formal concept analysis · 2017-11-28 · outline concept lattices elements of structure theory...

Outline Concept lattices Elements of structure theory Attribute logic Rough Sets

Formal Concept Analysis

Bernhard Ganter

Institut fur Algebra

TU Dresden

D-01062 Dresden

[email protected]

Description Logics Workshop Dresden 2008

Bernhard Ganter, Dresden University FCA@DL2008


Outline

1 Concept latticesData from a hospitalFormal definitionsMore examples

2 Elements of structure theoryContext arithmeticContext productOrder relation bond

3 Attribute logicThe base

4 Rough SetsTraditional and generalised



Data from a hospital

Interview data from a treatment of Anorexia nervosa

over

-sen

sitive

withdra

wn

self-c

onfiden

t

dutifu

l

cord

ial

diffi

cult

att

entive

easily

offen

ded

calm

appre

hen

sive

chatt

y

super

fici

al

sensitive

am

bitio

us

Myself × × × × × × × × × ×

My Ideal × × × × × × × ×

Father × × × × × × × × × × × ×

Mother × × × × × × × × × × ×

Sister × × × × × × × × × ×

Brother-in-law × × × × × × ×




A biplot of the interview data

uncomplicated

valiant

thick-skinned

complicated

difficult

apprehensive

over-sensitive

sympathetic

attentive

superficial

Brother-in-law

My Ideal

Myself

Sister

Father

Mother




The concept lattice of the interview data

Brother-in-lawMyIdeal

Mother

Sister

Father

Myself

ambitiouscordial

attentivedutiful

super ficial

self-confident

chatty

over-sensitivecalmsensitive

apprehensivedifficultwithdrawn

easily offended



Formal definitions

Unfolding data in a concept lattice

The basic procedure of Formal Concept Analysis:

Data is represented in a very basic data type, called a formal

context.

Each formal context is transformed into a mathematicalstructure called a concept lattice. The information containedin the formal context is preserved.

The concept lattice is the basis for further data analysis. Itmay be represented graphically to support communication, orit may be investigated with algebraic methods to unravel itsstructure.



More examples

An example about airlines . . .



More examples

. . . and its concept lattice



More examples

Formal contexts

A formal context (G , M, I ) consists of sets G , M and a binaryrelation I ⊆ G × M.

For A ⊆ G and B ⊆ M, define

A′ := {m ∈ M | g I m for all g ∈ A}

B ′ := {g ∈ G | g I m for all m ∈ B}.

The mappingsX → X ′′

are closure operators.



More examples

Formal concepts

(A, B) is a formal concept of (G , M, I ) iff

A ⊆ G , B ⊆ M, A′ = B, A = B ′.

A is the extent and B is the intent of (A, B).



More examples

Concept lattice

Formal concepts can be ordered by

(A1, B1) ≤ (A2, B2) : ⇐⇒ A1 ⊆ A2.

The set B(G , M, I ) of all formal concepts of (G , M, I ), with thisorder, is a complete lattice, called the concept lattice of(G , M, I ).



More examples

The basic theorem



More examples

Why complete lattices?

Complete lattices are closely related to closure operators andthereby to closure systems (also called Moore families).

Closure operators are closely related to propositional Horntheories. The formulas are implications, i.e., expressions of theform

A → B,

where A and B are sets of attributes.

Combining this, we may say that

Complete lattices are the algebraic structures corresponding topropositional Horn theories.



More examples

Why algebraic structures?

The mathematical approach is different from that of Logic.We ask different questions and thereby get other results.

We admire the theoretical and algorithmic strength of DL. Atthe same time, we can build on a rich and well-developedmathematical theory of lattices and ordered sets.

Structure theoretic aspects, such as decompositions,morphisms, representations and classifications can becombined with established visualisations.

We try to systematically build a mathematical theory, theparts of which are as simple and as general as possible.



More examples

The boolean lattice

What are the formal concepts of (G , G , 6=)?

They are precisely the pairs

(A, G \ A) A ⊆ G .

(G , G , 6=) therefore is the standard con-text of the power set lattice of G .

a b c d

a × × ×

b × × ×

c × × ×

d × × ×



Outline







Context arithmetic

Lattice constructions and context constructions

Many lattice properties and lattice constructions have nicecounterparts on the formal context side. For example,

the dual lattice corresponds to the transposed formal contextcomplete sublattices correspond to closed subrelationscongruence relations correspond to compatible subcontexts

tolerance relations correspond to block relationsdirect products correspond to context sumstensor products correspond to context products

Galois connections correspond to bonds

and so on.



Context arithmetic

Closed relations and block relations

A closed relation of (G , M, I ) is a subrelation C ⊆ I suchthat every formal concept of (G , M, C ) also is a concept of(G , M, I ).The closed relations correspond to the complete sublattices ofB(G , M, I ).

A block relation of (G , M, I ) is a superrelation R ⊇ I suchthat every extent of (G , M, R) is an extent of (G , M, I ) andevery intent of (G , M, R) is an intent of (G , M, I ).The block relations correspond to the complete tolerancerelations of B(G , M, I ).



Context arithmetic

The context sum

The concept lattice of the context sum

(G , M, I ) + (H, N, J)

is isomorphic to the direct product

B(G , M, I ) × B(H, N, J).

G

H J

I

M N



Context product

Context product and tensor product

The product of formal contexts (G , M, I ) and (H, N, J) is definedto be the context

(G , M, I ) × (H, N, J) := (G × H, M × N,∇),

where(g , h) ∇ (m, n) : ⇐⇒ g I m or h J n.

The concept lattice

B((G , M, I ) × (H, N, J))

of the product is the tensor product

B(G , M, I ) ⊗ B(H, N, J).



Context product

An example of a context product

×

× ×

××

×

×=

× × × ×× × × ×

× × × × ×× × × × ×

× × × ×× × × ×



Context product

Bonds and closed relations of the sum

R ⊆ G × N is a bond from (G , M, I ) to(H, N, J) if and only if

I ∪ R ∪ J ∪ (H × M)

is a closed relation of the context sum

(G , M, I ) + (H, N, J)

(assuming G ∩ H = ∅ = M ∩ N).

H J

G I R

M N



Order relation bond

The sublattice corresponding to the identity bond

The concept lattice of the formal contextmade from the identity bond is isomorphicto the order relation of B(G , M, I ), withthe componentwise order. G I

G I I

M M



Order relation bond

The order lattice of M3

0

a b c

1

0 ≤ 0

0 ≤ b 0 ≤ c

b ≤ b c ≤ c

b ≤ 1 c ≤ 1

1 ≤ 1

0 ≤ a

a ≤ a 0 ≤ 1

a ≤ 1



Order relation bond

The identity bond as a context product

Note that

G I

G I I

M M

=×

× (G , M, I )

In other words: The order relation of a lattice is isomorphic to thetensor product of the lattice with a three-element chain.



Order relation bond

Free distributive lattices

The tensor product of n three-element chains is the free

distributive lattice on n generators, FCD(n), also known asthe lattice of all monotone boolean functions in n variables.

The order relation of FCD(n) is isomorphic to the tensorproduct of FCD(n) and a three-element chain. Therefore

The free distributive lattice on n + 1 generators is isomorphicto the order relation of the free distributive lattice on n

generators.



Order relation bond

Free distributive lattices FCD(1), FCD(2), FCD(3)



Outline







The base

The stem base

Duquenne and Gigues have discovered that each implicationaltheory has a canonical base (with respect to the Armstrongderivation rules). We call this the stem base.

To find the stem base for a given formal context (G , M, I ) requiresthe notion of a pseudo-intent: P ⊆ M is a pseudo-intent iff

P is not an intent and

P contains the closure of every pseudo-intent that is properlycontained in P.

The stem base then is

{P → P” | P is a pseudo-intent}.



The base

Pairs of squares



The base

But did we consider all possible cases?

How can we decide if our selection of examples is complete?

A possible strategy is to prove that every implication that holds forthese examples, holds in general.

Compute the stem base of the context of examples, and

prove that these implications hold in general,

or find counterexamples and extend the example set.

This can nicely be organised in an algorithm, called attribute

exploration.



The base

Stem base of the example set

common edge → parallel, common vertex, common segment

common segment → parallel

parallel, common vertex, common segment → common edge

overlap, common vertex → parallel, common segment,common edge

overlap, parallel, common segment → common edge, commonvertex

overlap, parallel, common vertex → common segment,common edge

disjoint, common vertex → ⊥

disjoint, parallel, common segment → ⊥

disjoint, overlap → ⊥



The base

Two of the implications do not hold in general

common edge → parallel, common vertex, common segment

common segment → parallel

parallel, common vertex, common segment → common edge



overlap, parallel, common vertex → common segment,common edge

disjoint, common vertex → ⊥

disjoint, parallel, common segment → ⊥

disjoint, overlap → ⊥



The base

Conterexamples for the two implications


Counterexample:


Counterexample:



The base

A better choice of examples



Outline







Traditional and generalised

ZdzisÃlaw Pawlak’s rough sets

An approximation space (U,∼) is a set U with an equivalencerelation ∼ on U.

A subset A ⊆ U is rough if it is not a union of equivalence classes.




Approximations

A rough set A is represented (approximately) by a pair(R(A), R(A)), giving the

lower approximation R(A)

and the

upper approximation R(A)

of A.

In this way, a interval arithmetic for sets is established.




The lower approximation

R(A)




The upper approximation

R(A)




The algebra of approximating pairs

The algebra of approximating pairs (alsocalled rough sets) is well studied. It canbe obtained from the formal context shownhere. U 6∼

U 6∼ 6∼

U U

Note this formal context is the context product of theindiscernibility relation (u, u, 6∼) with the context of a threeelement chain. The lattice of rough sets therefore is the lattice ofall intervals of the power set of indiscernibility equivalence classes.It is a Stone lattice.




Operator pairs for generalised approximation

Meanwhile, rough sets have been generalised. Approximations areconsidered that do not rely on equivalence relations.Natural assumptions to keep are that

R is a kernel operator, and

R is a closure operator.

The family of approximating pairs

(R(A), R(A))

then generates (but not necessarily forms!) a sublattice of thedirect product of the lattice of kernels and the lattice of closures.




The approximation lattice

The lattice generated by such generalised approximations can becharacterised in general: With some technical cheating, the formalcontext is as given in the diagram, where

(G , M, I ) is a formal context whose extent-complements are the kernel sets,

the extents of (G , N, J) are the closures,and

m ⊥ n : ⇐⇒ m′ ∪ n′ = G .

G J

M I−1 ⊥

G N




A natural generalisation

From the general characterisation it can be derived that there is aunique generalisation having most of the nice properties of theclassical case. It occurs when the indiscernibility equivalencerelation is generalised to an indiscernibility quasi-order.

For an indiscernibility quasi-order the al-gebra of approximations gets a strikinglysimple form.

It is the context product of (G , G , 6≥) withthe context of the three element chain.

The concept lattice therefore is isomorphicto the order relation of the lattice of quasi-order ideals.

G 6≥

G 6≥ 6≥

G G




A non-symmetric indiscernibility?

We have arrived at a seemingly counter-intuitive conclusion: Themathematically most promising generalisation of the rough setapproach is obtained by replacing the indiscernibility relation by anot necessarily symmetric quasi-order.Does non-symmetric indiscernibility make any sense?

Yes, it does! The intuition that an attribute-based indiscernibilityshould be symmetric tacitely assumes that attributes are scalednominally, that is, with mutually exclusive attribute values.The generalised version is natural when subsumption betweenattribute values is permitted.




Thank you for your attention!


formal concept analysis · 2017-11-28 · outline concept lattices elements of structure theory...

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