A FCA perspective on Rough Set Theory Bernhard Ganter & - - PowerPoint PPT Presentation

Functional and ordinal dependencies Rough Set Theory Generalisations

A FCA perspective on Rough Set Theory

Bernhard Ganter & Christian Meschke

Institut f¨ ur Algebra Technische Universit¨ at Dresden

October 20, 2010

Functional and ordinal dependencies Rough Set Theory Generalisations

Survey

Functional and ordinal dependencies Indiscernibility Dependencies Rough Set Theory Approximations Lattices of approximations Generalisations Approximations from arbitrary kernel-closure pairs Contextual Representation

Functional and ordinal dependencies Rough Set Theory Generalisations

Indiscernibility by attribute values

g m w

• Let (G, M, W , I) a many-valued context.
• Let B ⊆ M be a set of attributes. One

defines (g, h) ∈ IndB :⇐ ⇒ ∀m ∈ B : m(g) = m(h).

• IndB is an indiscernibility equivalence

relation.

• D ⊆ M is functionally dependent on B if

IndB ⊆ IndD .

• In this case one says that the functional

dependency B → D is valid.

Functional and ordinal dependencies Rough Set Theory Generalisations

Indiscernibility by attribute values

g m w

• Let (G, M, W , I) a many-valued context.
• Let B ⊆ M be a set of attributes. One

defines (g, h) ∈ IndB :⇐ ⇒ ∀m ∈ B : m(g) = m(h).

• IndB is an indiscernibility equivalence

relation.

• D ⊆ M is functionally dependent on B if

IndB ⊆ IndD .

• In this case one says that the functional

dependency B → D is valid.

Functional and ordinal dependencies Rough Set Theory Generalisations

Indiscernibility by attribute values

B

h g 4 sick + 4 sick +

• Let (G, M, W , I) a many-valued context.
• Let B ⊆ M be a set of attributes. One

defines (g, h) ∈ IndB :⇐ ⇒ ∀m ∈ B : m(g) = m(h).

• IndB is an indiscernibility equivalence

relation.

• D ⊆ M is functionally dependent on B if

IndB ⊆ IndD .

• In this case one says that the functional

dependency B → D is valid.

Functional and ordinal dependencies Rough Set Theory Generalisations

Indiscernibility by attribute values

D

hot hot

B

h g 4 sick + 4 sick +

• Let (G, M, W , I) a many-valued context.
• Let B ⊆ M be a set of attributes. One

defines (g, h) ∈ IndB :⇐ ⇒ ∀m ∈ B : m(g) = m(h).

• IndB is an indiscernibility equivalence

relation.

• D ⊆ M is functionally dependent on B if

IndB ⊆ IndD .

• In this case one says that the functional

dependency B → D is valid.

Functional and ordinal dependencies Rough Set Theory Generalisations

Reducts

• Let C, D ⊆ M be sets of condition and of decision attributes.
• A subset R ⊆ C is called a reduct of C if

IndR ⊆ IndC . A reduct is called minimal if no proper subset is a reduct.

• One defines

Pos(C, D) := {g ∈ G | [g] IndC ⊆ [g] IndD},

• and calls R ⊆ C a D-relative reduct of C if

Pos(R, D) = Pos(C, D).

Functional and ordinal dependencies Rough Set Theory Generalisations

Reducts

• Let C, D ⊆ M be sets of condition and of decision attributes.
• A subset R ⊆ C is called a reduct of C if

IndR ⊆ IndC . A reduct is called minimal if no proper subset is a reduct.

• One defines

Pos(C, D) := {g ∈ G | [g] IndC ⊆ [g] IndD},

• and calls R ⊆ C a D-relative reduct of C if

Pos(R, D) = Pos(C, D).

Functional and ordinal dependencies Rough Set Theory Generalisations

Functional dependencies

• Consider to the data table (G, M, W , I) the formal context

(G × G, M, Ifun), where (g, h) Ifun m :⇐ ⇒ m(g) = m(h).

• In this formal context, for B ⊆ M it holds that

B′ = IndB .

• Remark: since (g, h)Ifun = (h, g)Ifun one is free to “reduce” the

context’s object set G × G to the set of all two-elementary subsets of G.

Functional and ordinal dependencies Rough Set Theory Generalisations

Functional dependencies

• Consider to the data table (G, M, W , I) the formal context

(G × G, M, Ifun), where (g, h) Ifun m :⇐ ⇒ m(g) = m(h).

• In this formal context, for B ⊆ M it holds that

B′ = IndB .

• Remark: since (g, h)Ifun = (h, g)Ifun one is free to “reduce” the

context’s object set G × G to the set of all two-elementary subsets of G.

Functional and ordinal dependencies Rough Set Theory Generalisations

Functional Dependencies

• Hence, for C, D ⊆ M the following two statements are

equivalent:

(a) the attribute implication C → D holds in (G × G, M, Ifun), (b) the functional depenceny C → D holds in (G, M, W , I).

• The minimal reducts are the minimal generators in the closure

system of the intents of (G, M, Ifun).

• The intents of (G, M, Ifun) are referred to as coreducts.
• The intent C ′′ contains precisely those attributes whose

vaulues (on total G) can be inferred from the values of C.

Functional and ordinal dependencies Rough Set Theory Generalisations

Functional Dependencies

• Hence, for C, D ⊆ M the following two statements are

equivalent:

(a) the attribute implication C → D holds in (G × G, M, Ifun), (b) the functional depenceny C → D holds in (G, M, W , I).

• The minimal reducts are the minimal generators in the closure

system of the intents of (G, M, Ifun).

• The intents of (G, M, Ifun) are referred to as coreducts.
• The intent C ′′ contains precisely those attributes whose

vaulues (on total G) can be inferred from the values of C.

Functional and ordinal dependencies Rough Set Theory Generalisations

Ordinal dependencies

• The functional dependency C → D holds in (G, M, W , I) iff

there is a function f : W C → W D such that for every object g ∈ G it holds that (c(g) | c ∈ C)

f

− → (d(g) | d ∈ D).

• It would be interesting if such a mapping f is “somehow”
• rder-preserving.
• Let us assume that for every attribute m ∈ M

(Wm, ≤m) is a partially ordered set with m[G] ⊆ Wm ⊆ W .

Functional and ordinal dependencies Rough Set Theory Generalisations

Ordinal dependencies

• The functional dependency C → D holds in (G, M, W , I) iff

there is a function f : W C → W D such that for every object g ∈ G it holds that (c(g) | c ∈ C)

f

− → (d(g) | d ∈ D).

• It would be interesting if such a mapping f is “somehow”
• rder-preserving.
• Let us assume that for every attribute m ∈ M

(Wm, ≤m) is a partially ordered set with m[G] ⊆ Wm ⊆ W .

Functional and ordinal dependencies Rough Set Theory Generalisations

Ordinal dependencies

• One says, the ordinal dependency C → D holds if all objects

g, h ∈ G satisfy

• ∀c ∈ C : c(g) ≤c c(h)
• =

⇒

• ∀d ∈ D : d(g) ≤d d(h)
• .
• Ordinal dependency implies functional dependency (regardless
• f the choice of the order relations ≤m).
• If one chooses

x ≤m y :⇐ ⇒ x = y we receive the functional dependencies again.

Functional and ordinal dependencies Rough Set Theory Generalisations

Ordinal dependencies

• The ordinal dependency C → D holds in (G, M, W , I) iff the

attribute implication C → D holds in the formal context (G × G, M, Iord), where (g, h) Iord m :⇐ ⇒ m(g) ≤m m(h).

• Finally: let us take a look at an example.

Functional and ordinal dependencies Rough Set Theory Generalisations

Ordinal dependencies

• The ordinal dependency C → D holds in (G, M, W , I) iff the

attribute implication C → D holds in the formal context (G × G, M, Iord), where (g, h) Iord m :⇐ ⇒ m(g) ≤m m(h).

• Finally: let us take a look at an example.

Functional and ordinal dependencies Rough Set Theory Generalisations

Paul the octopus

Functional and ordinal dependencies Rough Set Theory Generalisations

An example

Germany’s results at the FIFA World Cup 2010 and the predictions

• f Paul the octopus.

game goals result Paul’s

• utcome
• pponent

stage GER OPP predict. 1 Australia group stage 4 win win correct 2 Serbia group stage 1 loss loss correct 3 Ghana group stage 1 win win correct 4 England round of 16 4 1 win win correct 5 Argentina quarter-finals 4 win win correct 6 Spain semi-finals 1 loss loss correct 7 Uruguay 3rd place play-off 3 2 win win correct

• 7 objects,

G = {1, . . . , 7},

Functional and ordinal dependencies Rough Set Theory Generalisations

An example

Germany’s results at the FIFA World Cup 2010 and the predictions

• f Paul the octopus.

game goals result Paul’s

• utcome
• pponent

stage GER OPP predict. 1 Australia group stage 4 win win correct 2 Serbia group stage 1 loss loss correct 3 Ghana group stage 1 win win correct 4 England round of 16 4 1 win win correct 5 Argentina quarter-finals 4 win win correct 6 Spain semi-finals 1 loss loss correct 7 Uruguay 3rd place play-off 3 2 win win correct

• 7 objects,

G = {1, . . . , 7},

Functional and ordinal dependencies Rough Set Theory Generalisations

An example

Germany’s results at the FIFA World Cup 2010 and the predictions

• f Paul the octopus.

game goals result Paul’s

• utcome
• pponent

stage GER OPP predict. 1 Australia group stage 4 win win correct 2 Serbia group stage 1 loss loss correct 3 Ghana group stage 1 win win correct 4 England round of 16 4 1 win win correct 5 Argentina quarter-finals 4 win win correct 6 Spain semi-finals 1 loss loss correct 7 Uruguay 3rd place play-off 3 2 win win correct

• 7 attributes,

M = {opponent, stage, GER, OPP, result, predict., outcome}.

Functional and ordinal dependencies Rough Set Theory Generalisations

Example: functional dependencies

• pponent

stage

• utcome

result prediction OPP GER

The (modified) stem base of the formal context (G × G, M, Ifun):

→

• utcome
• pponent

→ stage, GER, OPP, result, prediction GER → result prediction stage, OPP → result, prediction result → prediction prediction → result stage, result → OPP stage, GER, OPP →

• pponent

Functional and ordinal dependencies Rough Set Theory Generalisations

Example: functional dependencies

• pponent

stage

• utcome

result prediction OPP GER

The proper-premise implications of the formal context (G × G, M, Ifun):

→

• utcome
• pponent

→ stage, OPP, GER, result, prediction GER, stage → OPP, opponent GER → prediction, result OPP, stage → prediction, result result, stage → OPP prediction, stage → OPP prediction → result result → prediction

Functional and ordinal dependencies Rough Set Theory Generalisations

Example: functional dependencies

• pponent

stage

• utcome

result prediction OPP GER

The minimal reducts: ∅, {OPP}, {GER}, {result}, {opponent}, {stage}, {OPP, GER}, {OPP, result}, {OPP, stage}, {prediction, stage}, {GER, stage}, {result, stage}, {prediction, OPP}, {prediction}.

Functional and ordinal dependencies Rough Set Theory Generalisations

Example: ordinal dependencies

Which are “the intrinsic” order relations ≤m?

game goals result Paul’s

• utcome
• pponent

stage GER OPP predict. 1 Australia group stage 4 win win correct 2 Serbia group stage 1 loss loss correct 3 Ghana group stage 1 win win correct 4 England round of 16 4 1 win win correct 5 Argentina quarter-finals 4 win win correct 6 Spain semi-finals 1 loss loss correct 7 Uruguay 3rd place play-off 3 2 win win correct

• For m ∈ {opponent, outcome} we choose the discrete order,

i.e., v ≤m w ⇐ ⇒ v = w.

Functional and ordinal dependencies Rough Set Theory Generalisations

Example: ordinal dependencies

Which are “the intrinsic” order relations ≤m?

group stage round of 16 quarter-finals semi-finals final 3rd place play-off (Wstage, ≤stage) 1 2 3 4 (WGER, ≤GER) 3 2 1 (WOPP, ≤OPP) loss win (Wpredict., ≤predict.) (Wresult, ≤result)

Functional and ordinal dependencies Rough Set Theory Generalisations

Example: ordinal dependencies

• pponent
• utcome

GER result prediction OPP stage

The (modified) stem base of the formal context (G × G, M, Iord):

→

• utcome
• pponent

→ stage, GER, OPP result, prediction GER → result, prediction result → prediction prediction → result

Functional and ordinal dependencies Rough Set Theory Generalisations

Example: ordinal dependencies

• pponent
• utcome

GER result prediction OPP stage

The proper-premise implications of the formal context (G × G, M, Iord):

→

• utcome
• pponent

→ prediction, OPP, GER, result, stage GER → prediction, result result → prediction prediction → result

Functional and ordinal dependencies Rough Set Theory Generalisations

Classical Rough Sets

U

• Let the set U be the universe, and
• let Θ be an indiscernibility equivalence

relation on U.

• The pair (U, Θ) is called an approxima-

tion space.

• For a subset X ⊆ U one defines

⌊X⌋ := {u ∈ U | u/Θ ⊆ X} to be the lower approximation of X.

• The pair (⌊X⌋ , ⌈X⌉) is called the rough

set approximation of X.

Functional and ordinal dependencies Rough Set Theory Generalisations

Classical Rough Sets

U/Θ

• Let the set U be the universe, and
• let Θ be an indiscernibility equivalence

relation on U.

• The pair (U, Θ) is called an approxima-

tion space.

• For a subset X ⊆ U one defines

⌊X⌋ := {u ∈ U | u/Θ ⊆ X} to be the lower approximation of X.

• The pair (⌊X⌋ , ⌈X⌉) is called the rough

set approximation of X.

Functional and ordinal dependencies Rough Set Theory Generalisations

Classical Rough Sets

U/Θ

• Let the set U be the universe, and
• let Θ be an indiscernibility equivalence

relation on U.

• The pair (U, Θ) is called an approxima-

tion space.

• For a subset X ⊆ U one defines

⌊X⌋ := {u ∈ U | u/Θ ⊆ X} to be the lower approximation of X.

• The pair (⌊X⌋ , ⌈X⌉) is called the rough

set approximation of X.

Functional and ordinal dependencies Rough Set Theory Generalisations

Classical Rough Sets

U/Θ

• Let the set U be the universe, and
• let Θ be an indiscernibility equivalence

relation on U.

• The pair (U, Θ) is called an approxima-

tion space.

• For a subset X ⊆ U one defines

⌊X⌋ := {u ∈ U | u/Θ ⊆ X} to be the lower approximation of X.

• The pair (⌊X⌋ , ⌈X⌉) is called the rough

set approximation of X.

Functional and ordinal dependencies Rough Set Theory Generalisations

Classical Rough Sets

U/Θ

• Let the set U be the universe, and
• let Θ be an indiscernibility equivalence

relation on U.

• The pair (U, Θ) is called an approxima-

tion space.

• For a subset X ⊆ U one defines

⌊X⌋ := {u ∈ U | u/Θ ⊆ X} to be the lower approximation of X.

• The pair (⌊X⌋ , ⌈X⌉) is called the rough

set approximation of X.

Functional and ordinal dependencies Rough Set Theory Generalisations

Classical Rough Sets

U/Θ

• Let the set U be the universe, and
• let Θ be an indiscernibility equivalence

relation on U.

• The pair (U, Θ) is called an approxima-

tion space.

• For a subset X ⊆ U one defines

⌈X⌉ := {u ∈ U | u/Θ ∩ X = ∅} to be the upper approximation of X.

• The pair (⌊X⌋ , ⌈X⌉) is called the rough

set approximation of X.

Functional and ordinal dependencies Rough Set Theory Generalisations

Classical Rough Sets

U/Θ

• Let the set U be the universe, and
• let Θ be an indiscernibility equivalence

relation on U.

• The pair (U, Θ) is called an approxima-

tion space.

• For a subset X ⊆ U one defines

⌈X⌉ := {u ∈ U | u/Θ ∩ X = ∅} to be the upper approximation of X.

• The pair (⌊X⌋ , ⌈X⌉) is called the rough

set approximation of X.

Functional and ordinal dependencies Rough Set Theory Generalisations

Classical Rough Sets

U/Θ

• Let the set U be the universe, and
• let Θ be an indiscernibility equivalence

relation on U.

• The pair (U, Θ) is called an approxima-

tion space.

• For a subset X ⊆ U one defines

⌈X⌉ := {u ∈ U | u/Θ ∩ X = ∅} to be the upper approximation of X.

• Thus, the universe U gets divided into

three “regions of membership to X”: positive region, uncertain region and negative region.

Functional and ordinal dependencies Rough Set Theory Generalisations

Lattices of approximations

• The pairs (⌊X⌋ , ⌈X⌉) with X ⊆ U form a lattice, where

(⌊X⌋ , ⌈X⌉) ≤ (⌊Y ⌋ , ⌈Y ⌉) :⇐ ⇒ ⌊X⌋ ⊆ ⌊Y ⌋ and ⌈X⌉ ⊆ ⌈Y ⌉ .

• The mapping

(⌊X⌋ , ⌈X⌉) →

• X ∁

,

• X ∁

=

• ⌈X⌉∁, ⌈X⌉∁

yields a pseudo-complementation in this lattice.

• Dually, the mapping

(⌊X⌋ , ⌈X⌉) →

• X ∁

,

• X ∁

=

• ⌊X⌋∁, ⌊X⌋∁

yields a dual pseudo-complementation.

Functional and ordinal dependencies Rough Set Theory Generalisations

Lattices of approximations

• The pairs (⌊X⌋ , ⌈X⌉) with X ⊆ U form a lattice, where

(⌊X⌋ , ⌈X⌉) ≤ (⌊Y ⌋ , ⌈Y ⌉) :⇐ ⇒ ⌊X⌋ ⊆ ⌊Y ⌋ and ⌈X⌉ ⊆ ⌈Y ⌉ .

• The lattice additionally equipped with these two

mappings forms a regular double Stone algebra.

• Furthermore, every regular double Stone algebra

is representable in such a way by an approxima- tion space (Comer).

• The lattice is a direct product of chains having

length one or two.

Functional and ordinal dependencies Rough Set Theory Generalisations

Lattices of approximations

• The pairs (⌊X⌋ , ⌈X⌉) with X ⊆ U form a lattice, where

(⌊X⌋ , ⌈X⌉) ≤ (⌊Y ⌋ , ⌈Y ⌉) :⇐ ⇒ ⌊X⌋ ⊆ ⌊Y ⌋ and ⌈X⌉ ⊆ ⌈Y ⌉ .

• The lattice additionally equipped with these two

mappings forms a regular double Stone algebra.

• Furthermore, every regular double Stone algebra

is representable in such a way by an approxima- tion space (Comer).

• The lattice is a direct product of chains having

length one or two.

Functional and ordinal dependencies Rough Set Theory Generalisations

Octopus example

• Let us take a look at the lattice of approximations to the

approximation space (G, IndB) for different B ⊆ M.

• For B = {opponent} it follows

(⌊X⌋B, ⌈X⌉B) = (X, X).

game goals result Paul’s

• utcome
• pponent

stage GER OPP predict. 1 Australia group stage 4 win win correct 2 Serbia group stage 1 loss loss correct 3 Ghana group stage 1 win win correct 4 England round of 16 4 1 win win correct 5 Argentina quarter-finals 4 win win correct 6 Spain semi-finals 1 loss loss correct 7 Uruguay 3rd place play-off 3 2 win win correct

Functional and ordinal dependencies Rough Set Theory Generalisations

Octopus example

• Let us take a look at the lattice of approximations to the

approximation space (G, IndB) for different B ⊆ M.

• For B = {opponent} it follows

(⌊X⌋B, ⌈X⌉B) = (X, X).

game goals result Paul’s

• utcome
• pponent

stage GER OPP predict. 1 Australia group stage 4 win win correct 2 Serbia group stage 1 loss loss correct 3 Ghana group stage 1 win win correct 4 England round of 16 4 1 win win correct 5 Argentina quarter-finals 4 win win correct 6 Spain semi-finals 1 loss loss correct 7 Uruguay 3rd place play-off 3 2 win win correct

Functional and ordinal dependencies Rough Set Theory Generalisations

Octopus example

• Let us take a look at the lattice of approximations to the

approximation space (G, IndB) for different B ⊆ M.

• For B = {outcome} we receive the following three approxi-

mations: (∅, ∅), (∅, G), (G, G).

game goals result Paul’s

• utcome
• pponent

stage GER OPP predict. 1 Australia group stage 4 win win correct 2 Serbia group stage 1 loss loss correct 3 Ghana group stage 1 win win correct 4 England round of 16 4 1 win win correct 5 Argentina quarter-finals 4 win win correct 6 Spain semi-finals 1 loss loss correct 7 Uruguay 3rd place play-off 3 2 win win correct

Functional and ordinal dependencies Rough Set Theory Generalisations

Octopus example

• Let us take a look at the lattice of approximations to the

approximation space (G, IndB) for different B ⊆ M.

• For B = {OPP} the equivalence relation IndB has one

singleton and two non-singleton equivalence classes.

game goals result Paul’s

• utcome
• pponent

stage GER OPP predict. 1 Australia group stage 4 win win correct 2 Serbia group stage 1 loss loss correct 3 Ghana group stage 1 win win correct 4 England round of 16 4 1 win win correct 5 Argentina quarter-finals 4 win win correct 6 Spain semi-finals 1 loss loss correct 7 Uruguay 3rd place play-off 3 2 win win correct

Functional and ordinal dependencies Rough Set Theory Generalisations

Octopus example

2 1 4 3 6 5 7 B = {OPP}

Functional and ordinal dependencies Rough Set Theory Generalisations

Octopus example

1 1 1 0 2

Functional and ordinal dependencies Rough Set Theory Generalisations

Octopus example

1 1 1 0 2

co ... contains one ca ... contains all co 0 ca 0 ¬(ca 0) ¬(co 0)

Functional and ordinal dependencies Rough Set Theory Generalisations

Octopus example

1 1 1 0 2

co 0 ca 0 ¬(ca 0) ¬(co 0) co 1 ca 1 ¬(ca 1) ¬(co 1) co 2 ca 2 ¬(ca 2) ¬(co 2)

Functional and ordinal dependencies Rough Set Theory Generalisations

More general approximation spaces

• Classical: approximation space (U, Θ) with corresponding

approximation operators ⌊·⌋Θ and ⌈·⌉Θ,

• Generalisation: a pair (⌊·⌋ , ⌈·⌉) of an arbitrary kernel

(interior) and an arbitrary closure operator on U.

• Hence, the approximations

(⌊X⌋ , ⌈X⌉) with X ⊆ U are pairs in K × C, where K is the kernel system belonging to ⌊·⌋ and C is the closure system belonging to ⌈·⌉.

Functional and ordinal dependencies Rough Set Theory Generalisations

Lattice of Approximations

• Problem: In general, the pairs

(⌊X⌋ , ⌈X⌉) with X ⊆ U do not form lattice, and even if they do, this lattice needs not to be a sublattice of K × C.

• Solution: Let ΞK,C denote the complete sublattice of K × C

that is generated by {(⌊X⌋ , ⌈X⌉) | X ⊆ U}.

Functional and ordinal dependencies Rough Set Theory Generalisations

Some Notions

• Pairs (K, C) ∈ ΞK,C are called approximations. Thereby
• K is the positive region,
• C \ K is the boundary region, and
• U \ C is the negative region.
• an element x ∈ U is called robust if

{x} ∈ K and U \ {x} ∈ C.

• We put

R := {x ∈ U | x robust}.

Functional and ordinal dependencies Rough Set Theory Generalisations

Note that ...

• ... infimum and supremum in K × C (and hence also in ΞK,C)

are of the following form:

• t∈T

(Kt, Ct) =

t∈T

Kt,

t∈T

Ct

• ,
• t∈T

(Kt, Ct) =

t∈T

Kt

• ,
• t∈T

Ct

• .

Functional and ordinal dependencies Rough Set Theory Generalisations

Characterisation

Theorem

A pair (K, C) ∈ K × C is an approximation from ΞK,C iff the following two properties are satisfied:

• K ⊆ C,
• (C \ K) ∩ R = ∅.

Functional and ordinal dependencies Rough Set Theory Generalisations

Contextual representation

• Let K = (U, W , R) and C = (U, V , J) be formal contexts,

such that K = {U \ A | A ∈ Ext(U, W , R)}, C = Ext(U, V , J).

• For A ⊆ U it follows that

⌊A⌋ = A∁RR∁, ⌈A⌉ = AJJ.

Functional and ordinal dependencies Rough Set Theory Generalisations

Contextual representation

W R−1 U U J V ⊥ ∗ Let the kernel-closure pair (K, C) be given by two formal contexts K = (U, W , R) and C = (U, V , J).

Theorem (Ganter 2008)

The lattice ΞK,C of approximations is isomorphic to the concept lattice of the formal context on the left. Thereby, w ⊥ v :⇐ ⇒ wR ∪ vJ = U, x ∗ y :⇐ ⇒ x = y and there are w ∈ W , v ∈ V with wR = U \ {x} = vJ ⇐ ⇒ x = y or x not robust.

Functional and ordinal dependencies Rough Set Theory Generalisations

Example: the classical case

• Let (U, Θ) be an approximation space.
• Then K = (U, U, Θ) = C are the representing context, where

Θ = (U × U) \ Θ.

• R = {x ∈ R | {x} = [x]Θ}
• x ⊥ y

⇐ ⇒ xΘ ∪ yΘ = U ⇐ ⇒ xΘ ∩ yΘ = ∅ ⇐ ⇒ xΘy.

• Representing context:

U Θ U U Θ U Θ ∗

Functional and ordinal dependencies Rough Set Theory Generalisations

Questions

∗ R−1 ⊥ J U W U V

• Meaning: What do the (parts of the) concepts of the

representing context mean?

• Are there interesting special cases?
• Note: Since there is no reduced labelling for kernel systems

the concept lattices are hard to read.

Functional and ordinal dependencies Rough Set Theory Generalisations

Questions

∗ R−1 ⊥ J U W U V

K ∁ P N C

• Meaning: What do the (parts of the) concepts of the

representing context mean?

• Are there interesting special cases?
• Note: Since there is no reduced labelling for kernel systems

the concept lattices are hard to read.

Functional and ordinal dependencies Rough Set Theory Generalisations

Example

firm nonfirm smooth nonsmooth round nonround apple × × × grapefruit × × × kiwi × × × plum × × × toy cube × × × egg × × × tennis ball × × ×

Functional and ordinal dependencies Rough Set Theory Generalisations

Example

tennis ball grapefruit kiwi apple plum egg toy cube nonround smooth nonfirm round nonsmooth firm

Functional and ordinal dependencies Rough Set Theory Generalisations

Example

• Given the formal context (G, M, I) from above.
• What is a meaningful choice for the representing contexts?
• Let us choose

K := (G, M, I) =: C.

• Hence, the granules are the attribute extents.
• For A ⊆ G it holds that

⌊A⌋ =

• mI ⊆A

mI and ⌈A⌉∁ =

• mI ⊆A∁

mI =

• A∁

.

Functional and ordinal dependencies Rough Set Theory Generalisations

Example

• Given the formal context (G, M, I) from above.
• What is a meaningful choice for the representing contexts?
• Let us choose

K := (G, M, I) =: C.

• Hence, the granules are the attribute extents.
• For A ⊆ G it holds that

⌊A⌋ =

• mI ⊆A

mI and ⌈A⌉∁ =

• mI ⊆A∁

mI =

• A∁

.

Functional and ordinal dependencies Rough Set Theory Generalisations

Example

• Given the formal context (G, M, I) from above.
• What is a meaningful choice for the representing contexts?
• Let us choose

K := (G, M, I) =: C.

• Hence, the granules are the attribute extents.
• For A ⊆ G it holds that

⌊A⌋ =

• mI ⊆A

mI and ⌈A⌉∁ =

• mI ⊆A∁

mI =

• A∁

.

Functional and ordinal dependencies Rough Set Theory Generalisations

Example

• Suppose that (G, M, I) is a training data set for machine

learning, ...

• ... and that by A we have a partition of G into
• a set of positive examples, A itself, and
• a set of negative examples A∁ = G \ A.
• The granules mI being used to fill up A and A∁ form

positive and negative hypothesis, ...

• ... and can be read from the representing context.

Functional and ordinal dependencies Rough Set Theory Generalisations

Example

• Note that the hypothesis need to be read in a disjunctive way.

If one wants to have conjunctions, too, one has to replace (G, M, I) by (G, Ext(G, M, I), ∈).

• An object g is robust iff there is an attribute m ∈ M with

mI = {g} .

• Hence, robust elements are characterised by having a unique

feature.

Functional and ordinal dependencies Rough Set Theory Generalisations

Example

apple grapefruit kiwi plum toy cube egg tennis ball firm nonfirm smooth nonsmooth round nonround firm × × × × × × × × nonfirm × × × smooth × × × × nonsmooth × × × × × × round × × × × × × nonround × × × × apple × × × × × × × × × × grapefruit × × × × × × × × × × kiwi × × × × × × × × × × plum × × × × × × × × × × toy cube × × × × × × × × × × egg × × × × × × × × × × tennis ball × × × × × × × × × ×

Functional and ordinal dependencies Rough Set Theory Generalisations

Example

Functional and ordinal dependencies Rough Set Theory Generalisations

Example

firm nonfirm nonfirm firm nonsmooth smooth round nonround smooth nonsmooth nonround round

Functional and ordinal dependencies Rough Set Theory Generalisations

Example

firm nonfirm nonfirm firm nonsmooth smooth round nonround smooth nonsmooth nonround round egg toy cube plum apple kiwi tennis ball grapefruit tennis ball grapefruit kiwi apple plum egg toy cube

Functional and ordinal dependencies Rough Set Theory Generalisations

Example

firm nonfirm nonfirm firm nonsmooth smooth round nonround smooth nonsmooth nonround round egg toy cube plum apple kiwi tennis ball grapefruit tennis ball grapefruit kiwi apple plum egg toy cube tennis ball egg toy cube plum kiwi grapefruit apple

Functional and ordinal dependencies Rough Set Theory Generalisations

Example

firm nonfirm nonfirm firm nonsmooth smooth round nonround smooth nonsmooth nonround round tennis ball egg toy cube plum kiwi grapefruit apple

Functional and ordinal dependencies Rough Set Theory Generalisations

Example

firm nonfirm nonfirm firm nonsmooth smooth round nonround smooth nonsmooth nonround round tennis ball egg toy cube plum kiwi grapefruit apple

Functional and ordinal dependencies Rough Set Theory Generalisations

Example

firm nonfirm nonfirm firm nonsmooth smooth round nonround smooth nonsmooth nonround round tennis ball egg toy cube plum kiwi grapefruit apple

Functional and ordinal dependencies Rough Set Theory Generalisations

Non-symmetric indiscernibility

Using the formal context (G, Ext(G, M, I), / ∈) instead of (G, M, I) as representing contexts is equivalent to using (G, G, ), where g ≤ h :⇐ ⇒ gI ⊇ hI is the object quasi-order on G.

Functional and ordinal dependencies Rough Set Theory Generalisations

Questions?

? ? ? ?