A general framework for comparing heterogeneous binary relations

28/08/2013
Auteurs : Julien Ah-Pine
OAI : oai:www.see.asso.fr:2552:4907
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A general framework for comparing heterogeneous binary relations

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A general framework for comparing heterogeneous binary relations Julien Ah-Pine (julien.ah-pine@eric.univ-lyon2.fr) University of Lyon - ERIC Lab GSI 2013 Paris 28/08/2013 J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 1 Outline 1 Introduction 2 Kendall’s general coefficient Γ 3 Another view of Kendall’s Γ Relational Matrices Reinterpreting Kendall’s Γ using RM The Weighted Indeterminacy Deviation Principle 4 Extending Kendall’s Γ for heterogeneous BR Heterogeneous BR A geometrical framework Similarities of order t > 0 5 A numerical example 6 Conclusion J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 2 Introduction Outline 1 Introduction 2 Kendall’s general coefficient Γ 3 Another view of Kendall’s Γ Relational Matrices Reinterpreting Kendall’s Γ using RM The Weighted Indeterminacy Deviation Principle 4 Extending Kendall’s Γ for heterogeneous BR Heterogeneous BR A geometrical framework Similarities of order t > 0 5 A numerical example 6 Conclusion J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 3 Introduction Binary relations (BR) A Binary Relation (BR) R over a finite set A = {a, . . . , i, j, . . . , n} of n items is a subset of A × A. If (i, j) ∈ R we say “i is in relation with j for R” and this is denoted iRj. J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 4 Introduction Binary relations (BR) A Binary Relation (BR) R over a finite set A = {a, . . . , i, j, . . . , n} of n items is a subset of A × A. If (i, j) ∈ R we say “i is in relation with j for R” and this is denoted iRj. Equivalence Relations (ER) are reflexive, symmetric and transitive BR. J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 4 Introduction Binary relations (BR) A Binary Relation (BR) R over a finite set A = {a, . . . , i, j, . . . , n} of n items is a subset of A × A. If (i, j) ∈ R we say “i is in relation with j for R” and this is denoted iRj. Equivalence Relations (ER) are reflexive, symmetric and transitive BR. Order Relations (OR) are of different types : preorders, partial orders and total (or linear or complete) orders. If ties and missing values : preorders (reflexive, transitive BR) If no tie but missing values : partial orders (reflexive, antisymmetric, transitive BR) If no tie and no missing value : total orders (reflexive, antisymmetric, transitive and complete BR) J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 4 Introduction Equivalence Relations and qualitative variables ER are related to qualitative or nominal categorical variables. J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 5 Introduction Equivalence Relations and qualitative variables ER are related to qualitative or nominal categorical variables. Example : Color of eyes x = a b c d e Brown Brown Blue Blue Green J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 5 Introduction Equivalence Relations and qualitative variables ER are related to qualitative or nominal categorical variables. Example : Color of eyes x = a b c d e Brown Brown Blue Blue Green X is the ER “has the same color of eyes than” and can be represented by a graph and its adjacency matrix (AM) denoted X such that ∀i, j : Xij = 1 if iXj and Xij = 0 otherwise : X =       a b c d e a 1 1 . . . b 1 1 . . . c . . 1 1 . d . . 1 1 . e . . . . 1       J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 5 Introduction Order Relations and quantitative variables OR are related to quantitative variables. J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 6 Introduction Order Relations and quantitative variables OR are related to quantitative variables. Example : Ranking of items x = a b c d e 1 2 4 3 5 J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 6 Introduction Order Relations and quantitative variables OR are related to quantitative variables. Example : Ranking of items x = a b c d e 1 2 4 3 5 X is the OR “has a lower rank than” and its AM X is again such that ∀i, j : Xij = 1 if iXj and Xij = 0 otherwise : X =       a b c d e a 1 1 1 1 1 b . 1 1 1 1 c . . 1 . 1 d . . 1 1 1 e . . . . 1       J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 6 Introduction How to compare the relationships between BR ? We are given two variables of measurements x and y of the same kind (both qualitative or both quantitative). J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 7 Introduction How to compare the relationships between BR ? We are given two variables of measurements x and y of the same kind (both qualitative or both quantitative). How can we measure the proximity between the BR underlying the two variables ? J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 7 Introduction How to compare the relationships between BR ? We are given two variables of measurements x and y of the same kind (both qualitative or both quantitative). How can we measure the proximity between the BR underlying the two variables ? How to deal with heterogeneity ? J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 7 Introduction How to compare the relationships between BR ? We are given two variables of measurements x and y of the same kind (both qualitative or both quantitative). How can we measure the proximity between the BR underlying the two variables ? How to deal with heterogeneity ? When ER have different number of categories and different distributions ? For example : x = (A, A, B, B, C) ; y = (D, D, D, D, E) J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 7 Introduction How to compare the relationships between BR ? We are given two variables of measurements x and y of the same kind (both qualitative or both quantitative). How can we measure the proximity between the BR underlying the two variables ? How to deal with heterogeneity ? When ER have different number of categories and different distributions ? For example : x = (A, A, B, B, C) ; y = (D, D, D, D, E) When OR are of different types ? For example : x = (1, 2, 4, 3, 5) ; y = (1, 1, 1, 4, 5) J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 7 Kendall’s general coefficient Γ Outline 1 Introduction 2 Kendall’s general coefficient Γ 3 Another view of Kendall’s Γ Relational Matrices Reinterpreting Kendall’s Γ using RM The Weighted Indeterminacy Deviation Principle 4 Extending Kendall’s Γ for heterogeneous BR Heterogeneous BR A geometrical framework Similarities of order t > 0 5 A numerical example 6 Conclusion J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 8 Kendall’s general coefficient Γ Kendall’s Γ coefficient In statistics, Kendall in [Kendall(1948)] proposed a general correlation coefficient in order to define a broad family of association measures between x and y : Γ(x, y) = i,j Xij Yij i,j X2 ij i,j Y2 ij (1) where X and Y are two square matrices derived from x and y. J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 9 Kendall’s general coefficient Γ Particular cases of Γ Particular cases of Γ given in [Vegelius and Janson(1982), Kendall(1948)]. Association measure Xij Tchuprow’s T n nx u − 1 if xi = xj −1 if xi = xj J-index px − 1 if xi = xj −1 if xi = xj Table: Particular cases of Γ as for ER nx u is the nb of items in category u of x and px is the nb of categories of x. J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 10 Kendall’s general coefficient Γ Particular cases of Γ Particular cases of Γ given in [Vegelius and Janson(1982), Kendall(1948)]. Association measure Xij Tchuprow’s T n nx u − 1 if xi = xj −1 if xi = xj J-index px − 1 if xi = xj −1 if xi = xj Table: Particular cases of Γ as for ER nx u is the nb of items in category u of x and px is the nb of categories of x. Association measure Xij Kendall’s τa 1 if xi < xj −1 if xi > xj Spearman’s ρa Xij = xi − xj Table: Particular cases of Γ as for OR For Spearman’s ρa, xi is the rank of item i. J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 10 Another view of Kendall’s Γ Outline 1 Introduction 2 Kendall’s general coefficient Γ 3 Another view of Kendall’s Γ Relational Matrices Reinterpreting Kendall’s Γ using RM The Weighted Indeterminacy Deviation Principle 4 Extending Kendall’s Γ for heterogeneous BR Heterogeneous BR A geometrical framework Similarities of order t > 0 5 A numerical example 6 Conclusion J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 11 Another view of Kendall’s Γ Relational Matrices Relational Matrices (RM) and some properties AM of BR have particular properties and they are more specifically called Relational Matrices (RM) by Marcotorchino in the Relational Analysis approach[Marcotorchino and Michaud(1979)]. J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 12 Another view of Kendall’s Γ Relational Matrices Relational Matrices (RM) and some properties AM of BR have particular properties and they are more specifically called Relational Matrices (RM) by Marcotorchino in the Relational Analysis approach[Marcotorchino and Michaud(1979)]. For instance, the relational properties of X can be expressed as linear equations of X : reflexivity, ∀i (Xii = 1) ; symmetry, ∀i, j (Xij − Xji = 0) ; antisymmetry, ∀i, j (Xij + Xji ≤ 1) ; complete (or total), ∀i = j (Xij + Xji ≥ 1) ; transitivity, ∀i, j, k (Xij + Xjk − Xik ≤ 1). J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 12 Another view of Kendall’s Γ Reinterpreting Kendall’s Γ using RM Reinterpreting Kendall’s Γ using RM There have been several works about using RM to reformulate association measures in order to better understand their differences [Marcotorchino(1984-85), Ghashghaie(1990), Najah Idrissi(2000), Ah-Pine and Marcotorchino(2010)]. J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 13 Another view of Kendall’s Γ Reinterpreting Kendall’s Γ using RM Reinterpreting Kendall’s Γ using RM There have been several works about using RM to reformulate association measures in order to better understand their differences [Marcotorchino(1984-85), Ghashghaie(1990), Najah Idrissi(2000), Ah-Pine and Marcotorchino(2010)]. In this work, we propose to reinterpret Kendall’s Γ in terms of RM and which emphasizes the so-called weighted indeterminacy deviation principle. J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 13 Another view of Kendall’s Γ Reinterpreting Kendall’s Γ using RM Reinterpreting Kendall’s Γ using RM There have been several works about using RM to reformulate association measures in order to better understand their differences [Marcotorchino(1984-85), Ghashghaie(1990), Najah Idrissi(2000), Ah-Pine and Marcotorchino(2010)]. In this work, we propose to reinterpret Kendall’s Γ in terms of RM and which emphasizes the so-called weighted indeterminacy deviation principle. 1 We give the definition of the opposite of an ER and of an OR. J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 13 Another view of Kendall’s Γ Reinterpreting Kendall’s Γ using RM Reinterpreting Kendall’s Γ using RM There have been several works about using RM to reformulate association measures in order to better understand their differences [Marcotorchino(1984-85), Ghashghaie(1990), Najah Idrissi(2000), Ah-Pine and Marcotorchino(2010)]. In this work, we propose to reinterpret Kendall’s Γ in terms of RM and which emphasizes the so-called weighted indeterminacy deviation principle. 1 We give the definition of the opposite of an ER and of an OR. 2 We introduce Λ, our formulation of Kendall’s Γ in terms of RM of BR, RM of opposites of BR and weighting schemes. J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 13 Another view of Kendall’s Γ Reinterpreting Kendall’s Γ using RM Reinterpreting Kendall’s Γ using RM There have been several works about using RM to reformulate association measures in order to better understand their differences [Marcotorchino(1984-85), Ghashghaie(1990), Najah Idrissi(2000), Ah-Pine and Marcotorchino(2010)]. In this work, we propose to reinterpret Kendall’s Γ in terms of RM and which emphasizes the so-called weighted indeterminacy deviation principle. 1 We give the definition of the opposite of an ER and of an OR. 2 We introduce Λ, our formulation of Kendall’s Γ in terms of RM of BR, RM of opposites of BR and weighting schemes. 3 We show how Λ yields to well-known association measures. J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 13 Another view of Kendall’s Γ Reinterpreting Kendall’s Γ using RM Reinterpreting Kendall’s Γ using RM There have been several works about using RM to reformulate association measures in order to better understand their differences [Marcotorchino(1984-85), Ghashghaie(1990), Najah Idrissi(2000), Ah-Pine and Marcotorchino(2010)]. In this work, we propose to reinterpret Kendall’s Γ in terms of RM and which emphasizes the so-called weighted indeterminacy deviation principle. 1 We give the definition of the opposite of an ER and of an OR. 2 We introduce Λ, our formulation of Kendall’s Γ in terms of RM of BR, RM of opposites of BR and weighting schemes. 3 We show how Λ yields to well-known association measures. 4 We explain the weighted indeterminacy deviation principle. J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 13 Another view of Kendall’s Γ Reinterpreting Kendall’s Γ using RM Opposite relation of an ER and of an OR We introduce the opposite relation X of an ER or an OR X. J. Ah-Pine (University of Lyon) A gen. fram. for compar. heterog. BR GSI 2013 / 14 Another view of Kendall’s Γ Reinterpreting Kendall’s Γ using RM Opposite relation of an ER and of an OR We introduce the opposite relation X of an ER or an OR X. If x is a categorical variable then X = X (the complement relation) : Xij