Triadic Factor Analysis Cynthia Glodeanu Institute of Algebra, TU - PowerPoint PPT Presentation

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Triadic Factor Analysis Cynthia Glodeanu Institute of Algebra, TU Dresden October 19, 2010.

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Outline Factor Analysis 1 Factor Analysis through Formal Concept Analysis Triadic Concept Analysis 2 Triadic Factor Analysis 3 Example

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Outline Factor Analysis 1 Factor Analysis through Formal Concept Analysis Triadic Concept Analysis 2 Triadic Factor Analysis 3 Example

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Outline Factor Analysis 1 Factor Analysis through Formal Concept Analysis Triadic Concept Analysis 2 Triadic Factor Analysis 3 Example

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Outline Factor Analysis 1 Factor Analysis through Formal Concept Analysis Triadic Concept Analysis 2 Triadic Factor Analysis 3 Example

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis The basic objective of Factor Analysis is to find underlying factors, which are responsible for the covariation among the observed variables. These factors are smaller in number than the number of observed variables. The factors are considered as new attributes, potentially more essential than the original ones.

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis The basic objective of Factor Analysis is to find underlying factors, which are responsible for the covariation among the observed variables. These factors are smaller in number than the number of observed variables. The factors are considered as new attributes, potentially more essential than the original ones.

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis The basic objective of Factor Analysis is to find underlying factors, which are responsible for the covariation among the observed variables. These factors are smaller in number than the number of observed variables. The factors are considered as new attributes, potentially more essential than the original ones.

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis New approach to Factor Analysis using formal concepts. A p × q binary matrix W is decomposed into the Boolean matrix product of a p × n and n × q binary matrix with n as small as possible. The Boolean matrix product : n � ( P ◦ Q ) ij := P il · Q lj . l =1

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis New approach to Factor Analysis using formal concepts. A p × q binary matrix W is decomposed into the Boolean matrix product of a p × n and n × q binary matrix with n as small as possible. The Boolean matrix product : n � ( P ◦ Q ) ij := P il · Q lj . l =1

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Definition F ⊆ B ( G , M , I ) such that � I = A × B ( A , B ) ∈F is called factorization . If F is minimal with respect to its cardinality then it is called optimal factorization . The elements of F are called (optimal) factors .

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Theorem (Belohlavek, Vychodil) Universality of concepts as factors For every W there is F ⊆ B ( G , M , I ) such that W = A F ◦ B F . Theorem (Belohlavek, Vychodil) Optimality of concepts as factors Let W = P ◦ Q for p × n and n × q binary matrices P and Q. Then there exists a set F ⊆ B ( G , M , I ) of formal concepts of W with |F| ≤ n such that for the p × |F| and |F| × q binary matrices A F and B F we have W = A F ◦ B F . Theorem (Belohlavek, Vychodil, Markowsky) The problem to find a decomposition W = P ◦ Q of an p × q binary matrix W into an p × n binary matrix P and a n × q binary matrix Q with n as small as possible is NP-hard and the corresponding decision problem is NP-complete.

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Theorem (Belohlavek, Vychodil) Universality of concepts as factors For every W there is F ⊆ B ( G , M , I ) such that W = A F ◦ B F . Theorem (Belohlavek, Vychodil) Optimality of concepts as factors Let W = P ◦ Q for p × n and n × q binary matrices P and Q. Then there exists a set F ⊆ B ( G , M , I ) of formal concepts of W with |F| ≤ n such that for the p × |F| and |F| × q binary matrices A F and B F we have W = A F ◦ B F . Theorem (Belohlavek, Vychodil, Markowsky) The problem to find a decomposition W = P ◦ Q of an p × q binary matrix W into an p × n binary matrix P and a n × q binary matrix Q with n as small as possible is NP-hard and the corresponding decision problem is NP-complete.

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Theorem (Belohlavek, Vychodil) Universality of concepts as factors For every W there is F ⊆ B ( G , M , I ) such that W = A F ◦ B F . Theorem (Belohlavek, Vychodil) Optimality of concepts as factors Let W = P ◦ Q for p × n and n × q binary matrices P and Q. Then there exists a set F ⊆ B ( G , M , I ) of formal concepts of W with |F| ≤ n such that for the p × |F| and |F| × q binary matrices A F and B F we have W = A F ◦ B F . Theorem (Belohlavek, Vychodil, Markowsky) The problem to find a decomposition W = P ◦ Q of an p × q binary matrix W into an p × n binary matrix P and a n × q binary matrix Q with n as small as possible is NP-hard and the corresponding decision problem is NP-complete.

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Outline Factor Analysis 1 Factor Analysis through Formal Concept Analysis Triadic Concept Analysis 2 Triadic Factor Analysis 3 Example

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Definition A triadic context ( K 1 , K 2 , K 3 , Y ) where Y ⊆ K 1 × K 2 × K 3 . The elements of K 1 , K 2 and K 3 are called (formal) objects, attributes and conditions , respectively. ( g , m , b ) ∈ Y : object g has attribute m under condition b .

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Definition A triadic concept ( triconcept ) is a triple ( A 1 , A 2 , A 3 ) with A i ⊆ K i , i ∈ { 1 , 2 , 3 } , that is maximal w.r.t component-wise set inclusion in satisfying A 1 × A 2 × A 3 ⊆ Y , i.e. for X i ⊆ K i , i ∈ { 1 , 2 , 3 } with X 1 × X 2 × X 3 ⊆ Y , the containments A i ⊆ X i , i ∈ { 1 , 2 , 3 } always imply ( A 1 , A 2 , A 3 ) = ( X 1 , X 2 , X 3 ). The components A 1 , A 2 and A 3 are called the extent , the intent , and the modus of ( A 1 , A 2 , A 3 ) respectively.

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Definition For { i , j , k } = { 1 , 2 , 3 } with j < k and for X ⊆ K i and Z ⊆ K j × K k , the ( i ) -derivation operators are defined by: Z �→ Z ( i ) := { a i ∈ K i | ( a i , a j , a k ) ∈ Y , ∀ ( a j , a k ) ∈ Z } , X �→ X ( i ) := { ( a j , a k ) ∈ K j × K k | ( a i , a j , a k ) ∈ Y , ∀ a i ∈ X } . Correspond to the derivation operators of the dyadic contexts: K (1) ( K 1 , K 2 × K 3 , Y (1) ) , := K (2) ( K 2 , K 1 × K 3 , Y (2) ) , := K (3) ( K 3 , K 1 × K 2 , Y (3) ) := where gY (1) ( m , b ) : ⇔ mY (2) ( g , b ) : ⇔ bY (3) ( g , m ) : ⇔ ( g , m , b ) ∈ Y .

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Definition For { i , j , k } = { 1 , 2 , 3 } with j < k and for X ⊆ K i and Z ⊆ K j × K k , the ( i ) -derivation operators are defined by: Z �→ Z ( i ) := { a i ∈ K i | ( a i , a j , a k ) ∈ Y , ∀ ( a j , a k ) ∈ Z } , X �→ X ( i ) := { ( a j , a k ) ∈ K j × K k | ( a i , a j , a k ) ∈ Y , ∀ a i ∈ X } . Correspond to the derivation operators of the dyadic contexts: K (1) ( K 1 , K 2 × K 3 , Y (1) ) , := K (2) ( K 2 , K 1 × K 3 , Y (2) ) , := K (3) ( K 3 , K 1 × K 2 , Y (3) ) := where gY (1) ( m , b ) : ⇔ mY (2) ( g , b ) : ⇔ bY (3) ( g , m ) : ⇔ ( g , m , b ) ∈ Y .

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Definition For { i , j , k } = { 1 , 2 , 3 } and for X i ⊆ K i , X j ⊆ K j and X k ⊆ K k the ( i , j , X k ) -derivation operators are defined by X i �→ X ( i , j , X k ) := { a j ∈ K j | ( a i , a j , a k ) ∈ Y , ∀ ( a i , a k ) ∈ X i × X k } , i X j �→ X ( i , j , X k ) := { a i ∈ K i | ( a i , a j , a k ) ∈ Y , ∀ ( a j , a k ) ∈ X j × X k } . j Correspond to the derivation operators of the dyadic contexts: K ij X k := ( K i , K j , Y ij X k ) where ( a i , a j ) ∈ Y ij X k : ⇐ ⇒ ( a i , a j , a k ) ∈ Y for all a k ∈ X k .

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Definition For { i , j , k } = { 1 , 2 , 3 } and for X i ⊆ K i , X j ⊆ K j and X k ⊆ K k the ( i , j , X k ) -derivation operators are defined by X i �→ X ( i , j , X k ) := { a j ∈ K j | ( a i , a j , a k ) ∈ Y , ∀ ( a i , a k ) ∈ X i × X k } , i X j �→ X ( i , j , X k ) := { a i ∈ K i | ( a i , a j , a k ) ∈ Y , ∀ ( a j , a k ) ∈ X j × X k } . j Correspond to the derivation operators of the dyadic contexts: K ij X k := ( K i , K j , Y ij X k ) where ( a i , a j ) ∈ Y ij X k : ⇐ ⇒ ( a i , a j , a k ) ∈ Y for all a k ∈ X k .

Factor Analysis Triadic Concept Analysis Triadic Factor Analysis Outline Factor Analysis 1 Factor Analysis through Formal Concept Analysis Triadic Concept Analysis 2 Triadic Factor Analysis 3 Example