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Factor Analysis Triadic Concept Analysis Triadic Factor Analysis
Factor Analysis Triadic Concept Analysis Triadic Factor Analysis
Triadic Factor Analysis Cynthia Glodeanu Institute of Algebra, TU - - PowerPoint PPT Presentation
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
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Factor Analysis Triadic Concept Analysis Triadic Factor Analysis
Outline
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Factor Analysis Factor Analysis through Formal Concept Analysis
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Triadic Concept Analysis
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Triadic Factor Analysis Example
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Factor Analysis Triadic Concept Analysis Triadic Factor Analysis
Outline
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Factor Analysis Factor Analysis through Formal Concept Analysis
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Triadic Concept Analysis
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Triadic Factor Analysis Example
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Factor Analysis Triadic Concept Analysis Triadic Factor Analysis
Outline
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Factor Analysis Factor Analysis through Formal Concept Analysis
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Triadic Concept Analysis
3
Triadic Factor Analysis Example
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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
- bserved variables.
These factors are smaller in number than the number of
- bserved variables.
The factors are considered as new attributes, potentially more essential than the original ones.
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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
- bserved variables.
These factors are smaller in number than the number of
- bserved variables.
The factors are considered as new attributes, potentially more essential than the original ones.
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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
- bserved variables.
These factors are smaller in number than the number of
- bserved variables.
The factors are considered as new attributes, potentially more essential than the original ones.
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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 : (P ◦ Q)ij :=
n
- l=1
Pil · Qlj.
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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 : (P ◦ Q)ij :=
n
- l=1
Pil · Qlj.
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Factor Analysis Triadic Concept Analysis Triadic Factor Analysis
Definition F ⊆ B(G, M, I) such that I =
- (A,B)∈F
A × B 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.
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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 = AF ◦ BF. 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 AF and BF we have W = AF ◦ BF. 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.
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Theorem (Belohlavek, Vychodil) Universality of concepts as factors For every W there is F ⊆ B(G, M, I) such that W = AF ◦ BF. 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 AF and BF we have W = AF ◦ BF. 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.
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Theorem (Belohlavek, Vychodil) Universality of concepts as factors For every W there is F ⊆ B(G, M, I) such that W = AF ◦ BF. 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 AF and BF we have W = AF ◦ BF. 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.
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Factor Analysis Triadic Concept Analysis Triadic Factor Analysis
Outline
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Factor Analysis Factor Analysis through Formal Concept Analysis
2
Triadic Concept Analysis
3
Triadic Factor Analysis Example
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Factor Analysis Triadic Concept Analysis Triadic Factor Analysis
Definition A triadic context (K1, K2, K3, Y ) where Y ⊆ K1 × K2 × K3. The elements of K1, K2 and K3 are called (formal) objects, attributes and conditions, respectively. (g, m, b) ∈ Y : object g has attribute m under condition b.
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Definition A triadic concept (triconcept) is a triple (A1, A2, A3) with Ai ⊆ Ki, i ∈ {1, 2, 3}, that is maximal w.r.t component-wise set inclusion in satisfying A1 × A2 × A3 ⊆ Y , i.e. for Xi ⊆ Ki, i ∈ {1, 2, 3} with X1 × X2 × X3 ⊆ Y , the containments Ai ⊆ Xi, i ∈ {1, 2, 3} always imply (A1, A2, A3) = (X1, X2, X3). The components A1, A2 and A3 are called the extent, the intent, and the modus of (A1, A2, A3) respectively.
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Definition For {i, j, k} = {1, 2, 3} with j < k and for X ⊆ Ki and Z ⊆ Kj × Kk, the (i)-derivation operators are defined by: Z → Z (i) := {ai ∈ Ki | (ai, aj, ak) ∈ Y , ∀(aj, ak) ∈ Z}, X → X (i) := {(aj, ak) ∈ Kj × Kk | (ai, aj, ak) ∈ Y , ∀ai ∈ X}. Correspond to the derivation operators of the dyadic contexts: K(1) := (K1, K2 × K3, Y (1)), K(2) := (K2, K1 × K3, Y (2)), K(3) := (K3, K1 × K2, Y (3)) where gY (1)(m, b) :⇔ mY (2)(g, b) :⇔ bY (3)(g, m) :⇔ (g, m, b) ∈ Y .
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Factor Analysis Triadic Concept Analysis Triadic Factor Analysis
Definition For {i, j, k} = {1, 2, 3} with j < k and for X ⊆ Ki and Z ⊆ Kj × Kk, the (i)-derivation operators are defined by: Z → Z (i) := {ai ∈ Ki | (ai, aj, ak) ∈ Y , ∀(aj, ak) ∈ Z}, X → X (i) := {(aj, ak) ∈ Kj × Kk | (ai, aj, ak) ∈ Y , ∀ai ∈ X}. Correspond to the derivation operators of the dyadic contexts: K(1) := (K1, K2 × K3, Y (1)), K(2) := (K2, K1 × K3, Y (2)), K(3) := (K3, K1 × K2, Y (3)) where gY (1)(m, b) :⇔ mY (2)(g, b) :⇔ bY (3)(g, m) :⇔ (g, m, b) ∈ Y .
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Definition For {i, j, k} = {1, 2, 3} and for Xi ⊆ Ki, Xj ⊆ Kj and Xk ⊆ Kk the (i, j, Xk)-derivation operators are defined by Xi → X (i,j,Xk)
i
:= {aj ∈ Kj | (ai, aj, ak) ∈ Y , ∀(ai, ak) ∈ Xi × Xk}, Xj → X (i,j,Xk)
j
:= {ai ∈ Ki | (ai, aj, ak) ∈ Y , ∀(aj, ak) ∈ Xj × Xk}. Correspond to the derivation operators of the dyadic contexts: Kij
Xk := (Ki, Kj, Y ij Xk)
where (ai, aj) ∈ Y ij
Xk :⇐
⇒ (ai, aj, ak) ∈ Y for all ak ∈ Xk.
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Factor Analysis Triadic Concept Analysis Triadic Factor Analysis
Definition For {i, j, k} = {1, 2, 3} and for Xi ⊆ Ki, Xj ⊆ Kj and Xk ⊆ Kk the (i, j, Xk)-derivation operators are defined by Xi → X (i,j,Xk)
i
:= {aj ∈ Kj | (ai, aj, ak) ∈ Y , ∀(ai, ak) ∈ Xi × Xk}, Xj → X (i,j,Xk)
j
:= {ai ∈ Ki | (ai, aj, ak) ∈ Y , ∀(aj, ak) ∈ Xj × Xk}. Correspond to the derivation operators of the dyadic contexts: Kij
Xk := (Ki, Kj, Y ij Xk)
where (ai, aj) ∈ Y ij
Xk :⇐
⇒ (ai, aj, ak) ∈ Y for all ak ∈ Xk.
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Factor Analysis Triadic Concept Analysis Triadic Factor Analysis
Outline
1
Factor Analysis Factor Analysis through Formal Concept Analysis
2
Triadic Concept Analysis
3
Triadic Factor Analysis Example
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Factor Analysis Triadic Concept Analysis Triadic Factor Analysis
Definition F ⊆ T(K) such that Y =
- (A,B,C)∈F
(A × B × C) is called a factorization of the triadic context K. If F is minimal with respect to its cardinality then it is called optimal
- factorization. The elements of F are called (optimal) factors.
Definition A Boolean 3d-matrix (shortly 3d-matrix) is a rectangular box Bp×q×r such that bijk ∈ {0, 1} for all i ∈ {1, . . . , p}, j ∈ {1, . . . , q}, k ∈ {1, . . . , r}.
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Definition F ⊆ T(K) such that Y =
- (A,B,C)∈F
(A × B × C) is called a factorization of the triadic context K. If F is minimal with respect to its cardinality then it is called optimal
- factorization. The elements of F are called (optimal) factors.
Definition A Boolean 3d-matrix (shortly 3d-matrix) is a rectangular box Bp×q×r such that bijk ∈ {0, 1} for all i ∈ {1, . . . , p}, j ∈ {1, . . . , q}, k ∈ {1, . . . , r}.
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For a factorization F = {(A1, B1, C1), . . . , (An, Bn, Cn)}: (AF)il := 1, if i ∈ Al 0, if i / ∈ Al , (BF)jl := 1, if j ∈ Bl 0, if j / ∈ Bl , (CF)kl := 1, if k ∈ Cl 0, if k / ∈ Cl .
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Definition For p × n, q × n and r × n binary matrices P, Q and R the Boolean 3d-matrix product (shortly 3d-product) is defined as a ternary operation: (P ◦ Q ◦ R)ijk :=
n
- l=1
Pil · Qjl · Rkl with i ∈ {1, . . . , p}, j ∈ {1, . . . , q} and k ∈ {1, . . . , r}.
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Alternative definitions: (P ∗ Q ◦ R)ijk :=
n
- l=1
(P ∗ Q)(ij)l · Rkl with (P ∗ Q)(ij)l := Pil · Qjl (P ◦ Q ∗ R)ijk :=
n
- l=1
Pil · (Q ∗ R)(jk)l with (Q ∗ R)(jk)l := Qjl · Rkl
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Alternative definitions: (P ∗ Q ◦ R)ijk :=
n
- l=1
(P ∗ Q)(ij)l · Rkl with (P ∗ Q)(ij)l := Pil · Qjl (P ◦ Q ∗ R)ijk :=
n
- l=1
Pil · (Q ∗ R)(jk)l with (Q ∗ R)(jk)l := Qjl · Rkl
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Theorem For every 3d-matrix B there is F ⊆ T(KB) such that: B = AF ◦ BF ◦ CF. Theorem Let K = (K1, K2, K3, Y ) be a triadic context and B the corresponding 3d-matrix such that B = P ◦ Q ◦ R for p × n, q × n and r × n binary matrices P, Q and R. Then there exists a set F ⊆ T(K) of triconcepts with |F| ≤ n such that for the p × |F|, q × |F| and r × |F| binary matrices AF, BF and CF we have B = AF ◦ BF ◦ CF.
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Theorem For every 3d-matrix B there is F ⊆ T(KB) such that: B = AF ◦ BF ◦ CF. Theorem Let K = (K1, K2, K3, Y ) be a triadic context and B the corresponding 3d-matrix such that B = P ◦ Q ◦ R for p × n, q × n and r × n binary matrices P, Q and R. Then there exists a set F ⊆ T(K) of triconcepts with |F| ≤ n such that for the p × |F|, q × |F| and r × |F| binary matrices AF, BF and CF we have B = AF ◦ BF ◦ CF.
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Algorithmic approach: In the triadic case the factorization problem is NP-hard as well Generalisation of the greedy approximation algorithm
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Factor Analysis Triadic Concept Analysis Triadic Factor Analysis
Algorithmic approach: In the triadic case the factorization problem is NP-hard as well Generalisation of the greedy approximation algorithm
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Factor Analysis Triadic Concept Analysis Triadic Factor Analysis
Algorithmic approach: In the triadic case the factorization problem is NP-hard as well Generalisation of the greedy approximation algorithm
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hostelworld hostels hostelbookers 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0: Nuevo S. × × × × × 1: Samay × × × × × × × × × × × × × 2: Oasis B. × × × × × × × × × × × × × × × × 3: One × × × × × × × × × × × × × × × × 4: Ole B. × × × × × × × × × × × × × × × × 5: Garden B. × × × × × × × × × × × × × × where K2 := {character, safety, location, staff, fun, cleanliness}.
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Factor Analysis Triadic Concept Analysis Triadic Factor Analysis K3 1: hostels 0: hostelworld 2: hostelbookers 2 1 5 3,4 5 2 1 2 1 K1 2 3 3,5 1 4 K2
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Optimal factorization: ({0, 1, 2, 5}, 2, K3), ({2, 3, 4, 5}, K2, 1), ({1, 2, 5}, {2, 3, 5}, K3), ({1, 2, 3, 4}, {1, 3, 5}, K3), (K1, {2, 3}, {1, 2}), ({1, 3, 4, 5}, {0, 1, 2, 3, 5}, 2), ({2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2}), ({2, 3, 4}, {0, 1, 3, 5}, {0, 1}).
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Optimal factorization: ({0, 1, 2, 5}, 2, K3) best location, ({2, 3, 4, 5}, K2, 1), ({1, 2, 5}, {2, 3, 5}, K3), ({1, 2, 3, 4}, {1, 3, 5}, K3), (K1, {2, 3}, {1, 2}), ({1, 3, 4, 5}, {0, 1, 2, 3, 5}, 2), ({2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2}), ({2, 3, 4}, {0, 1, 3, 5}, {0, 1}).
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Optimal factorization: ({0, 1, 2, 5}, 2, K3) best location, ({2, 3, 4, 5}, K2, 1) best deals according to the users from “hostels”, ({1, 2, 5}, {2, 3, 5}, K3), ({1, 2, 3, 4}, {1, 3, 5}, K3), (K1, {2, 3}, {1, 2}), ({1, 3, 4, 5}, {0, 1, 2, 3, 5}, 2), ({2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2}), ({2, 3, 4}, {0, 1, 3, 5}, {0, 1}).
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Optimal factorization: ({0, 1, 2, 5}, 2, K3) best location, ({2, 3, 4, 5}, K2, 1) best deals according to the users from “hostels”, ({1, 2, 5}, {2, 3, 5}, K3) excellent concerning location, staff and cleanliness, ({1, 2, 3, 4}, {1, 3, 5}, K3) excellent concerning safety, staff and cleanliness, (K1, {2, 3}, {1, 2}) excellent regarding location and staff, ({1, 3, 4, 5}, {0, 1, 2, 3, 5}, 2) best deals according to the users from the 3rd platform, ({2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2}), ({2, 3, 4}, {0, 1, 3, 5}, {0, 1}).
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f1 f2 f3 f4 f5 f6 f7 f8 Nuevo S. × × Samay × × × × × Oasis B. × × × × × × × One × × × × × × Ole B. × × × × × × Garden B. × × × × ×
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f1 f2 f3 f4 f5 f6 f7 f8 character × × × safety × × × × × location × × × × × × staff × × × × × × × fun × × cleanliness × × × × × × f1 f2 f3 f4 f5 f6 f7 f8 hostelworld × × × × hostels × × × × × × × hostelbookers × × × × × ×
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f1 f2 f3 f4 f5 f6 f7 f8 character × × × safety × × × × × location × × × × × × staff × × × × × × × fun × × cleanliness × × × × × × f1 f2 f3 f4 f5 f6 f7 f8 hostelworld × × × × hostels × × × × × × × hostelbookers × × × × × ×
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Factor Analysis Triadic Concept Analysis Triadic Factor Analysis