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83rd EWG-MCDA 2016 Choquet integral: distributions and decisions Vicen c Torra School of Informatics, University of Sk ovde Sk ovde, Sweden March 31-32 , 2016 Overview > Example Outline Overview Basics and objectives:

  1. Outline Aggregation and Choquet integral in MCDM • Aggregation functions and parameters – Choquet integral (generalization of the AM, WM, OWA): a measure ◦ Instead of weight(criteria) : w(security) ◦ We consider weight(set of criteria) : w(security,price,confort) – We can, of course, use w(security,price,confort)=w(security)+w(price)+w(confort) – but also ◦ w(security,price,confort) > w(security)+w(price)+w(confort) or ◦ w(security,price,confort) < w(security)+w(price)+w(confort) 83rd EWG-MCDA 2016 13 / 59

  2. Outline Aggregation and Choquet integral in MCDM • Aggregation functions and parameters – Choquet integral (generalization of the AM, WM, OWA): a measure ∗ And the level curves ? decision ? f 2 f 2 ( x 2 ) x 2 f 2 ( x 1 ) x 1 f 1 f 1 ( x 2 ) f 1 ( x 1 ) 83rd EWG-MCDA 2016 14 / 59

  3. Outline Preliminaries Non-additive (fuzzy) measures and the Choquet integral 83rd EWG-MCDA 2016 15 / 59

  4. 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) Finite case: µ ( A ∪ B ) = µ ( A ) + µ ( B ) for disjoint A , B Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 16 / 59

  5. 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) Finite case: µ ( A ∪ B ) = µ ( A ) + µ ( B ) for disjoint A , B • Probability and weights: µ ( X ) = 1 Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 16 / 59

  6. Definitions Outline Definitions: measures Non-additive (or fuzzy) measures. • ( X, A ) a measurable space, a non-additive measure µ on ( X, A ) is a set function µ : A → [0 , 1] satisfying the following axioms: (i) µ ( ∅ ) = 0 (ii) µ ( X ) ≤ ∞ (iii) A ⊆ B implies µ ( A ) ≤ µ ( B ) (monotonicity) • Weights: µ ( X ) = 1 Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 17 / 59

  7. Definitions Outline Definitions: measures Non-additive measures. Examples. Distorted probabilities • m : R + → R + a continuous and increasing function such that m (0) = 0 ; P be a probability. The following set function µ m is a non-additive measure: µ m,P ( A ) = m ( P ( A )) (1) Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 18 / 59

  8. Definitions Outline Definitions: measures Non-additive measures. Examples. Distorted probabilities • m : R + → R + a continuous and increasing function such that m (0) = 0 ; P be a probability. The following set function µ m is a non-additive measure: µ m,P ( A ) = m ( P ( A )) (1) • If m ( x ) = x 2 , then µ m ( A ) = ( P ( A )) 2 • If m ( x ) = x p , then µ m ( A ) = ( P ( A )) p (a) (b) (c) (d) Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 18 / 59

  9. Definitions Outline Definitions: measures Non-additive measures. Examples. Distorted probabilities • m : R + → R + a continuous and increasing function such that m (0) = 0 ; P be a probability. The following set function µ m is a non-additive measure: µ m,P ( A ) = m ( P ( A )) (2) Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 19 / 59

  10. Definitions Outline Definitions: measures Non-additive measures. Examples. Distorted probabilities • m : R + → R + a continuous and increasing function such that m (0) = 0 ; P be a probability. The following set function µ m is a non-additive measure: µ m,P ( A ) = m ( P ( A )) (2) Applications. • To represent interactions Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 19 / 59

  11. Definitions Outline Definitions: integrals Choquet integral (Choquet, 1954): • µ a non-additive measure, g a measurable function. The Choquet integral of g w.r.t. µ , where µ g ( r ) := µ ( { x | g ( x ) > r } ) : � ∞ � ( C ) gdµ := µ g ( r ) dr. (3) 0 Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 20 / 59

  12. Definitions Outline Definitions: integrals Choquet integral (Choquet, 1954): • µ a non-additive measure, g a measurable function. The Choquet integral of g w.r.t. µ , where µ g ( r ) := µ ( { x | g ( x ) > r } ) : � ∞ � ( C ) gdµ := µ g ( r ) dr. (3) 0 • When the measure is additive, this is the Lebesgue integral Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 20 / 59

  13. Definitions Outline Definitions: integrals Choquet integral (Choquet, 1954): • µ a non-additive measure, g a measurable function. The Choquet integral of g w.r.t. µ , where µ g ( r ) := µ ( { x | g ( x ) > r } ) : � ∞ � ( C ) gdµ := µ g ( r ) dr. (3) 0 • When the measure is additive, this is the Lebesgue integral (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 | f ( x ) = b i } { x | f ( x ) ≥ a i } x Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 20 / 59

  14. 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 ) } . Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 21 / 59

  15. Definitions Outline Definitions: measures Choquet integral: Properties: • When µ is additive, CI corresponds to the weighted mean • CI can represent min, max, mean, order statistics, ... • When µ is µ m,P ( A ) = m ( P ( A )) with m ( x ) = x p , CI µ m ( f ) ( a ) → max , ( b ) → median , ( c ) → min , ( d ) → mean (a) (b) (c) (d) Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 22 / 59

  16. Outline Preliminaries Classification and shapes of distributions 83rd EWG-MCDA 2016 23 / 59

  17. Classification Outline Classification Motivation: Another motivation: classification • Two classes defined in terms of normal distributions (obtained from real data or directly from the parameters of the distribution N ( µ, Σ) ). • An element x in R 2 Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 24 / 59

  18. Classification Outline Classification Motivation: Another motivation: classification • Two classes defined in terms of normal distributions (obtained from real data or directly from the parameters of the distribution N ( µ, Σ) ). • An element x in R 2 → where to classify x ? Two classes 8 6 table[,2] 4 2 0 −2 −2 0 2 4 6 8 table[,1] Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 24 / 59

  19. Classification Outline Classification Classification problems: Classification of x into Ω • x in a n -dimensional space ( i.e. , x ∈ R n ) • Set of k classes Ω = { ω 1 , . . . , ω k } Formalization: • Bayes’ maximum-a-posteriori (MAP) classification decision rule: assigns x to the class ω i s.t. the probability P ( ω i | x ) is maximized. I.e., (Bayes condition): P ( ω i | x ) = P ( x | ω i ) P ( ω i ) P ( x ) or, as P ( x ) is constant for all classes, d i ( x ) = P ( x | ω i ) P ( ω i ) � f ◦ d results into the same classification as for d (e.g. f = ln ) Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 25 / 59

  20. Classification Outline Classification Classification problems: Classes ω i generated from • covariance matrices Σ i • means ¯ x i → class-conditional probability-density function (Gaussian distribution) 1 x i ) T Σ − 1 (2 π ) m/ 2 | Σ i | 1 / 2 e − 1 2 ( x − ¯ ( x − ¯ x i ) P ( x | ω i ) = i Two classes with different correlations 10 table[,2] 5 0 0 5 10 table[,1] Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 26 / 59

  21. Classification Outline Classification Proposition: • Bayes’ maximum-a-posteriori (MAP) classification decision rule, when Σ i = Σ j , and P ( ω i ) = P ( ω j ) , is (Mahalanobis distance) x i ) T Σ − 1 ( x − ¯ d i ( x ) = − ( x − ¯ x i ) Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 27 / 59

  22. Classification Outline Classification Proposition. Bayes’ maximum-a-posteriori (MAP) classification • If Σ i = I for all i (the identity function) x i ) T ( x − ¯ x i || 2 d i ( x ) = − ( x − ¯ x i ) = −|| x − ¯ → Euclidean distance • If Σ i is diagonal (not necessarily equal to I ) m � ( σ 2 j ) − 1 ( x j − ¯ x ij ) 2 d i ( x ) = − j =1 → Weighted Euclidean distance (with weights equal to the inverse of the variances: p j = ( σ 2 j ) − 1 ) Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 28 / 59

  23. Shape of Distributions Outline Shape of distributions The class-conditional probability-density functions established above define level curves with the shape of an ellipse → circumference when variables are independent Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 29 / 59

  24. Shape of Distributions Outline Shape of distributions The class-conditional probability-density functions established above define level curves with the shape of an ellipse → circumference when variables are independent Two classes with different correlations 10 table[,2] 5 0 0 5 10 table[,1] Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 29 / 59

  25. Shape of Distributions Outline Shape of distributions The class-conditional probability-density functions established above define level curves with the shape of an ellipse → circumference when variables are independent Two classes with different correlations 10 table[,2] 5 0 0 5 10 table[,1] What about another shape / another distance ? Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 29 / 59

  26. Shape of Distributions Outline Shape of distributions The class-conditional probability-density functions established above define level curves with the shape of an ellipse → circumference when variables are independent Two classes with different correlations 10 table[,2] 5 0 0 5 10 table[,1] What about another shape / another distance ? What about using the Choquet integral here ? Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 29 / 59

  27. Shape of Distributions Outline Shape of distributions Why Choquet integral?: • Non-additive measures on a set X permit us to represent interactions between objects in X !! ... similar to covariances !! Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 30 / 59

  28. Shape of Distributions Outline Shape of distributions Why Choquet integral?: • Non-additive measures on a set X permit us to represent interactions between objects in X !! ... similar to covariances !! • Choquet integral integrates a function with respect to a non-additive measure Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 30 / 59

  29. Shape of Distributions Outline Shape of distributions Why Choquet integral?: • Non-additive measures on a set X permit us to represent interactions between objects in X !! ... similar to covariances !! • Choquet integral integrates a function with respect to a non-additive measure → can it be used to compute a distance / to define a distribution ? Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 30 / 59

  30. Shape of Distributions Outline Shape of distributions Why Choquet integral?: • Non-additive measures on a set X permit us to represent interactions between objects in X !! ... similar to covariances !! • Choquet integral integrates a function with respect to a non-additive measure → can it be used to compute a distance / to define a distribution ? → if so, what is the shape of the distribution ? Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 30 / 59

  31. Outline Choquet integral based distribution 83rd EWG-MCDA 2016 31 / 59

  32. CI distribution Outline Choquet integral based distribution: Definition Definition: • Y = { Y 1 , . . . , Y n } random variables; µ : 2 Y → [0 , 1] a non-additive measure and m a vector in R n . • The exponential family of Choquet integral based class-conditional probability-density functions is defined by: PC m ,µ ( x ) = 1 Ke − 1 2 CI µ (( x − m ) ◦ ( x − m )) where K is a constant that is defined so that the function is a probability, and where v ◦ w denotes the Hadamard or Schur ( v ◦ w ) = (elementwise) product of vectors v and w (i.e., ( v 1 w 1 . . . v n w n ) ). Notation: • We denote it by C ( m , µ ) . 83rd EWG-MCDA 2016 32 / 59

  33. CI distribution Outline Choquet integral based distribution: Examples • Shapes (level curves) 15.0 15.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q qq qq q q q q q qq qq q q q q q q q q q qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq q q q q q q q q q q q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q qq q qq q q q q q qqqqqqqqqq qqqqqqqqqq q q q q q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qq qq q q q q q q qq q q qq q q q q q q qq qq q q q qqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq q qq q q qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qqq qqq qqq qqq qqq qqq qqq qqq qqq qqq qqq qqq qqqq qqqqqqq qqqqqqq qqqq qqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 15.0 15.0 (-15.0,-15.0) (-15.0,-15.0) 15.0 15.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q qq qq q q q q q q q q q q q q q q qq qq q qq qq qq qq qq qq qq q q q q qq qq qq q qq qq q qqq qqqqqqqqqq qqqqqqqqqq qqq q q qq q qq q q q q qq q q q qq qqqqqqq qqqqqqq qq qq qq qq qq qq qq qq qq qq qqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqq qq qq qq qq qq qqqqqq qqqqqq qq qqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqq qqqq qq qqqq qqqq qq qq qq qqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqq qq qqqqqqqqqq qqqqqqqqqq qqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqq qqqqqqqqqqqq qqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 15.0 15.0 (-15.0,-15.0) (-15.0,-15.0) (a) µ A ( { x } ) = 0 . 1 and µ A ( { y } ) = 0 . 1 , (b) µ B ( { x } ) = 0 . 9 and µ B ( { y } ) = 0 . 9 , (c) µ C ( { x } ) = 0 . 2 and µ C ( { y } ) = 0 . 8 , and (d) µ D ( { x } ) = 0 . 4 and µ D ( { y } ) = 0 . 9 . 83rd EWG-MCDA 2016 33 / 59

  34. CI distribution Outline Choquet integral based distribution: Properties Proposition. Distribution and distance (Choquet distance): • If P ( w i ) = P ( w j ) holds for all i � = j , the decision rule is (max): − CI µ (( x − ¯ x i ) ⊗ ( x − ¯ x i )) Proposition: Distribution/distance and level curves: • The level curves of the Choquet integral in two variables X = { x, y } corresponds to an ellipse when µ ( { x } ) = 1 − µ ( { y } ) . → A natural result: we have an ellipse when µ ( { x } ) + µ ( { y } ) = 1 → i.e., when µ is a probability. This follows from the fact that the Choquet integral with a measure that is a probability is equivalent to a weighted mean. Then, similar results are obtained for larger dimensions. 83rd EWG-MCDA 2016 34 / 59

  35. CI distribution Outline Choquet integral based distribution: Properties Property: • The family of distributions N ( m , Σ ) in R n with a diagonal matrix Σ of rank n , and the family of distributions C ( m , µ ) with an additive measure µ with all µ ( { x i } ) � = 0 are equivalent. ( µ ( X ) is not necessarily here 1) 83rd EWG-MCDA 2016 35 / 59

  36. CI distribution Outline Choquet integral based distribution: Properties Property: • The family of distributions N ( m , Σ ) in R n with a diagonal matrix Σ of rank n , and the family of distributions C ( m , µ ) with an additive measure µ with all µ ( { x i } ) � = 0 are equivalent. ( µ ( X ) is not necessarily here 1) Corollary: • The distribution N ( 0 , I ) corresponds to C ( 0 , µ 1 ) where µ 1 is the additive measure defined as µ 1 ( A ) = | A | for all A ⊆ X . 83rd EWG-MCDA 2016 35 / 59

  37. CI distribution Outline Choquet integral based distribution: N vs. C Properties: • In general, the two families of distributions N ( m , Σ ) and C ( m , µ ) are different. • C ( m , µ ) always symmetric w.r.t. Y 1 and Y 2 axis. 15.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q qq qq q qq qq q qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q qq qq q q q qq qq qq q q qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qqq qqq qqq qqq qqq qqq qqq qqq qqq qqq qqq qqq qqqqqqq qqqqqqq qqqq qqqq qqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 15.0 (-15.0,-15.0) 83rd EWG-MCDA 2016 36 / 59

  38. CI distribution Outline Choquet integral based distribution: N vs. C Properties: • In general, the two families of distributions N ( m , Σ ) and C ( m , µ ) are different. • C ( m , µ ) always symmetric w.r.t. Y 1 and Y 2 axis. 15.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q qq qq q qq qq q qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q qq qq q q q qq qq qq q q qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qqq qqq qqq qqq qqq qqq qqq qqq qqq qqq qqq qqq qqqqqqq qqqqqqq qqqq qqqq qqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 15.0 (-15.0,-15.0) • A generalization of both: Choquet-Mahalanobis based distribution. – Mahalanobis: Σ represents some interactions – Choquet (measure): µ represents some interactions 83rd EWG-MCDA 2016 36 / 59

  39. Outline Choquet-Mahalanobis based distribution 83rd EWG-MCDA 2016 37 / 59

  40. CMI distribution Outline Choquet integral based distribution: generalized distance Definition: • Σ be a matrix, Σ − 1 = LL ∗ be the Cholesky decomposition of its inverse. • The Choquet-Mahalanobis integral is defined by x ) = CI µ ( v ⊗ w ) CMI µ, Σ ( x, ¯ (4) where v and w are the vectors defined by: x ) T L and w = L ∗ ( x − ¯ v = ( x − ¯ x ) , where v ⊗ w denotes the elementwise product of vectors v and w (i.e., ( v ⊗ w ) = ( v 1 w 1 . . . v n w n ) ). 83rd EWG-MCDA 2016 38 / 59

  41. CMI distribution Outline Choquet integral based distribution: generalized distance On the definition: • Well defined when Σ is a covariance matrix When Σ − 1 is a definite-positive matrix, the Cholesky descomposition is unique. This is the case when Σ is a covariance matrix valid for generating a probability- density function. 83rd EWG-MCDA 2016 39 / 59

  42. CMI distribution Outline Choquet integral based distribution: generalized distance On the definition: • Well defined when Σ is a covariance matrix When Σ − 1 is a definite-positive matrix, the Cholesky descomposition is unique. This is the case when Σ is a covariance matrix valid for generating a probability- density function. Proper generalization: • Generalization of both the Mahalanobis and the Choquet integral based distance. – The definition with Σ equal to the identity results into the Choquet integral of ( x − ¯ x ) ⊗ ( x − ¯ x ) with respect to µ . – The definition with µ corresponding to an additive probability µ ( A ) = 1 / | A | results into 1 /n of the Mahalanobis distance with respect to Σ . 83rd EWG-MCDA 2016 39 / 59

  43. CMI distribution Outline Choquet integral based distribution: Definition Definition: • Y = { Y 1 , . . . , Y n } random variables, µ : 2 Y → [0 , 1] a measure, m a vector in R n , and Q a positive-definite matrix. • The exponential family of Choquet-Mahalanobis integral based class- conditional probability-density functions is defined by: PCM m ,µ, Q ( x ) = 1 Ke − 1 2 CI µ ( v ◦ w ) where K is a constant that is defined so that the function is a probability, where LL T = Q is the Cholesky decomposition of the matrix Q , v = ( x − m ) T L , w = L T ( x − m ) , and where v ◦ w denotes the elementwise product of vectors v and w . Notation: • We denote it by CMI ( m , µ, Q ) . 83rd EWG-MCDA 2016 40 / 59

  44. CMI distribution Outline Choquet integral based distribution: Properties Property: • The distribution CMI ( m , µ, Q ) generalizes the multivariate normal distributions and the Choquet integral based distribution. In addition – A CMI ( m , µ, Q ) with µ = µ 1 corresponds to multivariate normal distributions, – A CMI ( m , µ, Q ) with Q = I corresponds to a CI ( m , µ ) . 83rd EWG-MCDA 2016 41 / 59

  45. CMI distribution Outline Choquet integral based distribution: Properties Graphically: • Choquet integral (CI distribution), Mahalobis distance (multivariate normal distribution), generalization (CMI distribution) Choquet−Mahalanobis Choquet Mahalanobis WM 83rd EWG-MCDA 2016 42 / 59

  46. CMI distribution Outline Choquet integral based distribution: Examples 1st Example: Interactions only expressed in terms of a measure. • No correlation exists between the variables. • CMI with σ 1 = 1 , σ 2 = 1 , ρ 12 = 0 . 0 , µ x = 0 . 01 , µ y = 0 . 01 . 83rd EWG-MCDA 2016 43 / 59

  47. CMI distribution Outline Choquet integral based distribution: Examples 2nd Example: Interactions only in terms of a covariance matrix. • CMI with σ 1 = 1 , σ 2 = 1 , ρ 12 = 0 . 9 , µ x = 0 . 10 , µ y = 0 . 90 . 83rd EWG-MCDA 2016 44 / 59

  48. CMI distribution Outline Choquet integral based distribution: Examples 3rd Example: Interactions both: covariance matrix and measure. • CMI with σ 1 = 1 , σ 2 = 1 , ρ 12 = 0 . 9 , µ x = 0 . 01 , µ y = 0 . 01 . 83rd EWG-MCDA 2016 45 / 59

  49. CMI distribution Outline Choquet integral based distribution: Properties More properties: Data not always acc. normality assumption ◦ spherical, elliptical distributions ◦ They generalize, respectively, N ( 0 , I ) and N ( m , Σ) 83rd EWG-MCDA 2016 46 / 59

  50. CMI distribution Outline Choquet integral based distribution: Properties More properties: Data not always acc. normality assumption ◦ spherical, elliptical distributions ◦ They generalize, respectively, N ( 0 , I ) and N ( m , Σ) • Neither CMI ( m , µ, Q ) ⊆ / ⊇ spherical / elliptical distributions. 83rd EWG-MCDA 2016 46 / 59

  51. CMI distribution Outline Choquet integral based distribution: Properties More properties: Data not always acc. normality assumption ◦ spherical, elliptical distributions ◦ They generalize, respectively, N ( 0 , I ) and N ( m , Σ) • Neither CMI ( m , µ, Q ) ⊆ / ⊇ spherical / elliptical distributions. Example: • Non-additive µ : CMI ( m , µ, Q ) not repr. spherical/elliptical 83rd EWG-MCDA 2016 46 / 59

  52. CMI distribution Outline Choquet integral based distribution: Properties More properties: Data not always acc. normality assumption ◦ spherical, elliptical distributions ◦ They generalize, respectively, N ( 0 , I ) and N ( m , Σ) • Neither CMI ( m , µ, Q ) ⊆ / ⊇ spherical / elliptical distributions. Example: • Non-additive µ : CMI ( m , µ, Q ) not repr. spherical/elliptical • No CMI for the following spherical distribution: Spherical distribution with density � 2 � r − r 0 f ( r ) = (1 /K ) e − σ , where r 0 is a radius over which the density is maximum, σ is a variance, and K is the normalization constant. 83rd EWG-MCDA 2016 46 / 59

  53. CMI distribution Outline Choquet integral based distribution: Properties More properties: (symmetry) • P ( x ) a C ( m , µ ) i.e., mean m = ( m 1 , . . . , m n ) and a fuzzy measure µ . Then, for all x ∈ R n and all i ∈ { 1 , . . . , n } P ( x 1 , . . . , x i − 1 , x i + m i , x i +1 , . . . , x n ) = P ( x 1 , . . . , x i − 1 , − x i + m i , x i +1 , . . . , x n ) . • P ( x ) a CMI ( m , µ, Q ) i.e., with mean m = ( m 1 , . . . , m n ) , a positive-definite diagonal matrix Q , and a fuzzy measure µ . Then, for all x ∈ R n and all i ∈ { 1 , . . . , n } P ( x 1 , . . . , x i − 1 , x i + m i , x i +1 , . . . , x n ) = P ( x 1 , . . . , x i − 1 , − x i + m i , x i +1 , . . . , x n ) . 83rd EWG-MCDA 2016 47 / 59

  54. CMI distribution Outline Choquet integral based distribution: Properties More properties: • P ( x ) a C ( m , µ ) i.e., with mean m = ( m 1 , . . . , m n ) . Then, for any fuzzy measure µ , ◦ the mean vector ¯ X = [ E [ X 1 ] , E [ X 2 ] , . . . , E [ X n ]] is m and ◦ Σ = [ Cov [ X i , X j ]] for i = 1 , . . . , n and j = 1 , . . . , n is zero for all i � = j and, thus, diagonal. • P ( x ) a CMI ( m , µ, Q ) i.e., with mean m = ( m 1 , . . . , m n ) . Then, for any fuzzy measure µ and any diagonal matrix Q , ◦ the mean vector ¯ X = [ E [ X 1 ] , E [ X 2 ] , . . . , E [ X n ]] is m and ◦ Σ = [ Cov [ X i , X j ]] for i = 1 , . . . , n and j = 1 , . . . , n is zero for all i � = j and thus, diagonal. 83rd EWG-MCDA 2016 48 / 59

  55. CMI distribution Outline Choquet integral based distribution: Properties More properties: • When Q is not diagonal, we may have Cov [ X i , X j ] � = Q ( X i , X j ) . 83rd EWG-MCDA 2016 49 / 59

  56. CMI distribution Outline Choquet integral based distribution: Properties More properties: If this type of data distinguishable from Normal ? 83rd EWG-MCDA 2016 50 / 59

  57. CMI distribution Outline Choquet integral based distribution: Properties More properties: If this type of data distinguishable from Normal ? Study: • Case of X = { x 1 , x 2 } • CMI (0 , µ ) with µ ( { x } ) = i/ 10 and µ ( { y } ) = i/ 10 for i = 1 , 2 , . . . , 9 83rd EWG-MCDA 2016 50 / 59

  58. CMI distribution Outline Choquet integral based distribution: Properties More properties: If this type of data distinguishable from Normal ? Study: • Case of X = { x 1 , x 2 } • CMI (0 , µ ) with µ ( { x } ) = i/ 10 and µ ( { y } ) = i/ 10 for i = 1 , 2 , . . . , 9 • Test: Normality test for CI-based distribution ◦ Normality of the marginals ◦ Normality of the multidimensional distribution 83rd EWG-MCDA 2016 50 / 59

  59. CMI distribution Outline Choquet integral based distribution: Properties More properties: Normality test for CI-based distribution • Normality of the marginals: Shapiro-Wilk test Marginal computed numerically integrate , uniroot function in R. Almost always the test is passed for samples of n = 100 data 83rd EWG-MCDA 2016 51 / 59

  60. CMI distribution Outline Choquet integral based distribution: Properties More properties: Normality test for CI-based distribution • Normality of the marginals: Shapiro-Wilk test Marginal computed numerically integrate , uniroot function in R. Almost always the test is passed for samples of n = 100 data • Marginals (left) of the bivariate CI ( 0 , µ ) , and the normal distribution (right) with the same variance. µ ( { x 1 } ) = 0 . 1 and µ ( { x 2 } ) = 0 . 1 0.15 0.20 0.10 0.15 marginalX2 normalDistr 0.10 0.05 0.05 0.00 0.00 −20 −10 0 10 20 −20 −10 0 10 20 x x 83rd EWG-MCDA 2016 51 / 59

  61. CMI distribution Outline Choquet integral based distribution: Properties More properties: Normality test for CI-based distribution • Normality of the marginals: Shapiro-Wilk test • Marginals (left) of CI ( 0 , µ ) , and (right) N same variance. (i) µ ( { x 1 } ) = 0 . 1 and µ ( { x 2 } ) = 0 . 1 ; (ii) µ ( { x 1 } ) = 0 . 1 and µ ( { x 2 } ) = 0 . 2 ; (iii) µ ( { x 1 } ) = 0 . 2 and µ ( { x 2 } ) = 0 . 1 ; (iv) µ ( { x 1 } ) = 0 . 9 and µ ( { x 2 } ) = 0 . 9 0.15 0.20 0.25 0.20 0.20 0.15 0.10 0.15 marginalX2 normalDistr marginalX2 normalDistr 0.15 0.10 0.10 0.10 0.05 0.05 0.05 0.05 0.00 0.00 0.00 0.00 −20 −10 0 10 20 −20 −10 0 10 20 −20 −10 0 10 20 −20 −10 0 10 20 x x x x 0.20 0.14 0.30 0.30 0.12 0.25 0.15 0.25 0.10 0.20 0.20 marginalX2 normalDistr 0.08 marginalX2 normalDistr 0.10 0.15 0.15 0.06 0.10 0.10 0.04 0.05 0.05 0.05 0.02 0.00 0.00 0.00 0.00 −20 −10 0 10 20 −20 −10 0 10 20 −20 −10 0 10 20 −20 −10 0 10 20 x x x x 83rd EWG-MCDA 2016 52 / 59

  62. CMI distribution Outline Choquet integral based distribution: Properties More properties: Normality test for CI-based distribution • Normality of the distribution: Mardia’s test based on skewness and kurtosis – Skewness test is passed. – Almost all distributions (in R 2 ) pass kurtosis test in experiments: ◦ CI (0 , µ ) distributions with µ ( { x } ) = i/ 10 and µ ( { y } ) = i/ 10 for i = 1 , 2 , . . . , 9 . ◦ Test only fails in (i) µ ( { x } ) = 0 . 1 and µ ( { y } ) = 0 . 1 , (ii) µ ( { x } ) = 0 . 2 and µ ( { y } ) = 0 . 1 . 83rd EWG-MCDA 2016 53 / 59

  63. Outline Summary 83rd EWG-MCDA 2016 54 / 59

  64. Summary Outline Summary Summary: • Definition of distributions based on the Choquet integral Integral for non-additive measures • Relationship with multivariate normal and spherical distributions 83rd EWG-MCDA 2016 55 / 59

  65. Summary Outline Summary Summary: • Definition of distributions based on the Choquet integral Integral for non-additive measures • Relationship with multivariate normal and spherical distributions Future work: • Study of the properties • Parameters determination from data ( µ, Q ) • Statistical tests 83rd EWG-MCDA 2016 55 / 59

  66. Summary Outline Summary • Level-dependent capacity (non-additive, fuzzy measure) Defined by S. Greco, B. Matarazzo, S. Giove (FSS, 2011) ◦ Level-dependent-based distribution (generalizes CI-based) 2 CI G − 1 µG (( x − ¯ x ) ⊗ ( x − ¯ x )) P ( x ) = 1 K e ◦ Example. Two perspectives of same level dependent CI. Defined by the same fuzzy measures µ 1 and µ 2 with intervals (0 , 3) for µ 1 , and (3 , 100) for µ 2 . µ 1 ( { x } ) = 0 . 05 and µ 1 ( { y } ) = 0 . 95 , and µ 2 ( { x } ) = 0 . 95 and µ 2 ( { y } ) = 0 . 05 83rd EWG-MCDA 2016 56 / 59

  67. Outline Thank you 83rd EWG-MCDA 2016 57 / 59

  68. References Outline References References: • V. Torra, Y. Narukawa, On a comparison between Mahalanobis distance and Choquet integral: the Choquet-Mahalanobis operator, Information Sciences 190 (2012) 56-63. • V. Torra, Distributions based on the Choquet integral and non- additive measures, RIMS Kokyuroku 1906 (2014) 136-143. • V. Torra, Some properties of Choquet integral based probability functions, Acta et Commentationes Universitatis Tartuensis de Mathematica 19:1 (2015) 35-47. http://dx.doi.org/10.12697/ACUTM.2015.19.04 83rd EWG-MCDA 2016 58 / 59

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