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Choquet integral: distributions and decisions Vicen c Torra School - - PowerPoint PPT Presentation

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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:

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83rd EWG-MCDA 2016

Choquet integral: distributions and decisions Vicen¸ c Torra School of Informatics, University of Sk¨

  • vde

Sk¨

  • vde, Sweden

March 31-32 , 2016

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Overview >Example Outline

Overview

Basics and objectives:

  • Distribution based on the Choquet integral

(for non-additive measures) Motivation:

  • Theory: Mathematical properties
  • Methodology: different ways to express interactions
  • Application: Decision (MCDM), classification,

statistical disclosure control (data privacy)

Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 1 / 59

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Outline

Outline

  • 1. Preliminaries
  • 2. Choquet integral based distribution
  • 3. Choquet-Mahalanobis based distribution
  • 4. Summary

83rd EWG-MCDA 2016 2 / 59

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Outline

Preliminaries Aggregation operators and the Choquet integral in Decision

83rd EWG-MCDA 2016 3 / 59

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Outline

MCDM: Aggregation for (numerical) utility functions

83rd EWG-MCDA 2016 4 / 59

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Outline

Aggregation for (numerical) utility functions

  • Decision, utility functions

83rd EWG-MCDA 2016 5 / 59

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Outline

Aggregation for (numerical) utility functions

  • Decision, utility functions

Alternatives = { Ford T, Seat 600, Simca 1000, VW, Citr.Acadiane }

83rd EWG-MCDA 2016 5 / 59

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Outline

Aggregation for (numerical) utility functions

  • Decision, utility functions

Alternatives = { Ford T, Seat 600, Simca 1000, VW, Citr.Acadiane } Criteria = { Seats, Security, Price, Comfort, trunk}

83rd EWG-MCDA 2016 5 / 59

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Outline

Aggregation for (numerical) utility functions

  • Decision, utility functions

Alternatives = { Ford T, Seat 600, Simca 1000, VW, Citr.Acadiane } Criteria = { Seats, Security, Price, Comfort, trunk} Decision making process:

83rd EWG-MCDA 2016 5 / 59

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Outline

Aggregation for (numerical) utility functions

  • Decision, utility functions

Alternatives = { Ford T, Seat 600, Simca 1000, VW, Citr.Acadiane } Criteria = { Seats, Security, Price, Comfort, trunk} Decision making process: Modelling=Criteria + Utilities, aggregation, selection

Number of Security Price Confort trunk seats Ford T 20 20 Seat 600 60 100 50 Simca 1000 100 30 100 50 70 VW Beetle 80 50 30 70 100 Citro¨ en Acadiane 20 40 60 40

83rd EWG-MCDA 2016 5 / 59

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Outline

Aggregation for (numerical) utility functions

  • Decision, utility functions

83rd EWG-MCDA 2016 6 / 59

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Outline

Aggregation for (numerical) utility functions

  • Decision, utility functions

Alternatives = { Ford T, Seat 600, Simca 1000, VW, Citr.Acadiane }

83rd EWG-MCDA 2016 6 / 59

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Outline

Aggregation for (numerical) utility functions

  • Decision, utility functions

Alternatives = { Ford T, Seat 600, Simca 1000, VW, Citr.Acadiane } Criteria = { Seats, Security, Price, Comfort, trunk}

83rd EWG-MCDA 2016 6 / 59

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Outline

Aggregation for (numerical) utility functions

  • Decision, utility functions

Alternatives = { Ford T, Seat 600, Simca 1000, VW, Citr.Acadiane } Criteria = { Seats, Security, Price, Comfort, trunk} Decision making process:

83rd EWG-MCDA 2016 6 / 59

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Outline

Aggregation for (numerical) utility functions

  • Decision, utility functions

Alternatives = { Ford T, Seat 600, Simca 1000, VW, Citr.Acadiane } Criteria = { Seats, Security, Price, Comfort, trunk} Decision making process: Modelling, aggregation = C, selection

Seats Security Price Comfort trunk C = AM Ford T 20 20 8 Seat 600 60 100 50 42 Simca 1000 100 30 100 50 70 70 VW 80 50 30 70 100 66

  • Citr. Acadiane

20 40 60 40 32

83rd EWG-MCDA 2016 6 / 59

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Outline

Aggregation for (numerical) utility functions

  • MCDM: Aggregation to deal with contradictory criteria

83rd EWG-MCDA 2016 7 / 59

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Outline

Aggregation for (numerical) utility functions

  • MCDM: Aggregation to deal with contradictory criteria
  • But there are occasions in which ordering is clear

when ai ≤ bi it is clear that a ≤ b E.g., Seats Security Price Comfort trunk C = AM Seat 600 60 100 50 42 Simca 1000 100 30 100 50 70 70

83rd EWG-MCDA 2016 7 / 59

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Outline

Aggregation for (numerical) utility functions

  • MCDM: Aggregation to deal with contradictory criteria
  • But there are occasions in which ordering is clear

when ai ≤ bi it is clear that a ≤ b E.g., Seats Security Price Comfort trunk C = AM Seat 600 60 100 50 42 Simca 1000 100 30 100 50 70 70 Aggregation operators are appropriate because they satisfy monotonicity

83rd EWG-MCDA 2016 7 / 59

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Outline

Aggregation for (numerical) utility functions

  • MCDM: Aggregation to deal with contradictory criteria
  • But there are occasions in which ordering is clear

when ai ≤ bi it is clear that a ≤ b E.g., Seats Security Price Comfort trunk C = AM Seat 600 60 100 50 42 Simca 1000 100 30 100 50 70 70 Aggregation operators are appropriate because they satisfy monotonicity

  • Pareto dominance:

Given two vectors a = (a1, . . . , an) and b = (b1, . . . , bn), we say that b dominates a when ai ≤ bi for all i and there is at least one k such that ak < bk.

83rd EWG-MCDA 2016 7 / 59

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Outline

Aggregation for (numerical) utility functions

  • Pareto set, Pareto frontier, or non dominance set:

Seats Security Price Comfort trunk C = AM Simca 1000 100 30 100 50 70 70 VW 80 50 30 70 100 66

  • Citr. Acadiane

20 40 60 40 32

  • Each one wins at least in one criteria to another one

83rd EWG-MCDA 2016 8 / 59

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Outline

Aggregation and Choquet integral in MCDM

  • Pareto set, Pareto frontier, or non dominance set:

Given a set of alternatives U represented by vectors u = (u1, . . . , un), the Pareto frontier is the set u ∈ U such that there is no other v ∈ U such that v dominates u. PF = {u|there is no v s.t. v dominates u}

  • Pareto optimal: an element u of the Pareto set

x1 f1(x2) f1(x1) f1 f2 f2(x2) f2(x1) x2

83rd EWG-MCDA 2016 9 / 59

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Outline

Aggregation and Choquet integral in MCDM

  • Decision making process:

83rd EWG-MCDA 2016 10 / 59

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Outline

Aggregation and Choquet integral in MCDM

  • Decision making process:

Modelling, aggregation, selection=order,first

83rd EWG-MCDA 2016 10 / 59

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Outline

Aggregation and Choquet integral in MCDM

  • Decision making process:

Modelling, aggregation, selection=order,first

  • The function of aggregation functions
  • Different aggregations lead to different orders (in the PF)

83rd EWG-MCDA 2016 10 / 59

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Outline

Aggregation and Choquet integral in MCDM

  • Decision making process:

Modelling, aggregation, selection=order,first

  • The function of aggregation functions
  • Different aggregations lead to different orders (in the PF)
  • Aggregation establishes which points are equivalent
  • Different aggregations, lead to different curves of points (level curves)

Ranking alt alt Consensus alt Criteria Satisfaction on: Price Quality Comfort FordT 206 0.2 0.8 0.3 0.7 0.7 0.8 FordT 206 FordT 206 0.35 0.72 0.72 0.35 ... ... ... ... ... ... x1 f1(x2) f1(x1) f1 f2 f2(x2) f2(x1) x2

83rd EWG-MCDA 2016 10 / 59

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Outline

Aggregation and Choquet integral in MCDM

  • Aggregation functions and different level curves
  • Arithmetic mean
  • Geometric mean, Harmonic mean, ...
  • Weighted mean
  • OWA, ...

83rd EWG-MCDA 2016 11 / 59

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Outline

Aggregation and Choquet integral in MCDM

  • Aggregation functions and different level curves
  • Arithmetic mean
  • Geometric mean, Harmonic mean, ...
  • Weighted mean
  • OWA, ...
  • Choquet integral (generalization of the AM, WM, OWA)

∗ to represent interactions between criteria ∗ non-independent criteria allowed

83rd EWG-MCDA 2016 11 / 59

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Outline

Aggregation and Choquet integral in MCDM

  • Aggregation functions and parameters

– Arithmetic mean: no parameters – Geometric mean, Harmonic mean, ...: : no parameters – Weighted mean: weighting vector – OWA, ...: weighting vector

83rd EWG-MCDA 2016 12 / 59

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Outline

Aggregation and Choquet integral in MCDM

  • Aggregation functions and parameters

– Arithmetic mean: no parameters – Geometric mean, Harmonic mean, ...: : no parameters – Weighted mean: weighting vector – OWA, ...: weighting vector

83rd EWG-MCDA 2016 12 / 59

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Outline

Aggregation and Choquet integral in MCDM

  • Aggregation functions and parameters

– Choquet integral (generalization of the AM, WM, OWA): a measure

83rd EWG-MCDA 2016 13 / 59

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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)

83rd EWG-MCDA 2016 13 / 59

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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)

83rd EWG-MCDA 2016 13 / 59

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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)
  • r
  • w(security,price,confort) < w(security)+w(price)+w(confort)

83rd EWG-MCDA 2016 13 / 59

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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 ?

x1 f1(x2) f1(x1) f1 f2 f2(x2) f2(x1) x2

83rd EWG-MCDA 2016 14 / 59

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Outline

Preliminaries Non-additive (fuzzy) measures and the Choquet integral

83rd EWG-MCDA 2016 15 / 59

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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

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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

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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

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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 µm is a non-additive measure: µm,P(A) = m(P(A)) (1)

Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 18 / 59

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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 µm is a non-additive measure: µm,P(A) = m(P(A)) (1)

  • If m(x) = x2, then µm(A) = (P(A))2
  • If m(x) = xp, then µm(A) = (P(A))p

(a) (b) (c) (d)

Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 18 / 59

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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 µm is a non-additive measure: µm,P(A) = m(P(A)) (2)

Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 19 / 59

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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 µm is a non-additive measure: µm,P(A) = m(P(A)) (2) Applications.

  • To represent interactions

Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 19 / 59

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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 µg(r) := µ({x|g(x) > r}): (C)

  • gdµ :=

∞ µg(r)dr. (3)

Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 20 / 59

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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 µg(r) := µ({x|g(x) > r}): (C)

  • gdµ :=

∞ µg(r)dr. (3)

  • When the measure is additive, this is the Lebesgue integral

Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 20 / 59

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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 µg(r) := µ({x|g(x) > r}): (C)

  • gdµ :=

∞ µg(r)dr. (3)

  • When the measure is additive, this is the Lebesgue integral

bi bi−1 ai ai−1 bi bi−1 x1 x1 x1 xN xN x {x|f(x) ≥ ai} {x|f(x) = bi} (a) (b) (c)

Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 20 / 59

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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, (C)

  • fdµ =

N

  • i=1

[f(xs(i)) − f(xs(i−1))]µ(As(i)), where f(xs(i)) indicates that the indices have been permuted so that 0 ≤ f(xs(1)) ≤ · · · ≤ f(xs(N)) ≤ 1, and where f(xs(0)) = 0 and As(i) = {xs(i), . . . , xs(N)}.

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

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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 µm,P(A) = m(P(A)) with m(x) = xp,

CIµm(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

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Outline

Preliminaries Classification and shapes of distributions

83rd EWG-MCDA 2016 23 / 59

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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 R2

Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 24 / 59

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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 R2

→ where to classify x?

−2 2 4 6 8 −2 2 4 6 8

Two classes

table[,1] table[,2]

Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 24 / 59

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Classification Outline

Classification

Classification problems: Classification of x into Ω

  • x in a n-dimensional space (i.e., x ∈ Rn)
  • Set of k classes Ω = {ω1, . . . , ωk}

Formalization:

  • Bayes’ maximum-a-posteriori (MAP) classification decision rule:

assigns x to the class ωi s.t. the probability P(ωi|x) is maximized. I.e., (Bayes condition):

P(ωi|x) = P(x|ωi)P(ωi) P(x)

  • r, as P(x) is constant for all classes,

di(x) = P(x|ωi)P(ωi) 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

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Classification Outline

Classification

Classification problems: Classes ωi generated from

  • covariance matrices Σi
  • means ¯

xi → class-conditional probability-density function (Gaussian distribution) P(x|ωi) = 1 (2π)m/2|Σi|1/2e−1

2(x−¯

xi)T Σ−1

i

(x−¯ xi)

5 10 5 10

Two classes with different correlations

table[,1] table[,2]

Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 26 / 59

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Classification Outline

Classification

Proposition:

  • Bayes’ maximum-a-posteriori (MAP) classification decision rule, when

Σi = Σj, and P(ωi) = P(ωj), is (Mahalanobis distance) di(x) = −(x − ¯ xi)TΣ−1(x − ¯ xi)

Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 27 / 59

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Classification Outline

Classification

  • Proposition. Bayes’ maximum-a-posteriori (MAP) classification
  • If Σi = I for all i (the identity function)

di(x) = −(x − ¯ xi)T(x − ¯ xi) = −||x − ¯ xi||2 → Euclidean distance

  • If Σi is diagonal (not necessarily equal to I)

di(x) = −

m

  • j=1

(σ2

j)−1(xj − ¯

xij)2 → Weighted Euclidean distance

(with weights equal to the inverse of the variances: pj = (σ2

j)−1)

Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 28 / 59

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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

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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

5 10 5 10

Two classes with different correlations

table[,1] table[,2]

Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 29 / 59

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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

5 10 5 10

Two classes with different correlations

table[,1] table[,2]

What about another shape / another distance ?

Vicen¸ c Torra; Choquet integral: distributions and decisions 83rd EWG-MCDA 2016 29 / 59

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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

5 10 5 10

Two classes with different correlations

table[,1] table[,2]

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

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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

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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

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SLIDE 61

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

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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

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Outline

Choquet integral based distribution

83rd EWG-MCDA 2016 31 / 59

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CI distribution Outline

Choquet integral based distribution: Definition

Definition:

  • Y = {Y1, . . . , Yn} random variables; µ : 2Y → [0, 1] a non-additive

measure and m a vector in Rn.

  • The exponential family of Choquet integral based class-conditional

probability-density functions is defined by: PCm,µ(x) = 1 Ke−1

2CIµ((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 (elementwise) product of vectors v and w (i.e., (v ◦ w) = (v1w1 . . . vnwn)). Notation:

  • We denote it by C(m, µ).

83rd EWG-MCDA 2016 32 / 59

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CI distribution Outline

Choquet integral based distribution: Examples

  • Shapes (level curves)

(-15.0,-15.0) 15.0 15.0 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qq qq qq qq qq qq qqq qqq qqq qqq qqqqqqq qqqqqqqqqqqqqqqqqq qqqqqqq qqq qqq qqq qqq qq qq qq qq qq qq q qq q qq q qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq q 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qq qq qqq qqq qqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqq qqq qqq qq qq qq qq (-15.0,-15.0) 15.0 15.0 q q qq q q q q q q q q q q 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 q q q q q q 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqq q q q q qq q q q q q q q q q 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 q q q q q 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq (-15.0,-15.0) 15.0 15.0 qqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqq qq qq q 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qq qqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqq qqq qq qq q q 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qq qq qqq qqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqq (-15.0,-15.0) 15.0 15.0 qqq qqqq qq qqqqqq qq qq qq qq q qq q qq q q 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qq q qq q qq qq qq qq qqqqqq qq qqqq 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqq qqqqqqqqqqqqqqqq qq qqqq qq qq qq qq q qq q q q 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qq q qq qq qq qq qqqq qq qqqqqqqqqqqqqqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq

(a) µA({x}) = 0.1 and µA({y}) = 0.1, (b) µB({x}) = 0.9 and µB({y}) = 0.9, (c) µC({x}) = 0.2 and µC({y}) = 0.8, and (d) µD({x}) = 0.4 and µD({y}) = 0.9.

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CI distribution Outline

Choquet integral based distribution: Properties

  • Proposition. Distribution and distance (Choquet distance):
  • If P(wi) = P(wj) holds for all i = j, the decision rule is (max):

−CIµ((x − ¯ xi) ⊗ (x − ¯ xi)) 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.

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CI distribution Outline

Choquet integral based distribution: Properties

Property:

  • The family of distributions N(m, Σ) in Rn with a diagonal matrix Σ
  • f rank n, and the family of distributions C(m, µ) with an additive

measure µ with all µ({xi}) = 0 are equivalent.

(µ(X) is not necessarily here 1)

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CI distribution Outline

Choquet integral based distribution: Properties

Property:

  • The family of distributions N(m, Σ) in Rn with a diagonal matrix Σ
  • f rank n, and the family of distributions C(m, µ) with an additive

measure µ with all µ({xi}) = 0 are equivalent.

(µ(X) is not necessarily here 1)

Corollary:

  • The distribution N(0, I) corresponds to C(0, µ1) where µ1 is the

additive measure defined as µ1(A) = |A| for all A ⊆ X.

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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. Y1 and Y2 axis.

(-15.0,-15.0) 15.0 15.0 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qq qq qq qq qq qq qqq qqq qqq qqq qqqqqqq qqqqqqqqqqqqqqqqqq qqqqqqq qqq qqq qqq qqq qq qq qq qq qq qq q qq q qq q qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq q 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qq qq qqq qqq qqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqq qqq qqq qq qq qq qq

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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. Y1 and Y2 axis.

(-15.0,-15.0) 15.0 15.0 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qq qq qq qq qq qq qqq qqq qqq qqq qqqqqqq qqqqqqqqqqqqqqqqqq qqqqqqq qqq qqq qqq qqq qq qq qq qq qq qq q qq q qq q qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq q 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 qq qq qqq qqq qqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqq qqq qqq qq qq qq qq

  • A generalization of both: Choquet-Mahalanobis based distribution.

– Mahalanobis: Σ represents some interactions – Choquet (measure): µ represents some interactions

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Outline

Choquet-Mahalanobis based distribution

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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

CMIµ,Σ(x, ¯ x) = CIµ(v ⊗ w) (4) where v and w are the vectors defined by: v = (x − ¯ x)TL and w = L∗(x − ¯ x), where v ⊗ w denotes the elementwise product of vectors v and w (i.e., (v ⊗ w) = (v1w1 . . . vnwn)).

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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.

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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.

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CMI distribution Outline

Choquet integral based distribution: Definition

Definition:

  • Y = {Y1, . . . , Yn} random variables, µ : 2Y → [0, 1] a measure, m a

vector in Rn, and Q a positive-definite matrix.

  • The exponential family of Choquet-Mahalanobis integral based class-

conditional probability-density functions is defined by: PCMm,µ,Q(x) = 1 Ke−1

2CIµ(v◦w)

where K is a constant that is defined so that the function is a probability, where LLT = Q is the Cholesky decomposition of the matrix Q, v = (x − m)TL, w = LT(x − m), and where v ◦ w denotes the elementwise product of vectors v and w. Notation:

  • We denote it by CMI(m, µ, Q).

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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 = µ1 corresponds to multivariate normal distributions, – A CMI(m, µ, Q) with Q = I corresponds to a CI(m, µ).

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CMI distribution Outline

Choquet integral based distribution: Properties

Graphically:

  • Choquet integral (CI distribution), Mahalobis distance (multivariate

normal distribution), generalization (CMI distribution)

Mahalanobis Choquet−Mahalanobis Choquet WM

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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 σ1 = 1, σ2 = 1, ρ12 = 0.0, µx = 0.01, µy = 0.01.

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CMI distribution Outline

Choquet integral based distribution: Examples

2nd Example: Interactions only in terms of a covariance matrix.

  • CMI with σ1 = 1, σ2 = 1, ρ12 = 0.9, µx = 0.10, µy = 0.90.

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CMI distribution Outline

Choquet integral based distribution: Examples

3rd Example: Interactions both: covariance matrix and measure.

  • CMI with σ1 = 1, σ2 = 1, ρ12 = 0.9, µx = 0.01, µy = 0.01.

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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)

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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.

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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

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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

f(r) = (1/K)e−

r−r0

σ

2

,

where r0 is a radius over which the density is maximum, σ is a variance, and K is the normalization constant.

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CMI distribution Outline

Choquet integral based distribution: Properties

More properties: (symmetry)

  • P(x) a C(m, µ) i.e., mean m = (m1, . . . , mn) and a fuzzy measure

µ. Then, for all x ∈ Rn and all i ∈ {1, . . . , n}

P(x1, . . . , xi−1, xi + mi, xi+1, . . . , xn) = P(x1, . . . , xi−1, −xi + mi, xi+1, . . . , xn).

  • P(x) a CMI(m, µ, Q) i.e., with mean m = (m1, . . . , mn), a

positive-definite diagonal matrix Q, and a fuzzy measure µ. Then, for all x ∈ Rn and all i ∈ {1, . . . , n}

P(x1, . . . , xi−1, xi + mi, xi+1, . . . , xn) = P(x1, . . . , xi−1, −xi + mi, xi+1, . . . , xn).

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CMI distribution Outline

Choquet integral based distribution: Properties

More properties:

  • P(x) a C(m, µ) i.e., with mean m = (m1, . . . , mn). Then, for any

fuzzy measure µ,

  • the mean vector ¯

X = [E[X1], E[X2], . . . , E[Xn]] is m and

  • Σ = [Cov[Xi, Xj]] 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 = (m1, . . . , mn). Then,

for any fuzzy measure µ and any diagonal matrix Q,

  • the mean vector ¯

X = [E[X1], E[X2], . . . , E[Xn]] is m and

  • Σ = [Cov[Xi, Xj]] for i = 1, . . . , n and j = 1, . . . , n is zero for all

i = j and thus, diagonal.

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CMI distribution Outline

Choquet integral based distribution: Properties

More properties:

  • When Q is not diagonal, we may have

Cov[Xi, Xj] = Q(Xi, Xj).

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CMI distribution Outline

Choquet integral based distribution: Properties

More properties: If this type of data distinguishable from Normal ?

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CMI distribution Outline

Choquet integral based distribution: Properties

More properties: If this type of data distinguishable from Normal ? Study:

  • Case of X = {x1, x2}
  • CMI(0, µ) with µ({x}) = i/10 and µ({y}) = i/10 for i = 1, 2, . . . , 9

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CMI distribution Outline

Choquet integral based distribution: Properties

More properties: If this type of data distinguishable from Normal ? Study:

  • Case of X = {x1, x2}
  • 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

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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

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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({x1}) = 0.1 and µ({x2}) = 0.1

−20 −10 10 20 0.00 0.05 0.10 0.15 0.20 x marginalX2 −20 −10 10 20 0.00 0.05 0.10 0.15 x normalDistr

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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({x1}) = 0.1 and µ({x2}) = 0.1; (ii) µ({x1}) = 0.1 and µ({x2}) = 0.2; (iii) µ({x1}) = 0.2 and µ({x2}) = 0.1; (iv) µ({x1}) = 0.9 and µ({x2}) = 0.9

−20 −10 10 20 0.00 0.05 0.10 0.15 0.20 x marginalX2 −20 −10 10 20 0.00 0.05 0.10 0.15 x normalDistr −20 −10 10 20 0.00 0.05 0.10 0.15 0.20 0.25 x marginalX2 −20 −10 10 20 0.00 0.05 0.10 0.15 0.20 x normalDistr −20 −10 10 20 0.00 0.05 0.10 0.15 0.20 x marginalX2 −20 −10 10 20 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 x normalDistr −20 −10 10 20 0.00 0.05 0.10 0.15 0.20 0.25 0.30 x marginalX2 −20 −10 10 20 0.00 0.05 0.10 0.15 0.20 0.25 0.30 x normalDistr

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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 R2) 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.

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slide-95
SLIDE 95

Outline

Summary

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slide-96
SLIDE 96

Summary Outline

Summary

Summary:

  • Definition of distributions based on the Choquet integral

Integral for non-additive measures

  • Relationship with multivariate normal and spherical distributions

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slide-97
SLIDE 97

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

slide-98
SLIDE 98

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)

P(x) = 1

Ke −1

2CIG µG((x−¯

x)⊗(x−¯ x))

  • Example. Two perspectives of same level dependent CI. Defined by the same fuzzy

measures µ1 and µ2 with intervals (0, 3) for µ1, and (3, 100) for µ2. µ1({x}) = 0.05 and µ1({y}) = 0.95, and µ2({x}) = 0.95 and µ2({y}) = 0.05

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slide-99
SLIDE 99

Outline

Thank you

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SLIDE 100

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

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SLIDE 101

Outline

Thank you

Slides at: http://www.mdai.cat/ifao/ http://www.mdai.cat/ifao/slides.php

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