Logical Representation of Dependencies of Items and the Complexity of Customer Sets

Abstract

The problem of discovering of frequent market baskets and association rules has been considered widely in literatures of data mining. In this study, by using the algebraic representation of market basket model, we propose a concept of logical constraints of items in an effort to detect the logical relationships hidden among them. Via the relationships of the propositional logics and logical constraints of items we propose also the concept of the complexity of customers. As a result we show that every set of customers can be characterized by a logical constraint and can be divided into different blocks that are characterized by quite simple logical constraints. In the natural way the complexity of a customer set is defined as the number of the blocks it contains.

Keywords

• Market basket model
• frequent product
• propositional logics
• lattice

1. INTRODUCTION

Great efforts have been made to discover the information hidden in the sets of market baskets and in the sets of customer transactions. The studies of customer market baskets (MB) and mining the association rules are important in various applications, for example, in decision making and strategy determination of retail economy ([1]). Therefore discovering of large itemsets and association rules attracts the interest of researchers (see [ 8,10 ]). One can notice that in their studies the researchers deal with the set of items (e.g. bread, milk,....) purchased by customers, but did not consider the quantity of each item. It would be interesting also if we know that 70% of customers buy bread and milk, but only 50% of customers buy 1 kg bread and 2 litter milk, while 1% of customers buy 10 kg bread and 1 litter milk. Similar example can be found for association rules. The reasons for quantitative analysis of transactions are evident.

In the previous studies (see [ 5, 6 ]) we have introduced a quantitative analysis of transactions. We are interested not only in the statement ”90% of customers who buy bread and milk also buy butter”, but in the fact ”90% of customers who buy 1 kg bread and 2 litter milk also buy 0,5 kg butter”. By dealing with the quantity of items our approach is somehow different of those in other studies (see [1]) and is suitable for a quantitative analysis of transactions.

Similarly to [ 5, 6 ], instead of itemsets we use market baskets or transactions. In the following we establish the relationships between the structure of the set of market baskets and propositional logics. In Section 2 we recall the algebraic approach in analysing the structure of the set of market baskets which was defined in [ 6 ]. In Section 3 the concept of the constraints of market baskets is defined and we show that every set of market baskets can be characterized by some logical constraint. In Section 4 we introduce the concept of complexity of the sets of market baskets, which we call by the complexity of the customer sets. All these are done via the relationship between the structure of the set of market baskets and propositional logics which are defined in this section. 2.

A GENERALIZATION OF THE MARKET BASKET MODEL

In this section we recall the concepts and results that are established in [ 5, 6 ]. For a finite set of items P = {p1, p2, ..., pn} we consider a market basket (MB) as a tube α = (α[1], α[2], ..., α[n]), where α[i] ∈ N is the quantity of pi in the basket α. The set of all MBs is denoted by Ω. For α, β ∈ Ω where α = (α[1], α[2], ..., α[n]), β = (β[1], β[2], ..., β[n]) we write α ≤ β if for all i = 1, 2, ..., n we have α[i] ≤ β[i]. hΩ, ≤i is a lattice with the natural partial order ≤ (see [ 4 ]). For a set A ⊆ Ω we denote

U (A) = {α ∈ Ω|∀β ∈ A : β ≤ α} and

L(A) = {α ∈ Ω|∀β ∈ A : α ≤ β}.

We denote also by sup(A) and inf (A) the smallest, and the biggest MB in U (A) and L(A), respectively.

For a set A ⊆ Ω and α ∈ Ω we denote by suppA(α) = |{β ∈ A|α ≤ β}| |A| the support of α in A. In word, suppA(α) denotes the rate of all market baskets that exceeds the given threshold (in the form of a sample market basket) α to A. The support of an market basket is a statistical index by which one can estimate the ”vogue” of α in the given group of customers A. Naturally, the market baskets of more support attract better the attention of the shop managers. Some semantic interpretations are needed to avoid the possible misunderstandings: If pi (the name of i-th item) is, for example, bread, then αi is the quantity of bread in the basket (of the customer) α. For α, β ∈ Ω where α = (α[1], α[2], ..., α[n]) and β = (β[1], β[2], ..., β[n]) we write γ = α ∪ β if γ[i] = max{α[i], β[i]} for all i = 1, 2, ..., n. We call α −→ β an association rule of β to α. By the confidence of α −→ β in a set of MBs A we mean the rate confA(α −→ β) = suppA(α ∪ β) suppA(α) For a set A ⊆ Ω, α ∈ Ω and 0 ≤ ε ≤ 1, α is ε-frequent MB, if suppA(α) ≥ ε. The set of all ε-frequent MBs is denoted by ΦεA. In our setting the Apriori Principle is stated as follows: a threshold 0 ≤ ε ≤ 1 there exist α1, α2, ..., αs ∈ Ω where s = |⌈Aε||A|⌉ such that

s ΦεA = [ L(αi).

i=1 We should remark that αi ≤ αj iff L(αi) ⊆ L(αj ). For a set of MBs A and a given threshold ε the basic ε- frequent set of MBs of A is the set of MBs α1, α2, ..., αs for which s i. ΦεA = Si=1 L(αi). ii. ∀i, j : 0 ≤ i, j ≤ s we have αi αj and αj αi For a given A, ε the basic ε- frequent set of MBs of A is ε unique, which we denote by SA. We have: Apriori Principle: For a set A ⊆ Ω, α, β ∈ Ω and 0 ≤ ε ≤ 1, if α ≤ β and β is ε-frequent then α is ε-frequent. The following example was considered in [ 5 ]: Theorem 3: For a set of items P , a threshold 0 ≤ ε ≤ 1 every set of MBs A ⊆ Ω has an unique basic ε- frequent set ε of MBs SA. of A are: Example 1: Consider a set of items P = {a, b, c} and a set of transactions A = {α, β, γ, δ}, where α = (2, 1, 0), β = (1, 1, 1), γ = (1, 0, 1), δ = (2, 2, 0). One can see that 3

and 4 the ε-frequent MBs for σ = (1, 1, 0), η = (1, 2, 0) we have suppA(σ) = 1 1 suppA(η) = . For the threshold ε =

4 2 1 ΦA2 = {(2, 1, 0), (1, 0, 1), (1, 1, 0), (2, 0, 0),

(0, 0, 1), (0, 1, 0), (1, 0, 0), (0, 0, 0)}.

Let us denote

ΦA,k = {α ∈ Ω|∃α1, α2, ..., αk ∈ A : α ≤ {α1, α2, ..., αk}}

One can remark that if k ≤ l then ΦA,k ⊇ ΦA,l and ΦεA = ΦA,k where k = ⌈ε|A|⌉ denotes the smallest integer that is greater or equal to ε|A|. We have Theorem 1 and Theorem 2 (see [ 5, 6 ] for the proof): Theorem 1: For a set of items P = {p1, p2, ..., pn}, a set of MBs A ⊆ Ω and a threshold 0 ≤ ε ≤ 1 an MB α ∈ Ω is ε-frequent iff there exist α1, α2, ..., αk ∈ A such that α ∈ L({α1, α2, ..., αk}) where k = ⌈ε|A|⌉.

By Theorem 1 in [ 5 ] we have proposed an algorithm that creates all ε-frequent MBs of a given set of transactions A in O |kA| . (m + 1)n running time.

Algorithm 1: (Creating all ε-frequent MBs of a given set of transactions A) Input: Set of items P , Set of MBs A ⊆ Ω and a threshold 0 ≤ ε ≤ 1.

Output: ΦεA.

Theorem 2: (Explicit representation of large MBs) For a set of items P = {p1, p2, ..., pn}, a set of MBs A ⊆ Ω and

An algorithm that creates the basic ε- frequent set of MBs O |kA| .m.n in running time for a given set of MBs A ⊆ Ω and a given threshold ε is proposed in [ 5 ]: Algorithm 2: (Creating the basic ε- frequent set of MBs SAε) Input: Set of items P , Set of MBs A ⊆ Ω and a threshold 0 ≤ ε ≤ 1.

Output:SAε One can remark that in the case of large amount of transactions A the basic ε - frequent set of MBs SAε can be generated much more quickly than the set of all ε-frequent set of MBs ΦεA.

Example 2: We continue the Example 1. For the set of transactions A Algorithm 2 generates the basic 21 - frequent 1 set of MBs SA2 = {ρ, θ} where ρ = (2, 1, 0), θ = (1, 0, 1). It means that the family of 21 - frequent set of MBs of A is 1 ΦA2 = L(ρ) ∪ L(θ).

As shown in [ 5, 6 ] we can find all associations with given confidence. For a set of items P , a set of MBs A ⊆ Ω and a threshold 0 ≤ ε ≤ 1 an association α −→ β is ε-confident if confA(α −→ β) ≥ ε. The set of all ε-confident associations of A is denoted by CAε. We have: Theorem 4: For a set of products P , a set of MBs A ⊆ Ω and 0 ≤ ε ≤ 1 an association α −→ β is ε-confident iff |U (α ∪ β) ∩ A|

≥ ε.

|U (α) ∩ A| A natural question for cross marketing, store layout, ...(see, for example, [1]) is to find all association rules with a given confidence. In our generalized model the following theorem shows in a sense an explicit representation of all association rules. More exactly, we show for a given MB α which set of MBs β may be associated to α with a given threshold of confidence. over P . We define the logical constraints of MBs (for short, constraint) as follows:

For MBs ρ, σ where ρ ≤ σ, let us denote

M (ρ, σ) = {η ∈ Ω|ρ ∪ η ≤ σ} It should be remarked that M (ρ, σ) can be represented explicitly. If ρ = (ρ1, ρ2, ..., ρs), σ = (σ1, σ2, ..., σs) then η = (η1, η2, ..., ηs) ∈ M (ρ, σ) if and only if max (ρi, ηi) = σi for all i = 1, 2, ..., s, i.e. ηi = σi in the case ρi σi and ηi ≤ σi in the case ρi = σi.

Theorem 5: (Explicit representation of association rules) For a set of items P = {p1, p2, ..., pn}, a set of MBs A ⊆ Ω, an MB α ∈ Ω and a threshold 0 ≤ ε ≤ 1 there exist α1, α2, ..., αk ∈ Ω such that ∀β ∈ Ω : α −→ β is ε-confident k association rule if and only if β ∈ Si=1 M (α, αi). It was showed in [ 5 ] that Theorem 5 in a sense gives an explicit presentation for association rules and by the following algorithm one can find all ε-confident association rules for given left side.

Algorithm 3: (Creating all ε- confident association rules α −→ β for given α) Input: A set of items P , a set of MBs A ⊆ Ω, a threshold 0 ≤ ε ≤ 1 and an MB α.

k Output: Si=1 M (α, αi).

Example 3: We continue the Example 1. For the set of MBs A (see Example 1), the MB σ = (1, 1, 0) and threshold ε = 12 we should find all MB η such that σ −→ η is ε- confident association rule. We can see U (σ) ∩ A = {(2, 1, 0), (1, 1, 1), (2, 2, 0)} and s := ⌈ε|U (α) ∩ A|⌉ = 2. By step 2 in Algorithm 3 we have k = 4 and α1 = (1, 1, 0), α2 = (2, 1, 0). The set of all MBs η such that σ −→ η is 21 confident association rule is

M (σ, α1) ∪ M (σ, α2) = {(1, 1, 0), (1, 0, 0), (0, 1, 0), (0, 0, 0), (2, 1, 0), (2, 0, 0)} As a result we can see t h′at besides′ the trivial association rules of the form σ −→ σ , where σ ≤ σ we got non-trivial association rules σ −→ (2, 1, 0) and σ −→ (2, 0, 0). In words, in the set of customers A more than 50% of customers that buy a and b also buy 2 a and 1 b items, as well more than 50% of customers who buy a and b also buy 2 a items.

LOGICAL CONSTRAINTS OF MARKET BASKETS

In this section we propose a concept of constraints of MBs which we call the logical constraints of MBs. The reason for our attempt is clear: The constraint (¬α) where α means the meat certainly holds with high support for the vegetarian customer groups, while the constraint (α ∧ β) −→ γ seemingly gains high support from the householder customers, if α, β and γ means milk, egg and flour respectively. In the same way one can easily see that γ −→ α∨β holds commonly for any customer groups.

Let us construct the logical constraints of MBs. For a set of items P = {p1, p2, ..., pn} let Ω be the set of all MBs 1. All α ∈ Ω are constraints. In this case π(α) = U (α) = {β ∈ Ω|α ≤ β} ⊆ Ω. 2. If α is a constraint then (¬α) is a constraint and π(¬α) = (π(α))c where by Ac we denote Ω \ A for A ⊆ Ω

3. If α, β are constraints then

(α ∨ β) is a constraint and π(α ∨ β) = π(α) ∪ π(β). (α ∧ β) is a constraint and π(α ∧ β) = π(α) ∩ π(β). 4. All constraints are constructed as in 1., 2. and 3. As usual, the parentheses are omitted where it causes no confusion. We call π(α) the set of supporting market baskets of α. Two constraints α, β are equivalent, noted by α ≡ β, if π(α) = π(β). A constraint is tautology if π(α) = Ω. The set of all constraints is denoted by C(Ω).

The following properties of propositions in propositional calculus hold also for the constraints: 1. If α, β, γ ∈ GI(Ω) are constraints then α ∨ β ≡ β ∨ α, α ∧ β ≡ β ∧ α α ∨ (β ∨ γ) ≡ (α ∨ β) ∨ γ, α ∧ (β ∧ γ) ≡ (α ∧ β) ∧ γ

3. If α, β are constraints then

¬(α ∧ β) ≡ ¬α ∨ ¬β and ¬(α ∨ β) ≡ ¬α ∧ ¬β 2. If α ∈ GI(Ω) is a constraint then ¬(¬α) ≡ α. 4. For α, β ∈ GI(Ω) the notation α → β is used also for ¬α ∨ β.

The above identities are always true. We call these identities the logical identities. It is easy to see that for a given A in the same way we can define πA(α) = π(α) ∩ A, which we call the relative set of supporting MBs of α. Similarly we say that two constraints α, β are relatively equivalent (in A), noted by α ≡A β, if πA(α) = πA(β). It is easy to verify the following theorem:

Theorem 6:

1. For any finite set of MBs A ⊆ Ω there is a constraint α∗A ∈ GI(Ω) such that π(α∗A) = A. 2. For all β, γ ∈ GI(Ω), β ≡A γ if and only if β ∧ α∗A ≡ γ ∧ α∗A.

Proof: 1) For any finite set of MBs A ⊆ Ω we find the constraint αA ∈ GI(Ω) such that π(α∗A) = A. If P = {p1, p2, ..., pn}, ∗ ρ = (ρ[1], ρ[2], ..., ρ[n]) ∈ Ω then let

One can see that

ρi+ = (ρ[1], ρ[2], ..., ρ[i] + 1, ..., ρ[n]).

n n {ρ} = π(ρ) \ [ π(ρi+) = π(ρ ∧ ^ ¬(ρi+)).

i=1 i=1 n α∗A = _ [ρ ∧ ^ ¬(ρi+)]

ρ∈A i=1 We have A = π(α∗A). 2) The assertion is proved easily by using the definitions. We have β ≡A γ ⇐⇒ πA(β) = πA(γ) ⇐⇒ π(β) ∩ A = π(γ) ∩ A ⇐⇒ β ∧ α∗A ≡ γ ∧ α∗A.

The proof is completed.

One can remark that there are two trivial cases: The first is the case, when α∗A is tautology. In this case ≡A coincides with ≡. This coincidence does not hold in general. We call a set of customers (transactions) complete if αA is tautology. The second one is the case when α∗A is tautologically false. For β ∈ GI(Ω) we denote βA = β ∧ α∗A.

Example 4: We continue the Example 1. Let P = {a, b, c} and a set of transactions A = {α, β, γ, δ}, where α = (2, 1, 0), β = (1, 1, 1), γ = (1, 0, 1), δ = (2, 2, 0). If a = ”Flour”, b = ”Egg”, c = ”Milk”, which can be identified by a = (1, 0, 0), b = (0, 1, 0), and c = (0, 0, 1), respectively, then π(a) = U ((1, 0, 0)) = {(x, y, z)|x ≥ 1}, πA(a) = {α, β, γ, δ} π(b) = U ((0, 1, 0)) = {(x, y, z)|y ≥ 1}, πA(b) = {α, β, δ} π(c) = U ((0, 0, 1)) = {(x, y, z)|z ≥ 1} πA(c) = {β, γ} In this case the constraint a ∧ b → c that may be interpreted as F lour ∧ Egg → M ilk, characterises those customers, who if buy Flour and Egg then must buy Milk. It is easy to see that the set of supporting MBs of this constraint is π(a∧b → c) = {(x, y, z)|x = 0 or y = 0 or z ≥ 1}. One also can see that in this case πA(a ∧ b → c) = π(a ∧ b → c) ∩ A = {β, γ}, i.e. (a ∧ b → c) ≡A c.

It is easily to see that the properties of propositions in propositional calculus (see [ 9 ]) hold also for the constraints in the given set of customers, but the converse is not always true. Although one can verify the following for α, β ∈ GI(Ω) and an arbitrary set of customers A: 1. (α ∨ β)A ≡A βA ∨ αA. 2. (α ∧ β)A ≡A βA ∧ αA.

3. (¬α)A ≡A ¬(αA).

One should distinguish ≡A and ≡.

THE COMPLEXITY OF THE CUSTOMER SETS In this section we propose a criteria for the complexity of the customer sets. The practical aspect of this attempt is clear: every shop manager want have the answer to the question how complex is their customer set. One can remark that the set of customers that contains only one customer is simple. An other simple customer set is the case when the transactions of the customer in the set (that may be a large mass) are ”similar” in some way. In our version the concept of complexity may be understood as following.

Let P = {p1, p2, ..., pn} be a a finite set of items and Ω be the set of MBs over P . We recall that U (α) = {β ∈ Ω|α ≤ β} for α ∈ Ω. We call a set B ⊆ Ω a block of customers if there are α1, α2, . . . , αm ∈ Ω, β1, β2, . . . , βn ∈ Ω such that m n B = \ U (αk) \ [ U (βk)

k=1 k=1 The block is denoted by [α1, α2, . . . , αm|β1, β2, . . . , βn]. We have the following simple theorem: Theorem 7: Let P = {p1, p2, ..., pn} be a a finite set of items and Ω be the set of all MBs over P .

1. Every γ ∈ Ω is a block, i.e. there are α1, α2, . . . , αm ∈ Ω, β1, β2, . . . , βn ∈ Ω such that {γ} = [α1, α2, . . . , αm| β1, β2, . . . , βn]. 2. Every A ⊆ Ω is union of some blocks, i.e. there are 0 ≤ k, α1 , α2k, . . . , αkmk ∈ Ω, β1k, β2k, . . . , βnk ∈ Ω such k k that

k A = [[αi1, αi2, . . . , αimi |β1i, β2i, . . . , βnii ]

i=1

We denote

k c(A) = min{k|∃Bi blocks, i = 1, . . . , k : A = [ Bi} i=1 k We call c(A) the complexity of A. If A = Si=1 Bi where k k = c(A) then we say that A = Si=1 Bi is a minimal representation of A by blocks. We should notice that a set A ⊆ Ω may have different minimal representations, even if we does not take in account of the permutation of blocks. Let us consider an example: Example 5: (Following the Example 4) As considered in Example 4 let α = (2, 1, 0), β = (1, 1, 1), γ = (1, 0, 1) and let θ = (1, 1, 2), λ = (1, 0, 2). One can verify

{γ} = U (γ) \ {U ((2, 0, 1)) ∪ U (β) ∪ U (λ)} and {β, γ} = U (γ) \ {U ((2, 0, 1)) ∪ U ((2, 1, 1)) ∪ U (θ) ∪ U (λ)} Thus we have c({β, γ}) = c({γ}) = 1. One can verify also that c({α, γ}) = 2.

We have also c({γ, θ, λ}) = 2 and one can verify that: {γ, θ, λ} = [γ|β, λ] ∪ [λ|(1, 2, 2), (1, 0, 3)] = [γ|β, (1, 0, 3)] ∪ [θ|(1, 2, 2), (1, 1, 3)]

We use propositional logics in finding the blocks of a given set of MBs. In propositional logics a full conjunctive clause is an expression of the form Vn

k=1 xk in which xk are all variables in either positive or negative form. A full disjunctive normal form (full DNF) is a disjunction of full conjunctive clauses. It is well known in propositional logics that all logical formulas can be transformed into full DNF(see, for example, [ 9 ]). More exactly, if α is a constraint of items (which is namely a logical formula) then by using simple transformations we can find the full DNF of α:

n mi ni α = _[ ^ βki ∧ ^ (¬γki)] i=1 k=1 k=1 where for all β ∈ Ω, β appear in every clause of α in either positive or negative form. One can verify that

mi ni U ([ ^ βki ∧ ^ (¬γki)]) = [β1i, . . . , βmii |γ1i, . . . , γnii ] k=1 k=1 is a block. By this in fact we have proved the following theorem.

Theorem 8:

1. There is an algorithm by that for any constraint of MBs α we can find the system of full customer blocks of U (α), i.e. we can find

{[αi1, αi2, . . . , αimi |β1i, β2i, . . . , βnii ]|i = 1, 2, . . . , n} where αi1, αi2, . . . , αimi , β1i, β2i, . . . , βnii are all MBs that appear in α, such that

k U (α) = [[αi1, αi2, . . . , αimi |β1i, β2i, . . . , βnii ]

i=1 2. The decomposition of U (α) into full customer blocks is unique. 3. The minimal representations of U (α) can be obtained by decomposition of U (α) into full customer blocks and by combining some full customer blocks into one to reduce the number of blocks. 4. The complexity of U (α) does not exceed the number of full clauses in the full DNF of α.

Proof: 1. The algorithm that is well known in propositional logics (see [ 9 ]) converts a constraint of MBs α into full DNF. By this algorithm we can find the system of full customer blocks of U (α). 2. This is a result in propositional logics (see [ 9 ]). 3. If

k U (α) = [[αi1, αi2, . . . , αimi |β1i, β2i, . . . , βnii ]

i=1 is a minimal representations of U (α) where, for example, some block [αi1, αi2, . . . , αimi |β1i, β2i, . . . , βnii ] is not full. Then using the equivalence X ≡ (X ∧ a) ∨ (X ∧ ¬a) we can insert into the block the missing item a. In result we have the decomposition of U (α) into full customer blocks, which, accordingly to 2., is unique. The reverse transformation converts the full DNF of α into the given minimal representation of U (α). 4. The proof is evident by definition of the complexity. The complexity of U (α) is the number of clauses in the minimal representation of α that does not exceed the number of full clauses in the full DNF of α.

Let us consider an example:

Example 6: (Following the Example 4) Let a = ”Flour”, b = ”Egg”, c = ”Milk”, which can be identified by a = (1, 0, 0), b = (0, 1, 0) and c = (0, 0, 1), respectively. The constraint α = (a ∧ b → c)(¬b → (a ∨ c)) characterises the set of all those customers, who if buy flour and egg then buy also milk, and if do not buy egg, then would buy flour or milk. Let us denote this set of customers by A, i.e. A = U (α). By using simple transformations we have the full DNF of α: α = (a ∧ b ∧ c) ∨ (¬a ∧ b ∧ c) ∨ (¬a ∧ b ∧ ¬c) ∨ (¬a ∧ ¬b ∧ c) ∨(a ∧ ¬b ∧ c) ∨ (a ∧ ¬b ∧ ¬c) The full customer blocks of A = U (α) is

A = U (α) = This means that A can be characterized as the union of three blocks of customers: the first block contains those customers who buy milk, the second block contains all customers who buy flour but do not buy eggs, and the third one is the block of all customers who buy eggs but do not buy flour. One can see that the complexity of A is 3 and the structure of A is clear. 5.

CONCLUSION

In this short paper by using the logical structure of the market baskets we have introduced the concept of constraints of market baskets. We showed that every set of market baskets can be characterized by some constraints. The relationships between the structure of the set of market baskets and propositional logics are discovered. Similarly to the well known methods and results in propositional logics we showed that every set of customers can be represented, in different ways, as union of some blocks, and the number of these blocks can be considered as a complexity of the given set of customers.

In fact, the shop managers have to deal with large amounts of itemsets, as well as with the large amounts of market baskets. Finding the efficient algorithms to discover the logical constraints of market baskets is always the problem of practical value.

One should remark that in the last part of this paper we define the complexity for really the set of market baskets, not for the set of customers, although we call it by the complexity of the customer sets. The difference is evident: some customers may buy the same market basket. In this sense the analysis of the customer sets requires more research.

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