The Empirical Variance of a Set of Fuzzy Intervals Jrme Fortin, - PowerPoint PPT Presentation

The Empirical Variance of a Set of Fuzzy Intervals Jérôme Fortin, Didier Dubois, Hélène Fargier Irit Université Paul Sabatier Toulouse III FUZZIEEE2005 Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 1 / 17

Introduction ◮ Context ◮ Except for some arithmetic operations, computing with fuzzy interval is not so simple Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 2 / 17

Introduction ◮ Context ◮ Except for some arithmetic operations, computing with fuzzy interval is not so simple ◮ Many authors propose to use interval analysis methods on α -cuts Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 2 / 17

Introduction ◮ Context ◮ Except for some arithmetic operations, computing with fuzzy interval is not so simple ◮ Many authors propose to use interval analysis methods on α -cuts ◮ Aim of the work ◮ Viewing a fuzzy interval as a pair of fuzzy bounds, each fuzzy bound modeled by a monotonic profile Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 2 / 17

Introduction ◮ Context ◮ Except for some arithmetic operations, computing with fuzzy interval is not so simple ◮ Many authors propose to use interval analysis methods on α -cuts ◮ Aim of the work ◮ Viewing a fuzzy interval as a pair of fuzzy bounds, each fuzzy bound modeled by a monotonic profile ◮ Apply interval analysis to the whole membership functions for locally monotonic functions Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 2 / 17

Introduction ◮ Context ◮ Except for some arithmetic operations, computing with fuzzy interval is not so simple ◮ Many authors propose to use interval analysis methods on α -cuts ◮ Aim of the work ◮ Viewing a fuzzy interval as a pair of fuzzy bounds, each fuzzy bound modeled by a monotonic profile ◮ Apply interval analysis to the whole membership functions for locally monotonic functions ◮ Extend this model for the computation of the variance of a set of fuzzy numbers Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 2 / 17

Introduction ◮ Context ◮ Except for some arithmetic operations, computing with fuzzy interval is not so simple ◮ Many authors propose to use interval analysis methods on α -cuts ◮ Aim of the work ◮ Viewing a fuzzy interval as a pair of fuzzy bounds, each fuzzy bound modeled by a monotonic profile ◮ Apply interval analysis to the whole membership functions for locally monotonic functions ◮ Extend this model for the computation of the variance of a set of fuzzy numbers ◮ Deduce a method to compute the variance of a single fuzzy interval, viewed as nested intervals Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 2 / 17

Example C = A ∗ B µ 1 B A 0.5 −1 −0.5 0.5 1 x Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 3 / 17

Example C = A ∗ B µ 1 B A 0.5 −1 −0.5 0.5 1 x For intervals, [ c − , c + ] = [ a − , a + ] ∗ [ b − , b + ] c − = min ( a − ∗ b − , a − ∗ b + , a + ∗ b − , a + ∗ b + ) c + = max ( a − ∗ b − , a − ∗ b + , a + ∗ b − , a + ∗ b + ) Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 3 / 17

Example C = A ∗ B µ 1 B A 0.5 −1 −0.5 0.5 1 x λ A − ( λ ) = 2 1 − λ A + ( λ ) = 2 λ B − ( λ ) = 2 − 1 B + ( λ ) 2 − λ 1 = Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 3 / 17

Example C = A ∗ B 1 A−B− 0.8 0.6 0.4 0.2 0 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 ( A − ∗ B − )( λ ) = λ 2 ∗ ( λ 2 − 1 ) Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 4 / 17

Example C = A ∗ B 1 A−B− A+B− 0.8 0.6 0.4 0.2 0 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 ( A + ∗ B − )( λ ) = ( 1 − λ 2 ) ∗ ( λ 2 − 1 ) Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 4 / 17

Example C = A ∗ B 1 A−B− A+B− A−B+ 0.8 0.6 0.4 0.2 0 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 ( A − ∗ B + )( λ ) = λ 2 − λ ) 2 ∗ ( 1 Non monotonic profile Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 4 / 17

Example C = A ∗ B 1 A−B− A+B− A−B+ A+B+ 0.8 0.6 0.4 0.2 0 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 ( A + ∗ B + )( λ ) = ( 1 − λ 2 − λ ) 2 ) ∗ ( 1 Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 4 / 17

Example C = A ∗ B 1 A−B− A+B− A−B+ A+B+ 0.8 C 0.6 0.4 0.2 0 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 ( A ∗ B ) − = min ( A − ∗ B − , A + ∗ B − , A − ∗ B + , A + ∗ B + ) ( A ∗ B ) + = max ( A − ∗ B − , A + ∗ B − , A − ∗ B + , A + ∗ B + ) Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 4 / 17

Generalization Question : Why is this method valid to compute the product of two fuzzy intervals? Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 5 / 17

Generalization Question : Why is this method valid to compute the product of two fuzzy intervals? ◮ Because product is monotonic in each argument when the other argument is a constant Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 5 / 17

Generalization Question : Why is this method valid to compute the product of two fuzzy intervals? ◮ Because product is monotonic in each argument when the other argument is a constant Methodology : ◮ Characterization of functions which reach their extrema on the vertices of their hyper-rectangular domain Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 5 / 17

Generalization Question : Why is this method valid to compute the product of two fuzzy intervals? ◮ Because product is monotonic in each argument when the other argument is a constant Methodology : ◮ Characterization of functions which reach their extrema on the vertices of their hyper-rectangular domain ◮ Direct extension of the classical interval computation to fuzzy intervals viewed as pairs of fuzzy bounds Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 5 / 17

Generalization Question : Why is this method valid to compute the product of two fuzzy intervals? ◮ Because product is monotonic in each argument when the other argument is a constant Methodology : ◮ Characterization of functions which reach their extrema on the vertices of their hyper-rectangular domain ◮ Direct extension of the classical interval computation to fuzzy intervals viewed as pairs of fuzzy bounds ◮ Extension to the computation of the fuzzy variance Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 5 / 17

Classical interval computation Given a function f from R n to R and n intervals [ x − i , x + i ] Goal : find the interval range of y = f ( x ) such that x ∈ X = × i [ x − i , x + i ] Definition (Configuration) X is the set of all configurations. H is the set of all extreme configurations: H = × i { x − i , x + i } ( | H | = 2 n ) Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 6 / 17

Monotony Definition (Global monotony) f is said to be globally monotonic if for each variable x i , f is either increasing or decreasing according to x i . Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 7 / 17

Monotony Definition (Global monotony) f is said to be globally monotonic if for each variable x i , f is either increasing or decreasing according to x i . Definition (Local monotony) f is said to be locally monotonic if for each variable x i , for all n-tuples ( a 1 , a 2 , ··· , a i − 1 , a i + 1 , ··· , a n ) ∈ R n − 1 the restricted function f ( a 1 , a 2 , ··· , a i − 1 , x i , a i + 1 , ··· , a n ) is monotonic. Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 7 / 17

Results Let f : R n − → R be a function, X = × i [ x − i , x + i ] the cartesian product of n intervals, and [ y − , y + ] = { f ( x ) | x ∈ X } Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 8 / 17

Results Let f : R n − → R be a function, X = × i [ x − i , x + i ] the cartesian product of n intervals, and [ y − , y + ] = { f ( x ) | x ∈ X } Proposition If f is locally monotonic, then f reaches its bounds on a extreme configuration: y − = min ω ∈ H ( f ( ω )) y + = max ω ∈ H ( f ( ω )) Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 8 / 17