IUKM 2019 - Nara, Japan Choquet integral in decision making and metric learning Vicen¸ c Torra Hamilton Institute, Maynooth University Ireland March 27, 2018
Outline Overview Basics and objectives: • Using Choquet integral in two types of applications decision and metric learning (reidentification) • Distances • and distribution (for non-additive measures) Vicen¸ c Torra; Choquet integral in decision making and metric learning IUKM 2019 - Nara, Japan 1 / 62
Outline Outline 1. Preliminaries • Choquet integral: mathematical perspective ◦ Non-additive measures ◦ Now we need an integral • Choquet integral: Application perspective ◦ Aggregation operators and CI in decision: MCDM ◦ Aggregation operators and CI in reidentification: risk assessment ◦ Zooming out 2. Distances in classification (filling the gaps) 3. Distributions IUKM 2019 - Nara, Japan 2 / 62
Outline Choquet integral: a mathematical introduction IUKM 2019 - Nara, Japan 3 / 62
Outline Non-additive measures IUKM 2019 - Nara, Japan 4 / 62
Definitions Outline Definitions: measures Additive measures. • ( X, A ) a measurable space; then, a set function µ is an additive measure if it satisfies (i) µ ( A ) ≥ 0 for all A ∈ A , (ii) µ ( X ) ≤ ∞ (iii) for every countable sequence A i ( i ≥ 1) of A that is pairwise disjoint (i.e,. A i ∩ A j = ∅ when i � = j ) ∞ ∞ � � µ ( A i ) = µ ( A i ) i =1 i =1 Vicen¸ c Torra; Choquet integral in decision making and metric learning IUKM 2019 - Nara, Japan 5 / 62
Definitions Outline Definitions: measures Additive measures. • ( X, A ) a measurable space; then, a set function µ is an additive measure if it satisfies (i) µ ( A ) ≥ 0 for all A ∈ A , (ii) µ ( X ) ≤ ∞ (iii) for every countable sequence A i ( i ≥ 1) of A that is pairwise disjoint (i.e,. A i ∩ A j = ∅ when i � = j ) ∞ ∞ � � µ ( A i ) = µ ( A i ) i =1 i =1 Finite case: µ ( A ∪ B ) = µ ( A ) + µ ( B ) for disjoint A , B Vicen¸ c Torra; Choquet integral in decision making and metric learning IUKM 2019 - Nara, Japan 5 / 62
Definitions Outline Definitions: measures Additive measures. Example: • Lebesgue measure. Unique measure λ s.t. λ ([ a, b ]) = b − a for every finite interval [ a, b ] Vicen¸ c Torra; Choquet integral in decision making and metric learning IUKM 2019 - Nara, Japan 6 / 62
Definitions Outline Definitions: measures Additive measures. Example: • Lebesgue measure. Unique measure λ s.t. λ ([ a, b ]) = b − a for every finite interval [ a, b ] • Probability. When µ ( X ) = 1 . Vicen¸ c Torra; Choquet integral in decision making and metric learning IUKM 2019 - Nara, Japan 6 / 62
Definitions Outline Definitions: measures Additive measures. Example: • Lebesgue measure. Unique measure λ s.t. λ ([ a, b ]) = b − a for every finite interval [ a, b ] • Probability. When µ ( X ) = 1 . • Or just price ... A B Vicen¸ c Torra; Choquet integral in decision making and metric learning IUKM 2019 - Nara, Japan 6 / 62
Definitions Outline Definitions: measures • Non-additive measures ◦ ( X, A ) a measurable space, a non-additive (fuzzy) measure µ on ( X, A ) is a set function µ : A → [0 , 1] satisfying the following axioms: (i) µ ( ∅ ) = 0 , µ ( X ) = 1 (boundary conditions) (ii) A ⊆ B implies µ ( A ) ≤ µ ( B ) (monotonicity) Vicen¸ c Torra; Choquet integral in decision making and metric learning IUKM 2019 - Nara, Japan 7 / 62
Definitions Outline Definitions: measures • Non-additive measures ◦ ( X, A ) a measurable space, a non-additive (fuzzy) measure µ on ( X, A ) is a set function µ : A → [0 , 1] satisfying the following axioms: (i) µ ( ∅ ) = 0 , µ ( X ) = 1 (boundary conditions) (ii) A ⊆ B implies µ ( A ) ≤ µ ( B ) (monotonicity) • Naturally, additivity implies monotonicity ◦ E.g., B = A ∪ C (with A ∩ C = ∅ ) then µ ( B ) = µ ( A )+ µ ( C ) ≥ µ ( A ) ◦ But in non-additive measures, we allow µ ( B = A ∪ C ) <µ ( A ) + µ ( C ) µ ( B = A ∪ C ) >µ ( A ) + µ ( C ) As e.g., µ ( B ) = 0 . 5 < µ ( A ) + µ ( C ) = 0 . 3 + 0 . 4 = 0 . 7 A way to represent interactions Vicen¸ c Torra; Choquet integral in decision making and metric learning IUKM 2019 - Nara, Japan 7 / 62
Definitions Outline Definitions: measures • Non-additive measures. Price ◦ When we have a discount, for disjoints A and B , we have µ ( A ∪ B ) < µ ( A ) + µ ( B ) but µ ( A ∪ B ) ≥ µ ( A ) • There quite a large number of families of measures Vicen¸ c Torra; Choquet integral in decision making and metric learning IUKM 2019 - Nara, Japan 8 / 62
Definitions Outline Definitions: measures • Non-additive measures. Distorted probabilities ◦ m : R + → R + a continuous and increasing function such that m (0) = 0 ; P be a probability. µ m,P ( A ) = m ( P ( A )) (1) ◦ If m ( x ) = x p , then µ m ( A ) = ( λ ( A )) p (a) (b) (c) (d) ◦ Used in economics: Prospect theory (Kahneman and Tversky, 1979). Small probabilities tend to be overestimated, while large ones, underestimated. Vicen¸ c Torra; Choquet integral in decision making and metric learning IUKM 2019 - Nara, Japan 9 / 62
Definitions Outline Definitions: measures • Non-additive measures. Distorted Lebesgue ◦ m : R + → R + a continuous and increasing function such that m (0) = 0 ; λ be the Lebesgue measure. µ m ( A ) = m ( λ ( A )) (2) Vicen¸ c Torra; Choquet integral in decision making and metric learning IUKM 2019 - Nara, Japan 10 / 62
Definitions Outline Definitions: measures • Non-additive measures. Distorted Lebesgue ◦ m : R + → R + a continuous and increasing function such that m (0) = 0 ; λ be the Lebesgue measure. µ m ( A ) = m ( λ ( A )) (2) ◦ If m ( x ) = x 2 , then µ m ( A ) = ( λ ( A )) 2 ◦ If m ( x ) = x p , then µ m ( A ) = ( λ ( A )) p (a) (b) (c) (d) Vicen¸ c Torra; Choquet integral in decision making and metric learning IUKM 2019 - Nara, Japan 10 / 62
Definitions Outline Definitions: measures • Non-additive measures. A large number of families ◦ Sugeno λ -measures: µ ( A ∪ B ) = µ ( A )+ µ ( B )+ λµ ( A ) µ ( B ) ( λ > − 1 ) ◦ For P a non empty set of probability measures, the upper and lower probabilities ⊲ ¯ P ( A ) = sup P ∈P P ( A ) ⊲ P ( A ) = inf P ∈P P ( A ) (dual in the sense: ¯ P ( A ) = 1 − P ( A c ) ) • m-dimensional distorted probabilities (NT/NT, 2005, 2011, 2012, 2018) Unconstrained fuzzy measures DP Vicen¸ c Torra; Choquet integral in decision making and metric learning IUKM 2019 - Nara, Japan 11 / 62
Outline Now we need an integral IUKM 2019 - Nara, Japan 12 / 62
Definitions Outline Definitions: integrals • Additive measure: the way you add areas does not change 1 results (a) (b) (c) b i b i a i b i − 1 b i − 1 a i − 1 x 1 x N x 1 x N x 1 x { x | f ( x ) = b i } { x | f ( x ) ≥ a i } • Riemann integral (a) vs Lebesgue integral (c) ◦ Riemann sum: � I ∈C f ( x ( I )) ∗ µ ( I ) ( C non-overlapping collection, x ( I ) an element of I ) ◦ Lebesgue sum: � a i ∈ Range ( f ) ( a i − a i − 1 ) µ (Γ( a i )) where Γ( a ) := { x | f ( x ) ≥ a } 1 Well, if it is calculable IUKM 2019 - Nara, Japan 13 / 62
Definitions Outline Definitions: integrals • Lebesgue integral � ∞ � fdµ := µ f ( r ) dr 0 where µ f ( r ) = µ ( { x | f ( x ) ≥ r } ) IUKM 2019 - Nara, Japan 14 / 62
Definitions Outline Definitions: integrals • Choquet integral (Choquet, 1954): ◦ µ a non-additive measure, f a measurable function. The Choquet integral of f w.r.t. µ , where µ f ( r ) := µ ( { x | f ( x ) > r } ) : � ∞ � ( C ) fdµ := µ f ( r ) dr. 0 IUKM 2019 - Nara, Japan 15 / 62
Definitions Outline Definitions: integrals • Choquet integral (Choquet, 1954): ◦ µ a non-additive measure, f a measurable function. The Choquet integral of f w.r.t. µ , where µ f ( r ) := µ ( { x | f ( x ) > r } ) : � ∞ � ( C ) fdµ := µ f ( r ) dr. 0 • Properties. ◦ When the measure is additive, this is the Lebesgue integral (standard integral) IUKM 2019 - Nara, Japan 15 / 62
Definitions Outline Definitions: integrals Choquet integral. Discrete version • µ a non-additive measure, f a measurable function. The Choquet integral of f w.r.t. µ , N � � ( C ) fdµ = [ f ( x s ( i ) ) − f ( x s ( i − 1) )] µ ( A s ( i ) ) , i =1 where f ( x s ( i ) ) indicates that the indices have been permuted so that 0 ≤ f ( x s (1) ) ≤ · · · ≤ f ( x s ( N ) ) ≤ 1 , and where f ( x s (0) ) = 0 and A s ( i ) = { x s ( i ) , . . . , x s ( N ) } . IUKM 2019 - Nara, Japan 16 / 62
Definitions Outline Definitions: integrals • Choquet integral. Example: ◦ Distorted probability µ m ( A ) = m ( P ( A )) (with m (0) = 0 , m (1) = 1 ) CI µ m ( f ) : ( a ) → max , ( b ) → median , ( c ) → min , ( d ) → mean (expectation) (a) (b) (c) (d) ◦ Upper and lower probabilities: bounds for expectations CI P ( f ) ≤ inf P E P ( f ) ≤ sup P E P ( f ) ≤ CI ¯ P ( f ) IUKM 2019 - Nara, Japan 17 / 62
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