-The field of research is the creating of fuzzy concept lattices based on fuzzy data “objects-properties”. Our contribution is the account of existential relations on the set of observed and/or measured properties, i.e. “properties existence constraints”. Two most well-known approaches to creating of fuzzy concepts lattices are considered: the one-sided threshold and fuzzy closure methods. It is shown that for the more popular one-sided threshold method, potential violations of properties existence constraints in the concept lattice are countered by the rational threshold cut method, previously developed for extracting crisp formal concepts from fuzzy initial data. However, this way is fundamentally unacceptable for the fuzzy closure method. For this case, the idea of special preliminary processing of the initial data is put forward - the “normalization” of the fuzzy set of properties for each object in the training sample. The practical importance of the study is to increase an adequacy of Fuzzy Formal Concept Analysis.
data
mining, formal concepts analysis, fuzzy formal concept analysis
I.
INTRODUCTION
One of the most powerful methods of data mining for the
last two decades is the Formal Concepts Analysis (FCA)
[
In FCA intent and extent are associated with the relation between the set of objects
and set of properties , : × → { ,
}. The tuple ( , , ) is usually set in the form of the object-properties reflection table and is referred to as a formal context (FC). FC induces Galois operators “↑” and “↓”. Formal concept is defined by the bicluster ( , ), which is formed ⊆ (extent) and ⊆ (intent). This bicluster ( , ) satisfies ↑= and ↓= , where { ∈ |∀ ↑= { ∈ |∀ ∈ : ( , ) = } and ↓ = ∈ : ( , ) = }. The set consists of all formal contexts, that is extracted from FC, and is ordered by extent (or intent) inclusion. This set forms the complete lattice and is called the lattice of formal concepts.
On the one hand fuzzy FCA (FFCA) is adaptation to the
FCA elasticity of “human concepts” proved by psychologists
in the sense that the question of applicability of the concept
to the object is the question of degree and not the question of
“yes”/”no”. People work productively in the conditions of
such granulation of opinions [
Sergeу Smirnov Institute for the Control of Complex Systems
Samara Federal Scientific Center of RAS
×
of “fuzzy FCA” is related to the choice of the scale = [
development of
FCA is related to the understanding of its hypothetical-deductive
PROPERTIES EXISTENCE CONSTRAINTS
According to [
content and two (and only two) existential relations for this set: conditionality С: × → { , } , when it is established beforehand that having the property , every object necessarily has the property (although the converse
may be incorrect), i.e. С( , ) ↔ ∀ ∈ : ∈ { } ↑→ ∈ { } ↑ reflexive, not symmetric Conditionality transitive; incompatibility : × → { , }, when it is predetermined that having the property , every object obviously does not have property , and on the contrary, i.e. ( , ) ↔ ∀ ∈ : ∈ { } ↑→ ∉ { } ↑. The relation antireflexive, symmetric and not transitive, but characterized by the so-called “transitivity relative to conditionality”, which means ∀ , , ∈ : С( , ) ∧ ( , ) → ( , ).
It should be noted that the relation of conditionality generates a reflexive, symmetrical and transitive relation of inter-conditionality in a set of properties: ( , ) ↔ С( , ) ∧ С( , ).
Conditionality and incompatibility relations limit set or, in other words, co-existence of properties for training selection objects.
According to the subject's a priori
hypotheses any object ∈ can only have a “normal”
Copyright © 2020 for this paper by its authors.
subset of the set of measurable properties
[
The “one-sidedness” of the method is that the fuzzy FC is interpreted asymmetrically as an aggregate of fuzzy sets over the universe , each of which describes one of the FC objects. In other words, each ∈ in the FC is represented as a fuzzy set { ( , 1)/ 1, ( , 2)/ 2, . . . , ( , )/ }, where
= | |, ( , ) – the degree of verity of the
assertion “the property ∈ is inherent in the object ”.
The usual (crisp) formal concept is defined in this case using
the threshold value
∈ [
}
At the second step, the methodological complex of the classical FCA is used to derive crisp concepts from ( , , ( )) and construct their lattice.
The content of the third step is the transformation of each received crisp concept ( , ) into a fuzzy one ( , ). The intent is a fuzzy set over the universe . The estimate of the truth ( ) of an object ∈ belong to determined by the degree to which it has all the properties in the content , or rather, by the assessment of the joint truth (intersection) of fuzzy judgments “to an object ∈ inherent in the property
∈ ” for all properties of . It is usually proposed to evaluate this degree of membership by a is is min-conjunction: ( ) = { min ∈ ( , ) , 0,
if ; in other cases.
}
It is easy to verify that between such fuzzy concepts the same partial order
will be preserved as between clear concepts obtained at the intermediate step of the one-way threshold method. If we additionally require ∀ ∈ : = ∅ → ( ) = 1, we see that the fuzzy concepts constructed in the described way — the biclusters ( , ) with a clear • • • content and fuzzy extent — form a complete lattice of fuzzy concepts.
consideration, lattice”),
The method forms fuzzy concepts with fuzzy extents and fuzzy intents. An algebraic structure is introduced into called a full residual lattice (“division = 〈 ,∧,∨,⊗, → ,0,1〉 such that: 〈 ,∧,∨ ,0,1〉 is a complete lattice with the smallest element 0 and the largest element 1; 〈 ,⊗ ,1〉 is a commutative monoid; fuzzy conjunction
⊗ and fuzzy implication → satisfy the conjugacy condition ⊗ ≤ ↔ ≤ → .
The authors of [
Evaluating this formalization, following [
Continuing the review of the fuzzy closure method, we For fuzzy sets ↓∈ are defined as
∈ и ∈ , fuzzy sets ↑∈ и ↑ ( ) = ⋀ ∈ ( ( ) → ( , )), ↓ ( ) = ⋀ ∈ ( ( ) → ( , )).
↑ ( ) indicates the degree of verity that property characterizes all objects in a fuzzy set . Similarly, ↓ ( ) indicates the verity that all properties in a fuzzy set are inherent in object .
A pair( , ) ∈ × is a fuzzy formal concept if ↑ =
and ↓= . The set of all fuzzy formal concepts extracted from fuzzy FC is partially ordered by the inclusion of fuzzy intents (or, equally, fuzzy extents) and forms a complete lattice of fuzzy concepts.
According to [
IV.
ACCEPT THE PROPERTIES EXISTENCE CONSTRAINTS
It is obvious (see section 2) that for crisp formal concepts
extracted from this fuzzy FC the “natural” criterion for
accounting for PEC is the normality of sets of properties that
determine the content of the constructed concepts [
more common condition for accounting for PEC (the “fundamental” criterion): object generated according to any fuzzy concept of a given fuzzy FC must be characterized by a normal set of properties. A. Situation when using the one-side threshold method
It is easy to see that to account for PEC in the one-side
threshold method, when α-approximation of fuzzy relation
instead of the standard α-sections use the method of rational
α-section [
B. Normalization of formal context for use fuzzy closure method
The method in which the construction of fuzzy concepts uses the fuzzy closure operator does not use threshold values and the contents of the output concepts are fuzzy subsets – defined elements of the set . So due to the fact that the PEC requirement is formulated in the language of ordinary sets (objects in the domain of interest can only have normal subsets of measurable properties ), there is a need to find a connection between the power set elements 2 and set .
It is known that in the case = [
Now the requirement to carry out the “fundamental” criteria for PEC accounting can be applied to the right side (1): PEC will be carried out if all sets ({ } ↑) are normal. Note that in real FC for each ∈ the number of different “summands” in (1) is finite.
Finally, to account to the PEC we can offer this effective method of preprocessing of fuzzy FC ( , , ) for constructing the lattice of fuzzy formal concepts: according to the available PEC all normal subsets of the set of measured properties are detected; relational is being normalized: for each ∈ the right part (1) is taken as the initial fuzzy set of properties except for “summands” where the set of properties ({ } ↑) is not normal.
When constructing a formal algorithm that solves the problem of PEC accounting for the case of fuzzy closure, we should take into account some features of the crisp sets that are part of the decomposition (1) and features of the structure of the PEC themselves that allow optimizing the filtering process of normal sets, such as: properties, then ∀ 2 < 1 the set ({ } ↑) 2 will be discarded too. If ({ } ↑) 1 was discarded as a set that does not contain at least one property from some group of inter-conditionality properties then ∀ 2 > 1 the set ({ } ↑) 2 will be discarded too.
V.
Realized research allow to specify ways to combine the deductive achievements of FFCA in the construction of lattices of formal concepts with understanding the role of the second - a priori - aspect of the common hypotheticaldeductive nature of FCA. Specifically, we propose ways to account for existential relations on a set of observed and/or measured properties which provide derivation of correct fuzzy concepts.
A promising task is to explore the influence of properties existence constraints on qualitative and quantitative characteristics fuzzy concept lattices depending on the parameters of these constraints and parameters of initial formal concepts.
Practical value of research results is to improve the
adequacy of application of FCA. In particular it is used in
the construction of fuzzy formal concepts the important role
of which is highly appreciated in different applications (see
for example [