=Paper=
{{Paper
|id=Vol-1624/paper12
|storemode=property
|title=Conceptual Methods for Identifying Needs of Mobile Network Subscribers
|pdfUrl=https://ceur-ws.org/Vol-1624/paper12.pdf
|volume=Vol-1624
|authors=Dmitry Palchunov,Gulnara Yakhyaeva,Ekaterina Dolgusheva
|dblpUrl=https://dblp.org/rec/conf/cla/PalchunovYD16
}}
==Conceptual Methods for Identifying Needs of Mobile Network Subscribers==
https://ceur-ws.org/Vol-1624/paper12.pdf
Conceptual Methods for Identifying Needs of Mobile
Network Subscribers
Dmitry Palchunov1,2, Gulnara Yakhyaeva2, Ekaterina Dolgusheva2
1
Sobolev Institute of Mathematics, Novosibirsk, Russian Federation
palch@math.nsc.ru
2
Novosibirsk State University, Novosibirsk, Russian Federation
gul_nara@mail.ru, cralya_cher9299@mail.ru
Abstract. The paper is devoted to methods for identifying payment plans and
services by mobile operators which are the best for the given subscribers. We
base our research on the model-theoretic approach to domain formalization. We
use Formal Concept Analysis for processing the mobile subscriber data. An On-
tological Model of the domain “Mobile Networks” is constructed in the scope
of this research. The Ontological Model of the domain is constructed by inte-
gration of data extracted from depersonalized subscriber profiles. The signature
of this Ontological Model contains unary predicates which describe subscriber
behavior and features of payment plans and services. We consider formal con-
texts where objects are subscriber models and attributes are formulas of predi-
cate logic. We investigate concept lattices and association rules of these formal
contexts. Knowledge about optimal payment plans and services for a given sub-
scriber is generated automatically with the help of the association rules.
Keywords: mobile networks, mobile network subscribers, formal context, con-
cept lattice of formal context, association rules, ontology, ontological model.
1 Introduction
Mobile connection is a very important part of our life. Mobile operators provide the
possibility to be in touch for people in different countries. Operators provide access to
USSD-applications and to the Internet.
Mobile operators develop various payment plans and services to satisfy their cli-
ents' needs. However it is difficult for mobile network subscribers to get up-to-date
information about new payment plans or services. Mobile operators send SMS mes-
sages to inform clients about news. But it is very expensive to inform all subscribers
about every small change or update of services. A possible solution of this problem is
sending personal recommendations about services and payment plans that could be
useful for a given subscriber.
A visualization approach based on a graph of calls made by subscribers was used
in [1] for mining behavior patterns of mobile network subscribers. A behavior pattern
discovered during the graph exploration resulted in developing and applying a new
payment plan. Development of methods for increasing the number of subscribers
using services by a mobile network is studied in [2]. An algorithm called Frequent
Pattern-Growth Strategy is used for mining patterns in how subscribers use mobile
network services. Optimization strategies are suggested by experts based on series of
‘frequent’ sets.
Formal Concept Analysis is a well-known formalism in data analysis and
knowledge engineering, see recent surveys [3, 4]. Formal Concept Analysis is used to
develop user behavior templates [5, 6]. These results are applied to planning and run-
ning marketing campaigns.
Association rules for optimizing structures of menus for accessing mobile network
services were constructed in [7]. The Apriori algorithm was used in [8] to develop
association rules patterns in services visited during a single subscriber session. Today
we have more effective algorithms for mining association rules, e.g. see [9].
Fuzzy concept lattices were first introduced in [10]. Papers [11-13] are devoted to
definitions of fuzzy transaction, support and confidence of fuzzy association rules.
The authors of [11] used an algorithm developed in [14] for building sets of fuzzy
rules which describe dependencies between popular telecom services provided by
mobile networks in Taiwan.
Our research is devoted to methods for identifying payment plans and services
which would be optimal for a given mobile network subscriber. Such knowledge al-
lows mobile operator to make really useful recommendations for subscribers.
We base our research on the model-theoretic approach to domain formalization
[15-18]. We use methods and techniques of Formal Concept Analysis for processing
the mobile subscriber data. Now a lot of attention is paid to the relationships between
FCA and models of knowledge representation and processing [19].
The ontological model of the domain “Mobile Networks” is constructed by integra-
tion of data extracted from depersonalized subscriber profiles. The signature of this
ontological model contains unary predicates which describe subscriber behavior and
features of payment plans and services. To generate meaningful recommendation of
alternative services and payment plans, we define formal contexts where objects are
subscriber models, and attributes are formulas of predicate logic. We investigate con-
cept lattices and association rules of these formal contexts to get high-quality recom-
mendation. To do this, we consider extensions of attribute sets of formal contexts.
In [20] extensions of infinite attribute sets were considered, it was suggested to use
concept descriptions of bounded depth. In [21] a new approach to reduce the number
of attributes was presented.
In this paper we consider finite extensions of the initial finite context. We use in-
terrelation between axiomatizable classes and FCA [22]. Section 2.1 is devoted to
isomorphisms between lattices of relatively axiomatizable classes of one-element
models and lattices of formal concepts of formal contexts generated by these classes.
Section 2.2 describes extensions of such formal contexts having distributive concept
lattices.
The main purpose of this paper is to develop methods of identifying payment plans
and services which would be optimal for the given mobile network subscriber. To do
this, firstly, we construct Case Model based on the known information about behavior
patterns of mobile network subscribers (Section 2.2). We represent the Case Model as
a relatively axiomatizable class of one-element models. On the base of this Case
Model we define a formal context.
Secondly, we move from the Case Model to Ontology Model (Section 3.1). We
construct the set of ontological projections which is the basis of extensions of attribute
set of the formal context under consideration (Section 3.2).
And finally we mine association rules with high confidence and support in the ex-
tended formal context. Computer experiments show that the methods presented in the
paper allow us to find association rules which can be used for recommendations.
2 Case Model
2.1 Relatively axiomatizable classes and formal contexts
Here we introduce some definitions and results on the relationship between relatively
axiomatizable classes and formal contexts. The main result of this section is Proposi-
tion 2 which is necessary for proofs of Propositions 4 and 5 in Section 2.2. The proofs
of the statements are based on [22].
An algebraic system (a model) is a tuple 𝔄 = 𝐴; 𝑃! , . . . , 𝑃! , 𝑓! , . . . , 𝑓! , 𝑐! , . . . , 𝑐! ,
where the set 𝔄 = 𝐴 is called universe, 𝑃! , . . . , 𝑃! are predicates defined on the set
𝐴, 𝑓! , . . . , 𝑓! are functions defined on the set 𝐴 and 𝑐! , . . . , 𝑐! are constants. The tuple
𝜎 = 𝑃! , . . . , 𝑃! , 𝑓! , . . . , 𝑓! , 𝑐! , . . . , 𝑐! is called signature of the algebraic system 𝔄.
Denote by 𝐹𝑉(𝜑) the set of all free variables of a formula 𝜑. A formula having no
free variables is called sentence. For a signature 𝜎 we denote:
𝐹(𝜎) ⇋ {𝜑 | 𝜑 𝑖𝑠 𝑎 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑖𝑔𝑛𝑎𝑡𝑢𝑟𝑒 𝜎},
𝐹! 𝜎 ⇋ 𝜑 𝜑 ∈ 𝐹 𝜎 𝑎𝑛𝑑 𝐹𝑉 𝜑 = {𝑥}},
𝑆(𝜎) ⇋ {𝜑 | 𝜑 𝑖𝑠 𝑎 𝑠𝑒𝑛𝑡𝑒𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑖𝑔𝑛𝑎𝑡𝑢𝑟𝑒 𝜎} and
𝐾(𝜎) ⇋ {𝔄 | 𝔄 𝑖𝑠 𝑎 𝑚𝑜𝑑𝑒𝑙 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑖𝑔𝑛𝑎𝑡𝑢𝑟𝑒 𝜎}.
Here 𝐹𝑉 𝜑 = {𝑥} means that each formula 𝜑 ∈ 𝐹! 𝜎 has just one free variable,
which is the fixed variable 𝑥.
Consider a signature 𝜎 and a model 𝔄 ∈ 𝐾(𝜎). For a sentence 𝜓 ∈ 𝑆(𝜎) we denote
𝔄 ⊨ 𝜓 if 𝜓 is true in the model 𝔄. For a formula 𝜑(𝑥! , … , 𝑥! ) ∈ 𝐹(𝜎) we write
𝔄 ⊨ 𝜑 if 𝔄 ⊨ ∀𝑥! … ∀𝑥! 𝜑(𝑥! , … , 𝑥! ).
Definition 1. Let 𝐾 ⊆ 𝐾(𝜎). For a formula 𝜑 ∈ 𝐹(𝜎) we denote 𝐾 ⊨ 𝜑 if 𝔄 ⊨ 𝜑
for any 𝔄 ∈ 𝐾. For a set of formulas 𝛤 ⊆ 𝐹(𝜎) we denote 𝐾 ⊨ 𝛤 if 𝔄 ⊨ 𝜑 for any
𝔄 ∈ 𝐾 and 𝜑 ∈ 𝛤. For a set of formulas 𝛤 ⊆ 𝐹(𝜎) we denote
𝐾 𝛤 ⇋ 𝐾! 𝛤 ⇋ 𝔄 ∈ 𝐾 𝜎 𝔄 ⊨ 𝜑 𝑓𝑜𝑟 𝑎𝑛𝑦 𝜑 ∈ 𝛤 }.
A class 𝐾 ⊆ 𝐾 𝜎 is called axiomatizable if there exists a set 𝛤 ⊆ 𝑆(𝜎) such that
𝐾 = { 𝔄 ∈ 𝐾(𝜎) | 𝔄 ⊨ 𝛤}.
For the aims of our research we need to generalize the notion of relatively axio-
matizable class [22] to the case of arbitrary sets of formulas 𝛥.
Definition 2. Let 𝐾, 𝐾! ⊆ 𝐾 𝜎 and 𝛥 ⊆ 𝐹(𝜎). We say that the class 𝐾! is axio-
matizable in the class 𝐾 relatively to the set of formulas 𝛥 if there exists a set 𝛤 ⊆ 𝛥
such that 𝐾! = {𝔄 ∈ 𝐾 | 𝔄 ⊨ 𝛤}.
Notice that the class 𝐾! ⊆ 𝐾 𝜎 is axiomatizable if and only if 𝐾! is axiomatizable
in the class 𝐾 = 𝐾 𝜎 relatively to the set of formulas 𝛥 = 𝑆(𝜎).
Definition 3. For 𝐾 ⊆ 𝐾(𝜎) and 𝛥 ⊆ 𝐹(𝜎) we denote
𝔹 𝐾, 𝛥 ⇋ 𝐾! 𝐾! 𝑖𝑠 𝑎𝑥𝑖𝑜𝑚𝑎𝑡𝑖𝑧𝑎𝑏𝑙𝑒 𝑖𝑛 𝐾 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝑙𝑦 𝑡𝑜 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑓𝑜𝑟𝑚𝑢𝑙𝑎𝑠 𝛥}
and 𝑇! (𝐾) ⇋ {𝜑 ∈ 𝛥 | 𝐾 ⊨ 𝜑}. The set of formulas 𝑇! (𝐾) is call 𝜟-type of 𝐾.
Note that 𝐾! ∈ 𝔹(𝐾, 𝛥) if and only if 𝐾! = {𝔄 ∈ 𝐾| 𝔄 ⊨ 𝑇! (𝐾! )}.
For each class 𝐾 ⊆ 𝐾(𝜎) and set 𝛥 ⊆ 𝑆(𝜎) we consider the formal context
(𝐾, 𝛥, ⊨), with derivation operator ()′ [23].
Remark 1. Let 𝐾 ⊆ 𝐾(𝜎), 𝛥 ⊆ 𝐹(𝜎) and 𝐴 ⊆ 𝐾. Then 𝐴′ = 𝑇! (𝐾).
For a formal context G, M, I by 𝔙 G, M, I we denote the lattice of formal con-
cepts of the formal context G, M, I .
Proposition 1. Let 𝐾 ⊆ 𝐾(𝜎) , 𝛥 ⊆ 𝐹(𝜎) , 𝐴 ⊆ 𝐾 and 𝐵 ⊆ 𝛥 . Then 𝐴, 𝐵 ∈
𝔅 𝐾, 𝛥, ⊨ if and only if 𝐴 is axiomatizable in the class 𝐾 relatively to the set of for-
mulas 𝛥 and 𝐵 = 𝑇! (𝐴).
Proof. (⇒) Let 𝐴, 𝐵 ∈ 𝔅 𝐾, 𝛥, ⊨ . Then 𝐵 = 𝐴′, so 𝐵 = 𝑇! (𝐴) by Remark 1.
We have 𝐴 = 𝐵′, hence 𝐴 = {𝔄 ∈ 𝐾 | 𝔄 ⊨ 𝐵} and 𝐵 ⊆ 𝛥. Therefore, by Definition 2,
the class 𝐴 is axiomatizable in the class 𝐾 relatively to the set of formulas 𝛥.
(⇐) Let the class 𝐴 be axiomatizable in the class 𝐾 relatively to the set of formu-
las 𝛥 and 𝐵 ⊆ 𝛥. So there exists 𝛤 ⊆ 𝛥 such that 𝐴 = {𝔄 ∈ 𝐾 | 𝔄 ⊨ 𝛤}. Then in the
formal context (𝐾, 𝛥, ⊨) we have 𝛤′ = 𝐴 . So 𝐴′′ = 𝐴 . The set 𝐵 = 𝑇! (𝐴) , thus
𝐵 = 𝐴′ by Remark 1. Therefore, 𝐴, 𝐵 ∈ 𝔅 𝐾, 𝛥, ⊨ .
Corollary 1. Let 𝐾, 𝐾! ⊆ 𝐾 𝜎 and 𝛥 ⊆ 𝐹(𝜎).
1. 𝐾! ∈ 𝔹(𝐾, 𝛥) if and only if 𝐾! , 𝑇! (𝐾! ) ∈ 𝔅 𝐾, 𝛥, ⊨ .
2. 𝐾! = 𝐾! ′′ if and only if 𝐾! is axiomatizable in the class 𝐾 relatively to the set
of formulas 𝛥.
Therefore, the classes which are axiomatizable in a class 𝐾 relatively to a set of
formulas Δ are exactly extents of the formal concepts of the formal context (𝐾, 𝛥, ⊨).
We consider 𝔹(𝐾, 𝛥) as a set ordered by inclusion ⊆. So 𝔹(𝐾, 𝛥) is a lattice.
Proposition 2. The lattices 𝔅(𝐾, 𝛥, ⊨) and 𝔹(𝐾, 𝛥) are isomorphic, i.e.,
𝔅(𝐾, 𝛥, ⊨) ≅ 𝔹(𝐾, 𝛥), for any 𝐾 ⊆ 𝐾(𝜎) and 𝛥 ⊆ 𝐹(𝜎).
Proof. Let us consider the mapping ℎ: 𝔅(𝐾, 𝛥, ⊨) → 𝔹(𝐾, 𝛥) defined as follows:
ℎ 𝐴, 𝐵 = 𝐴 for any 𝐴, 𝐵 ∈ 𝔅(𝐾, 𝛥, ⊨) . By Proposition 1 for any 𝐴, 𝐵 ∈
𝔅(𝐾, 𝛥, ⊨) we have ℎ 𝐴, 𝐵 = 𝐴 ∈ 𝔹(𝐾, 𝛥). For each 𝐴 ∈ 𝔹(𝐾, 𝛥) it is true that
𝐴, 𝑇! (𝐴) ∈ 𝔅(𝐾, 𝛥, ⊨), so ℎ 𝐴, 𝑇! (𝐴) = 𝐴. Thus the mapping ℎ is onto.
For any 𝐴! , 𝐵! , 𝐴! , 𝐵! ∈ 𝔅(𝐾, 𝛥, ⊨) we have: 𝐴! , 𝐵! ≤ 𝐴! , 𝐵! iff
𝐴! ⊆ 𝐴! . Hence the mapping ℎ preserves the partial order.
Therefore, the mapping ℎ is an isomorphism.
2.2 Description of the Case Model
Further we consider signatures consisting of a finite set of unary predicate symbols,
i.e. 𝜎 =< 𝑃! , … , 𝑃! >. We consider the set 𝛥 ⊆ 𝑆(𝜎) for different signatures 𝜎 which
means that the original signature is enriched by new unary predicate symbols. From a
model-theoretic point of view we may assume that there is some covering signature
𝜎 ! and all considered signatures are its subsets.
Consider a finite set 𝐴 = 𝑒! , … , 𝑒! of subscribers of a given mobile network and
fix a signature 𝜎 = 𝜎ℙ ∪ 𝜎ℚ where 𝜎ℙ is a set of personal characteristics of subscrib-
ers and 𝜎ℚ is a set of payments plans, services and options. Each of these sets has a
hierarchical structure. There are more details about the signatures 𝜎ℙ and 𝜎ℚ below.
For each subscriber 𝑒! we know which characteristics (presented by signature predi-
cates from 𝜎) are true and which characteristics are false. Thus, for each subscriber 𝑒!
there is a one-element model 𝒆𝒊 = 𝑒! , 𝜎 which is called a case of the domain 𝕄.
Consider the Case Model 𝔄 = 𝐴, 𝜎 defined by a set of cases 𝒆𝟏 , … , 𝒆𝒏 [20].
On the model 𝔄 for each signature predicate 𝑃 ∈ 𝜎 and for every element 𝑒 ∈ 𝐴 we
have 𝔄 ⊨ 𝑃 𝑒 if and only if the predicate 𝑃 𝑥 is true in the model (case) 𝒆 (i.e.,
𝒆 ⊨ 𝑃 𝑥 ). Here 𝒆 ⊨ 𝑃 𝑥 means that ⊨ 𝑥𝑃 𝑥 . On the base of the Case Model
𝔄 = 𝐴, 𝜎 in the section 3.5 we will define the ontological model.
Denote by 𝐾𝔄 = 𝒆𝟏 , … , 𝒆𝒏 the class of cases (one-element models) generated by
the set of subscribers 𝑒! , … , 𝑒! .
Note that 𝐾𝔄 = 𝑒 ; 𝜎 | 𝑒 ∈ 𝐴 𝑎𝑛𝑑 𝑒 ; 𝜎 ≤ 𝔄 . Here the notation 𝒆 =
𝑒 ; 𝜎 ≤ 𝔄 means that the model 𝒆 is a submodel of the model 𝔄. Recall that in
pure predicate signature each subset of a model is the universe of its submodel.
Here we consider different sets of formulas Δ ⊆ 𝐹! (𝜎). In particular, we consider
Δ! = 𝑃 𝑥 | 𝑃 ∈ 𝜎 ⊆ 𝐹! (𝜎). Denote by 𝐶𝔄! = 𝐾𝔄 , ∆, ⊨ the formal context having
the set of objects 𝐾𝔄 , the set of attributes Δ and the incidence relation ⊨. Denote
𝐶𝔄! = 𝐾𝔄 , ∆! , ⊨ .
∆⊆ 𝐹! (𝜎) is a set of properties of the cases 𝒆 ∈ 𝐾𝔄 , which are definable by formu-
las of the signature 𝜎. When we change the set ∆ we change the set of attributes of the
formal context keeping fixed the set of objects 𝐾𝔄 . Reductions and expansions of
formal contexts were studied in [25].
Let us consider two formal contexts 𝐶! = 𝐺, 𝑀! , 𝐼 and 𝐶! = 𝐺, 𝑀! , 𝐼 . Suppose
that 𝑀! ⊆ 𝑀! , 𝐴 ⊆ 𝐺 and 𝐴 = 𝐴′′ в 𝐶! . Then 𝐴 = 𝐴′′ in 𝐶! .
We define a mapping 𝑖: 𝔅(𝐺, 𝑀! , 𝐼) → 𝔅(𝐺, 𝑀! , 𝐼) as follows: 𝑖 𝐴, 𝐵! = (𝐴, 𝐵! ),
where 𝐴 ⊆ 𝐺, 𝐵! ⊆ 𝑀! , 𝐵! ⊆ 𝑀! , 𝐴 = 𝐴′′, 𝐵! = 𝐴′ in the context 𝐶! and 𝐵! = 𝐴′ in
the context 𝐶! .
Remark 2. The mapping 𝑖: 𝔅(𝐺, 𝑀! , 𝐼) → 𝔅(𝐺, 𝑀! , 𝐼) is an isomorphic embedding
of the lattice 𝔅(𝐺, 𝑀! , 𝐼) into the lattice 𝔅(𝐺, 𝑀! , 𝐼).
Next consider an arbitrary signature 𝜎! and an arbitrary class 𝐾! ⊆ 𝐾(𝜎! ).
Remark 3. Let ∆⊆ 𝐹(𝜎! ) and 𝜑! , … , 𝜑! ∈ 𝐹(𝜎! ). Then the mapping
𝑖: 𝔅(𝐾! , ∆, ⊨) → 𝔅(𝐾! , ∆ ∪ 𝜑! & … &𝜑! , ⊨) is an isomorphism of lattices.
Corollary 2. а) The sets of association rules of the formal contexts (𝐾! , ∆, ⊨) and
(𝐾! , ∆ ∪ 𝜑! & … &𝜑! , ⊨) coincide up to the substitution of the formula
𝜑! & … &𝜑! by the set 𝜑! , … , 𝜑! .
b) The sets of attribute implications of the formal contexts (𝐾! , ∆, ⊨) и (𝐾! , ∆ ∪
𝜑! & … &𝜑! , ⊨) coincide up to the substitution of the formula 𝜑! & … &𝜑! by the
set 𝜑! , … , 𝜑! .
Corollary 3. If ∆, ∆! ⊆ 𝐹(𝜎! ), ∆⊆ ∆! and the set ∆! \∆ consists of some conjunc-
tions of formulas from ∆ then the sets of attribute implications as well as the sets of
association rules of the formal contexts (𝐾! , ∆, ⊨) and (𝐾! , ∆! , ⊨) coincide up to the
substitution of the conjunctions from ∆! \∆ by the corresponding sets of formulas.
Let us go back to the formal context (𝐾𝔄 , ∆! , ⊨).
Remark 4. Let 𝑃! , 𝑃! ∈ 𝜎 . Then the mapping
𝑖: 𝔅(𝐾𝔄 , ∆! , ⊨) → 𝔅(𝐾𝔄 , ∆! ∪ 𝑃! (𝑥) ∨ 𝑃! (𝑥) , ⊨) is an isomorphic embedding of
lattices; in the general case this mapping is not an isomorphism. Moreover, in the
general case 𝔅(𝐾𝔄 , ∆! , ⊨) ≇ 𝔅 𝐾𝔄 , ∆! ∪ 𝑃! (𝑥) ∨ 𝑃! (𝑥) , ⊨ .
Corollary 4. In the general case if we add a disjunction 𝑃! 𝑥 ∨ … ∨ 𝑃! (𝑥) to
the set of formulas ∆! , where 𝑃! , … , 𝑃! ∈ 𝜎, then the set of association rules of the
formal context (𝐾𝔄 , ∆! , ⊨) will be changed.
Denote ∆∨! = ∆! ∪ 𝑃! 𝑥 ∨ … ∨ 𝑃! (𝑥) , | 𝑃! ∈ 𝜎 .
We will be adding disjunctions of signature predicates into the set ∆! for improv-
ing association rules based on an algorithm for subscribers’ behavior prediction. It
means that we will consider the set of formulas ∆∨! instead of the set of formulas ∆!
and the formal context (𝐾𝔄 , ∆∨! , ⊨) instead of the formal context (𝐾𝔄 , ∆! , ⊨).
Definition 4. We say that a set of formulas 𝛥 ⊆ 𝐹(𝜎! ) is closed under disjunction
if (𝜑 ∨ 𝜓) ∈ 𝛥 for any 𝜑, 𝜓 ∈ 𝛥.
Proposition 3. Let 𝐾 ⊆ 𝐾𝔄 and ∆⊆ 𝐹(𝜎). If the set of formulas ∆ is closed under
disjunction then the lattice 𝔹(𝐾, ∆) is distributive.
Proof. Assume that ∆⊆ 𝐹(𝜎) and 𝐾! , 𝐾! ∈ 𝔹(𝐾, ∆). Then 𝐾! , 𝐾! ⊆ 𝐾 and there ex-
ist Γ! , Γ! ∈ ∆ such that 𝐾! = 𝔄 ∈ 𝐾 | 𝔄 ⊨ Γ! and 𝐾! = 𝔄 ∈ 𝐾 | 𝔄 ⊨ Γ! .
Denote Γ! = Γ! ∪ Γ! and Γ! = 𝜑 ∨ 𝜓) | 𝜑 ∈ Γ! 𝑎𝑛𝑑 𝜓 ∈ Γ! . Then 𝐾! ∩ 𝐾! =
𝔄 ∈ 𝐾 | 𝔄 ⊨ Γ! , hence 𝐾! ∩ 𝐾! ∈ 𝔹(𝐾, ∆).
Let 𝔄 ∈ 𝐾. Then 𝔄 is a one-element model. Therefore for any 𝜑 ∈ Γ! and 𝜓 ∈ Γ!
we have: 𝔄 ⊨ 𝜑 ∨ 𝜓 ⇔ 𝔄 ⊨ ∀𝑥! … ∀𝑥! 𝜑(𝑥! , … , 𝑥! ∨ 𝜓(𝑥! , … , 𝑥! )) ⇔
⇔ 𝔄 ⊨ ∀𝑥! … ∀𝑥! 𝜑 𝑥! , … , 𝑥! 𝑜𝑟 𝔄 ⊨ ∀𝑥! … ∀𝑥! 𝜓 𝑥! , … , 𝑥! ⇔
⇔ 𝔄 ⊨ 𝜑 𝑜𝑟 𝔄 ⊨ 𝜓, where 𝐹𝑉 𝜑 ∨ 𝜓 = 𝑥! , … , 𝑥! .
Assume that 𝔄 ∈ 𝐾! ∪ 𝐾! , then 𝔄 ∈ 𝐾! 𝑜𝑟 𝔄 ∈ 𝐾! ⇒
⇒ 𝔄 ∈ 𝐾 𝑎𝑛𝑑 𝔄 ⊨ Γ! 𝑜𝑟 𝔄 ∈ 𝐾 𝑎𝑛𝑑 𝔄 ⊨ Γ! ⇒
⇒ 𝔄 ∈ 𝐾 𝑎𝑛𝑑 𝔄 ⊨ Γ! 𝑜𝑟 𝔄 ⊨ Γ! ⇒ 𝔄 ∈ 𝐾 𝑎𝑛𝑑 𝔄 ⊨ Γ! .
Next, suppose that 𝔄 ⊭ Γ! and 𝔄 ⊭ Γ! . So there exist 𝜑 ∈ Γ! and 𝜓 ∈ Γ! such that
𝔄 ⊭ 𝜑 and 𝔄 ⊭ 𝜓. Then 𝔄 ⊭ 𝜑 ∨ 𝜓 , so 𝔄 ⊭ Γ! .
Thus, if 𝔄 ⊨ Γ! then 𝔄 ⊨ Γ! 𝑜𝑟 𝔄 ⊨ Γ! . Hence, if 𝔄 ∈ 𝐾 and 𝔄 ⊨ Γ! then
𝔄 ∈ 𝐾! or 𝔄 ∈ 𝐾! , so 𝔄 ∈ 𝐾! ∪ 𝐾! .
Therefore, 𝐾! ∪ 𝐾! = 𝔄 ∈ 𝐾 | 𝔄 ⊨ Γ! and 𝐾! ∪ 𝐾! ∈ 𝔹(𝐾, ∆).
We proved that 𝐾! ∩ 𝐾! , 𝐾! ∪ 𝐾! ∈ 𝔹(𝐾, ∆) for any 𝐾! , 𝐾! ∈ 𝔹(𝐾, ∆). Hence,
the lattice 𝔹(𝐾, ∆) is distributive.
Proposition 4. The lattice of formal concepts 𝔅(𝐾𝔄 , ∆∨! , ⊨) is distributive.
Proof: in virtue of Proposition 2 and Proposition 3.
However the initial formal context (𝐾𝔄 , ∆! , ⊨) does not have this good property.
Remark 5. In the general case the lattice of formal concepts 𝔅(𝐾𝔄 , ∆! , ⊨) is not
distributive. It means that there exists a class 𝐾𝔄 such that the lattice 𝔅(𝐾𝔄 , ∆! , ⊨) is
not distributive.
Remark 6. Let ∆ ⊆ 𝐹! 𝜎 , ∆! ⊆ ∆ and the set ∆\∆! consists of some conjunctions
of formulas from ∆! . Then there exists a class 𝐾𝔄 such that the lattice (𝐾𝔄 , ∆, ⊨) is
not distributive.
For the set of all formulas the situation is better.
Proposition 5. 1) The lattice of formal concepts 𝔅(𝐾𝔄 , 𝐹! 𝜎 , ⊨) is distributive.
2) The lattice of formal concepts 𝔅(𝐾𝔄 , 𝐹 𝜎 , ⊨) is distributive.
Proof: in virtue of Proposition 2 and Proposition 3.
Association rule mining for the original context 𝐾𝔄 , ∆! , ⊨ does not produce a lot
of rules with high confidence. A lot of various payment plans and services exist, and
commonly more than one service can be useful for the subscriber. The service that
will be preferred by the user depends on many factors. Some of these factors can
change time to time. So we cannot detect such factors in scope of formal context
𝐾𝔄 , ∆! , ⊨ because the context is based on long users’ history.
Moreover, mobile operator can suggest 2-3 possible services and the subscriber
may select himself the most useful service. That is why it makes sense to add disjunc-
tions of signature predicates to ∆! and use context 𝐾𝔄 , ∆∨! , ⊨ on next steps.
There are two problems with association rules that were mined using formal con-
text (𝐾𝔄 , ∆∨! , ⊨). First of all some of rules have high confidence, but their conclusions
are disjunctions of meaningfully nonrelated services. Such association rules could not
be used for recommendations. It will be looking like spam for mobile network sub-
scribers. So experts should process all rules and select only meaningful rules. Second,
processing the whole formal context (𝐾𝔄 , ∆∨! , ⊨) is very laborious computational pro-
cedure.
To solve these problems we are moving from the Case Model 𝔄 = 𝐴, 𝜎 to the
Ontological Model 𝔄, 𝑇 ! , 𝑇 ! , 𝑇 ! of the domain. We add new unary predicates to the
signature 𝜎 to describe meaning of payment plans and services. Using new predicates
(from the signature 𝜎ℝ ) we generate automatically meaningful disjunctions of original
predicates from the signature 𝜎.
3 Ontological Model of the domain
3.1 Ontology
Ontological Model of the domain consists of four parts [15]:
(1) The domain ontology, i.e. description of the structure and the meaning of the
domain concepts.
(2) General knowledge and domain regularities, sentences which are true for every
case.
(3) The set of cases from the domain, that we consider in the given moment. This
is empirical knowledge about the domain; the set of cases that we are looking at in
this article is represented by the model 𝔄 = 𝐴, 𝜎 .
(4) Estimated and probabilistic knowledge: probabilistic and confidential esti-
mates, fuzzy values of sentences [16].
In this section we describe construction of the domain ontology.
From a model-theoretic point of view the domain ontology construction consists
of description of the signature and creation of a set of axioms that describe the mean-
ing of the concepts of the domain [17, 18]. To define the signature 𝜎𝕄 of the domain
𝕄 = “Mobile networks” we consider two sets of attributes: 𝜎ℙ , the set of individual
subscriber’s features and 𝜎ℚ , the set of various payment plans and services.
The set of attributes 𝜎ℙ , “Individual subscribers’ feature” consists of two parts:
𝜎ℙ! , “payment plans” and 𝜎ℙ! , “accrual”. Every part 𝜎ℙ! (𝑖 = 1, 2) consists of two
subparts, such as 𝜎ℙ!! , “traffic (and accrual) without roaming inside operator net-
work”,…, 𝜎!!" , “traffic (and accrual) in common roaming”, … . Each of the listed
signatures consists of more detailed categories, e.g., 𝜎!!!! , “Traffic SMS without
roaming inside operator network”. Every category 𝜎!!"# contains finite number of
! ! !
signature symbols 𝑃!!" , … . , 𝑃!!" . For example, 𝑃!!! 𝑥 = “Traffic of SMS without
roaming inside network for subscriber 𝑥 is not more than 50 SMS in month” and
!
𝑃!!" 𝑥 = “Traffic of SMS without roaming inside network for subscriber 𝑥 is more
than 50 SMS in month”.
Signature 𝜎ℚ consists of two parts: 𝜎ℚ! and 𝜎ℚ! . Part 𝜎ℚ! is “payment plans”, it has
hierarchical structure and consists of symbols of unary predicates. Each unary predi-
cate describes the presence or absence of connected payment plan for subscribers.
Signature 𝜎ℚ! , “services and options”, consists of symbols of unary predicates. Each
unary predicate describes the presence or absence of connected service or option.
To describe the domain ontology, we define a finite set of ontological axioms
𝒜𝓍! ⊆ 𝐹! (𝜎ℙ ∪ 𝜎ℚ ). We introduce the following axioms.
Axioms of hyponym-hyperonym. Hierarchical structure of the signature 𝜎ℚ! is
represented by axioms such as:
! ! !
(𝑄!"# 𝑥 → 𝑄!" 𝑥 ) 𝑎𝑛𝑑 (𝑄!" (𝑥) → 𝑄! (𝑥)).
Axioms of completeness. For each predicates inside every class 𝜎ℙ!"# and class
𝜎ℚ! for a given subscriber there must be at least one true predicate. The schemes of
such axioms are the following:
∨ 𝑃 𝑥 | 𝑃(𝑥) ∈ 𝜎ℙ!"# 𝑎𝑛𝑑 ∨ 𝑃 𝑥 | 𝑃(𝑥) ∈ 𝜎ℚ! .
Axioms of including. For example, if payment plan 𝑡 contains “more than 100 free
SMS” then it contains “more 50 free SMS”. The schemes of such axioms are the fol-
lowing:
! !
(𝑄!"#! 𝑥 → 𝑄!"#! 𝑥 ), where 𝑛! < 𝑛! .
Next we construct an extension 𝜎ℙ ∪ 𝜎ℚ of the signature 𝜎𝕄 by additional unary
predicates that describe properties of payment plans and interests of subscribers. For
that we introduce two types of concepts:
1) Concepts 𝜎ℝ . This is a set of features for different payment plans, services, and
options. For example, amount of free calls time, volume of SMS package or of Inter-
net package and etc. With the help of 𝜎ℝ we can give formal definition for payment
plans and services, i.e., formal definition of predicates of the signature 𝜎ℚ .
2) Concepts 𝜎𝕀 describing subscriber’s interests, e.g., reducing the costs of calls,
SMS, etc.
Concepts from 𝜎ℝ ∪ 𝜎𝕀 are used for automation of construction formulas as at-
tributes in formal contexts for association rules mining. Notice that the pair 𝜎𝕄 , 𝒜𝓍
forms the ontology of the domain 𝕄 [18].
In the next step we introduce a new set of axioms 𝒜𝓍! ⊆ 𝐹! (𝜎𝕄 ) and call it the
domain axioms. This set will be used for describing various characteristics of pay-
ment plans and services provided at present moment of time by a mobile network.
Among other things, these axioms relate personal parameters of a subscriber. The
range of parameters contains subscriber traffics denoted by predicates from 𝜎ℙ! and
payments denoted by predicates from 𝜎ℙ! , with regard to activated payment plans
from 𝜎ℚ! and services from 𝜎ℚ! .
Axioms 𝒜𝓍! are true for any case from the domain, and the same statement is true
for ontological axioms as well. However, there is a difference between ontological
and domain axioms, as the second ones might change over time. Consider the follow-
ing formula as an example of a domain axiom:
(𝑄! 𝑥 → ¬𝑄! 𝑥 ), 𝑤ℎ𝑒𝑟𝑒 𝑄! ∈ 𝜎ℚ! , 𝑄! ∈ 𝜎ℚ! .
This formula declares the following: if a subscriber has payment plan 𝑄! activat-
ed, then service 𝑄! cannot be activated for this subscriber. Note that a mobile network
company can naturally change its decision for not supporting simultaneously the pre-
cise payment plan along with the specific service, at any moment.
3.2 Ontological projections
In order to automate development of the formula set Δ for the sake of finding associa-
tion rules, we use the Ontological Model of the domain.
Definition 4. An Ontological Model of a domain is a tuple 𝔄, 𝑇 ! , 𝑇 ! , 𝑇 ! , where
𝑇 is an analytical theory of the domain, 𝑇 ! is a theory of the domain, and 𝑇 ! is a
!
fuzzy theory of the model 𝔄𝕄 .
The analytical theory 𝑇 ! of the domain under consideration is axiomatized by the
sentences 𝒜𝓍! which are axioms of the domain ontology. A theory 𝑇 ! of the domain
is axiomatized by the axioms 𝒜𝓍! of the domain.
Formula definitions of predicates from 𝜎ℚ (which present payment plans and ser-
vices) are defined by construction of ontological projection.
Definition 5. Consider the Ontological Model 𝔄, 𝑇 ! , 𝑇 ! , 𝑇 ! , let 𝑄 ∈ 𝜎ℚ . Denote
𝑆! = 𝜑 ∈ 𝐹! 𝜎ℝ | 𝑇 ! ⊢ 𝑄 𝑥 → 𝜑 𝑥 .
An ontological projection of the predicate 𝑄 on the signature 𝜎ℝ is the formula
!
𝜑! ℝ 𝑥 = & 𝑃 𝑥 | 𝑃 ∈ 𝜎ℝ 𝑎𝑛𝑑 𝑃(𝑥) ∈ 𝑆! .
A projection of the predicate 𝑄 on the set of formulas 𝐹! 𝜎ℝ is the formula
!
𝜓! ℝ 𝑥 = &𝑆! = & 𝜑 𝑥 | 𝜑(𝑥) ∈ 𝑆! .
Let us consider the formal context 𝐶𝔄! = 𝐾𝔄 , ∆, ⊨ . We search association rules
with the following requirements:
1) а) the premise of the association rule is included in the set ∆↾𝜎ℙ or
b) the premise of the association rule is included in the set ∆↾ 𝜎ℙ ∪ 𝜎ℚ ;
2) a) the conclusion of the association rule belongs to the set ∆↾ 𝜎ℚ or
b) the conclusion of the association rule belongs to the set ∆↾ 𝜎ℝ or
c) the conclusion of the association rule belongs to the set ∆↾ 𝜎ℝ ∪ 𝜎𝕀 ;
3) the support and the confidence of the rules are higher than specified limits.
Notice that the set of association rules of the formal context 𝐶𝔄! = 𝐾𝔄 , ∆, ⊨ is in-
cluded in the fuzzy theory 𝑇 ! of the model 𝔄𝕄 [26, 27].
Then the software system automatically processes obtained association rules. For
example, consider an association rule with one-element conclusion 𝑃 belonging to 𝜎ℝ .
This rule will be transformed into association rule with the same premise, but the
conclusion of the new rule will be one-variable formula from Δ which is a disjunction
of all predicates 𝑄! ∈ 𝜎ℚ such that P belongs to the ontological projection of 𝑄! .
4 Software Implementation
Using the results of the presented investigation, we have developed a software for
mining association rules in the formal context (𝐾𝔄 , ∆∨! , ⊨). We have found out that
adding predicates from 𝜎ℝ to the formal context gives us the possibility to find associ-
ation rules with high confidence and support. Conclusions of such rules are trans-
formed into disjunctions of predicates from 𝜎ℚ with the help of the operator of ontol-
ogy projection. Obtained association rules seem to be useful for mobile network com-
panies. The software processes the impersonal data for more than 10 million subscrib-
ers. This is information for one month of mobile network using by subscribers.
The set of characteristics of subscribers contains more than 90 different items1:
1) Personal features of subscriber,
2) Attributes that describe calls made by subscriber,
3) Attributes that describe the mode of using Internet
4) Attributes that describe the mode of using SMS,
5) Attributes that describe the mode of using MMS,
6) Attributes that describe the mode of using LBS (Location Based Services),
7) List of mobile services that were connected to subscriber,
8) Payment plan that is used by subscriber.
The total amount of services that can be connected to subscriber is more than 90.
The total count of different payment plans is more than 1200.
Thus, we have more than 10 million objects and nearly 1400 attributes. Part of at-
tributes is quantitative, most part of attributes (more than 1200) are binary.
Let us notice that attributes of connected payment plans, services and personal at-
tributes are always filled. That is why we use only quantitative attributes for density
!
calculation. We calculate data density as follows: , where P is the number of non-
!∗!
zero subscribers’ attributes, M is the total number of subscribers, N is the number of
quantitative attributes. For our data the data density is equal to 0.043.
The data is stored in a file with Basket format. Basket is one of standard formats
for storing data of “objects-attributes” type in R.
Let us consider an example of association rules which have conclusions consisting
of payment plans providing access to the Internet. The predicate 𝑃(𝑥) ∈ 𝜎ℝ denotes
that subscriber’s payment plan includes unlimited Internet traffic of the special kind1.
The payment plans having the unlimited Internet traffic of this kind are 𝑄! , 𝑄! , 𝑄! ∈
𝜎ℚ! , where 𝑄! is “Unlimited”, 𝑄! is “United”, and 𝑄! is “Online”. These payment
plans provide unlimited access to Internet with different connection speed and differ-
ent price. Formally, in terms of ontological projections, it means that 𝑃(𝑥) ∈ 𝑆!! ,
𝑃(𝑥) ∈ 𝑆!! , 𝑃(𝑥) ∈ 𝑆!! and 𝑃 𝑥 ∉ 𝑆! for every 𝑄 ∈ 𝜎ℚ \ 𝑄! , 𝑄! , 𝑄! .
Mined association rules have premises with various sets of personal features of
subscribers from 𝜎ℙ and the conclusion 𝑃 𝑥 . The automatically chosen rules have
rather high confidence and support (see examples 1 and 2, table 1).
After that the predicate 𝑃 𝑥 is substituted by the equivalent disjunction
(𝑄! ∨ 𝑄! ∨ 𝑄! ) in the conclusions of the association rules. Table 1 shows that substi-
tuting the disjunction (𝑄! ∨ 𝑄! ∨ 𝑄! ) by any of these predicates 𝑄! notably decreases
both confidence and support of the association rules.
Thus, the new association rules generated by the algorithm in the extended formal
context (𝐾𝔄 , ∆∨! , ⊨) have higher support and confidence as compared to rules with the
same premise which may be found in the original formal context (𝐾𝔄 , ∆∨! , ⊨).
Table 1. Examples of association rules1.
Rule Support Confidence
Example 1 𝑷𝟏 , … , 𝑷𝒏 → 𝑷 11% 91%
𝑃! , … , 𝑃! → 𝑄! 6% 50%
𝑃! , … , 𝑃! → 𝑄! 3% 23%
𝑃! , … , 𝑃! → 𝑄! 2% 24%
!!
Example 2 𝑷!!𝟏 , … , 𝑷𝒍 → 𝑷 11% 89%
𝑃!!! , … , 𝑃!!! → 𝑄! 4% 35%
𝑃!!! , … , 𝑃!!! → 𝑄! 5% 38%
1
Due to NDA, the details of the attribute list and characteristics 𝑃! , 𝑄! cannot be given. So in
the examples below, the real names of characteristics 𝑃! and 𝑄! have been changed.
𝑃!!! , … , 𝑃!!! → 𝑄! 2% 18%
!!!
Example 3 𝑷!!!
𝟏 , … , 𝑷𝒍 →𝝋 10% 82%
!!! !!!
𝑃! , … , 𝑃! → 𝑄! 5% 40%
𝑃!!!! , … , 𝑃!!!! → 𝑄! 4% 38%
𝑃!!!! , … , 𝑃!!!! → 𝑄! 0.01% 0.08%
𝑃!!!! , … , 𝑃!!!! → 𝑇! 4% 45%
𝑃!!!! , … , 𝑃!!!! → 𝑇! 1% 5%
𝑃!!!! , … , 𝑃!!!! → 𝑇! 0.01% 0.06%
𝑃!!!! , … , 𝑃!!!! → 𝑇! 1% 3%
𝑃!!!! , … , 𝑃!!!! → (𝑇! ∨ 𝑇! ∨ 𝑇! ∨ 𝑇! ) 6% 51%
∨
If we would process association rules just in the formal context (𝐾𝔄 , ∆! , ⊨) without
using the signature 𝜎ℝ , then many conclusions of mined rules will be non-meaningful
disjunctions. Let us consider Example 3 in Table 1. Here 𝜑 = (𝑄! ∨ 𝑄! ∨ 𝑄! ∨ 𝑇! ∨
𝑇! ∨ 𝑇! ∨ 𝑇! ) ∈ ∆∨! , services 𝑄! provide unlimited Internet, and services 𝑇! provide
unlimited SMS. Here 𝑇! is “unlimited free SMS for month with a fixed price”, 𝑇! is
“1000 free SMS for month with a fixed price”, 𝑇! is “unlimited cheap SMS”, and 𝑇!
is “discount for SMS, using with special conditions”. The confidence of the associa-
tion rule 𝑃!!!! , … , 𝑃!!!! → 𝜑 is high enough. The value is much greater than the confi-
dence of rules 𝑃!!!! , … , 𝑃!!!! → 𝑄! and 𝑃!!!! , … , 𝑃!!!! → 𝑇! , but the mobile operator
cannot use this association rule for recommendations, because it contains non-related
services 𝑄! and 𝑇! in the conclusion. However, if we consider the association rule
𝑃!!!! , … , 𝑃!!!! → (𝑇! ∨ 𝑇! ∨ 𝑇! ∨ 𝑇! ), we can see that this rule has low confidence.
5 Conclusion
The paper is devoted to methods for identifying payment plans and services by
mobile operators which would be most useful for the given mobile network subscrib-
ers. We use the Case Model 𝐾𝔄 , 𝜎 for mobile subscriber’s behavior description. The
Case Model is based on depersonalized subscribers’ data provided by mobile opera-
tor. Objects (elements of the model) are mobile subscribers. The signature of the Case
Model consists of unary predicates. These predicates describe individual subscriber’s
features (accruals, traffics) or features of payment plans and services. We construct
the formal context (𝐾𝔄 , ∆, ⊨) based on the Case Model. Then we mine association
rules describing payment plans and services that are commonly used by subscribers
with given features. After that we consider the formal context (𝐾𝔄 , ∆! , ⊨). Our exper-
iments show that interesting association rules have low confidence values in this con-
text. That is why they cannot be used by mobile operator for any recommendations.
To improve association rules quality we deal with an extension of this formal con-
text, the formal context 𝐾𝔄 , ∆!! , ⊨ . Using this context we can find association rules
with high confidence. However, a big part of mined rules have conclusions which are
disjunctions of non-related services, e.g. ‘Song instead of Beep’ and ‘Unlimited Inter-
net’. That is why such association rules could not be used for recommendations
Finally, we consider enriched signature 𝜎𝕄 instead of the signature 𝜎 to find se-
mantically useful disjunctions. Signature 𝜎𝕄 contains predicates that describe specific
features of payment plans and services. Using the formal context (𝐾𝔄 , ∆!𝕄 , ⊨) we
compute association rules such that their conclusions are predicates of the signature
𝜎ℝ . We transform the obtained association rules into association rules of the formal
context 𝐾𝔄 , ∆!! , ⊨ using the Ontological Model 𝐾𝔄 , 𝑇 ! , 𝑇 ! , 𝑇 ! . We substitute the
predicate in the conclusion of an association rule by disjunction of predicates of the
initial signature 𝜎. As the result we obtain association rules of the formal context
𝐾𝔄 , ∆!! , ⊨ . These rules have high confidence and support, and the conclusions of
these rules are completely meaningful for the mobile network operator as well as for
mobile network subscribers. Mined association rules allow making recommendations
for customers who will be interested in information about these services and tariffs.
Acknowledgment The reported study was partially supported by Russian Founda-
tion for Basic Research, research project No. 14-07-00903-a.
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