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 Ontological Model of the domain “Mobile Networks” is constructed in the scope of this research. The Ontological Model of the domain is constructed by integration 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 contexts where objects are subscriber models and attributes are formulas of predicate logic. We investigate concept lattices and association rules of these formal contexts. Knowledge about optimal payment plans and services for a given subscriber is generated automatically with the help of the association rules.
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 clients' 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 messages 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 [
Formal Concept Analysis is a well-known formalism in data analysis and
knowledge engineering, see recent surveys [
Association rules for optimizing structures of menus for accessing mobile network
services were constructed in [
Fuzzy concept lattices were first introduced in [
Our research is devoted to methods for identifying payment plans and services which would be optimal for a given mobile network subscriber. Such knowledge allows mobile operator to make really useful recommendations for subscribers.
We base our research on the model-theoretic approach to domain formalization
[
The ontological model of the domain “Mobile Networks” is constructed by integration 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 concept lattices and association rules of these formal contexts to get high-quality recommendation. To do this, we consider extensions of attribute sets of formal contexts.
In [
In this paper we consider finite extensions of the initial finite context. We use
interrelation between axiomatizable classes and FCA [
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 extended 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 2.1
Case Model
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
Proposition 2 which is necessary for proofs of Propositions 4 and 5 in Section 2.2. The proofs
of the statements are based on [
An algebraic system (a model) is a tuple = ; !, . . . , !, !, . . . , !, !, . . . , ! , where the set = is called universe, !, . . . , ! are predicates defined on the set , !, . . . , ! are functions defined on the set and !, . . . , ! are constants. The tuple = !, . . . , !, !, . . . , !, !, . . . , ! is called signature of the algebraic system .
Denote by () the set of all free variables of a formula . A formula having no free variables is called sentence. For a signature we denote: () ⇋ { | ℎ }, ! ⇋ ∈ = {}}, () ⇋ { | ℎ } and () ⇋ { | ℎ }.
Here = {} means that each formula ∈ ! has just one free variable, which is the fixed variable .
Consider a signature and a model ∈ (). For a sentence ∈ () we denote ⊨ if is true in the model . For a formula (!, … , !) ∈ () we write ⊨ if ⊨ ∀! … ∀!(!, … , !).
Definition 1. Let ⊆ (). For a formula ∈ () we denote ⊨ if ⊨ for any ∈ . For a set of formulas ⊆ () we denote ⊨ if ⊨ for any ∈ and ∈ . For a set of formulas ⊆ () we denote
⇋ ! ⇋ ∈ ⊨ ∈ }.
A class ⊆ is called axiomatizable if there exists a set ⊆ () such that = { ∈ () | ⊨ }.
For the aims of our research we need to generalize the notion of relatively
axiomatizable class [
Definition 2. Let , ! ⊆ and ⊆ (). We say that the class ! is axiomatizable in the class relatively to the set of formulas if there exists a set ⊆ such that ! = { ∈ | ⊨ }.
Notice that the class ! ⊆ is axiomatizable if and only if ! is axiomatizable in the class = relatively to the set of formulas = ().
Definition 3. For ⊆ () and ⊆ () we denote , ⇋ ! ! ℎ } and !() ⇋ { ∈ | ⊨ }. The set of formulas !() is call -type of .
Note that ! ∈ (, ) if and only if ! = { ∈ | ⊨ !(!)}.
For each class ⊆ () and set ⊆ () we consider the formal context
(, , ⊨), with derivation operator ()′ [
Remark 1. Let ⊆ (), ⊆ () and ⊆ . Then ′ = !().
For a formal context cepts of the formal context
we denote the lattice of formal con
Proposition 1. Let ⊆ () , ⊆ () , ⊆ and ⊆ . Then , ∈ , , ⊨ if and only if is axiomatizable in the class relatively to the set of formulas and = !().
Proof. (⇒) Let , ∈ , , ⊨ . Then = ′, so = !() by Remark 1. We have = ′, hence = { ∈ | ⊨ } and ⊆ . Therefore, by Definition 2, the class is axiomatizable in the class relatively to the set of formulas .
(⇐) Let the class be axiomatizable in the class relatively to the set of formulas and ⊆ . So there exists ⊆ such that = { ∈ | ⊨ }. Then in the formal context (, , ⊨) we have ′ = . So ′′ = . The set = !() , thus = ′ by Remark 1. Therefore, , ∈ , , ⊨ .
Corollary 1. Let , ! ⊆ and ⊆ (). 1. ! ∈ (, ) if and only if !, !(!) ∈ , , ⊨ .
2. ! = !′′ if and only if ! is axiomatizable in the class relatively to the set of formulas .
Therefore, the classes which are axiomatizable in a class relatively to a set of formulas Δ are exactly extents of the formal concepts of the formal context (, , ⊨).
We consider (, ) as a set ordered by inclusion ⊆. So (, ) is a lattice.
Proposition 2. The lattices (, , ⊨) and (, ) are isomorphic, i.e., (, , ⊨) ≅ (, ), for any ⊆ () and ⊆ ().
Proof. Let us consider the mapping ℎ: (, , ⊨) → (, ) defined as follows: ℎ , = for any , ∈ (, , ⊨) . By Proposition 1 for any , ∈ (, , ⊨) we have ℎ , = ∈ (, ) . For each ∈ (, ) it is true that , !() ∈ (, , ⊨), so ℎ , !() = . Thus the mapping ℎ is onto.
For any !, ! , !, ! ∈ (, , ⊨) we have: ! ⊆ !. Hence the mapping ℎ preserves the partial order.
!, ! ≤ !, ! iff 2.2
Description of the Case Model Further we consider signatures consisting of a finite set of unary predicate symbols, i.e. =< !, … , ! >. We consider the set ⊆ () for different signatures 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 ! and all considered signatures are its subsets.
Consider a finite set = !, … , ! of subscribers of a given mobile network and fix a signature = ℙ ∪ ℚ where ℙ is a set of personal characteristics of subscribers and ℚ is a set of payments plans, services and options. Each of these sets has a hierarchical structure. There are more details about the signatures ℙ and ℚ below. For each subscriber ! we know which characteristics (presented by signature predicates from ) are true and which characteristics are false. Thus, for each subscriber ! there is a one-element model = ! , which is called a case of the domain .
Consider the Case Model = , defined by a set of cases , … , [
Denote by = , … , the class of cases (one-element models) generated by the set of subscribers !, … , ! .
Note that = ; | ∈ ; ≤ . Here the notation = ; ≤ means that the model is a submodel of the model . 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 ⊆ !(). In particular, we consider Δ! = | ∈ ⊆ !(). Denote by ! = , ∆, ⊨ the formal context having the set of objects , the set of attributes Δ and the incidence relation ⊨. Denote ! = , ∆!, ⊨ .
∆⊆ !() is a set of properties of the cases ∈ , which are definable by
formulas of the signature . When we change the set ∆ we change the set of attributes of the
formal context keeping fixed the set of objects . Reductions and expansions of
formal contexts were studied in [
Let us consider two formal contexts ! = , !, and ! = , !, . Suppose that ! ⊆ !, ⊆ and = ′′ в !. Then = ′′ in !.
We define a mapping : (, !, ) → (, !, ) as follows: , ! = (, !), where ⊆ , ! ⊆ !, ! ⊆ !, = ′′, ! = ′ in the context ! and ! = ′ in the context !.
Remark 2. The mapping : (, !, ) → (, !, ) is an isomorphic embedding of the lattice (, !, ) into the lattice (, !, ).
Next consider an arbitrary signature ! and an arbitrary class ! ⊆ (!).
Remark 3. Let ∆⊆ (!) and !, … , ! ∈ (!). Then the mapping : (!, ∆, ⊨) → (!, ∆ ∪ !& … &! , ⊨) is an isomorphism of lattices.
Corollary 2. а) The sets of association rules of the formal contexts (!, ∆, ⊨) and (!, ∆ ∪ !& … &! , ⊨) coincide up to the substitution of the formula !& … &! by the set !, … , ! .
b) The sets of attribute implications of the formal contexts (!, ∆, ⊨) и (!, ∆ ∪ !& … &! , ⊨) coincide up to the substitution of the formula !& … &! by the set !, … , ! .
Corollary 3. If ∆, ∆!⊆ (!), ∆⊆ ∆! and the set ∆!\∆ consists of some conjunctions of formulas from ∆ then the sets of attribute implications as well as the sets of association rules of the formal contexts (!, ∆, ⊨) and (!, ∆!, ⊨) coincide up to the substitution of the conjunctions from ∆!\∆ by the corresponding sets of formulas.
Remark 4. Let !, ! ∈ . Then the mapping : (, ∆!, ⊨) → (, ∆! ∪ !() ∨ !() , ⊨) is an isomorphic embedding of lattices; in the general case this mapping is not an isomorphism. Moreover, in the general case (, ∆!, ⊨) ≇ , ∆! ∪ !() ∨ !() , ⊨ .
Corollary 4. In the general case if we add a disjunction ! ∨ … ∨ !() to the set of formulas ∆!, where !, … , ! ∈ , then the set of association rules of the formal context (, ∆!, ⊨) will be changed.
Denote ∆∨!= ∆! ∪ ! ∨ … ∨ !() , | ! ∈ .
We will be adding disjunctions of signature predicates into the set ∆! for improving 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 (, ∆!, ⊨) instead of the formal context (, ∆!, ⊨).
∨
Definition 4. We say that a set of formulas ⊆ (!) is closed under disjunction if ( ∨ ) ∈ for any , ∈ .
Proposition 3. Let ⊆ and ∆⊆ (). If the set of formulas ∆ is closed under disjunction then the lattice (, ∆) is distributive.
Proof. Assume that ∆⊆ () and !, ! ∈ (, ∆). Then !, ! ⊆ and there exist Γ!, Γ! ∈ ∆ such that ! = ∈ | ⊨ Γ! and ! = ∈ | ⊨ Γ! .
Denote Γ! = Γ! ∪ Γ! and Γ! = ∨ ) | ∈ Γ! ∈ Γ! . Then ! ∩ ! = ∈ | ⊨ Γ! , hence ! ∩ ! ∈ (, ∆).
Let ∈ . Then is a one-element model. Therefore for any ∈ Γ! and ∈ Γ! we have: ⊨ ∨ ⇔ ⊨ ∀! … ∀! (!, … , ! ∨ (!, … , !)) ⇔ ⇔ ⊨ ∀! … ∀! !, … , ! ⊨ ∀! … ∀! !, … , ! ⇔ ⇔ ⊨ ⊨ , where ∨ = !, … , ! .
Assume that ∈ ! ∪ ! , then ∈ ! ∈ ! ⇒
⇒ ∈ ⊨ Γ! ∈ ⊨ Γ! ⇒ ⇒ ∈ ⊨ Γ! ⊨ Γ! ⇒ ∈ ⊨ Γ!.
Next, suppose that ⊭ Γ! and ⊭ Γ!. So there exist ∈ Γ! and ∈ Γ! such that ⊭ and ⊭ . Then ⊭ ∨ , so ⊭ Γ!.
Thus, if ⊨ Γ! then ⊨ Γ! ⊨ Γ! . Hence, if ∈ and ⊨ Γ! then ∈ ! or ∈ !, so ∈ ! ∪ ! .
Therefore, ! ∪ ! = ∈ | ⊨ Γ! and ! ∪ ! ∈ (, ∆).
We proved that ! ∩ ! , ! ∪ ! ∈ (, ∆) for any !, ! ∈ (, ∆). Hence, the lattice (, ∆) is distributive.
Proposition 4. The lattice of formal concepts (, ∆∨!, ⊨) is distributive. Proof: in virtue of Proposition 2 and Proposition 3.
However the initial formal context (, ∆!, ⊨) does not have this good property.
Remark 5. In the general case the lattice of formal concepts (, ∆!, ⊨) is not distributive. It means that there exists a class such that the lattice (, ∆!, ⊨) is not distributive.
Remark 6. Let ∆ ⊆ ! , ∆!⊆ ∆ and the set ∆\∆! consists of some conjunctions of formulas from ∆!. Then there exists a class such that the lattice (, ∆, ⊨) is not distributive.
For the set of all formulas the situation is better.
Proposition 5. 1) The lattice of formal concepts (, ! , ⊨) is distributive. 2) The lattice of formal concepts (, , ⊨) is distributive.
Proof: in virtue of Proposition 2 and Proposition 3.
Association rule mining for the original context , ∆!, ⊨ 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 , ∆!, ⊨ 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 disjunctions of signature predicates to ∆! and use context , ∆∨!, ⊨ on next steps.
There are two problems with association rules that were mined using formal context (, ∆∨!, ⊨). 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 subscribers. So experts should process all rules and select only meaningful rules. Second, processing the whole formal context (, ∆∨!, ⊨) is very laborious computational procedure.
To solve these problems we are moving from the Case Model = , to the Ontological Model , !, !, ! of the domain. We add new unary predicates to the signature to describe meaning of payment plans and services. Using new predicates (from the signature ℝ) we generate automatically meaningful disjunctions of original predicates from the signature .
Ontological Model of the domain
Ontology
Ontological Model of the domain consists of four parts [
(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 = , .
(4) Estimated and probabilistic knowledge: probabilistic and confidential
estimates, fuzzy values of sentences [
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
meaning of the concepts of the domain [
The set of attributes ℙ, “Individual subscribers’ feature” consists of two parts: ℙ!, “payment plans” and ℙ!, “accrual”. Every part ℙ! ( = 1, 2) consists of two subparts, such as ℙ!!, “traffic (and accrual) without roaming inside operator network”,…, !!", “traffic (and accrual) in common roaming”, … . Each of the listed signatures consists of more detailed categories, e.g., !!!!, “Traffic SMS without roaming inside operator network”. Every category !!"# contains finite number of signature symbols !!!", … . , !!!" . For example, !!!! = “Traffic of SMS without roaming inside network for subscriber is not more than 50 SMS in month” and !!!" = “Traffic of SMS without roaming inside network for subscriber is more than 50 SMS in month”.
Signature ℚ consists of two parts: ℚ!and ℚ!. Part ℚ!is “payment plans”, it has hierarchical structure and consists of symbols of unary predicates. Each unary predicate describes the presence or absence of connected payment plan for subscribers. Signature ℚ!, “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 ! ⊆ !(ℙ ∪ ℚ). We introduce the following axioms.
Axioms of hyponym-hyperonym. Hierarchical structure of the signature ℚ! is represented by axioms such as:
(!!"# → !!" ) (!!"() → !()).
Axioms of completeness. For each predicates inside every class ℙ!"# and class ℚ! for a given subscriber there must be at least one true predicate. The schemes of such axioms are the following:
∨ | () ∈ ℙ!"# ∨ | () ∈ ℚ! .
Axioms of including. For example, if payment plan contains “more than 100 free SMS” then it contains “more 50 free SMS”. The schemes of such axioms are the following:
(!!"!# → !!"!# ), where ! < !.
Next we construct an extension ℙ ∪ ℚ of the signature by additional unary predicates that describe properties of payment plans and interests of subscribers. For that we introduce two types of concepts:
1) Concepts ℝ. 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 Internet package and etc. With the help of ℝ we can give formal definition for payment plans and services, i.e., formal definition of predicates of the signature ℚ.
2) Concepts describing subscriber’s interests, e.g., reducing the costs of calls, SMS, etc.
Concepts from ℝ ∪ are used for automation of construction formulas as
attributes in formal contexts for association rules mining. Notice that the pair ,
forms the ontology of the domain [
In the next step we introduce a new set of axioms ! ⊆ !() and call it the domain axioms. This set will be used for describing various characteristics of payment 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 ℙ! and payments denoted by predicates from ℙ!, with regard to activated payment plans from ℚ!and services from ℚ!.
Axioms ! 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 following formula as an example of a domain axiom:
(! → ¬! ), ℎ ! ∈ ℚ!, ! ∈ ℚ!.
This formula declares the following: if a subscriber has payment plan ! activated, then service ! cannot be activated for this subscriber. Note that a mobile network company can naturally change its decision for not supporting simultaneously the precise 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 association rules, we use the Ontological Model of the domain.
Definition 4. An Ontological Model of a domain is a tuple , !, !, ! , where ! is an analytical theory of the domain, ! is a theory of the domain, and ! is a fuzzy theory of the model .
The analytical theory ! of the domain under consideration is axiomatized by the sentences ! which are axioms of the domain ontology. A theory ! of the domain is axiomatized by the axioms ! of the domain.
Formula definitions of predicates from ℚ (which present payment plans and services) are defined by construction of ontological projection.
Definition 5. Consider the Ontological Model , !, !, ! , let ∈ ℚ. Denote ! = ∈ ! ℝ | ! ⊢ → .
An ontological projection of the predicate on the signature ℝ is the formula !!ℝ = & | ∈ ℝ () ∈ ! .
A projection of the predicate on the set of formulas ! ℝ is the formula !!ℝ = &! = & | () ∈ ! .
Let us consider the formal context ! = , ∆, ⊨ . We search association rules with the following requirements: 1) а) the premise of the association rule is included in the set ∆↾ℙ or b) the premise of the association rule is included in the set ∆↾ ℙ ∪ ℚ ; 2) a) the conclusion of the association rule belongs to the set ∆↾ ℚ or b) the conclusion of the association rule belongs to the set ∆↾ ℝ or c) the conclusion of the association rule belongs to the set ∆↾ ℝ ∪ ; 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 ! = , ∆, ⊨ is
included in the fuzzy theory ! of the model [
Then the software system automatically processes obtained association rules. For example, consider an association rule with one-element conclusion belonging to ℝ. 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 such that P belongs to the ontological projection of !. 4
Software Implementation Using the results of the presented investigation, we have developed a software for mining association rules in the formal context (, ∆∨!, ⊨). We have found out that adding predicates from ℝ to the formal context gives us the possibility to find association rules with high confidence and support. Conclusions of such rules are transformed into disjunctions of predicates from ℚ with the help of the operator of ontology projection. Obtained association rules seem to be useful for mobile network companies. The software processes the impersonal data for more than 10 million subscribers. 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 attributes is quantitative, most part of attributes (more than 1200) are binary.
Let us notice that attributes of connected payment plans, services and personal attributes 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 () ∈ ℝ 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!, where ! is “Unlimited”, ! is “United”, and ! is “Online”. These payment plans provide unlimited access to Internet with different connection speed and different price. Formally, in terms of ontological projections, it means that () ∈ !!, () ∈ !!, () ∈ !! and ∉ ! for every ∈ ℚ\ !, !, ! .
Mined association rules have premises with various sets of personal features of subscribers from ℙ and the conclusion . The automatically chosen rules have rather high confidence and support (see examples 1 and 2, table 1).
After that the predicate is substituted by the equivalent disjunction (! ∨ ! ∨ !) in the conclusions of the association rules. Table 1 shows that substituting the disjunction (! ∨ ! ∨ !) by any of these predicates ! notably decreases both confidence and support of the association rules.
Thus, the new association rules generated by the algorithm in the extended formal context (, ∆∨!, ⊨) have higher support and confidence as compared to rules with the same premise which may be found in the original formal context (, ∆∨!, ⊨). 1 Due to NDA, the details of the attribute list and characteristics !, ! cannot be given. So in the examples below, the real names of characteristics ! and ! have been changed. Example 3 !!!, … , !!! → ! !!!, … , !!! → !!!!, … , !!!! → ! !!!!, … , !!!! → ! !!!!, … , !!!! → ! !!!!, … , !!!! → ! !!!!, … , !!!! → ! !!!!, … , !!!! → ! !!!!, … , !!!! → ! 2% 10% 5% 4% 82% 40% 38% !!!!, … , !!!! → (! ∨ ! ∨ ! ∨ !) 6% 51%
If we would process association rules just in the formal context (, ∆∨!, ⊨) without using the signature ℝ, then many conclusions of mined rules will be non-meaningful disjunctions. Let us consider Example 3 in Table 1. Here = (! ∨ ! ∨ ! ∨ ! ∨ ! ∨ ! ∨ !) ∈ ∆∨! , services ! provide unlimited Internet, and services ! provide unlimited SMS. Here ! is “unlimited free SMS for month with a fixed price”, ! is “1000 free SMS for month with a fixed price”, ! is “unlimited cheap SMS”, and ! is “discount for SMS, using with special conditions”. The confidence of the association rule !!!!, … , !!!! → is high enough. The value is much greater than the confidence of rules !!!!, … , !!!! → ! and !!!!, … , !!!! → ! , but the mobile operator cannot use this association rule for recommendations, because it contains non-related services ! and ! in the conclusion. However, if we consider the association rule !!!!, … , !!!! → (! ∨ ! ∨ ! ∨ !), 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 subscribers. We use the Case Model , for mobile subscriber’s behavior description. The Case Model is based on depersonalized subscribers’ data provided by mobile operator. 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 (, ∆, ⊨) 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 (, ∆!, ⊨). Our experiments show that interesting association rules have low confidence values in this context. 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 context, the formal context , ∆!!, ⊨ . 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 Internet’. That is why such association rules could not be used for recommendations
Finally, we consider enriched signature instead of the signature to find semantically useful disjunctions. Signature contains predicates that describe specific features of payment plans and services. Using the formal context (, ∆!, ⊨) we compute association rules such that their conclusions are predicates of the signature ℝ. We transform the obtained association rules into association rules of the formal context , ∆!!, ⊨ using the Ontological Model , !, !, ! . We substitute the predicate in the conclusion of an association rule by disjunction of predicates of the initial signature . As the result we obtain association rules of the formal context , ∆!!, ⊨ . 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 Foundation for Basic Research, research project No. 14-07-00903-a.