Aggregation on finite ordinal scales by scale independent functions

Aggregation on finite ordinal scales by scale independent functions

Jean-Luc Marichal ^∗ and Radko Mesiar ^∗∗

∗ University of Luxembourg, Luxembourg

∗∗ Slovak Technical University, Bratislava (Slovakia)

IPMU 2004 – p.1

What is a finite ordinal scale ?

What is a finite ordinal scale ?

Two equivalent approaches :

IPMU 2004 – p.2

What is a finite ordinal scale ?

Two equivalent approaches :

• Symbolical approach Finite chain ( S, 4 ₎

S = { s ¹ ≺ s ² ≺ · · · ≺ s _k }

What is a finite ordinal scale ?

Two equivalent approaches :

• Symbolical approach Finite chain ( S, 4 ₎

S = { s ¹ ≺ s ² ≺ · · · ≺ s _k }

r r r r r -

s ¹ s ² · · · s _k S

IPMU 2004 – p.2

What is a finite ordinal scale ?

Two equivalent approaches :

• Symbolical approach Finite chain ( S, 4 ₎

S = { s ¹ ≺ s ² ≺ · · · ≺ s _k }

r r r r r -

s ¹ s ² · · · s _k S

Example: evaluation of a product by a consumer

What is a finite ordinal scale ?

Two equivalent approaches :

• Numerical approach

Strictly increasing sequence of real numbers (defined up to order)

IPMU 2004 – p.3

What is a finite ordinal scale ?

Two equivalent approaches :

• Numerical approach

Strictly increasing sequence of real numbers (defined up to order)

r r r r r -

4 5 7 8 10

r r r r r -

−3 1 . 5 14 . 2 58 263

rating benchmarks ^%

Equivalence between both definitions

IPMU 2004 – p.4

Equivalence between both definitions

∃ isomorphism ^f : S → E ⊆ R such that

s _i 4 _{s j ⇔ f (s i )} 6 _{f (s j )}

Equivalence between both definitions

∃ isomorphism ^f : S → E ⊆ R such that

s _i 4 _{s j ⇔ f (s i )} 6 _{f (s j )}

...and defined up to an automorphism ^φ of ^E : f ⁰ = φ ◦ f

IPMU 2004 – p.4

Equivalence between both definitions

∃ isomorphism ^f : S → E ⊆ R such that

s _i 4 _{s j ⇔ f (s i )} 6 _{f (s j )}

...and defined up to an automorphism ^φ of ^E : f ⁰ = φ ◦ f

r r r r r - S

s 1 s 2 s 3 s 4 s 5

¡ ¡

¡ ª

@ @

@ R

f f ⁰ = φ ◦ f

Generally ^E = R or ^E = ]0 , 1[

IPMU 2004 – p.5

Generally ^E = R or ^E = ]0 , 1[

What if ^E = [0 , 1] ?

Generally ^E = R or ^E = ]0 , 1[

What if ^E = [0 , 1] ? Let (S, 4 ₎ be a finite ordinal scale

IPMU 2004 – p.5

Generally ^E = R or ^E = ]0 , 1[

What if ^E = [0 , 1] ? Let (S, 4 ₎ be a finite ordinal scale

f ( s _i ) = 0 ⇒ i = 1

Generally ^E = R or ^E = ]0 , 1[

What if ^E = [0 , 1] ? Let (S, 4 ₎ be a finite ordinal scale

f ( s _i ) = 0 ⇒ i = 1 f ( s _i ) = 1 ⇒ i = | S |

IPMU 2004 – p.5

Generally ^E = R or ^E = ]0 , 1[

What if ^E = [0 , 1] ? Let (S, 4 ₎ be a finite ordinal scale

f ( s _i ) = 0 ⇒ i = 1 f ( s _i ) = 1 ⇒ i = | S |

We assume that ^f : S → E is endpoint-preserving

Generally ^E = R or ^E = ]0 , 1[

What if ^E = [0 , 1] ? Let (S, 4 ₎ be a finite ordinal scale

f ( s _i ) = 0 ⇒ i = 1 f ( s _i ) = 1 ⇒ i = | S |

We assume that ^f : S → E is endpoint-preserving

Consider two independent ordinal scales ( S, 4 _{S )} and ( T, 4 _{T )}

• If ^E = ]0 , 1[ , those scales have nothing in common

• If E = [0, 1] , those scales have fixed endpoints

IPMU 2004 – p.5

Aggregation on finite ordinal scales

Aggregation on finite ordinal scales

(a ¹ , . . . , a _n ) ∈ S ⁿ 7→ a ∈ S

IPMU 2004 – p.6

Aggregation on finite ordinal scales

(a ¹ , . . . , a _n ) ∈ S ⁿ 7→ a ∈ S

Define G : S ⁿ → S

( a ¹ , . . . , a _n ) ∈ S ⁿ 7→ G ( a ¹ , . . . , a _n ) ∈ S

Aggregation on finite ordinal scales

(a ¹ , . . . , a _n ) ∈ S ⁿ 7→ a ∈ S

Define G : S ⁿ → S

( a ¹ , . . . , a _n ) ∈ S ⁿ 7→ G ( a ¹ , . . . , a _n ) ∈ S

Example: n = 2 and _| S | = 3

a ² \ a ¹ s ¹ s ² s ³ s ¹ s ² s ¹ s ³ s ² s ¹ s ¹ s ³ s 3 s 3 s 2 s 2

IPMU 2004 – p.6

Aggregation on finite ordinal scales

(a ¹ , . . . , a _n ) ∈ S ⁿ 7→ a ∈ S

Define G : S ⁿ → S

( a ¹ , . . . , a _n ) ∈ S ⁿ 7→ G ( a ¹ , . . . , a _n ) ∈ S

Example: n = 2 and _| S | = 3

a ² \ a ¹ s ¹ s ² s ³

s ¹ s ² s ¹ s ³

s ² s ¹ s ¹ s ³

Aggregation on finite ordinal scales

Alternative approach

IPMU 2004 – p.7

Aggregation on finite ordinal scales

Alternative approach

Scale independent function F : E ⁿ → E

( x ¹ , . . . , x _n ) ∈ E ⁿ 7→ F ( x ¹ , . . . , x _n ) ∈ E

Aggregation on finite ordinal scales

Alternative approach

Scale independent function F : E ⁿ → E

( x ¹ , . . . , x _n ) ∈ E ⁿ 7→ F ( x ¹ , . . . , x _n ) ∈ E

Example: F = median

median(0 . 1 , 0 . 3 , 0 . 6) = 0 . 3

IPMU 2004 – p.7

Aggregation on finite ordinal scales

A ( E ) = automorphism group of ^E

Aggregation on finite ordinal scales

A ( E ) = automorphism group of ^E

Definition (Marichal and Roubens 1993)

F : E ⁿ → E is an invariant function if, for any ^φ ∈ A(E ) , we have

F [ φ ( x ¹ ) , . . . , φ ( x _n )] = φ [ F ( x ¹ , . . . , x _n )]

IPMU 2004 – p.8

Aggregation on finite ordinal scales

A ( E ) = automorphism group of ^E

Definition (Marichal and Roubens 1993)

F : E ⁿ → E is an invariant function if, for any ^φ ∈ A(E ) , we have

F [ φ ( x ¹ ) , . . . , φ ( x _n )] = φ [ F ( x ¹ , . . . , x _n )]

Examples:

median : yes

Aggregation on finite ordinal scales

A ( E ) = automorphism group of ^E

Definition (Marichal and Roubens 1993)

F : E ⁿ → E is an invariant function if, for any ^φ ∈ A(E ) , we have

F [ φ ( x ¹ ) , . . . , φ ( x _n )] = φ [ F ( x ¹ , . . . , x _n )]

Examples:

median : yes

arithmetic mean : no !

IPMU 2004 – p.8

Description of invariant functions

Description of invariant functions

Definition (Mesiar 2003, Bartlomiejczyk & Drewniak 2004) A subset ^I ⊆ E ⁿ is invariant if

x ∈ I ⇒ φ(x) ∈ I ∀ φ ∈ A(E )

IPMU 2004 – p.9

Description of invariant functions

Definition (Mesiar 2003, Bartlomiejczyk & Drewniak 2004) A subset ^I ⊆ E ⁿ is invariant if

x ∈ I ⇒ φ(x) ∈ I ∀ φ ∈ A(E )

I invariant is minimal if @ ^J invariant and ^J Ã ^I .

Description of invariant functions

Definition (Mesiar 2003, Bartlomiejczyk & Drewniak 2004) A subset ^I ⊆ E ⁿ is invariant if

x ∈ I ⇒ φ(x) ∈ I ∀ φ ∈ A(E ) I invariant is minimal if @ ^J invariant and ^J Ã ^I . Example: E = [0, 1] and n = 2

IPMU 2004 – p.9

Description of invariant functions

Definition (Mesiar 2003, Bartlomiejczyk & Drewniak 2004) A subset ^I ⊆ E ⁿ is invariant if

x ∈ I ⇒ φ(x) ∈ I ∀ φ ∈ A(E ) I invariant is minimal if @ ^J invariant and ^J Ã ^I . Example: E = [0, 1] and n = 2

6

¡ ¡ ¡ ¡

r r

x ²

Description of invariant functions

Proposition (Ovchinnikov 1998, Mesiar 2003)

F : E ⁿ → E is an invariant function if and only if, for any minimal invariant subset I ⊆ E ⁿ ,

IPMU 2004 – p.10

Description of invariant functions

Proposition (Ovchinnikov 1998, Mesiar 2003)

F : E ⁿ → E is an invariant function if and only if, for any minimal invariant subset I ⊆ E ⁿ ,

• either F | _I = inf E (if inf E ∈ E )

Description of invariant functions

Proposition (Ovchinnikov 1998, Mesiar 2003)

F : E ⁿ → E is an invariant function if and only if, for any minimal invariant subset I ⊆ E ⁿ ,

• either F | _I = inf E (if inf E ∈ E )

• or F | _I = sup E (if sup E ∈ E )

IPMU 2004 – p.10

Description of invariant functions

Proposition (Ovchinnikov 1998, Mesiar 2003)

F : E ⁿ → E is an invariant function if and only if, for any minimal invariant subset I ⊆ E ⁿ ,

• either F | _I = inf E (if inf E ∈ E )

• or F | _I = sup E (if sup E ∈ E )

• or ∃ i ∈ {1 , . . . , n } such that ^F | _I = P _i (coord. proj.)

Scale independent vs. invariant functions

IPMU 2004 – p.11

Scale independent vs. invariant functions

Proposition

F : E ⁿ → E is invariant

E ⁿ −−−→ ^F E

Scale independent vs. invariant functions

Proposition

F : E ⁿ → E is invariant m

E ⁿ −−−→ ^F E

IPMU 2004 – p.11

Scale independent vs. invariant functions

Proposition

F : E ⁿ → E is invariant m

∀ ( S, 4 ₎ , ∃ G : S ⁿ → S ,

E ⁿ −−−→ ^F E

n G

Scale independent vs. invariant functions

Proposition

F : E ⁿ → E is invariant m

∀ ( S, 4 ₎ , ∃ G : S ⁿ → S ,

∀ f : S → E ,

E ⁿ −−−→ ^F E

f

x







x







f

S ⁿ −−−→ ^G S

IPMU 2004 – p.11

Scale independent vs. invariant functions

Proposition

F : E ⁿ → E is invariant m

∀ ( S, 4 ₎ , ∃ G : S ⁿ → S ,

∀ f : S → E ,

F [f (a ¹ ), . . . , f (a _n )] = f [G(a ¹ , . . . , a _n )]

E ⁿ −−−→ ^F E

f

x







x







f

n G

Scale independent vs. invariant functions

Proposition

F : E ⁿ → E is invariant m

∀ ( S, 4 ₎ , ∃ G : S ⁿ → S ,

∀ f : S → E ,

F [f (a ¹ ), . . . , f (a _n )] = f [G(a ¹ , . . . , a _n )]

E ⁿ −−−→ ^F E

f

x







x







f

S ⁿ −−−→ ^G S G represents ^F in ( S, 4 ₎

IPMU 2004 – p.11

Scale independent vs. invariant functions

Proposition

F : E ⁿ → E is invariant m

∀ ( S, 4 ₎ , ∃ G : S ⁿ → S ,

∀ f : S → E ,

F [f (a ¹ ), . . . , f (a _n )] = f [G(a ¹ , . . . , a _n )]

E ⁿ −−−→ ^F E

f

x







x







f

n G

Scale independent vs. invariant functions

Proposition

F : E ⁿ → E is invariant m

∀ ( S, 4 ₎ , ∃ G : S ⁿ → S ,

∀ f : S → E ,

F [f (a ¹ ), . . . , f (a _n )] = f [G(a ¹ , . . . , a _n )]

E ⁿ −−−→ ^F E

f

x







x







f

S ⁿ −−−→ ^G S G represents ^F in ( S, 4 ₎

G is uniquely determined : ^G ( a ) = f ^{− 1} ( F [ f ( a )])

IPMU 2004 – p.11

Example

On R ² :

Example

On R ² :

- 6

¡ ¡ ¡ ¡ ¡ ¡ ¡

x ¹ x 2

x ¹

x 2

F (x ¹ , x ² ) = x ¹ ∧ x ²

IPMU 2004 – p.12

Example

On R ² :

- 6

¡ ¡ ¡ ¡ ¡ ¡ ¡

x ¹ x 2

x ¹

x 2

- 6

¡ ¡ ¡ ¡ ¡ ¡ ¡

a ¹ a 2

s s s

s s s

s s s

s 1 s 2 s 3

s ¹ s 2

s ³

s ¹ s ¹ s ¹ s ¹

s 1

s ² s 2

s ² s 3

F (x ¹ , x ² ) = x ¹ ∧ x ² G(a ¹ , a ² ) = a ¹ ∧ a ²

Example

On R ² :

- 6

¡ ¡ ¡ ¡ ¡ ¡ ¡

x ¹ x 2

x ¹

x 2

- 6

¡ ¡ ¡ ¡ ¡ ¡ ¡

a ¹ a 2

s s s

s s s

s s s

s 1 s 2 s 3

s ¹ s 2

s ³

s ¹ s ¹ s ¹ s ¹

s 1

s ² s 2

s ² s 3

F (x ¹ , x ² ) = x ¹ ∧ x ² G(a ¹ , a ² ) = a ¹ ∧ a ²

F and ^G are always isomorphic !

IPMU 2004 – p.12

Important remark

Considering ^G : S ⁿ → S is not equivalent to considering

F : E ⁿ → E invariant

Important remark

Considering ^G : S ⁿ → S is not equivalent to considering F : E ⁿ → E invariant

Example: n = 2 and ^E = R

IPMU 2004 – p.13

Important remark

Considering ^G : S ⁿ → S is not equivalent to considering F : E ⁿ → E invariant

Example: n = 2 and ^E = R

• 4 invariant functions ^F : R ² → R

Important remark

Considering ^G : S ⁿ → S is not equivalent to considering F : E ⁿ → E invariant

Example: n = 2 and ^E = R

• 4 invariant functions ^F : R ² → R

F = min, max, P ¹ , P ²

IPMU 2004 – p.13

Important remark

Considering ^G : S ⁿ → S is not equivalent to considering F : E ⁿ → E invariant

Example: n = 2 and ^E = R

• 4 invariant functions ^F : R ² → R

F = min, max, P ¹ , P ²

• | S | ^|S|

discrete functions ^G : S ² → S

Important remark

Considering ^G : S ⁿ → S is not equivalent to considering F : E ⁿ → E invariant

Example: n = 2 and ^E = R

• 4 invariant functions ^F : R ² → R

F = min, max, P ¹ , P ²

• | S | ^|S|

discrete functions ^G : S ² → S

Note: If ^F invariant then

F (x ¹ , . . . , x _n ) ∈ { x ¹ , . . . , x _n } ∪ {inf E, sup E }

IPMU 2004 – p.13

Continuous invariant functions

Continuous invariant functions

Proposition (Ovchinnikov 1998, Marichal 2002) F : E ⁿ → E is a continuous invariant function iff

IPMU 2004 – p.14

Continuous invariant functions

Proposition (Ovchinnikov 1998, Marichal 2002) F : E ⁿ → E is a continuous invariant function iff

• either F ≡ inf E (if inf E ∈ E )

Continuous invariant functions

Proposition (Ovchinnikov 1998, Marichal 2002) F : E ⁿ → E is a continuous invariant function iff

• either F ≡ inf E (if inf E ∈ E )

• or ^F ≡ sup E (if sup E ∈ E )

IPMU 2004 – p.14

Continuous invariant functions

Proposition (Ovchinnikov 1998, Marichal 2002) F : E ⁿ → E is a continuous invariant function iff

• either F ≡ inf E (if inf E ∈ E )

• or ^F ≡ sup E (if sup E ∈ E )

• or ∃ a lattice polynomial L : E ⁿ → E such that F = L

Continuous invariant functions

Proposition (Ovchinnikov 1998, Marichal 2002) F : E ⁿ → E is a continuous invariant function iff

• either F ≡ inf E (if inf E ∈ E )

• or ^F ≡ sup E (if sup E ∈ E )

• or ∃ a lattice polynomial L : E ⁿ → E such that F = L

Example of lattice polynomial : ^L ( x ) = ( x ¹ ∨ x ² ) ∧ x ³

IPMU 2004 – p.14

Continuous invariant functions

Proposition (Ovchinnikov 1998, Marichal 2002) F : E ⁿ → E is a continuous invariant function iff

• either F ≡ inf E (if inf E ∈ E )

• or ^F ≡ sup E (if sup E ∈ E )

• or ∃ a lattice polynomial L : E ⁿ → E such that F = L

Example of lattice polynomial : ^L ( x ) = ( x ¹ ∨ x ² ) ∧ x ³

Smooth discrete functions

IPMU 2004 – p.15

Smooth discrete functions

Definition (Godo and Sierra 1988)

G : S ⁿ → S is smooth if,

Smooth discrete functions

Definition (Godo and Sierra 1988) G : S ⁿ → S is smooth if,

∀ a, b ∈ S ⁿ , with ^a = b except on ⁱ th coordinate

IPMU 2004 – p.15

Smooth discrete functions

Definition (Godo and Sierra 1988) G : S ⁿ → S is smooth if,

∀ a, b ∈ S ⁿ , with ^a = b except on ⁱ th coordinate

where |ind( a _i ) − ind( b _i )| = 1

Smooth discrete functions

Definition (Godo and Sierra 1988) G : S ⁿ → S is smooth if,

∀ a, b ∈ S ⁿ , with ^a = b except on ⁱ th coordinate where |ind( a _i ) − ind( b _i )| = 1

⇒ ^{¯ ¯} _¯ ind[G(a)] − ind[G(b)] ^{¯ ¯} _¯ 6 ₁

IPMU 2004 – p.15

Smooth discrete functions

Definition (Godo and Sierra 1988) G : S ⁿ → S is smooth if,

∀ a, b ∈ S ⁿ , with ^a = b except on ⁱ th coordinate where |ind( a _i ) − ind( b _i )| = 1

⇒ ^{¯ ¯} _¯ ind[G(a)] − ind[G(b)] ^{¯ ¯} _¯ 6 ₁

Example: n = 2 and | S | = 3

a ² \ a ¹ s ¹ s ² s ³ s ¹ s ² s ¹ s ³

s s s s

Smooth discrete functions

Definition (Godo and Sierra 1988) G : S ⁿ → S is smooth if,

∀ a, b ∈ S ⁿ , with ^a = b except on ⁱ th coordinate where |ind( a _i ) − ind( b _i )| = 1

⇒ ^{¯ ¯} _¯ ind[G(a)] − ind[G(b)] ^{¯ ¯} _¯ 6 ₁

Example: n = 2 and | S | = 3

a ² \ a ¹ s ¹ s ² s ³ s ¹ s ² s ¹ s ³ s 2 s 1 s 1 s 3

s ³ s ³ s ² s ² Not smooth !

IPMU 2004 – p.15

Smooth discrete functions

Proposition

An invariant function is continuous iff it is represented

only by smooth discrete functions

A more general case

Two finite ordinal scales (S, 4 _{S )} and (T, 4 _{T )} ( a ¹ , . . . , a _n ) ∈ S ⁿ 7→ a ∈ T

IPMU 2004 – p.17

A more general case

Two finite ordinal scales (S, 4 _{S )} and (T, 4 _{T )} ( a ¹ , . . . , a _n ) ∈ S ⁿ 7→ a ∈ T

Define G : S ⁿ → T

( a ¹ , . . . , a _n ) ∈ S ⁿ 7→ G ( a ¹ , . . . , a _n ) ∈ T

A more general case

Two finite ordinal scales (S, 4 _{S )} and (T, 4 _{T )} ( a ¹ , . . . , a _n ) ∈ S ⁿ 7→ a ∈ T

Define G : S ⁿ → T

( a ¹ , . . . , a _n ) ∈ S ⁿ 7→ G ( a ¹ , . . . , a _n ) ∈ T

Example: n = 2 , | S | = 3 , and | T | = 5 a ² \ a ¹ s ¹ s ² s ³

s ¹ t ⁴ t ¹ t ² s 2 t 4 t 3 t 5

s ³ t ² t ¹ t ¹

IPMU 2004 – p.17

A more general case

Alternative approach:

Scale independent function ^F : E ⁿ → R

A more general case

Alternative approach:

Scale independent function ^F : E ⁿ → R Definition

F : E ⁿ → R is weakly invariant if, for any φ ∈ A(E ) , there is a strictly increasing ^ψ _φ : R → R such that

F [ φ ( x 1 ) , . . . , φ ( x _n )] = ψ _φ [ F ( x 1 , . . . , x _n )]

IPMU 2004 – p.18

Description of weakly invariant functions

Description of weakly invariant functions

Proposition

F : E ⁿ → R is weakly invariant if and only if, for any minimal invariant subset ^I ⊆ E ⁿ , there exist

IPMU 2004 – p.19

Description of weakly invariant functions

Proposition

F : E ⁿ → R is weakly invariant if and only if, for any minimal invariant subset ^I ⊆ E ⁿ , there exist

i) i ∈ {1, . . . , n }

Description of weakly invariant functions

Proposition

F : E ⁿ → R is weakly invariant if and only if, for any minimal invariant subset ^I ⊆ E ⁿ , there exist

i) i ∈ {1, . . . , n }

ii ) g : R → R, constant or strictly monotonic

IPMU 2004 – p.19

Description of weakly invariant functions

Proposition

F : E ⁿ → R is weakly invariant if and only if, for any minimal invariant subset ^I ⊆ E ⁿ , there exist

i) i ∈ {1, . . . , n }

ii ) g : R → R, constant or strictly monotonic such that

F | _I = g ◦ P _i

Description of weakly invariant functions

Proposition

F : E ⁿ → R is weakly invariant if and only if, for any minimal invariant subset ^I ⊆ E ⁿ , there exist

i) i ∈ {1, . . . , n }

ii ) g : R → R, constant or strictly monotonic such that

F | _I = g ◦ P _i (+ some extra conditions)

IPMU 2004 – p.19

Scale independent vs. weakly invariant functions

Scale independent vs. weakly invariant functions

Proposition

F : E ⁿ → R is weakly invariant

E ⁿ −−−→ ^F R

IPMU 2004 – p.20

Scale independent vs. weakly invariant functions

Proposition

F : E ⁿ → R is weakly invariant m

E ⁿ −−−→ ^F R

Scale independent vs. weakly invariant functions

Proposition

F : E ⁿ → R is weakly invariant m

∀ ( S, 4 ₎ , ∃ ( T, 4 _{) & G : S} ⁿ _{→ T} ,

E ⁿ −−−→ ^F R

S ⁿ −−−→ ^G T

IPMU 2004 – p.20

Scale independent vs. weakly invariant functions

Proposition

F : E ⁿ → R is weakly invariant m

∀ ( S, 4 ₎ , ∃ ( T, 4 _{) & G : S} ⁿ _{→ T} ,

∀ f : S → E , ∃ g _f : T → R ^,

E ⁿ −−−→ ^F R

f

x







x







g

G

Scale independent vs. weakly invariant functions

Proposition

F : E ⁿ → R is weakly invariant m

∀ ( S, 4 ₎ , ∃ ( T, 4 _{) & G : S} ⁿ _{→ T} ,

∀ f : S → E , ∃ g _f : T → R ^,

F [ f ( a 1 ) , . . . , f ( a _n )] = g _f [ G ( a 1 , . . . , a _n )]

E ⁿ −−−→ ^F R

f

x







x







g

S ⁿ −−−→ ^G T

IPMU 2004 – p.20

Scale independent vs. weakly invariant functions

Proposition

F : E ⁿ → R is weakly invariant m

∀ ( S, 4 ₎ , ∃ ( T, 4 _{) & G : S} ⁿ _{→ T} ,

∀ f : S → E , ∃ g _f : T → R ^,

F [ f ( a 1 ) , . . . , f ( a _n )] = g _f [ G ( a 1 , . . . , a _n )]

E ⁿ −−−→ ^F R

f

x







x







g

G

Scale independent vs. weakly invariant functions

Proposition

F : E ⁿ → R is weakly invariant m

∀ ( S, 4 ₎ , ∃ ( T, 4 _{) & G : S} ⁿ _{→ T} ,

∀ f : S → E , ∃ g _f : T → R ^,

F [ f ( a 1 ) , . . . , f ( a _n )] = g _f [ G ( a 1 , . . . , a _n )]

E ⁿ −−−→ ^F R

f

x







x







g

S ⁿ −−−→ ^G T G represents F in (S, 4 ₎

G and ^g _f are uniquely determined

IPMU 2004 – p.20

Continuous weakly invariant functions

Continuous weakly invariant functions

Proposition (Yanovskaya 1989, Marichal 2002)

F : E ⁿ → R is continuous and weakly invariant if and only if there exist

IPMU 2004 – p.21

Continuous weakly invariant functions

Proposition (Yanovskaya 1989, Marichal 2002)

F : E ⁿ → R is continuous and weakly invariant if and only if there exist

i ) L : E ⁿ → E , lattice polynomial

Continuous weakly invariant functions

Proposition (Yanovskaya 1989, Marichal 2002)

F : E ⁿ → R is continuous and weakly invariant if and only if there exist

i ) L : E ⁿ → E , lattice polynomial ii) g : E → R, constant

or continuous and strictly monotonic

IPMU 2004 – p.21

Continuous weakly invariant functions

Proposition (Yanovskaya 1989, Marichal 2002)

F : E ⁿ → R is continuous and weakly invariant if and only if there exist

i ) L : E ⁿ → E , lattice polynomial ii) g : E → R, constant

or continuous and strictly monotonic such that

F = g ◦ L

Continuous weakly invariant functions

Proposition

A continuous weakly invariant function is represented only by smooth discrete functions

IPMU 2004 – p.22

Continuous weakly invariant functions

Proposition

A continuous weakly invariant function is represented only by smooth discrete functions

The converse is false !!!