Extended methodology for deriving formal concepts

Two methodologies for formal concepts derivation are considered: the classical one, which focuses on the posterior analysis of the object’s properties of the studied knowledge domain, and non-classical, the cornerstone of which is the a priori formation of the set of measured object’s properties and the determination of existential relations on this set. In the article, firstly, a position is fixed in the technological chain of the target transformation of the source data, where the difference between considered methodologies shows itself. Secondly, the commonality of these two approaches is established in the aspect of the unity of their hypothetical-deductive basis. In this case, the cognitive activity of the subject is expressed first in a priori and then in a posteriori conceptual scaling of the measured properties. The work demonstrates the need for the joint use of the considered methodologies at processing incomplete and inconsistent empirical data about studied knowledge domain. The intermediate consolidation of these methodologies is possible only on the basis of multi-valued logic.


Introduction
As known, the standard protocol for the presentation of empirical information in various problems of data analysis is the table "objects-properties" [1][2][3]: (G * , M, V, I), (1) where G * = {gi}i = 1,…, r, r = G *   1 -is the set of observed objects: G The derivation of formal concepts from (1) is a variation of the clustering problem (more precisely, biclusterization [4,5], when clusters -formal concepts -retain an object-attribute description of the data). This task involves a preliminary transformation (1) into a single-valued formal context (G * , M, I), (2) where I  G * M -the binary relation described by the incidence matrix "object-properties", each element of which is the truth estimate of basic semantic proposition (BSP) about the studied KD: bij = «оbject gi  G * has the property mj  M». Therefore, the formal concepts derived from (2)biclusters -will retain information about objects with a similar composition of properties and will be in relation of a partial order by nesting of compositions. The fundamental role of such data analysis is 2 that its results correspond to the definition of a concept and the generalization relation on the set of concepts in philosophy and classical logic [6,7].
In [8], the difference in the methodologies for deriving formal concepts is determined by the orientation either to objects or to properties. This affects the "activation patterns" and "cognitive resource" of the subjects of the study. However, an analysis of [8] and the study that inspired this paper [9] leads to the conclusion that the existing difference in these methodologies appears exclusively at the stage of transformation (1) to (2).
The purpose of this article is to establish a fundamental commonality of the two existing methodologies for deriving formal concepts and to reason the need for their joint use at processing incomplete and inconsistent empirical data about studied KD.

Key points of competing approaches
From the standpoint of measurement theory [10,11], transformation of (1) to (2) should not cause problems.
Indeed, each real measurement procedure (measuring tool, instrument) has a sensitivity threshold, a limited dynamic range. Therefore, for each value domain of the measured properties, None  Vj is valid, where the constant None is the result of the measurement, which can be described as "no information" (but not in the sense of "measurement was not performed"), "outside the dynamic range", or following [12], "the object and the measurement procedure are not semantically combined". Then naturally we have Nevertheless, each of the considered methodologies in its own way "furnishes and extends" the transition from (1) to (2).

A posterior analysis of the objects properties of the training sample
The basis of both methodologies is the use of very popular formal concept analysis (FCA) proposed by R. Wille [13]. FCA "in the narrow sense" derives formal concepts from context (2) using the following notation and models: • Galois operators ,  (general notation "'") for the context (G * , M, I): • -(X) = X ' = {mjmj  M, gi  X: I(gi, mj) = True} -common objects properties making up •for a set of objects X, the set of their common attributes X' describes the similarity of objects from the set X, and the closed set X' is a cluster of similar objects; At the same time, undoubtedly, FCA "in the narrow sense" -an integral and classical methodology for deriving formal concepts (see, for example, [14][15][16][17]) which oriented in the terminology of [8] on "objects". Within the framework of the proposed construct of differences between the considered methodologies, this expresses in the following: • G * objects are independent of the subject in the sense that there are no assumptions regarding their appearance in the problem, specifically, in (1); • a posteriori subjective (that is, determined by some goals of the knowing subject) granulation of information is allowed in (1) using conceptual scales [14,18,19] (its elementary 3 manifestation is the expansion of the measured properties set M). Only after this the transition to (2) is made based on (3). If we leave aside the issue of fuzzy conceptual scales [19,20], then said peculiarities completely characterize FCA as one of the most powerful methods of data mining.

A priori formation of a system of measured properties
In a "property-oriented" methodology, the subject a priori forms a system of measured properties (SMP), determining not only the composition of the set M, but also the existential relations (or the properties existence constraints [9,21]) on this set: Relation E is antireflexive, symmetrical, and non-transitive, but is characterized by the so-called transitivity concerning conditionality, i.e.
In such conditions, the object of the training sample can have only a normal subset of the measured properties set [8,9]. A subset of the measured properties of Y  M is normal if and only if it is closed and compatible: • Y is closed, if it contains all the properties conditioned by any element of Y, i.e. mj  Y: Within the framework of the used construct of differences between methodologies, a priori formation of the system of measured properties during transformation of (1) in (2) supplements (3) by checking the set of inherent properties of the object for normality. If this check yields a negative result, either the results of measuring the object properties are qualified as unreliable, and therefore corresponding lines in (1) and (2) are discarded, or the need to reconsider the notions of KD, which were a priori expressed by the subject in the description of the SMP, is recognized.
It should be noted that a "property-oriented" methodology does not exclude a posteriori conceptual scaling of properties. But such scaling changes the system of measured properties both in terms of composition and in terms of existential relationships between properties [18]. These changes should be reflected in the modification of the SMP as a tuple (M, E, C), which will be used to check the set of inherent properties of the object for normality.

The unity of the hypothetical-deductive basis of methodologies
About a unified deductive basis of the considered methodologies for the derivation of formal concepts -FCA "in the narrow sense" -was said above (q.v. subsection 2.1). The nature of hypotheses, which  are produced as a result of the cognitive efforts of the subject and are tested in solving problems of deriving formal concepts, is unified as well.

Cognitive activity of the subject
The decision to measure some property mx in the objects of the studied KD is a rational consequence of the subjective assumption about the presence in the KD of objects possessing this property [22,23]. In conceptual terms, this means putting forward a hypothesis that the considered KD is characterized by the formal concept It is these 2 M of various hypothetical formal concepts that are studied in the classical methodology for deriving formal concepts from the context (2).
A priori formation of SMP, which distinguishes the competing methodology, should also be considered as hypothesizing about the conceptual structure of the studied KD. First, the formation of hypothetical concepts occurs according to the above rule with the usual replenishment of SMP with measured properties. Secondly, the introduction of existential relations on the set of measured properties limits the permissible conceptual description of the KD.
For all that, in the aspect of a priori formation of hypothetical concepts by the subject, the existential relations of the measured properties are not a cause, but a consequence.
Indeed, in classical logic, there are two possibilities for the formation of new concepts on the basis of existing ones [6,7]: • the division of an existing concept with the formation of new concepts replacing it is associated with the enumeration of all disjoint parts of its extent on a single basis -the attribute (the measured property), which is changing within the scope of the divisible extent (instead of division, the opposite action can be considered -a generalization [7]); • the restriction of the concept means the introduction of a new concept, the extent of which is part of the extent of the original concept, and the intent is distinguished by an additional attribute (a new measured property), inherent only in that part of the objects conceivable in the original concept that forms the extent of the new concept introduced; Let property mx be the basis of division of a hypothetical concept. Note that, in reality, the fact of measuring the property mx corresponds to set of hypothetical formal concepts that include the property mx in its intent. However, for only one of them, mx will be a distinctive (and not inherited according to the generalization relation on many concepts) property [14,22]. We are talking about the division of this particular concept. Therefore, it is assumed that mx is the basis of division of this particular concept.
«Modification» of mx within the confines of divisible extent {…, mx} (here «…» -replaces set of inherited properties from more general hypothetical formal concepts) cannot be interpreted otherwise than as a transition in the domain Vx from one subset of property values to another, provided that these subsets do not intersect. Easy to see that a subjectively constructed disjunctive coverage of a domain Vx: In the considered example the "restricting" property mx cannot be interpreted otherwise than as an indication of the selection principle of the extent {mk} part, when the choice is based on different values of the property mk [23]. Therefore, for the domain of the property mk where Vk(mx) is the domain part of the measured property mk, the values from which characterize the objects that make up the extent part of the original concept, that was enumerated at its restriction. This means that the restriction of the concept ({mk}, {mk}) is realized by the simplest ordinal conceptual scaling [18] of property mk, when its domain Vk is subjectively covered by only two sets: the domain Vk itself and its strict part Vk(mx)  Vk.
In general, the property mx  M is introduced in the SMP to restrict some existing hypothetical concept (Mq, Mq), Mq  M, Mq = {mk}k = 1,…, q, q  1. As explained in [24] this is realized by the simplest ordinal scaling of the nonstrict part of the properties from the intent of the restricted concept: Mr  Mq, and then the presence of the property mx in the object will determine the presence of the Mq properties in it.
Thus, both considered methodologies imply a priori cognitive activity of the subject associated with the formation of hypothetical concepts describing the unknown KD. In the "object-oriented" methodology, the subject predetermines the greatest variability of the potential results of formal concepts derivation. The "property-oriented" methodology is associated with the subjective narrowing of this variability. In this case, the researcher carries out the detailed elaboration of primary a priori hypotheses by means of two (and only two) types of a priori conceptual scaling of the properties to be measured -nominal and ordinal.

The structure of the system of measured properties
We have made several attempts to construct an adequate SMP that provides an effective solution to the problem of transition from (1) to (2) (see, for example, [23,25,26]). Finally, in [27], it was proposed to transform the "natural" description of the SMP in the form of a set of measured properties with two binary relations specified on it (which can be easily represented as a graph in which vertices represent properties, edges represent incompatibility, and arcs represent conditionality of properties) into a set of intersecting substructures, homogeneous in the form of existential conjugation of member-properties. It was shown in [27] that this model allows one to directly correlate the internal arrangement of such substructures with the pragmatically important concept of the normal subset of measured properties.

Combination of methodologies
The combination of the considered methodologies is understood as the necessity and the sufficiency of their joint use for the derivation of formal concepts. The analysis allows us to argue that at transforming (1) into (2) this combination is reduced to taking into account the results of both a priori and a posteriori conceptual scaling of the measured properties.

Genesis and processing of the "soft" formal context
The need to combine the methodological orientations "on objects" and "on properties" arises if the empirical information (1) is inconsistent, which results in inapplicability of formula (3). The source of contradictions and incompleteness of data in (1) can be a posteriori fuzzy granulation of empirical information [19,20] and / or taking into account the realities of accumulation of empirical data when (1) takes the form of a generalized table "object-properties" (GTOP) [25,26].
Then (3) is forcedly replaced by data (1) transformation on the basis of a suitable multi-valued logic, for example, fuzzy or vectorial [28], which makes the "object-property" correspondence in (2) accordingly fuzzy or non-strict. The methods developed by the FCA for the derivation of formal concepts are not applicable to such "soft" contexts. To analyze such formal contexts, FCA extensions are used, based on various interpretations of the target task and the corresponding preprocessing of the source data, and for example: • -section method of a non-strict formal context for deriving crisp formal concepts in a nonstrict FCA [26]; • -section method of a fuzzy formal context for deriving fuzzy formal concepts [29,30], when the formal context is interpreted as a collection of fuzzy properties sets, each of which describes a separate object of the training sample (since one-sided preference is given to objects, the second name of the method is known as "asymmetric threshold scheme"); • an approach which is using the closure operator of fuzzy set [31] uses a fuzzy formal context "as is" (i.e., without approximation and preference to objects or properties). This theoretically and computationally complex method even in the opinion of its authors induces only theoretical interest so far as it generates a huge amount of fuzzy formal concepts even for small-sized sparse fuzzy contexts. Thus, modern effective methods for deriving formal concepts from "soft" formal contexts are based on preliminary -approximation of the "soft" correspondence "object-properties" when setting the threshold of confidence to the source data. And it is precisely this task that determines the situation when the combination of both considered methodologies may be required.

Defuzzification and normalization of formal context
The defuzzification of the "soft" formal context, carried out on the basis of the standard procedure of -section of the correspondence "objects-properties", generally gives the context (2), where some objects of the training sample have properties set which are not normal [25,26].
The question of the need to normalize the obtained unambiguous context (as well as the choice of the confidence threshold in the initial data during its defuzzification) refers to a posteriori cognitive acts of the subject in the derivation of formal concepts. The violation of the normality of properties sets of at least one object in the obtained context (and, consequently, the contradiction of empirical data with a priori hypotheses), can be considered by the subject as a consequence of the incompleteness and inconsistency of the given information and demand to normalize the formal context, giving up the least reliable data. It is in this case that it is necessary to involve models and methods of the concept derivation methodology which orients "on properties".
The question of how to perform the correct defuzzification and normalization of the formal context, taking into account the foresaid, can be solved in different ways. In [25,26], an approach was critically analyzed, which provides an accurate determination of the area of confidence threshold values to the initial data. These values provide the required result of -approximation of the "soft" correspondence of "object-properties". Instead, a productive heuristics was proposed, which in general terms can be stated as follows: • the subject is free to choose a threshold (the choice of the -threshold is arbitrary); • the corresponding to threshold  and, in general, the inadmissible composition of the properties of each object of the desired single-valued FC should be successively shortened by cutting off at each step a properties set that violates the constraints of its existence ("normality"); • the cut-off mechanism consists in local tightening of the confidence threshold for properties to be "cut off" from an object within the SMP substructures that are homogeneous in the form of existential conjugation of member-properties [27].

Conclusion
The analysis clearly shows that the methodologies for deriving formal concepts are immersed in the common hypothetical-deductive theory of formal concept analysis, an essential element of which is the reflection of the cognitive activity of the subject at the formation stage of a single-valued formal context representing empirical information about the studied knowledge domain. This activity is expressed in a priori and a posteriori granulation respectively of ideal and real informational entities, which is carried out using subjectively constructed conceptual scales. Figure 2 shows the matching steps of data analysis in the derivation of formal concepts: Figure 2. Formation of a single-valued formal context in extended methodology for deriving formal concepts.
• 1, 4, 9 -the perception by a subject (at a certain stage of problem solving) of the result of any of subsequent steps of the analysis (for example, the 4th step can be the perception of the result of the 12th -the set of formal concepts (SFC); to exclude the pile-up of connections, these transitions are depicted only within the selected cognitive acts); • 2, 5 -formation/editing of SMP and GTOP, respectively, including conceptual scaling of measured properties (inverse to the subject impacts are associated with informing about the impossibility of performing the selected action); 8 measurement procedures and the reliability of the measurement series in GTOP and the standard alpha-section of the resulting "soft" formal context with an extremely soft confidence threshold; • 8 -identification of the IFC with the formal context, which is used further for the derivation of formal concepts -the operating formal context (OFC); • 10 -setting a subjective confidence threshold in the BSP and updating the OFC, by converting the IFC into the OFC with marking all BSP, the truth estimates of which does not reach the set threshold, as false and the rest -as true; • 11 -transformation of the OFC, during which the contradiction of empirical data with the a priori hypotheses about the desired FC is eliminated, namely, the properties sets of the measurement objects are normalized by setting local confidence thresholds that ensure the assignment of the least reliable BSP, previously considered true, to false (inverse to the subject impact is associated with informing about the impossibility of performing this action); • 12 -extraction of a set of formal concepts SFC from the OFC. Since the functionality of the most famous data analysis systems based on FCA (Fuzzy FCA-Wizard, Python FCA Tool, Galicia, ToscanaJ, ConExp [29,32]) is not complete (in light of the above), it is necessary to develop the corresponding algorithmic base and computer resources. Our efforts at that point are focused on the development of the tool of ontological data analysis OntoWorker [33].