in numerical data

Transcription

1 Mining bi-sets in numerical data Jérémy Besson, Céline Robardet, Luc De Raedt and Jean-François Boulicaut Institut National des Sciences Appliquées de Lyon - France Albert-Ludwigs-Universitat Freiburg - Germany

2 Outline Motivation

3 Mining numerical data Example: Gene expression data analysis What are the sets of genes that are simultaneousely over expressed in some biological situations?

4 Principle O P M(i,j) denotes the value of property j P for the object i O NBS defines a sub-matrix S of M s. t. the absolute value of the difference between the maximum value and the minimum value on S is less or equal to ǫ. Furthermore, none object or property can be added to the bi-set without violating this constraint.

5 The formal definition Definition (Numerical bi-sets) Given a real value ǫ, (X,Y ) is a NBS iff () (2) X O, Y P Max i X, j Y M(i,j) Min i X, j Y M(i,j) ǫ y Y, Max i X, j Y {y} M(i,j) Min i X, j Y {y} M(i,j) > ǫ x X, Max i X {x}, j Y M(i,j) Min i X {x}, j Y M(i,j) > ǫ

6 An example p p 2 p 3 p 4 p 5 o o o o ((o,o 2, o 3,o 4 ), (p 5 )) ((o 3,o 4 ), (p 4,p 5 )) ((o 4 ), (p, p 5 )) ((o,o 2, o 3,o 4 ), (p 3 )) ((o 4 ), (p, p 2 )) ((o 2 ), (p 2, p 3,p 4 )) ((o,o 2 ), (p 4 )) ((o ), (p, p 2,p 3, p 4 )) ((o,o 2, o 3 ),(p,p 2, p 3 )) o4 o3 o2 op p2 p3 p4 Data NBS NBS 2 p5

7 Definition (Specialization and monotonicity) Our specialization relation on bi-sets denoted is defined as follows: (X,Y ) (X 2,Y 2 ) iff X X 2 and Y Y 2. The constraints are respectively anti-monotonic and monotonic w.r.t.

8 Let W ǫ be the whole collection of NBS patterns for ǫ. Each NBS pattern (X,Y) from W ǫ is maximal w.r.t.. If there exists a bi-set (X,Y ) with similar values (belonging to an interval of size ǫ), then there exists a NBS (X,Y ) from W ǫ such that (X,Y ) (X,Y )

9 When ǫ increases, the size of NBS pattern increases too, whereas some new NBS patterns which are not extensions of previous one can appear. The collection of numerical bi-sets is paving the dataset.

10 DR-Miner Lattice of the whole collection of bi-set: ((, ),(G,M)) A sublattice (( G, M ),( G, M )) ( G, M) ( G, M) UB Cr (( G, M ),( G, M )) s G \ G, t G, Z o (s, M ) Z o (t, M ) + δ and s M \ M, t M, Z a (s, G ) Z a (t, G ) + δ UB Cd (( G, M )( G, M )) ( x G, Z o (x, M ) α) and ( y M Z a (y, G ) α)

11 NBS-Miner M is a real valued matrix, C a conjunction of monotonic and anti-monotonic constraints on 2 O 2 P and ǫ is a positive value. NBS-Miner Generate((, ), (O, P)) End NBS-Miner Generate(L) Let L = (( O, P ), ( O, P )) L Prop(L) If Prune(L) then If ( O, P ) ( O, P ) then (L, L 2) Enum(L,Choose(L)) J. Besson, C. Robardet, Generate(L L. De Raedt, J-F ) Boulicaut Mining bi-sets in numerical data

12 DR-Miner Pruning: if UB Cr (, ) or UB Cd (, ) are not satisfied, the sublattice (, ) is pruned Propagation (x \ ): if UB Cr (, \ {x}) is not satisfied then the sublattice is modified in ( {x}, ) if UB Cd ( {x}, ) is not satisfied then the sublattice is modified in (, \ {x}) Enumeration: we choose x \ ( {x}, ) (, \ {x})

13 DR-Miner \ {x} Propagation Enumeration {x} Pruning

14 Figure: Examples of extracted NBS

15 90 80 mean area number of NBS epsilon epsilon Figure: Mean area of the NBS w.r.t. ǫ

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