Aggregation functions and information fusion. Modeling decisions - PowerPoint PPT Presentation

Aggregation functions • Goals of data aggregation ( goals of the area ): ◦ Formalization of the aggregation process ◦ Definition of new functions (and impossibility results!) ◦ Selection of functions (methods to decide which is the most appropriate function in a given context) ◦ Parameter determination Vicen¸ c Torra; Modeling decisions 2017 14 / 91

Aggregation functions • Goals of data aggregation ( goals of the area ): ◦ Formalization of the aggregation process ◦ Definition of new functions (and impossibility results!) ◦ Selection of functions (methods to decide which is the most appropriate function in a given context) ◦ Parameter determination ◦ Study of existing methods: ◦ Caracterization of functions ◦ Determination of the modeling capabilities of the functions ◦ Relation between operators and parameters (how parameters influence the result: dictatorship?, sensitivity to data → index) . Vicen¸ c Torra; Modeling decisions 2017 14 / 91

Aggregation: II. on the definition of aggregation functions Vicen¸ c Torra; Modeling decisions 2017 15 / 91

Aggregation functions Definition of aggregation functions: • 1. Definition from properties: E.g., results consistent with changes of scale: C ( ra 1 , . . . , ra n ) = r C ( a 1 , . . . , a n ) • 2. Heuristic definition: from functions to properties E.g., after some testing we decide to use xxxxx : properties? • 3. Definition from examples E.g., find C that approximates the examples properties function examples Vicen¸ c Torra; Modeling decisions 2017 16 / 91

Aggregation functions • 1. Definition from properties properties − → function Vicen¸ c Torra; Modeling decisions 2017 17 / 91

Aggregation functions • 1. Definition from properties properties − → function • Some alternatives (1.a) Expressing properties as equations: functional equations Vicen¸ c Torra; Modeling decisions 2017 17 / 91

Aggregation functions • 1. Definition from properties properties − → function • Some alternatives (1.a) Expressing properties as equations: functional equations (1.b) Aggregation of a 1 , a 2 , . . . , a N ∈ D , as the datum c which is at a minimum distance from a i : � c { d ( c, a i ) } , C ( a 1 , a 2 , . . . , a N ) = arg min a i d is a distance over D . Vicen¸ c Torra; Modeling decisions 2017 17 / 91

Aggregation functions • Functional equations (case 1.a). ◦ What is a functional equation ? • Equations where the unknown are functions ◦ Example. Cauchy equation (a well known functional equation): φ ( x + y ) = φ ( x ) + φ ( y ) ◦ find φ ! Vicen¸ c Torra; Modeling decisions 2017 18 / 91

Aggregation functions • Functional equations (case 1.a). ◦ What is a functional equation ? • Equations where the unknown are functions ◦ Example. Cauchy equation (a well known functional equation): φ ( x + y ) = φ ( x ) + φ ( y ) ◦ find φ ! ◦ Solution: For continuous φ , φ ( x ) = αx for an arbitrary value for α Vicen¸ c Torra; Modeling decisions 2017 18 / 91

Aggregation functions • Functional equations (case 1.a). Example I in aggregation. ◦ Distribute s euros among m projects according to the opinion of N experts · · · · · · Proj 1 Proj 2 Proj j Proj m x 1 x 1 x 1 x 1 · · · · · · E 1 1 2 m j x 2 x 2 x 2 x 2 E 2 · · · · · · 1 2 j m . . . . . . . . . . . . x i x i x i x i E i · · · · · · 1 2 j m . . . . . . . . . . . . x N x N x N x N · · · · · · E N 1 2 j m DM f 1 ( x 1 ) f 2 ( x 2 ) · · · f j ( x j ) · · · f m ( x m ) Vicen¸ c Torra; Modeling decisions 2017 19 / 91

Aggregation functions • Functional equations (case 1.a). Example I in aggregation. ◦ The general solution of the system (Prop. 3.11) for m > 2 f j : [0 , s ] N → R + for j = { 1 , · · · , m } m m � � x j = s implies that f j ( x j ) = s j =1 j =1 f j ( 0 ) = 0 for j = 1 , · · · , m Vicen¸ c Torra; Modeling decisions 2017 20 / 91

Aggregation functions • Functional equations (case 1.a). Example I in aggregation. ◦ The general solution of the system (Prop. 3.11) for m > 2 f j : [0 , s ] N → R + for j = { 1 , · · · , m } m m � � x j = s implies that f j ( x j ) = s j =1 j =1 f j ( 0 ) = 0 for j = 1 , · · · , m is given by N � f 1 ( x ) = f 2 ( x ) = · · · = f m ( x ) = f (( x 1 , x 2 , . . . , x N )) = α i x i , i =1 where α 1 , · · · , α N are nonnegative constants satisfying � N i =1 α i = 1 , but are otherwise arbitrary. Vicen¸ c Torra; Modeling decisions 2017 20 / 91

Aggregation functions • Functional equations (case 1.a). Example II in aggregation. ◦ (Prop. 4.17) An operator C is separable in terms of a unique monotone increasing g (that is, C ( a 1 , . . . , a N ) = g ( a 1 ) ◦ · · · ◦ g ( a N ) ) (with ◦ continuous, associative, and cancellative) Vicen¸ c Torra; Modeling decisions 2017 21 / 91

Aggregation functions • Functional equations (case 1.a). Example II in aggregation. ◦ (Prop. 4.17) An operator C is separable in terms of a unique monotone increasing g (that is, C ( a 1 , . . . , a N ) = g ( a 1 ) ◦ · · · ◦ g ( a N ) ) (with ◦ continuous, associative, and cancellative) N � C ( a 1 , . . . , a N ) = φ − 1 ( φ ( g ( a i ))) i =1 and satisfies unanimity C ( a, . . . , a ) = a if and only if it is of the form (quasi-arithmetic mean) N C ( a 1 , . . . , a N ) = φ − 1 ( 1 � φ ( a i )) . N i =1 Vicen¸ c Torra; Modeling decisions 2017 21 / 91

Aggregation functions • Functional equations (case 1.a). Quasi-arithmetic means Name Generator function C ( a 1 , . . . , a N ) � N i =1 x i Arithmetic mean φ ( x ) = x N � � N N i =1 x i Geometric mean φ ( x ) = log x N N Harmonic mean φ ( x ) = 1 /x � N 1 i =1 xi � � N i =1 x 2 φ ( x ) = x 2 Root-mean-square i N � � N i =1 x α α φ ( x ) = x α Root-mean-power i N � � N i =1 e xi φ ( x ) = e x � Exponential mean log N �� − 1 � � N i =1 c 1 /xi � φ ( x ) = c 1 /x Radical mean log c N i =1 x xi � N m s.t. m m = φ ( x ) = x x Basis-exponential mean i N i =1 x 1 /xi � N m s.t. m 1 /m = φ ( x ) = x 1 /x Basis-radical mean i N Vicen¸ c Torra; Modeling decisions 2017 22 / 91

Aggregation functions • Functional equations (case 1.a). Example III in aggregation. ◦ (Prop. 4.20) An operator C is separable in terms of a unique monotone increasing g (that is, C ( a 1 , . . . , a N ) = g ( a 1 ) ◦ · · · ◦ g ( a N ) ) (with ◦ continuous, associative, and cancellative) N � C ( a 1 , . . . , a N ) = φ − 1 ( φ ( g ( a i ))) i =1 and satisfies unanimity C ( a, . . . , a ) = a and positive homogeneity C ( ra 1 , . . . , ra N ) = r C ( a 1 , . . . , a N ) Vicen¸ c Torra; Modeling decisions 2017 23 / 91

Aggregation functions • Functional equations (case 1.a). Example III in aggregation (cont.) ◦ if and only if C is either the root-mean-power N C ( a 1 , . . . , a N ) = ( 1 � a α i ) 1 /α N i =1 with parameter α � = 0 ( RMP α ) or the geometric mean N � a i ) 1 /N . C ( a 1 , . . . , a N ) = ( i =1 Vicen¸ c Torra; Modeling decisions 2017 24 / 91

Aggregation functions • Functional equations (case 1.a). Example III in aggregation (cont.) ◦ if and only if C is either the root-mean-power N C ( a 1 , . . . , a N ) = ( 1 � a α i ) 1 /α N i =1 with parameter α � = 0 ( RMP α ) or the geometric mean N � a i ) 1 /N . C ( a 1 , . . . , a N ) = ( i =1 • Remember Ex. 4.21. Assessing performance of Java runtime systems. Only RMP and GM are acceptable !! (Note: lim α → 0 RMP α = GM ) • Root-mean-powers, known as r th power mean, generalized mean. Vicen¸ c Torra; Modeling decisions 2017 24 / 91

Aggregation functions • Functional equations (case 1.a). Example IV in aggregation. ◦ (Prop. 4.24) When we add the equation of reciprocity C (1 /a 1 , . . . , 1 /a N ) = 1 / C ( a 1 , . . . , a N ) the only operator C satisfying all conditions is the geometric mean N � a i ) 1 /N . C ( a 1 , . . . , a N ) = ( i =1 Vicen¸ c Torra; Modeling decisions 2017 25 / 91

Aggregation functions • Functional equations (case 1.a). Root-mean-powers ( r th power mean, generalized mean) Name α C ( a 1 , . . . , a N ) � N i =1 x i Arithmetic mean α = 1 N � � N i =1 x 2 Root-mean-square α = 2 i N N Harmonic mean α = − 1 � N 1 i =1 xi Vicen¸ c Torra; Modeling decisions 2017 26 / 91

Aggregation functions • Example (case (b)): Consider the following expression � c { d ( c, a i ) } , C ( a 1 , a 2 , . . . , a N ) = arg min a i where a i are numbers from R and d is a distance on D . Then, Vicen¸ c Torra; Modeling decisions 2017 27 / 91

Aggregation functions • Example (case (b)): Consider the following expression � c { d ( c, a i ) } , C ( a 1 , a 2 , . . . , a N ) = arg min a i where a i are numbers from R and d is a distance on D . Then, 1. When d ( a, b ) = ( a − b ) 2 , C is the arithmetic mean I.e., C ( a 1 , a 2 , . . . , a N ) = � N i =1 a i /N . 2. When d ( a, b ) = | a − b | , C is the median I.e., the median of a 1 , a 2 , . . . , a N is the element which occupies the central position when we order a i . 3. When d ( a, b ) = 1 iff a = b , C is the plurality rule (mode or voting). I.e., C ( a 1 , a 2 , . . . , a N ) selects the element of R with a largest frequency among elements in ( a 1 , a 2 , . . . , a N ) . Vicen¸ c Torra; Modeling decisions 2017 27 / 91

Aggregation functions • 2. Heuristic definition: from functions to properties function − → properties ◦ We can use functional equations for this purpose: characterizations of functions ◦ We have propositions with “if and only if” eq1, eq2, eq3 if and only if aggr. function so, given a function we know the fundamental properties of the function. E.g., geometric mean: a fundamental property is reciprocity • Note: characterizations are not unique Vicen¸ c Torra; Modeling decisions 2017 28 / 91

Aggregation: III. from the weighted mean to fuzzy integrals Vicen¸ c Torra; Modeling decisions 2017 29 / 91

Aggregation: III. from the weighted mean to fuzzy integrals 1. An example Vicen¸ c Torra; Modeling decisions 2017 30 / 91

Aggregation: example Example. A and B teaching a tutorial+training course w/ constraints • The total number of sessions is six. • Professor A will give the tutorial, which should consist of about three sessions; three is the optimal number of sessions; a difference in the number of sessions greater than two is unacceptable. • Professor B will give the training part, consisting of about two sessions. • Both professors should give more or less the same number of sessions. A difference of one or two is half acceptable; a difference of three is unacceptable. Vicen¸ c Torra; Modeling decisions 2017 31 / 91

Aggregation: example Example. Formalization • Variables ◦ x A : Number of sessions taught by Professor A ◦ x B : Number of sessions taught by Professor B • Constraints ◦ the constraints are translated into ⋆ C 1 : x A + x B should be about 6 ⋆ C 2 : x A should be about 3 ⋆ C 3 : x B should be about 2 ⋆ C 4 : | x A − x B | should be about 0 ◦ using fuzzy sets, the constraints are described ... Vicen¸ c Torra; Modeling decisions 2017 32 / 91

Aggregation: example Example. Formalization • Constraints ◦ if fuzzy set µ 6 expresses “about 6,” then, we evaluate “ x A + x B should be about 6” by µ 6 ( x A + x B ) . → given µ 6 , µ 3 , µ 2 , µ 0 , ◦ Then, given a solution pair ( x A , x B ) , the degrees of satisfaction: ⋆ µ 6 ( x A + x B ) ⋆ µ 3 ( x A ) ⋆ µ 2 ( x B ) ⋆ µ 0 ( | x A − x B | ) Vicen¸ c Torra; Modeling decisions 2017 33 / 91

Aggregation: example Example. Formalization • Membership functions for constraints µ 2 µ 0 µ 3 µ 6 1 2 3 4 5 6 7 Vicen¸ c Torra; Modeling decisions 2017 34 / 91

Aggregation: example Example. Application alternative Satisfaction degrees Satisfaction degrees ( x A , x B ) ( µ 6 ( x A + x B ) , µ 3 ( x A ) , C 1 C 2 C 3 C 4 µ 2 ( x B ) , µ 0 ( | x A − x B | ) ) (2 , 2) ( µ 6 (4) , µ 3 (2) , µ 2 (2) , µ 0 (0) ) 0 0 . 5 1 1 (2 , 3) ( µ 6 (5) , µ 3 (2) , µ 2 (3) , µ 0 (1) ) 0 . 5 0 . 5 0 . 5 0 . 5 (2 , 4) ( µ 6 (6) , µ 3 (2) , µ 2 (4) , µ 0 (2) ) 1 0 . 5 0 0 . 5 (3 . 5 , 2 . 5) ( µ 6 (6) , µ 3 (3 . 5) , µ 2 (2 . 5) , µ 0 (1) ) 1 0 . 5 0 . 5 0 . 5 (3 , 2) ( µ 6 (5) , µ 3 (3) , µ 2 (2) , µ 0 (1) ) 0 . 5 1 1 0 . 5 (3 , 3) ( µ 6 (6) , µ 3 (3) , µ 2 (3) , µ 0 (0) ) 1 1 0 . 5 1 Vicen¸ c Torra; Modeling decisions 2017 35 / 91

Aggregation: III. from the weighted mean to fuzzy integrals 2. WM, OWA, and WOWA operators Vicen¸ c Torra; Modeling decisions 2017 36 / 91

Aggregation: WM, OWA, and WOWA operators • Operators ◦ Weighting vector (dimension N ): v = ( v 1 ...v N ) iff v i ∈ [0 , 1] and � i v i = 1 ◦ Arithmetic mean (AM : R N → R ): AM ( a 1 , ..., a N ) = (1 /N ) � N i =1 a i ◦ Weighted mean (WM: R N → R ): WM p ( a 1 , ..., a N ) = � N i =1 p i a i ( p a weighting vector of dimension N ) ◦ Ordered Weighting Averaging operator (OWA: R N → R ): N � OWA w ( a 1 , ..., a N ) = w i a σ ( i ) , i =1 where { σ (1) , ..., σ ( N ) } is a permutation of { 1 , ..., N } s. t. a σ ( i − 1) ≥ a σ ( i ) , and w a weighting vector. Vicen¸ c Torra; Modeling decisions 2017 37 / 91

Aggregation: WM, OWA, and WOWA operators Example. Application • Let us consider the following situation: ◦ Professor A is more important than Professor B ◦ The number of sessions equal to six is the most important constraint (not a crisp requirement) ◦ The difference in the number of sessions taught by the two professors is the least important constraint WM with p = ( p 1 , p 2 , p 3 , p 4 ) = (0 . 5 , 0 . 3 , 0 . 15 , 0 . 05) . Vicen¸ c Torra; Modeling decisions 2017 38 / 91

Aggregation: WM, OWA, and WOWA operators Example. Application • WM with p = ( p 1 , p 2 , p 3 , p 4 ) = (0 . 5 , 0 . 3 , 0 . 15 , 0 . 05) . alternative Aggregation of the Satisfaction degrees WM ( x A , x B ) WM p ( C 1 , C 2 , C 3 , C 4 ) (2 , 2) WM p (0 , 0 . 5 , 1 , 1) 0.35 (2 , 3) WM p (0 . 5 , 0 . 5 , 0 . 5 , 0 . 5) 0.5 (2 , 4) WM p (1 , 0 . 5 , 0 , 0 . 5) 0.675 (3 . 5 , 2 . 5) WM p (1 , 0 . 5 , 0 . 5 , 0 . 5) 0.75 (3 , 2) WM p (0 . 5 , 1 , 1 , 0 . 5) 0.725 (3 , 3) WM p (1 , 1 , 0 . 5 , 1) 0.925 Vicen¸ c Torra; Modeling decisions 2017 39 / 91

Aggregation: WM, OWA, and WOWA operators Example. Application • Compensation: how many values can have a bad evaluation • One bad value does not matter: OWA with w = (1 / 3 , 1 / 3 , 1 / 3 , 0) (lowest value discarded) alternative Aggregation of the Satisfaction degrees OWA ( x A , x B ) OWA w ( C 1 , C 2 , C 3 , C 4 ) (2 , 2) OWA w (0 , 0 . 5 , 1 , 1) 0.8333 (2 , 3) OWA w (0 . 5 , 0 . 5 , 0 . 5 , 0 . 5) 0.5 (2 , 4) OWA w (1 , 0 . 5 , 0 , 0 . 5) 0.6666 (3 . 5 , 2 . 5) OWA w (1 , 0 . 5 , 0 . 5 , 0 . 5) 0.6666 (3 , 2) OWA w (0 . 5 , 1 , 1 , 0 . 5) 0.8333 (3 , 3) OWA w (1 , 1 , 0 . 5 , 1) 1.0 Vicen¸ c Torra; Modeling decisions 2017 40 / 91

Aggregation: WM, OWA, and WOWA operators • Weighted Ordered Weighted Averaging WOWA operator (WOWA : R N → R ): WOWA p , w ( a 1 , ..., a N ) = � N i =1 ω i a σ ( i ) where ω i = w ∗ ( � j ≤ i p σ ( j ) ) − w ∗ ( � j<i p σ ( j ) ) , with σ a permutation of { 1 , ..., N } s. t. a σ ( i − 1) ≥ a σ ( i ) , and w ∗ a nondecreasing function that interpolates the points { ( i/N, � j ≤ i w j ) } i =1 ,...,N ∪ { (0 , 0) } . w ∗ is required to be a straight line when the points can be interpolated in this way. Vicen¸ c Torra; Modeling decisions 2017 41 / 91

Aggregation: WM, OWA, and WOWA operators • Construction of the w ∗ quantifier (a) (b) ( ) w w N N w w 2 2 ! 1 w 1 w 1 ! 1 � 0 � p � (1) p � (1) 1 = N 1 = N ::: 1 = N 0 0 0 0 p p p p p � (1) � (1) � (2) � ( N ) � (1) • Rationale for new weights ( ω i , for each value a i ) in terms of p and w . ◦ If a i is small, and small values have more importance than larger ones, increase p i for a i (i.e., ω i ≥ p σ ( i ) ). (the same holds if the value a i is large and importance is given to large values) ◦ If a i is small, and importance is for large values, ω i < p σ ( i ) (the same holds if a i is large and importance is given to small values). Vicen¸ c Torra; Modeling decisions 2017 42 / 91

Aggregation: WM, OWA, and WOWA operators • The shape of the function w ∗ gives importance ◦ (a) to large values ◦ (b) to medium values ◦ (c) to small values ◦ (d) equal importance to all values (a) (b) (c) (d) Vicen¸ c Torra; Modeling decisions 2017 43 / 91

Aggregation: WM, OWA, and WOWA operators Example. Application • Importance for constraints as given above: p = (0 . 5 , 0 . 3 , 0 . 15 , 0 . 05) • Compensation as given above: w = (1 / 3 , 1 / 3 , 1 / 3 , 0) (lowest value discarded) → WOWA with p and w . alternative Aggregation of the Satisfaction degrees WOWA ( x A , x B ) WOWA p , w ( C 1 , C 2 , C 3 , C 4 ) (2 , 2) WOWA p , w (0 , 0 . 5 , 1 , 1) 0.4666 (2 , 3) WOWA p , w (0 . 5 , 0 . 5 , 0 . 5 , 0 . 5) 0.5 (2 , 4) WOWA p , w (1 , 0 . 5 , 0 , 0 . 5) 0.8333 (3 . 5 , 2 . 5) WOWA p , w (1 , 0 . 5 , 0 . 5 , 0 . 5) 0.8333 (3 , 2) WOWA p , w (0 . 5 , 1 , 1 , 0 . 5) 0.8 (3 , 3) WOWA p , w (1 , 1 , 0 . 5 , 1) 1.0 Vicen¸ c Torra; Modeling decisions 2017 44 / 91

Aggregation: WM, OWA, and WOWA operators • Properties ◦ The WOWA operator generalizes the WM and the OWA operator. ◦ When p = (1 /N . . . 1 /N ) , OWA WOWA p , w ( a 1 , ..., a N ) = OWA w ( a 1 , ..., a N ) for all w and a i . ◦ When w = (1 /N ... 1 /N ) , WM WOWA p , w ( a 1 , ..., a N ) = WM p ( a 1 , ..., a N ) for all p and a i . ◦ When w = p = (1 /N ... 1 /N ) , AM WOWA p , w ( a 1 , ..., a N ) = AM ( a 1 , ..., a N ) Vicen¸ c Torra; Modeling decisions 2017 45 / 91

Aggregation: III. from the weighted mean to fuzzy integrals 3. Choquet integral Vicen¸ c Torra; Modeling decisions 2017 46 / 91

Choquet integrals • In the WM, a single weight is used for each element I.e., p i = p ( x i ) (where, x i is the information source that supplies a i ) → when we consider a set A ⊂ X , weight of A ??? Vicen¸ c Torra; Modeling decisions 2017 47 / 91

Choquet integrals • In the WM, a single weight is used for each element I.e., p i = p ( x i ) (where, x i is the information source that supplies a i ) → when we consider a set A ⊂ X , weight of A ??? . . . fuzzy measures µ ( A ) Vicen¸ c Torra; Modeling decisions 2017 47 / 91

Choquet integrals • Example. ◦ We need to evaluate students (who is best?) using marks in three subjects X= { Mathematics, Physics, Literature } ( M , P , L ) ◦ p M = 0 . 4 , p P = 0 . 4 , p L = 0 . 2 . In the WM, a single weight is used for each element I.e., p i = p ( x i ) (where, x i is the information source that supplies a i ) → when we consider a set A ⊂ X , weight of A ??? ◦ p(Mathematics, Physics) ? Vicen¸ c Torra; Modeling decisions 2017 48 / 91

Choquet integrals • Example. ◦ We need to evaluate students (who is best?) using marks in three subjects X= { Mathematics, Physics, Literature } ( M , P , L ) ◦ p M = 0 . 4 , p P = 0 . 4 , p L = 0 . 2 . In the WM, a single weight is used for each element I.e., p i = p ( x i ) (where, x i is the information source that supplies a i ) → when we consider a set A ⊂ X , weight of A ??? ◦ p(Mathematics, Physics) ? . . . fuzzy measures µ ( A ) Vicen¸ c Torra; Modeling decisions 2017 48 / 91

Choquet integrals • fuzzy measures µ ( A ) : X ⊂ X = { M, P, L } P M L Vicen¸ c Torra; Modeling decisions 2017 49 / 91

Choquet integrals • fuzzy measures µ ( A ) Formally, ◦ Fuzzy measure ( µ : ℘ ( X ) → [0 , 1] ), a set function satisfying (i) µ ( ∅ ) = 0 , µ ( X ) = 1 (boundary conditions) (ii) A ⊆ B implies µ ( A ) ≤ µ ( B ) (monotonicity) Vicen¸ c Torra; Modeling decisions 2017 50 / 91

Choquet integrals • fuzzy measures µ ( A ) Formally, ◦ Fuzzy measure ( µ : ℘ ( X ) → [0 , 1] ), a set function satisfying (i) µ ( ∅ ) = 0 , µ ( X ) = 1 (boundary conditions) (ii) A ⊆ B implies µ ( A ) ≤ µ ( B ) (monotonicity) ◦ Similar to probability or standard (additive) measures, but additivity condition is removed replaced by monotonicity Vicen¸ c Torra; Modeling decisions 2017 50 / 91

Choquet integrals • fuzzy measures µ ( A ) Formally, ◦ Fuzzy measure ( µ : ℘ ( X ) → [0 , 1] ), a set function satisfying (i) µ ( ∅ ) = 0 , µ ( X ) = 1 (boundary conditions) (ii) A ⊆ B implies µ ( A ) ≤ µ ( B ) (monotonicity) ◦ Similar to probability or standard (additive) measures, but additivity condition is removed replaced by monotonicity ◦ Why? to represent redundancy and support (for A ∩ B = ∅ ) µ ( A ∪ B ) < µ ( A ) + µ ( B ) µ ( A ∪ B ) > µ ( A ) + µ ( B ) Vicen¸ c Torra; Modeling decisions 2017 50 / 91

Choquet integrals • Fuzzy measures µ ( A ) : Example with X = { M, P, L } 1. Boundary conditions: µ ( ∅ ) = 0 , µ ( { M, P, L } ) = 1 2. Relative importance of scientific versus literary subjects: µ ( { M } ) = µ ( { P } ) = 0 . 45 , µ ( { L } ) = 0 . 3 3. Redudancy between mathematics and physics: µ ( { M, P } ) = 0 . 5 < µ ( { M } ) + µ ( { P } ) 4. Support between literature and scientific subjects: µ ( { M, L } ) = µ ( { P, L } ) = 0 . 9 > µ ( { P } ) + µ ( { L } ) = 0 . 45 + 0 . 3 = 0 . 75 µ ( { M, L } ) = µ ( { P, L } ) = 0 . 9 > µ ( { M } ) + µ ( { L } ) = 0 . 45 + 0 . 3 = 0 . 75 Vicen¸ c Torra; Modeling decisions 2017 51 / 91

Choquet integrals • Now, we have a fuzzy measure µ ( A ) then, how aggregation proceeds? ⇒ fuzzy integrals as e.g. the Choquet integral Vicen¸ c Torra; Modeling decisions 2017 52 / 91

Choquet integrals • In WM, we combine a i w.r.t. weights p i . → a i is the value supplied by information source x i . Formally Vicen¸ c Torra; Modeling decisions 2017 53 / 91

Choquet integrals • In WM, we combine a i w.r.t. weights p i . → a i is the value supplied by information source x i . Formally ◦ X = { x 1 , . . . , x N } is the set of information sources ◦ f : X → R + the values supplied by the sources → then a i = f ( x i ) Thus, N N � � WM p ( a 1 , ..., a N ) = p i a i = p i f ( x i ) = WM p ( f ( x 1 ) , ..., f ( x N )) i =1 i =1 Vicen¸ c Torra; Modeling decisions 2017 53 / 91

Choquet integrals • Choquet integral of f w.r.t. µ (alternative notation, CI µ ( a 1 , . . . , a N ) /CI µ ( f ) ) N � � ( C ) fdµ = [ f ( x s ( i ) ) − f ( x s ( i − 1) )] µ ( A s ( i ) ) , i =1 where s in f ( x s ( i ) ) is a permutation so that f ( x s ( i − 1) ) ≤ f ( x s ( i ) ) for i ≥ 1 , f ( x s (0) ) = 0 , and A s ( k ) = { x s ( j ) | j ≥ k } and A s ( N +1) = ∅ . • Alternative expressions (Proposition 6.18): N � � ( C ) fdµ = f ( x σ ( i ) )[ µ ( A σ ( i ) ) − µ ( A σ ( i − 1) )] , i =1 N � � ( C ) fdµ = f ( x s ( i ) )[ µ ( A s ( i ) ) − µ ( A s ( i +1) )] , i =1 where σ is a permutation of { 1 , . . . , N } s.t. f ( x σ ( i − 1) ) ≥ f ( x σ ( i ) ) , where A σ ( k ) = { x σ ( j ) | j ≤ k } for k ≥ 1 and A σ (0) = ∅ Vicen¸ c Torra; Modeling decisions 2017 54 / 91

Choquet integrals • Different equations point out different aspects of the CI fdµ = � N (6.1) ( C ) � i =1 [ f ( x s ( i ) ) − f ( x s ( i − 1) )] µ ( A s ( i ) ) , 0 a s (1) a s (2) a s (3) a s (4) a s (5) µ ( A s (2) ) µ ( A s (1) ) = { x s (1) , · · · , x s ( N ) } µ ( A s (4) ) = { x s (4) , · · · , x s ( N ) } fdµ = � N (6.2) ( C ) � i =1 f ( x σ ( i ) )[ µ ( A σ ( i ) ) − µ ( A σ ( i − 1) )] , Vicen¸ c Torra; Modeling decisions 2017 55 / 91

Choquet integrals � • fdµ = (for additive measures) (6.5) � x ∈ X f ( x ) µ ( { x } ) (6.6) � R i =1 b i µ ( { x | f ( x ) = b i } ) (6.7) � N i =1 ( a i − a i − 1 ) µ ( { x | f ( x ) ≥ a i } ) (6.8) � N � � i =1 ( a i − a i − 1 ) 1 − µ ( { x | f ( x ) ≤ a i − 1 } ) (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 • Among (6.5), (6.6) and (6.7), only (6.7) satisfies internality. Vicen¸ c Torra; Modeling decisions 2017 56 / 91

Choquet integrals • Properties of CI ◦ Horizontal additive because CI µ ( f ) = CI µ ( f ∧ c ) + CI µ ( f + c ) ( f = ( f ∧ c ) + f + c is a horizontal additive decomposition of f ) where, f + c is defined by (for c ∈ [0 , 1] ) � 0 if f ( x ) ≤ c f + c = f ( x ) − c if f ( x ) > c. f + f ∧ c f c c Vicen¸ c Torra; Modeling decisions 2017 57 / 91

Choquet integrals • Definitions ( X a reference set, f, g functions f, g : X → [0 , 1] ) ◦ f < g when, for all x i , f ( x i ) < g ( x i ) ◦ f and g are comonotonic if, for all x i , x j ∈ X , f ( x i ) < f ( x j ) imply that g ( x i ) ≤ g ( x j ) ◦ C is comonotonic monotone if and only if, for comonotonic f and g , f ≤ g imply that C ( f ) ≤ C ( g ) ◦ C is comonotonic additive if and only if, for comonotonic f and g , C ( f + g ) = C ( f ) + C ( g ) • Characterization. Let C satisfy the following properties ◦ C is comonotonic monotone ◦ C is comonotonic additive ◦ C (1 , . . . , 1) = 1 Then, there exists µ s.t. C ( f ) is the CI of f w.r.t. µ . Vicen¸ c Torra; Modeling decisions 2017 58 / 91

Choquet integrals • Properties ◦ WM, OWA and WOWA are particular cases of CI. ⋆ WM with weighting vector p is a CI w.r.t. µ p ( B ) = � x i ∈ B p i ⋆ OWA with weighting vector w is a CI w.r.t. µ w ( B ) = � | B | i =1 w i ⋆ WOWA with w.v. p and w is a CI w.r.t. µ p , w ( B ) = w ∗ ( � x i ∈ B p i ) ◦ Any CI with a symmetric measure is an OWA operator. ◦ Any CI with a distorted probability is a WOWA operator. ◦ Let A be a crisp subset of X ; then, the Choquet integral of A with respect to µ is µ ( A ) . Here, the integral of A corresponds to the integral of its characteristic function, or, in other words, to the integral of the function f A defined as f A ( x ) = 1 if and only if x ∈ A . Vicen¸ c Torra; Modeling decisions 2017 59 / 91

Aggregation: III. from the weighted mean to fuzzy integrals 4. Weighted minimum and maximum Vicen¸ c Torra; Modeling decisions 2017 60 / 91

Weighted Minimum and Weighted Maximum • Possibilistic weighting vector (dimension N ): v = ( v 1 ...v N ) iff v i ∈ [0 , 1] and max i v i = 1 . • Weighted minimum (WMin: [0 , 1] N → [0 , 1] ): WMin u ( a 1 , ..., a N ) = min i max( neg ( u i ) , a i ) (alternative definition can be given with v = ( v 1 , . . . , v N ) where v i = neg ( u i ) ) • Weighted maximum (WMax: [0 , 1] N → [0 , 1] ): WMax u ( a 1 , ..., a N ) = max i min( u i , a i ) Vicen¸ c Torra; Modeling decisions 2017 61 / 91

Weighted Minimum and Weighted Maximum • Only operators in ordinal scales (max, min, neg ) are used in WMax and WMin . • neg is completely determined in an ordinal scale Proposition 6.36. Let L = { l 0 , . . . , l r } with l 0 < L l 1 < L · · · < L l r ; then, there exists only one function, neg : L → L , satisfying (N1) if x < L x ′ then neg ( x ) > L neg ( x ′ ) for all x, x ′ in L . (N2) neg ( neg ( x )) = x for all x in L . This function is defined by neg ( x i ) = x r − i for all x i in L • Properties. For u = (1 , . . . , 1) ◦ WMIN u = min ◦ WMAX u = max Vicen¸ c Torra; Modeling decisions 2017 62 / 91

Aggregation: III. from the weighted mean to fuzzy integrals 5. Sugeno integral Vicen¸ c Torra; Modeling decisions 2017 63 / 91

Sugeno integral • Sugeno integral of f w.r.t. µ (alternative notation, SI µ ( a 1 , . . . , a N ) /SI µ ( f ) ) � ( S ) fdµ = max i =1 ,N min( f ( x s ( i ) ) , µ ( A s ( i ) )) , where s in f ( x s ( i ) ) is a permutation so that f ( x s ( i − 1) ) ≤ f ( x s ( i ) ) for i ≥ 2 , and A s ( k ) = { x s ( j ) | j ≥ k } . • Alternative expression (Proposition 6.38): max min( f ( x σ ( i ) ) , µ ( A σ ( i ) )) , i where σ is a permutation of { 1 , . . . , N } s.t. f ( x σ ( i − 1) ) ≥ f ( x σ ( i ) ) , where A σ ( k ) = { x σ ( j ) | j ≤ k } for k ≥ 1 Vicen¸ c Torra; Modeling decisions 2017 64 / 91

Sugeno integral • Graphical interpretation of Sugeno integrals (a) (b) ( ) f ( x ) f ( x ) s ( i ) � ( A ) s ( i ) R ( S ) f d� f ( x ) f ( x ) s ( i ) � ( A ) s ( i ) � ( A ) Vicen¸ c Torra; Modeling decisions 2017 65 / 91

Sugeno integral • Properties ◦ WMin and WMax are particular cases of SI ⋆ WMax with weighting vector u is a SI w.r.t. µ wmax ( A ) = max a i ∈ A u i . u ⋆ WMin with weighting vector u is a SI w.r.t. µ wmin ( A ) = 1 − max a i / ∈ A u i . u Vicen¸ c Torra; Modeling decisions 2017 66 / 91

Fuzzy integrals • Fuzzy integrals that generalize Choquet and Sugeno integrals ◦ The fuzzy t-conorm integral ◦ The twofold integral Vicen¸ c Torra; Modeling decisions 2017 67 / 91

Aggregation: IV. fuzzy measures Vicen¸ c Torra; Modeling decisions 2017 68 / 91

Fuzzy measures • Definition: (i) µ ( ∅ ) = 0 , µ ( X ) = 1 (boundary conditions) (ii) A ⊆ B implies µ ( A ) ≤ µ ( B ) (monotonicity) • Difficulty: ◦ 2 | X | − 2 values (because µ ( ∅ ) = 0 , µ ( X ) = 1 ) • Solution: Families of fuzzy measures (to reduce complexity) Vicen¸ c Torra; Modeling decisions 2017 69 / 91

Fuzzy measures • Examples of families: ◦ ⊥ -Decomposable Fuzzy Measures ( ⊥ a t-conorm) µ ( A ∪ B ) = µ ( A ) ⊥ µ ( B ) . Therefore, for a given A : ⊥ x i ∈ A v ( x i ) ◦ Sugeno λ -measures (for λ > − 1 ) µ ( A ∪ B ) = µ ( A ) + µ ( B ) + λµ ( A ) µ ( B ) ⊥ -decomposable for ⊥ ( x, y ) = min (1 , x + y + λxy ) . ◦ Extensively used in computer vision applications Vicen¸ c Torra; Modeling decisions 2017 70 / 91

Fuzzy measures • Examples of families: ◦ Distorted probabilities ( P : probability, f : increasing function) µ ( A ) = f ( P ( A )) ◦ Well known in economics (decision) ◦ m -dimensional distorted probability X 1 , . . . , X m , P i , f µ ( A ) = f ( P 1 ( A ∩ X 1 ) , P 2 ( A ∩ X 2 ) , · · · , P m ( A ∩ X m )) . Vicen¸ c Torra; Modeling decisions 2017 71 / 91

Aggregation: V. preference relations (MCDM: social choice) Vicen¸ c Torra; Modeling decisions 2017 72 / 91

Aggregation for preference relations • Social choice ◦ studies voting rules, and how the preferences of a set of people can be aggregated to obtain the preference of the set. Vicen¸ c Torra; Modeling decisions 2017 73 / 91

Aggregation for preference relations • Given preference relations, how aggregation is built? ◦ Formalization of preferences with > an = (preference, indiference) ◦ F ( R 1 , R 2 , . . . , R N ) to denote aggregated preference Vicen¸ c Torra; Modeling decisions 2017 74 / 91

Aggregation for preference relations • Given preference relations, how aggregation is built? ◦ Formalization of preferences with > an = (preference, indiference) ◦ F ( R 1 , R 2 , . . . , R N ) to denote aggregated preference ◦ Problems (I): consider ⋆ R 1 : x > y > z ⋆ R 4 : y > z > x ⋆ R 5 : z > x > y → simple majority rule: u > v if most prefer u to v ⋆ x > y, y > z, z > x (intransitive!!: x > y, y > z but not x > z ) ◦ Problems (II): → Arrow impossibility theorem Vicen¸ c Torra; Modeling decisions 2017 74 / 91

Aggregation for preference relations • Given preference relations, how aggregation is built? • Axioms of Arrow impossibility theorem C0 Finite number of voters and more than one Number of alternatives more or equal to three C1 Universality: Voters can select any total preorder 1 C2 Transitivity: The result is a total preorder C3 Unanimity: If all agree on x better than y , then x better than y in the social preference C4 Independence of irrellevant alternatives: the social preference of x and y only depends on the preferences on x and y C5 No-dictatorship: No voter can be a dictatorship • There is no function F that satisfies all C0-C5 axioms 1 either x � y or y � x Vicen¸ c Torra; Modeling decisions 2017 75 / 91