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

1 0.5 µ A x B 1 0.5 −1 −0.5

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 3 / 17

Example C = A∗ B

1 0.5 µ A x B 1 0.5 −1 −0.5

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

1 0.5 µ A x B 1 0.5 −1 −0.5

A−(λ)

=

λ

2

A+(λ)

=

1− λ

2

B−(λ)

=

λ

2 − 1

B+(λ)

=

1 2 −λ

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 3 / 17

Example C = A∗ B

0.2 0.4 0.6 0.8 1 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 A−B−

(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

0.2 0.4 0.6 0.8 1 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 A−B− A+B−

(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

0.2 0.4 0.6 0.8 1 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 A−B− A+B− A−B+

(A− ∗ B+)(λ) = λ

2 ∗( 1 2 −λ)

Non monotonic profile

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 4 / 17

Example C = A∗ B

0.2 0.4 0.6 0.8 1 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 A−B− A+B− A−B+ A+B+

(A+ ∗ B+)(λ) = (1− λ

2)∗( 1 2 −λ)

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 4 / 17

Example C = A∗ B

0.2 0.4 0.6 0.8 1 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 A−B− A+B− A−B+ A+B+ C

(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 Rn 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| = 2n)

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 xi, f is either increasing or decreasing according to xi.

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 xi, f is either increasing or decreasing according to xi.

Definition (Local monotony)

f is said to be locally monotonic if for each variable xi, for all n-tuples

(a1,a2,··· ,ai−1,ai+1,··· ,an) ∈ Rn−1 the restricted function

f(a1,a2,··· ,ai−1,xi,ai+1,··· ,an) is monotonic.

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 7 / 17

Results

Let f : Rn −

→ R be a function, X = ×i[x−

i ,x+ i ] the cartesian product

• f 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 : Rn −

→ R be a function, X = ×i[x−

i ,x+ i ] the cartesian product

• f 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− = minω∈H (f(ω)) y+ = maxω∈H (f(ω))

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 8 / 17

Results

Let f : Rn −

→ R be a function, X = ×i[x−

i ,x+ i ] the cartesian product

• f n intervals, and [y−,y+] = {f(x)|x ∈ X }

Proposition

If f is locally monotonic, and ∀j ∈ E1, f is increasing with respect to xj and ∀j ∈ E2 f decreasing with respect to xj then y− = minω∈H

• f(ω)| ∀j ∈ E1,ωj = x−

j

∀j ∈ E2,ωj = x+

j

• y+ = maxω∈H
• f(ω)| ∀j ∈ E1,ωj = x+

j

∀j ∈ E2,ωj = x−

j

• Jérôme Fortin, Didier Dubois, Hélène Fargier

The Empirical Variance of a Set of Fuzzy Intervals 8 / 17

Profiles

Definition (Profile)

A profile is a function Φ from [0,1] to R A profile can be viewed as a genuine fuzzy extension of a real number. We call left profile of I (denoted I−) the profile defined as follows: I− :

[0,1] − → R λ − →

I−(λ) = inf{x|µI(x) ≥ λ, x ≥ s−}

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 9 / 17

Profiles

Definition (Profile)

A profile is a function Φ from [0,1] to R A profile can be viewed as a genuine fuzzy extension of a real number. Let I be an USC fuzzy interval. We call left profile of I (denoted I−) the profile defined as follows: I− :

[0,1] − → R λ − →

I−(λ) = inf{x|µI(x) ≥ λ, x ≥ s−}

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 9 / 17

Profiles

Definition (Profile)

A profile is a function Φ from [0,1] to R A profile can be viewed as a genuine fuzzy extension of a real number. Let I be an USC fuzzy interval. We call left profile of I (denoted I−) the profile defined as follows: I− :

[0,1] − → R λ − →

I−(λ) = inf{x|µI(x) ≥ λ, x ≥ s−} We call right profile of I (denoted I+) the profile defined as following: I+ :

[0,1] − → R λ − →

I+(λ) = sup{x|µI(x) ≥ λ,x ≤ s+}

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 9 / 17

Profiles

Definition (Profile)

A profile is a function Φ from [0,1] to R A profile can be viewed as a genuine fuzzy extension of a real number. Let I be an USC fuzzy interval. We call left profile of I (denoted I−) the profile defined as follows: I− :

[0,1] − → R λ − →

I−(λ) = inf{x|µI(x) ≥ λ, x ≥ s−} We call right profile of I (denoted I+) the profile defined as following: I+ :

[0,1] − → R λ − →

I+(λ) = sup{x|µI(x) ≥ λ,x ≤ s+}

1 0.5 x A A λ

− + Α (λ) Α (λ)

− +

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 9 / 17

Fuzzy configurations

A fuzzy interval Xi is viewed as a pair of (increasing , decreasing) profiles (X −

i ,X + i ). It is viewed as the interval of profiles

{Φ,X −

i

≤ Φ ≤ X +

i }. A fuzzy configuration is a vector of monotonic

profiles. With n fuzzy intervals X1,··· ,Xn, we note

H the set of fuzzy extreme

configurations:

H = ×i{X −

i ,X + i }.

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 10 / 17

Fuzzy configurations

A fuzzy interval Xi is viewed as a pair of (increasing , decreasing) profiles (X −

i ,X + i ). It is viewed as the interval of profiles

{Φ,X −

i

≤ Φ ≤ X +

i }. A fuzzy configuration is a vector of monotonic

profiles. With n fuzzy intervals X1,··· ,Xn, we note

H the set of fuzzy extreme

configurations:

H = ×i{X −

i ,X + i }.

We extend functions to profile as follows:

• f(Ω)
• (λ) = f
• Ω(λ)
• Jérôme Fortin, Didier Dubois, Hélène Fargier

The Empirical Variance of a Set of Fuzzy Intervals 10 / 17

Fuzzy configurations

A fuzzy interval Xi is viewed as a pair of (increasing , decreasing) profiles (X −

i ,X + i ). It is viewed as the interval of profiles

{Φ,X −

i

≤ Φ ≤ X +

i }. A fuzzy configuration is a vector of monotonic

profiles. With n fuzzy intervals X1,··· ,Xn, we note

H the set of fuzzy extreme

configurations:

H = ×i{X −

i ,X + i }.

We extend functions to profile as follows:

• f(Ω)
• (λ) = f
• Ω(λ)
• Remarks:

◮ algebraic properties of real numbers extend to profiles (not to

fuzzy intervals).

◮ the result of one computation on monotonic profiles is not

necessary monotonic (e.g. product example)

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 10 / 17

Main results: interval analysis methods based on enumeration of configurations extend to fuzzy intervals and the enumeration of fuzzy configurations.

Proposition

Let f : Rn −

→ R be a locally monotonic function with respect to each

xi and Z = f(X1 ··· ,Xn) then Z − = infΩ∈

Hf(Ω)

Z + = supΩ∈

Hf(Ω)

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 11 / 17

Main results: interval analysis methods based on enumeration of configurations extend to fuzzy intervals and the enumeration of fuzzy configurations.

Proposition

Let f : Rn −

→ R be a locally monotonic function with respect to each

xi and Z = f(X1 ··· ,Xn) then Z − = infΩ∈

Hf(Ω)

Z + = supΩ∈

Hf(Ω)

et si ∀j ∈ E1, f est croissante par rapport à xj and if ∀j ∈ E1, f is increasing with respect to xj and ∀j ∈ E2 f decreasing with respect to xj then Y − = minΩ∈

H

• f(Ω)| ∀j ∈ E1,Ωj = X −

j

∀j ∈ E2,Ωj = X +

j

• Y + = maxΩ∈

H

• f(Ω)| ∀j ∈ E1,Ωj = X +

j

∀j ∈ E2,Ωj = X −

j

• Jérôme Fortin, Didier Dubois, Hélène Fargier

The Empirical Variance of a Set of Fuzzy Intervals 11 / 17

Applications of profiles

◮ Arithmetic operations which are not globally monotonic on the

whole real line: product and division

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 12 / 17

Applications of profiles

◮ Arithmetic operations which are not globally monotonic on the

whole real line: product and division

◮ Scheduling problems of the fuzzy PERT type

Functions which compute the latest starting dates and floats are locally but not globally monotonic

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 12 / 17

Applications of profiles

◮ Arithmetic operations which are not globally monotonic on the

whole real line: product and division

◮ Scheduling problems of the fuzzy PERT type

Functions which compute the latest starting dates and floats are locally but not globally monotonic

◮ Fuzzy weighted average fwa(w1,··· ,wn,x1,··· ,xn) = ∑i wi∗xi ∑i wi

◮ fwa(.) locally monotonic ◮ we can construct a linear subset of optimal configurations

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 12 / 17

Extension to the computation of the fuzzy variance

V(x1,··· ,xn) = ∑i(xi−e(x1,···,xn))2

n−1

◮ Variance on intervals: existing results (Ferson, Kreinovitch, et al.)

It is not a globally monotonic function.

◮ Polynomial algorithm for the greatest lower bound ◮ Exponential algorithm for the least upper bound

The variance function reaches its upper bound on a vertex of its domain, not the lower bound.

◮ Variance of fuzzy intervals

The same algorithms are still valid, but must be applied to partial profiles defined on elements of a partition of [0,1] defined by some break-points. We present the minimization problem only.

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 13 / 17

Minimization of variance on intervals

X1

2

X X3 E E E

1 2

1 2 3 4 5 7 8 6 ◮ Straightforward computation of the interval average E

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 14 / 17

Minimization of variance on intervals

X1

2

X X3 E E E

1 2

1 2 3 4 5 7 8 6 ◮ Straightforward computation of the interval average E ◮ Partitioning of the average interval using endpoints of overlapping

intervals

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 14 / 17

Minimization of variance on intervals

X1

2

X X3 E E E

1 2

1 2 3 4 5 7 8 6 ◮ Straightforward computation of the interval average E ◮ Partitioning of the average interval using endpoints of overlapping

intervals

◮ For each element Ej of the partition if V is reached for a tuple

x = (x1,··· ,xn) of average e(x) ∈ Ej, then xi =

• e

si e ∈ Xi x+

i

if e > Xi x−

i

if e < Xi

with e(x) = e{xi|xi = x+

i ou x− i }

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 14 / 17

Minimization of variance on intervals

X1

2

X X3 E E E

1 2

1 2 3 4 5 7 8 6 ◮ Straightforward computation of the interval average E ◮ Partitioning of the average interval using endpoints of overlapping

intervals

◮ For each element Ej of the partition if V is reached for a tuple

x = (x1,··· ,xn) of average e(x) ∈ Ej, then xi =

• e

si e ∈ Xi x+

i

if e > Xi x−

i

if e < Xi

with e(x) = e{xi|xi = x+

i ou x− i }

◮ If e(x) ∈ Ej then x is candidate for minimizing the variance

• therwise, the variance can’t be attained for a tuple of average in

Ej.

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 14 / 17

Minimal variance on fuzzy intervals

1 x 0.5 C E D

◮ Straightforward computation of the fuzzy average E using fuzzy

arithmetic.

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 15 / 17

Minimal variance on fuzzy intervals

1 x 0.5 C E D

◮ Straightforward computation of the fuzzy average E using fuzzy

arithmetic.

◮ Decomposition of the unit interval into subintervals such that the

end-points of the α-cuts are in a prescribed ranking.

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 15 / 17

Minimal variance on fuzzy intervals

1 x 0.5 C E D

◮ Straightforward computation of the fuzzy average E using fuzzy

arithmetic.

◮ Decomposition of the unit interval into subintervals such that the

end-points of the α-cuts are in a prescribed ranking.

◮ Partitioning of the area under the fuzzy average into areas

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 15 / 17

Minimal variance on fuzzy intervals

1 x 0.5 C E D

◮ Straightforward computation of the fuzzy average E using fuzzy

arithmetic.

◮ Decomposition of the unit interval into subintervals such that the

end-points of the α-cuts are in a prescribed ranking.

◮ Partitioning of the area under the fuzzy average into areas ◮ For each area Aj, define the (partial) fuzzy configuration

candidate for minimizing the variance:

◮ I− if I− > Aj

Aj I−

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 15 / 17

Minimal variance on fuzzy intervals

1 x 0.5 C E D

◮ Straightforward computation of the fuzzy average E using fuzzy

arithmetic.

◮ Decomposition of the unit interval into subintervals such that the

end-points of the α-cuts are in a prescribed ranking.

◮ Partitioning of the area under the fuzzy average into areas ◮ For each area Aj, define the (partial) fuzzy configuration

candidate for minimizing the variance:

◮ I− if I− > Aj ◮ I+ if I+ < Aj

Aj I+

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 15 / 17

Minimal variance on fuzzy intervals

1 x 0.5 C E D

◮ Straightforward computation of the fuzzy average E using fuzzy

arithmetic.

◮ Decomposition of the unit interval into subintervals such that the

end-points of the α-cuts are in a prescribed ranking.

◮ Partitioning of the area under the fuzzy average into areas ◮ For each area Aj, define the (partial) fuzzy configuration

candidate for minimizing the variance:

◮ I− if I− > Aj ◮ I+ if I+ < Aj ◮ E(Aj) otherwise

Aj I− I+

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 15 / 17

Minimal variance on fuzzy intervals

1 x 0.5 C E D

◮ Straightforward computation of the fuzzy average E using fuzzy

arithmetic.

◮ Decomposition of the unit interval into subintervals such that the

end-points of the α-cuts are in a prescribed ranking.

◮ Partitioning of the area under the fuzzy average into areas ◮ For each area Aj, define the (partial) fuzzy configuration

candidate for minimizing the variance:

◮ Calculation of the partial average profile e(Aj) obtained from the

partial fuzzy configuration.

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 15 / 17

Minimal variance on fuzzy intervals

1 x 0.5 C E D

◮ Straightforward computation of the fuzzy average E using fuzzy

arithmetic.

◮ Decomposition of the unit interval into subintervals such that the

end-points of the α-cuts are in a prescribed ranking.

◮ Partitioning of the area under the fuzzy average into areas ◮ For each area Aj, define the (partial) fuzzy configuration

candidate for minimizing the variance:

◮ Calculation of the partial average profile e(Aj) obtained from the

partial fuzzy configuration.

◮ Test whether e(Aj) overlaps area Aj; possible update of the

partitioning of the unit interval to account for that part of e(Aj) lying in Aj (= potential partial optimal profile).

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 15 / 17

Application to the variance of a single symmetric fuzzy interval

We define the potential variance of a fuzzy interval I as the interval variance of an infinity of nested intervals

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 16 / 17

Application to the variance of a single symmetric fuzzy interval

We define the potential variance of a fuzzy interval I as the interval variance of an infinity of nested intervals

Proposition

Let I be a symmetric fuzzy interval centred in 0. The maximal potential variance of I is obtained as: V = R 1

I+(λ)−I−(λ)

2

2dλ

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 16 / 17

Application to the variance of a single symmetric fuzzy interval

We define the potential variance of a fuzzy interval I as the interval variance of an infinity of nested intervals

Proposition

Let I be a symmetric fuzzy interval centred in 0. The maximal potential variance of I is obtained as: V = R 1

I+(λ)−I−(λ)

2

2dλ

The proof computes the interval variance of a set of n α-cuts equally distributed in the interval (0,1], and then its limit when n goes to infinity.

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 16 / 17

Application to the variance of a single symmetric fuzzy interval

We define the potential variance of a fuzzy interval I as the interval variance of an infinity of nested intervals

Proposition

Let I be a symmetric fuzzy interval centred in 0. The maximal potential variance of I is obtained as: V = R 1

I+(λ)−I−(λ)

2

2dλ

The proof computes the interval variance of a set of n α-cuts equally distributed in the interval (0,1], and then its limit when n goes to

• infinity. We conjecture the same result may hold in the general case.

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 16 / 17

Conclusion

◮ Fuzzy interval = Pair of profiles (genuine fuzzy real numbers)

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 17 / 17

Conclusion

◮ Fuzzy interval = Pair of profiles (genuine fuzzy real numbers) ◮ Easier computation of the exact result for many functions

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 17 / 17

Conclusion

◮ Fuzzy interval = Pair of profiles (genuine fuzzy real numbers) ◮ Easier computation of the exact result for many functions ◮ Easy to implement for piecewise linear possibility distributions

and for L− R parametrized fuzzy interval (if we can bound the number of break-points)

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 17 / 17

Conclusion

◮ Fuzzy interval = Pair of profiles (genuine fuzzy real numbers) ◮ Easier computation of the exact result for many functions ◮ Easy to implement for piecewise linear possibility distributions

and for L− R parametrized fuzzy interval (if we can bound the number of break-points)

◮ Same complexity than in the classical interval computation

(multiplied by the number of subsets in the partition of [0,1] wich appears durring the computation)

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 17 / 17

Conclusion

◮ Fuzzy interval = Pair of profiles (genuine fuzzy real numbers) ◮ Easier computation of the exact result for many functions ◮ Easy to implement for piecewise linear possibility distributions

and for L− R parametrized fuzzy interval (if we can bound the number of break-points)

◮ Same complexity than in the classical interval computation

(multiplied by the number of subsets in the partition of [0,1] wich appears durring the computation)

◮ Intermediary steps involves anomalous profiles that disappear in

the end of the computation. These profiles cannot be viewed as membership functions.

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 17 / 17

Conclusion

◮ Fuzzy interval = Pair of profiles (genuine fuzzy real numbers) ◮ Easier computation of the exact result for many functions ◮ Easy to implement for piecewise linear possibility distributions

and for L− R parametrized fuzzy interval (if we can bound the number of break-points)

◮ Same complexity than in the classical interval computation

(multiplied by the number of subsets in the partition of [0,1] wich appears durring the computation)

◮ Intermediary steps involves anomalous profiles that disappear in

the end of the computation. These profiles cannot be viewed as membership functions.

◮ Application of interval analysis to compute the variance of a

single fuzzy interval

Jérôme Fortin, Didier Dubois, Hélène Fargier The Empirical Variance of a Set of Fuzzy Intervals 17 / 17