=Paper=
{{Paper
|id=Vol-1638/Paper95
|storemode=property
|title=Data formation and processing in formal concept analysis: subjective aspects
|pdfUrl=https://ceur-ws.org/Vol-1638/Paper95.pdf
|volume=Vol-1638
|authors=Dmitriy E. Samoilov,Sergey V. Smirnov
}}
==Data formation and processing in formal concept analysis: subjective aspects ==
https://ceur-ws.org/Vol-1638/Paper95.pdf
Data Science
DATA FORMATION AND PROCESSING IN FORMAL
CONCEPT ANALYSIS: SUBJECTIVE ASPECTS
D.E. Samoilov1,2, S.V. Smirnov1
1
Institute for the Control of Complex Systems, Russian Academy of Science,
2
Samara National Research University, Samara, Russia
Abstract. The paper gives a brief overview of the subjective aspects of data
formation and processing in Formal Concept Analysis. It is shown that the fun-
damental cognitive scaling procedure that allows a different interpretation, in-
troduces new information into the analysis and the analysis is not correct in the
general case without paying the proper attention to this information. The rela-
tionship between the objects properties that arises from the use of various types
of scales and that need to be noted, is considered.
Keywords: Formal Concept Analysis, scaling, properties existence constraints
Citation: Samoilov DE, Smirnov SV. Data formation and processing in Formal
Concept Analysis: subjective aspects. CEUR Workshop Proceedings, 2016;
1638: 806-812. DOI: 10.18287/1613-0073-2016-1638-806-812
Introduction
For more than three decades the Formal Concept Analysis (FCA) is being developed
successfully at the intersection of applied mathematics and computer science [1-7].
FCA has made a significant contribution and will continue to stimulate the developing
of data mining, data representation and other parts of computer science due to the
classical (Aristotelian) approach to the concept as the fundamental mental entity de-
fined by the volume and content as well as to the basis of algebraic lattices theory.
FCA cognitive character appears in the account of the researcher's different axiologi-
cal systems. The outline of FCA subjective aspects and its application in data analysis
is the scope of this article. But the main focus is concentrated on primary data scaling.
We believe that the genesis of so-called “properties’ existence constraints” [8, 9], with-
out which the FCA problems solution is incorrect [10], is often determined by scaling
procedures [11, 12]. The occurrence of various restrictions of properties’ existence
constraints as a result of subjective selection of scales type is investigated.
Information Technology and Nanotechnology (ITNT-2016) 806
Data Science Samoilov DE, Smirnov SV. Data formation…
1 Subjective aspects of classical FCA
1.1 Basic definitions and models
FCA has to deal with mass encountered practical applications that require the object-
attributive data analysis. Classical FCA is focused on processing of binary data as a
set of truth values of basic semantic proposition bgm = “g object has m property”. It
uses the following symbols and models:
• K = (G*, M, I) – formal context where G* is a set of investigated knowledge do-
main’s objects (KD) comes in the researcher's view (i.e. the “learning sample” of
KD objects), M – set of objects’ measured properties, I – relation between the ob-
jects and their properties - a set of assessments ||bgm|| ∈ {True, False};
• Galois operators ϕ, ω (a common notation “ ' ”) for the context K:
• ϕ(X) = X ' = {mm ∈ M, ∀g ∈ X ((g, m) ∈ I)} - common objects' properties
X ⊆ G *;
• ω(Y) = Y ' = {gg ∈ G*, ∀m ∈ Y ((g, m) ∈ I)} – objects that have all the
properties of the Y ⊆ M;
• for a set of objects X, the set of their common properties X ' is the description
of the objects’ similarity from the set X, and the closed set X '' is a cluster of similar
objects;
• (X, Y) – formal concept where X ⊆ G* is extension, Y ⊆ M is intention, X = Y ',
Y = X ';
• В(K) –set of all formal concepts of K;
• (В(K), ≤) – concept's lattice where (X1, Y1) ≤ (X2, Y2), if X1 ⊆ X2 (or Y1 ⊇ Y2).
The subjective aspect of K context formation is manifested in cognitive asymmetry of
“objects” and “properties”: formally the objects G* are independent from the re-
searcher's KD, while the properties of M are the result of KD hypotheses production
maid by the subject and it is based on his current target system, his a priori knowledge
and his resource capabilities.
1.2 Formal concept's set reduction
The FCA results presentation for subsequent analysis may be difficult because of the
large number of detectable concepts. Two main ways of relevant formal concepts’
selection are developed for the reduction of the set В(K).
The support for multiple properties Y ⊆ M for a given context K is
supp(Y) = Y '/G*.
The set Y ⊆ M is called a frequent set of properties, if supp(Y) ≥ minsupp ∈ [0, 1].
If there are frequent concepts saved in the lattice only (their content is frequent sets of
properties), the lattice will be reduced to the so-called “iceberg concepts” [13, 14].
Information Technology and Nanotechnology (ITNT-2016) 807
Data Science Samoilov DE, Smirnov SV. Data formation…
The more granular approach is based on the identification in the В(K) the concepts
that are resistant to the support volume’s changes in the objects’ learning sample
[15, 16].
The stability index of the formal concept (X, Y) is determined by
σ(X, Y) = {Z ⊆ X | Z ' = Y}/2X.
The concept (X, Y) is considered to be stable when the σ(X, Y) ≥ σmin ∈ [0, 1], and the
lattice reducing means that the most stable formal concepts will be stored there only.
It is obvious that the subjective nature of the thresholds choice for the properties' vari-
ety support as well as for the concepts’ sustainability index is not associated with the
involvement of the additional information (knowledge) about the KD in the analysis.
1.3 Properties subsets’ implications
The implication on formal context’s properties subsets K is a dependence A → B,
A, B ⊆ M, provided that all objects with properties A, also have all the properties of B,
i.e. A ' ⊆ B '. Partial implication in the context K is distinguished by the lack of support
in the objects’ learning sample [1].
Entered into the FCA the partial implication’s confidence index makes it possible to
extend the set of relevant empirical regularities with condition of subjectively choos-
ing of the reliability threshold. But it doesn’t accompanied by using of additional data
about KD.
2 Subjective aspects of conceptual scaling
The basic form of empirical information about KD is an “object-properties” table,
which is treated in the FCA as a multi-valued context (G*, M, V, I). Here G* and M
have been already defined, V – is the property values’ set, and I - is the ternary rela-
tion between G*, M and V (I ⊆ G*×M×V) defined for all pairs from G*×M.
To reducing the many-valued context to a binary form, the conceptual scaling as a
fundamental cognitive procedure is applied [1, 11]. It informally means the subjective
construction of value domain’s “coverage” of each property of multi-valued context,
i.e., the formation of new KD objects’ distinctive properties that are measured in sub-
jectively formed scales.
The property scale m ∈ M is a binary context Sm = (Gm, Mm, Im). Here Gm is the scale
values, Mm – is the KD objects’ new properties that are entered by the scale, Im – is a
relation between the scale values and the new properties introducing the specific of
the KD subjective perception by its researcher.
We will show that subject enters qualitatively new information about KD into the
analysis while it implements a conceptual scaling. FCA practical application becomes
problematic without taking this information into account (these problems were dis-
cussed in [10]).
Information Technology and Nanotechnology (ITNT-2016) 808
Data Science Samoilov DE, Smirnov SV. Data formation…
2.1 Using of the nominal scale
The most common scaling reception is the use of the nominal scale [11, 17]. Table 1
gives an example of such scale.
Table 1. Scale of men’s growth
Growth, cm Low Average High
< 168 ×
168-175 ×
> 175 ×
The covering of the original values’ domain of scaling property is strictly disjunctive in
this case; items’ fuzzy scale can be a model of a more complex approach to this prob-
lem.
It is obvious that E pair incompatibility [8, 10] (for example, E(Low, High)) is inher-
ent by introduced nominal scale KD objects’ properties in either embodiment. It is
new essential information about KD that the researcher adds to the existing data in the
original multi-valued context.
2.2 Other types of scales
Specific areas of conceptual explorations - such as sociology [18] or machine vision
[19] - typically characterized by self-built complex types of scales.
We will show the effects of using of other types of scales on the examples from [20].
These examples do not embrace all of the possible methods of expression of the re-
searcher’s subjective perception of KD.
The ordinal scale should be used to preserve the values ordering in the domain of
multi-valued property.
So, the domain of multi-valued properties named “Financial position” (FP) can be de-
scribed by the following expressions (from “difficult” to “safe”) [20]:
1. not enough money even for food;
2. enough money for food, but not enough to buy clothes and shoes;
3. have enough money for clothes and shoes, but can’t afford the purchase of house-
hold appliances;
4. enough money to buy household appliances, but not enough to buy a new car;
5. enough money for everything, except the expensive acquisitions such as an apart-
ment, a house;
6. do not feel financial difficulties, could buy an apartment, a house, etc., if necessary.
The researchers will have table 2 as the most natural scale for this multi-valued prop-
erty.
Information Technology and Nanotechnology (ITNT-2016) 809
Data Science Samoilov DE, Smirnov SV. Data formation…
Table 2. Financial position scale
FP1 FP2 FP3 FP4 FP5 FP6
1 ×
2 × ×
3 × × ×
4 × × × ×
5 × × × × ×
6 × × × × × ×
This scaling sets the binary conditionality relation between newly introduced proper-
ties: С [8, 10]: i < k ↔ С(FPk, FPi).
Nowadays, the scales with division and ordering become very popular. They are de-
scribed in [20] to a closed question like “Do you feel safe?” (S). The response options
are:
1. definitely yes;
2. rather yes;
3. rather no;
4. definitely no.
The subjective understanding of this domain of values can be expressed by double
ordering scale (table 3).
Table 3. Safety scale
S1 S2 S3 S4
1 × ×
2 ×
3 ×
4 × ×
In this example, the researcher expands the available empirical evidence about KD
subjectively by entering the following binary relations between the newly introduced
properties:
• E = {(S1, S3), (S1, S4), (S2, S3), (S2, S4)};
• C = {(S1, S2), (S4, S3)}.
Conclusion
Fundamental subjective aspect of the FCA is an axiological basis of formation of the
initial data about KD. This aspect reveals itself in the formation of measurement pro-
cedures’ set.
Information Technology and Nanotechnology (ITNT-2016) 810
Data Science Samoilov DE, Smirnov SV. Data formation…
Subjectively established thresholds of different indicators are generally used to gener-
ate equivalence classes for the results and are directly interpreted in the terms of FCA.
Fundamental cognitive scaling procedure, on the other hand, is associated with the
subject’s introduction of additional information about studied KD. This information
should be taken into account at the stage of binary formal context formation [10] and
it has a significant effect on derivable formal concepts' structure.
Of course, the genesis of the existence limits of the properties is not exhausted by the
researcher’s subjective actions during scales’ designing for the property values of
objects seen in the learning sample. Subject’s a priori knowledge relevant to the re-
searched KD is the source of these restrictions in general.
Acknowledgements
The work was made on “Models and methods for the formation of concepts’ coherent
system in collective decision-making processes” within the government mandate to
the Institute for the Control of Complex Systems of Russian Academy of Science for
2016, as well as with the support from state program of the Samara University com-
petitiveness improvement among the world's leading research and education centers
for 2013-2020.
References
1. Ganter B, Wille R. Formal Concept Analysis. Mathematical foundations. Springer Berlin-
Heidelberg, 1999.
2. Carpineto C, Romano G. Concept Data Analysis: Theory and Applications. Wiley, 2004.
3. Ganter B, Obiedkov S. Conceptual Exploration. Springer, 2016.
4. Begriffliche Wissensverarbeitung. Methoden und Anwendungen. Eds.: B. Ganter,
R. Wille. Springer Berlin-Heidelberg, 2000.
5. Priss U, Szathmary L. Preface to the Special Issue on Concept Lattices and Their Applica-
tions - CLA 2012. Annals of Mathematics and Artificial Intelligence, 2014; 72(1): 1-2.
6. Ignatov DI. Introduction to Formal Concept Analysis and Its Applications in Infor-
mation Retrieval and Related Fields. In: P. Braslavski, N Karpov, M. Worring,
Y. Volkovich, D.I. Ignatov (Eds.): Information Retrieval (Revised Selected Papers 8th
Russian Summer School, RuSSIR 2014. Nizhniy Novgorod, Russia, August 18-22, 2014).
Springer International Publishing, 2015: 42-141.
7. Formal Concept Analysis Homepage. URL : http://www.upriss.org.uk/fca/fca.html.
8. Lammari N, Metais E. Building and maintaining ontologies: a set of algorithms. Da-
ta & Knowledge Engineering, 2004; 48(2): 155-176.
9. Pronina VA, Shipilina LB. Using the relationships between attributes to build domain on-
tology [In Russian]. Control Science, 2009; 1: 27-32.
10. Semenova VA, Smirnov SV. Intelligent analysis of incomplete data for building formal
ontologies. CEUR Workshop Proceedings, 2016; 1638: 796-805. DOI: 10.18287/1613-
0073-2016-1638-796-805
11. Ganter B, Wille R. Conceptual scaling. In: Applications of Combinatorics and Graph The-
ory to the Biological and Social Sciences. Ed.: F. Roberts. Springer Verlag, New York,
1989: 139-167.
Information Technology and Nanotechnology (ITNT-2016) 811
Data Science Samoilov DE, Smirnov SV. Data formation…
12. Belohlavek R, Konecny J. Scaling, Granulation, and Fuzzy Attributes in Formal Concept
Analysis. The IEEE International Conference on Fuzzy Systems (London, UK, July 23-26,
2007): 918-923.
13. Stumme G, Taouil R, Bastide Y, Pasqier N, Lakhal L. Computing Iceberg Concept Lattic-
es with Titanic. Journal on Knowledge and Data Engineering, 2002; 42(2): 189-222.
14. Nehme K, Valtchev P, Rouane MH, Godin R. On Computing the Minimal Generator Fam-
ily for Concept Lattices and Icebergs. In: B. Ganter and R. Godin (Eds.): ICFCA 2005,
LNCS 3403, 2005: 192–207.
15. Kusnetsov SO. Stability as an estimate validity of hypotheses derived on the operational
similarities basis. Scientific and technical information (Series 2), 1990; 12: 21-29. [In Rus-
sian]
16. Kusnetsov SO. On stability of a formal concept. Annals of Mathematics and Artificial In-
telligence, 2007; 49(1-4): 101-115.
17. Zagoruyko NG. Applied methods of data and knowledge analysis [In Russian]. Novosi-
birsk: Sobolev Institute of Mathematics, SB RAS, 1999.
18. Freeman L. Cliques, Galois Lattices, and the Structure of Human Social Groups. Social
Networks, 1996; 18: 173-187.
19. Kazanskiy NL, Popov SB. Machine Vision System for Singularity Detection in Monitoring
the Long Process. Optical Memory and Neural Networks (Information Optics), 2010;
19(1): 23-30.
20. Ignatov DI, Kononychina ON. Formal concept lattices for data analysis in sociological in-
terrogations. Integrated models and soft computation in Artificial Intelligence: Proc. of 5th
Int. Conf. (Kolomna, Russia, 2009, May 20-30). Vol. 1. Moscow: “Fizmathlit” Publisher,
2009: 230-240. [In Russian]
Information Technology and Nanotechnology (ITNT-2016) 812