A Transaction-based Neighbourhood-driven ... - Semantic Scholar

A Transaction-based Neighbourhood-driven Approach to Quantifying Interestingness of Association Rules B. Shekar

Rajesh Natarajan

Quantitative Methods and Information Systems Area Indian Institute of Management Bangalore Bannerghatta Road, Bangalore – 560 076 Karnataka, INDIA. [email protected]

Information Technology and Systems Group Indian Institute of Management Lucknow Prabandh Nagar, Lucknow-226 013 Uttar Pradesh, INDIA. [email protected]

Abstract

item interactions and the relationships arising from them. It should be noted that we consider only the transactions containing customer purchases. We introduce and intuitively evolve three measures for capturing various aspects of item-relatedness namely, MI (a modification of confidence), CI and SI. We present an illustrative example. The three item-relatedness measures are then combined to evolve a total relatedness coefficient (TR). We then develop an interestingness coefficient (IC) using the basic relationship - relatedness and interestingness are mutually inverse concepts. We compare IC with two commonly used objective measures of interestingness and bring out its intuitiveness.

In this paper, we present a data-driven approach for ranking association rules (ARs) based on interestingness. The occurrence of unrelated or weakly related item-pairs in an AR is interesting. In the retail market-basket context, items may be related through various relationships arising due to mutual interaction, ‘substitutability’ and ‘complementarity.’ Item-relatedness is a composite of these relationships. We introduce three relatedness measures for capturing relatedness between item-pairs. These measures use the concept of function embedding to appropriately weigh the relatedness contributions due to complementarity and substitutability between items. We propose an interestingness coefficient by combining the three relatedness measures. We compare this with two objective measures of interestingness and show the intuitiveness of the proposed interestingness coefficient.

1. Introduction An important problem in association rule (AR) mining is the generation of a large number of mined rules. Usage of interestingness measures to rank ARs is a solution. Objective measures [1, 12, 13] quantify the interestingness of a rule in terms of rule structure and the underlying data used in rule generation. Subjective measures [2, 3, 4, 5, 6, 8, 10, 18], in addition incorporate views of the user while evaluating interestingness of patterns. Here, we consider the retail market-basket context. ARs reveal items that are likely to be purchased together. ARs that contain weakly related or unrelated items are interesting. This is because frequent co-occurrence of such items is unexpected. We present a data-driven approach to evaluating interestingness. The paper is organized as follows. In the following section, we adopt a functional point of view to classify

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2. Item Relatedness Any item possesses many attributes or properties. A subset of these properties implies a function – the purpose for which the item is manufactured. We consider only this primary function. In addition, an item may also possess many secondary functions. We define a simple function, as a function that cannot be sub-divided i.e. it is atomic. A compound function is one that is realized by combining one or more simple or compound functions. A scenario or situation may consist of many such functions. A set of items possessing these functions to varying extents satisfies the scenario either partially or completely. A combination of many related scenarios constitutes a context, for example, a household context, an industrial context, etc. Interacting functions give rise to relationships between the items. Any two items can be related through many such relationships. Item-relatedness is a composite of these relationships. Relatedness is influenced by two attributes, strength of a relationship and cardinality of the set of relationships. Relatedness is directly proportional to the strength and also to the cardinality. Consider two items x and y. When they are brought together in a context, their primary functions may interact in one of the following ways.

2.1. Complementarity

2.3. Non-dependence

Two items may be termed complementary if they together find application. The primary function of item x may need the primary function of item y when they are used together. This interaction between the two primary functions gives rise to a new compound function f. For example, a knife and a fork can be considered complementary to each other. The attribute of sharpness enables a knife to cut objects. A fork is needed to hold the object. Absence of any one of the items usually inhibits the effective execution of the compound function. We identify two shades of complementarity; intrinsic and flexible. The primary functions of two items, say x and y, may be inextricably bound together. Also, x and y may not serve any meaningful function independently. The compound function f, implied by the primary functions of items x and y can be realized if and only if the two items function together. We refer to this type of complementarity as intrinsic complementarity. For example, consider the case of a 0.5 mm clutch pencil and 0.5 mm leads. Though these items are sold separately in a retail store they are intrinsically complementary to each other. Neither of them can be used separately for writing on a piece of paper. Consider another situation in which the primary functions of two items not only interact with each other but also combine with primary functions of other items to imply different functions. For example, items x and y may interact with item z in one situation to imply function f1. In another scenario, the same two items may interact with item w to imply function f2. Flexible complementarity is the ability of the primary functions of two items to serve multiple functions. In general, complementary items tend to be purchased together in purchase transactions.

In case of non-dependence, the primary functions of x and y do not interact with each other to serve any other compound function. Primary functions of x and y, however, may separately find application in different domains and different contexts. Since their functions do not interact, the likelihood of being used together in the same or similar contexts is quite low. Non-dependent items are likely to be from different domains, for example, kitchen knife and 0.5 mm lead. A combination of non-dependent items is unlikely to be purchased with a frequency higher than the product of the expected frequencies of the individual items. Relatedness between two items increases if complementarity and substitutability increases. A pair of items can take shades of complementarity and substitutability depending on the context. Therefore, on some occasions, we do find substitutes being purchased together in the same transaction. Two items can exhibit different relationships depending on the function demanded and the context. In this paper, we are concerned only with the primary functions of items.

2.2. Substitutability An item functionally substitutes another item if it serves the other’s function to a significant extent. Thus, a pen can substitute a pencil as far as the function of writing on paper is concerned. Substitutes have similar or closely related primary functions. In general, substitutes belong to the same generic category. Generally, substitution is with respect to one single function in a particular context. Although substitutes are not used together, they are strongly related. Two items that substitute each other are likely to possess many similar or closely related attributes. If two items are substitutes to each other, then they are not likely to be purchased in the same transaction. However, two substitutable items are likely to be bought with identical or similar items that go with them, in separate transactions.

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3. Measures for Item Relatedness We clarify some of the assumptions regarding customer purchasing behaviour that we make in our approach to item relatedness. First, we assume that items are purchased mainly for their primary function. Hence, we are concerned mainly with primary functions of items. Next, we assume that customers purchase items mainly for self or family consumption over a short period. Consequently, item usage patterns get reflected in customer purchasing behaviour. Hence, we draw conclusions about relatedness of items by examining customer transactions. Consider a database of retail market-basket transactions. Let I be the set of items available for sale in the retail market. I= {a, b, c, …}. Database D consists of a set of transactions, T. Each transaction t consists of a set of items purchased by a customer on a single buying instance at a retail store. Thus, x∈t⇒x∈I; t⊆I. Let x and y be two items occurring in an association rule whose relatedness is of interest to us. This means that t xy t

≥ minsup where |t| is the total number of transactions

in the database. Let tx represent the set of transactions that contains item x but not y; ty represent the set consisting of transactions containing item y but not x, and txy represent the set consisting of transactions containing both x and y. Each transaction in tx, ty and txy might contain items other than x and y accordingly. The remaining transactions

contain neither of the two items. In order to compute relatedness between items x and y, we need to examine txy and neighbourhood items in tx, ty and txy. Other items purchased with items x and y, can also contribute substantially to their relatedness if they are similar. This is because x and y might find applicability with other items in many more contexts than all by themselves. This increases their usefulness and relatedness. Let Z be the set of items/item-sets purchased together with x and y. In particular, an item-set z∈ Z if t xyz txy

≥ minsup where minsup is user-specified threshold.

The item-set z and the set Z are called co-occurring neighbour of {x,y} and co-occurring neighbourhood respectively. Co-occurring neighbours are likely to be used together with items {x,y}. Therefore, a relationship exists between co-occurring neighbours and item-pair {x,y}. Similarly, let M and N be item-sets that are purchased with items x and y in transaction sets tx and ty respectively. Then, M∩N is the set of item-sets that are purchased with each of x and y separately. We call set M∩N the non co-occurring neighbourhood of {x,y} and each item w∈M∩N the non co-occurring neighbour. It should be noted that “neighbourhood”, as defined above, represents the item-set frequently purchased along with a specific item-pair. A context, on the other hand, is a combination of many related scenarios each of which may refer to a set of items. A context is not specific to any particular item-pair. All neighbours of {x,y} need not occur in a context containing {x,y}. Here again we assume that items comprising sets M and N occur in respective transactions in significant numbers. In addition, transaction sets tx and ty are also assumed to be significant.

3.1. MI (Mutual Interaction) A customer is likely to purchase items that are used together in the same transaction. Thus, two items x and y, are related if they are purchased together in large numbers. The more the co-occurrence, greater is the relatedness. Co-occurrence of the two items implies that the function of item x and the function of items y interact with each other in a useful way. This gives rise to a compound function. However, the frequency of item x in transactions containing y need not equal the frequency of item y in transactions containing x. The presence of item x in transactions containing y can be stronger than the presence of item y in transactions containing x. This aspect is not captured if we consider only txy. We try to capture this aspect through mutual interaction measure, MI, given by: M

I

 f ( xy ) f ( xy )  = 0 .5 ×  +  f ( y)   f (x)

(1)

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where, f(xy) is the total number of transactions containing items x and y together i.e. |txy|; f(x) is the total number of transactions in which item x occurs, either alone or along with other items which is |tx|+|txy|; f(y) is the total number of transactions in which item y occurs which is |ty|+|txy|. From (1), it can be seen that MI equals the average confidence of ARs x→y and y→x. In other words, MI gives the predictive ability of the presence of one item given the other. Thus, MI gives us a numerical estimate of the mutual interaction of items x and y in the database. It does not consider the presence of other items that may occur with x, y, or both. MI can take a value in the range [0,1]. When items x and y are never purchased with each other, then MI takes a value of 0. If the two items are intrinsically complementary to each other (x always occurring with y and vice versa) then the values of f(xy), f(x) and f(y) become equal and MI takes a value of 1. Suppose x and y are perfectly substitutable with respect to their primary functions. This is reflected in their never being purchased together in transactions. Then, the value of f(xy) approaches 0 and MI also approaches 0. It should be noted that MI is computed by considering the value of f(xy) only. On the other hand, the statistical measure correlation [11] considers both the presence and absence of items while computing linear dependency between items. A large positive correlation coefficient does not distinguish between presence (of both the items) and absence. Either of f(xy), f(¬x¬y) might be large.

3.2. Function Embedding The concept of function embedding helps us in deciding the relevant relationships to be included while calculating relatedness between items. Consider P={x, a, b, c}. We do not consider transactions containing P while evaluating its subsets as frequent/infrequent. While calculating its relatedness we need to assign a larger weight to it as compared to its frequently purchased subsets {x, a}, {x, b} and {x, a, b}. Transaction data decides the relevance of a particular function. Consider subset {x, a, c} of P. This subset does not imply a relevant function since it does not feature in the frequently purchased market baskets. Therefore, while assigning a weight to P we do not consider subset {x, a, c}. The implied function of a set assigns a weight of 1 to it. In addition, each of its subsets purchased in separate transactions and in significant numbers assigns a weight of 1. Thus, set P is assigned a weight of 4. Sets {x, a}, {x, b} and {x, a, b} each contribute a weight of 1 to it. Similarly, {x, a, b} is assigned a weight of 3. It should be noted that while calculating the weight of P, {x, a, b}’s contribution is 1 (and not 3). This is because {x, a, b} implies one function different from its subsets. The functions of its subsets {x, a} and {x, b} have already been taken into account by their contributions to the weight of P. By assigning a weight of 1 to {x, a, b} we

avoid double counting the weights of {x, a} and {x, b}. Note that the number of non-empty subsets of P is 15.

3.3. CI (Intensity of Complementarity) Two items {x, y} are complementary if they are used together in significant numbers. Other items purchased with {x, y} give an indication of the different relationships that may exist between x and y. Complementarity of {x,y} increases if this pair can be used together with other items in different situations. Note that an item-set {a, b} and its subset {a} can both be considered as co-occurring neighbours of {x, y} if and only if both are individually purchased with {x, y} in significant numbers. While considering the significance of the occurrence of set {a} along with {x, y} we do not consider its occurrence in set {a, b}. This is because we consider sets {a} and {a, b} as separate sets that serve different functions. Let Z be the set of all co-occurring neighbours of {x, y}. Note that each element of Z can be either a single item or a set of items. The intensity of complementarity is given by: CI = 0 if |Z|=0 i.e. Z=φ 1 = × ∑ [Wt ( z ) × f ( xyz )] (Otherwise) (2) f ( xy) z∈Z where, Wt(z) is the weight of the itemset z. The weight of item set z is assigned as follows. Wt(z)=1+p, where p is the number of subsets of z that occur in Z. Thus, CI measures the intensity of flexible complementarity only and not intrinsic complementarity.

3.4. SI (Intensity of Substitutability) Item-sets in the non co-occurrence neighbourhood of {x,y} contribute to this relatedness between them. Consider an item w present in the non co-occurrence neighbourhood of {x,y}. Thus, items x and y can substitute each other in the presence of item set w∈(M∩N). If there are many such cases of x and y substituting each other with respect to different item sets from their non co-occurring neighbourhood the relatedness of {x, y} increases. This is because items x and y can substitute each other with respect to many functions. Common attributes between x and y in turn imply greater relatedness. We try to capture this aspect through measure SI. SI is given by: if (M∩N)=φ SI = 0 Wt ( w) × 1 − α w − β w = (Otherwise) (3)

∑

w∈M ∩ N

where, α w =

[

]

f ( yw) − f ( xyw) f ( xw) − f ( xyw) and β w = . f ( x ) − f ( xy ) f ( y ) − f ( xy)

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αW, βw ≥Sig, where Sig is a user-defined threshold. Wt(w) is the weightage of the item set w. Wt(w)=1+v; where v is the number of subsets of w∈M∩N. The weight of an item-set gives an indication of the number of useful functions implied by w. αW gives the proportion of transactions containing x (and not y) with item set w in it. Similarly, βw gives the proportion of transactions containing y (and not x) with w. αw, βw ≥Sig ensures that item-set w occurs in a significant portion of tx and ty. |αw- βw| gives the deviation of this proportionate occurrence of item-set w with items x and y in tx and ty respectively. Intuitively, we can expect substitutes to interact in a similar manner with respect to items they substitute. Therefore, if x and y substitute each other perfectly with respect to w, αw and βw can be expected to be nearly equal and |αw-βw| will take a value close to 0. At this point the deviation is the lowest and we should get highest value for substitutability contribution i.e. weight of the item-set w. Greater the deviation |αw- βw|, lower the relatedness. Therefore, we subtract |αw- βw| from 1 to give the contribution of w to relatedness due to substitutability. We have used two filters to ensure that substitutability contributions from non co-occurring neighbours are significant. The first filter ensures that the size of tx is tx is significant. This is done by checking if t x + | t xy | greater than a threshold, say, minsup. If it is less, then we can say that none of the non co-occurring neighbours of the set {x,y} will contribute significantly to substitutability. Then, SI can be directly assigned a value of 0. The second filter considers the frequency of non cooccurring neighbours of x in tx. αw, βw ≥Sig ensures that item-set w occurs in a significant portion of tx and ty of the database.

3.5. Total Relatedness (TR) Coefficient Total relatedness coefficient TR is the sum of the values of the three relatedness measures. Thus, TR= MI + CI +SI (4) The rationale for the summation is as follows. Relatedness between x and y is a composite of relationships existing between them. Stronger and larger number of relationships implies greater relatedness. Therefore, TR includes both the strength of relationships and the number of relationships between x and y. As can be observed from Equation 1, MI reveals the strength of only one relationship i.e. mutual interaction between x and y. Its value varies in the range [0, 1]. CI and SI bring out the intensity of complementarity and substitutability between the two items. Complementarity is due to the interaction between item sets present in {x,y}’s cooccurring neighbourhood and {x,y}. On the other hand, substitutability is a consequence of the

Table 1. A sample data set Transaction

Nos.

Transaction

Nos.

Transaction

Nos.

Transaction

Nos.

Transaction

Nos.

Transaction

Nos.

{a,d,f}

3

{a,b}

2

{a,e}

2

{b,c,e,f}

4

{b,e,f}

3

{c,d,e}

4

{d,f}

1

{a,b,c}

4

{a,e,f}

3

{b,c,e}

4

{a,b,c,d}

3

{c.d}

5

{b,e}

2

{a,b,d}

4

{a,f}

2

{b,d}

3

{b,f}

1

Table 2. Relatedness values for item-pairs occurring in association rules Sr. No.

Pair

Z*

M∩ ∩N *

MI

CI

SI

TR

1

{a, b}

{{c}, {d}, {c, d}}

{{e}, {e, f}}

0.4993

1.3077

2.6705

4.4775

2

{e, f}

{{a}, {b}, {b, c}}

{{a}, {b}}

0.5214

1.4

1.8571

3.7785

3

{b, e}

{{c}, {f}, {c, f}}

{a}

0.5121

1.4615

0.8954

2.869

4

{b, c}

{{a}, {a, d}, {e}, {e, f}}

{d}

0.5625

1.4667

0.575

2.6042

5

{c, e}

{{b}, {d}, {b, f}}

{φ}

0.5227

1.3333

0

1.856

*Z and M∩N are the respective co-occurring and non co-occurring neighbourhoods. interaction between {x,y}’s non co-occurring neighbourhood and {x,y}. Each item set that interacts with {x, y} contributes to one instance of a relationship. The strength of the relationship is revealed by the frequency of occurrence of the item set. Therefore, CI and SI are assigned values depending on the cardinality and strength of relationships between x and y. Consequently, values of CI and SI may exceed 1. Table 1 shows every transaction and the number of occurrences in the database. Each transaction, a subset of the set of six items {a, b, c, d, e, f}, represents purchases by a customer during a single buying instance at a mall. Though the data set given in Table 1 is small, with respect to both transactions and items, it is sufficient to demonstrate the efficacy of the relatedness measures. Item b that occurs in 30 transactions is the most frequently occurring item. {b, c} is the most frequent 2-itemset with 15 occurrences. The most frequent 3-itemset {b, c, e} occurs in 8 transactions. ARs were mined using a minimum support threshold of 15% and a minimum confidence threshold of 50%. This gave rise to ten ARs satisfying the minimum support and minimum confidence constraints. Relatedness measures MI, CI and SI were computed for those item pairs that occurred in at least one AR. The first filter (checking for the significance of tx and ty) was not applied while calculating SI. However, the second filter (αw, βw ≥ 0.10) was applied during the computation of SI. The results of the computations are given in Table 2.

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4. From Relatedness to Interestingness TR reveals the total relatedness of an item-pair based on information contained in transactions. Relatedness contributions due to substitutability and complementarity components for two unrelated items such as {beer, diaper} may not be high. Hence, TR will be assigned a low value. On the other hand, two related items may be expected to have similar or closely related properties. Many complementarity and substitutability relationships may arise due to these closely related properties. Hence two related items, such as {bread, butter}, can be expected to have a high TR value. From the relatedness perspective, a rule containing {beer, diaper} would be deemed more interesting than a rule containing {bread, butter}. However, in general, an AR contains more than one item-pair. Therefore, we need one consolidated value that represents the AR rather than any one of its itempairs. It can be intuitively argued that the relatedness of a set of items is driven by the least related item-pair in it [6]. Let {a1, a2, …, an}→{ b1, b2, …, bm} represent an AR. (m+n)C2 item pairs can be formed from this AR. Let the least related item-pair among these (m+n)C2 item-pairs be {x,y}and its total relatedness coefficient be TR(x,y). Since relatedness and interestingness are opposing notions, the interestingness of an AR can be given by: (m+ n) C2 × k IC = (5) TR ( x, y ) In Equation 5, we have weighted the interestingness coefficient by the number of item-pairs in the rule.

Table 3. Comparison of IC with Conviction (V) and Interest (Int) Sr. No.

Association Rule

Support

Confidence

TR*

IC

Conviction (V)

Interest (Int)

1

{c, e}→{b}

0.16

0.6667

1.8560

1.6164

1.2000

1.1100

2

{b, e}→{c}

0.16

0.6154

1.8560

1.6164

1.3520

1.2821

3

{b, c}→{e}

0.16

0.5333

1.8560

1.6164

1.2000

1.2121

4

{c}→{e}

0.24

0.50

1.8560

0.5388

1.12

1.1364

5

{e}→{c}

0.24

0.5454

1.8560

0.5388

1.144

1.1364

6

{b}→{c}

0.30

0.50

2.6042

0.3840

1.04

1.0417

7

{c}→{b}

0.30

0.625

2.6042

0.3840

1.0667

1.04167

8

{e}→{b}

0.26

0.5909

2.869

0.3486

0.9778

0.9848

9

{f}→{e}

0.20

0.588

3.7785

0.2647

1.3600

1.3368

10

{a}→{b}

0.26

0.5652

4.4775

0.2233

0.92

0.9420

*TR value of the least related item-pair in a rule is displayed when the rule has more than two items. Consider two ARs: AR1 and AR2. Let the least-related item pair in both rules be the same. If AR1 has more items than AR2, then AR1 can be more interesting than AR2. This is because AR1 brings out more relationships with the most interesting pair. This intuition leads us to weighing the interestingness of an AR by the cardinality of item-pairs in it. In Equation 5, k is the constant of proportionality. The importance given to the cardinality of item-pairs in a rule can be altered by assigning an appropriate value to ‘k’. A value of k 1 indicates complementarity effects between a and b while Interest < 1 indicates substitutability effects. Items a and b are conditionally independent when Interest = 1. Consider rule {e}→{b}. It has an Int value of 0.9848 and an IC value of 0.3486. From the economic interpretation of Int, items e and b should be substitutes. However, from Table 2 we see that CI value of {e, b} (1.4615) is greater than its SI value (0.8954). Thus, items e and b are also complementary. This aspect is not brought out by Int. Consider rule {a}→{b}. It has an Int-value of 0.9420 while IC assigns a value of 0.2233 to it. The Int value indicates that a and b are substitutes for each other. This is confirmed by the SI component (2.6705) of the TR coefficient for {a, b}. There also exists some degree of complementarity between items a and b. This again is not brought out by Int. Another observation pertaining to rule {c,e}→{b} is as follows. IC assigns the highest value of 1.6164 to it, whereas the values assigned by V and Int are 1.200 and 1.11 respectively. We cannot infer from the V and Int values that the rule {c, e}→{b} is the most interesting one of the ten rules.

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V and Int do not explicitly account for the number of item pairs in a rule or give additional weightage to the most interesting pair in a rule. This might result in counter-intuitive results as shown in the following example. Consider the rule {c,e}→{b} and its sub-rule {c}→{b}. We note that the item pair {c,e} in {c,e}→{b} is less related than {c,b}. Also, {c,e}→{b} has three item pairs while {c}→{b} has only one. Therefore, according to IC rule {c,e}→{b} (1.6164) is more interesting than {c}→{b} (0.3840). V values for {c,e}→{b} and {c}→{b} are 1.20 and 1.0667 respectively. This means that the items in the first rule are more dependent on each other and hence less interesting. Similarly, Int values for {c,e}→{b} and {c}→{b} are 1.110 and 1.04167 respectively. This seems to suggest that item c and item b are more independent and hence rule {c}→{b} is more interesting. This seems to be counter-intuitive as {c}→{b} is contained in {c,e}→{b}. In addition, it conveys less knowledge about the domain than {c,e}→{b}. All items in the antecedent and consequent are considered while computing V and Int. This can lead to the masking of the contribution of the least related itempair. V and Int do not consider substitutability and complementarity relationships explicitly. Therefore, interestingness rankings based on IC values are more intuitive than those due to V and Int.

6. Discussions Objective measures like support [1, 12], confidence [1, 12], conviction [12, 15] and interest [12, 15, 16] have been extensively used in data mining studies. Hilderman and Hamilton [13] survey seventeen interestingness measures from the data mining literature. Tan, Kumar and Srivatsava [12] have described several key properties that can be used to select the right objective measure for a given application. Our approach is different from the traditional approaches [12, 13]. Jaroszewicz and Simovici [17] have proposed a measure that is a generalization of many conditional and unconditional classical measures. Omiecinski [7] has proposed three alternative interestingness measures for associations: any-confidence, all-confidence and bond. Our work is quite different from these two studies on two counts. We consider classical association rules in a retail-market basket context. Secondly, in the studies mentioned, relationships have not explicitly been taken into account. Meo [14] has proposed a new model to evaluate dependencies in data mining problems. Our study also focuses on dependencies between items. However, we try to discern interestingness of ARs by using relatedness based on relationships between item-pairs. Brijs, et al. [19] have introduced a micro-economic integerprogramming model for product selection (PROFSET).

Our approach considers the simultaneous existence of complementarity and substitutability unlike Brijs, et al. [19]. Substitution rules were introduced by Teng and others [20]. According to them, an item-set X is a substitute for Y if X and Y are negatively correlated along with the existence of a negative AR (X→¬Y) in mined rule set. On the other hand, in our approach, the degree of substitutability (a component of relatedness) of an item pair {x, y} is computed by identifying item-sets that occur in {x, y}’s non co-occurring neighbourhood. Other relationships like flexible complementarity and mutual interaction have also been considered in our approach. Dong and Li [9] have presented a method of evaluating interestingness of ARs in terms of neighbourhood-based unexpectedness. The neighbourhood of a rule is defined in terms of a distance function on rules. In our work, we consider both co-occurring and non co-occurring neighbours of items. We also compare our interestingness coefficient (IC) with two objective measures of interestingness Conviction (V) and Interest (Int). It is seen that IC compares favourably with V and Int in ranking ARs. IC also takes into account aspects of relatedness not accounted for by V and Int, thus making it more intuitively appealing. IC treats the antecedent and consequent of an AR in a symmetric fashion. As a part of our future work, we propose to examine the feasibility of generalizing IC to account for directionality of ARs. In addition, we also propose to apply the suggested framework on real-life datasets.

7. References [1] A. A. Freitas. On Rule Interestingness Measures. Knowledge-Based Systems, 12, 1999, pp. 309-315. [2] A. Silberschatz, and A. Tuzhilin. What makes Patterns Interesting in Knowledge Discovery Systems. IEEE Transactions on Knowledge and Data Engineering, 8(6), 1996, pp. 970-974. [3] B. Liu, W. Hsu, L. Mun, and H. Lee. Finding Interesting Patterns Using User Expectations. IEEE Transactions on Knowledge and Data Engineering, 11(6), 1999, pp. 817832. [4] B. Liu, W. Hsu, S. Chen, and Y. Ma. Analyzing the Subjective Interestingness of Association Rules. IEEE Intelligent Systems, 15(5), 2000, pp. 47-55. [5] B. Padmanabhan, and A. Tuzhilin. Unexpectedness as a Measure of Interestingness in Knowledge Discovery. Decision Support Systems, 27(3), 1999, pp. 303-318. [6] B. Shekar, and R. Natarajan. A Framework for Evaluating Knowledge-based Interestingness of Association Rules. Fuzzy Optimization and Decision Making, 3(2), June 2004, pp. 157-185. [7] E. R. Omiecinski. Alternative Interest Measures for Mining Associations in Databases. IEEE Transactions on

Proceedings of the Fourth IEEE International Conference on Data Mining (ICDM’04) 0-7695-2142-8/04 $ 20.00 IEEE

Knowledge and Data Engineering, 15(1), Jan/Feb 2003, pp. 57-69. [8] G. Adomavicius, and A. Tuzhilin. Discovery of Actionable Patterns in Databases: The Action Hierarchy Approach. Proceedings of the Third International Conference on Knowledge Discovery and Data Mining(KDD –1997), AAAI, 1997, pp. 111-114. [9] G. Dong, and J. Li. Interestingness of Discovered Association Rules in Terms of Neighborhood-Based Unexpectedness. Proceedings of the Second Pacific-Asia Conference on Knowledge Discovery and Data Mining, 1998, pp.72-86. [10] J. F. Roddick, and S. Rice. What’s Interesting About Cricket? – On Thresholds and Anticipation in Discovered Rules. SIGKDD Explorations, 3(1), 2001, pp. 1-5. [11] P. Tan, and V. Kumar. Interestingness Measures for Association Patterns: A Perspective. KDD'2000 Workshop on Postprocessing in Machine Learning and Data Mining, Boston, MA, August 2000. [12] P. Tan, V. Kumar, V. and J. Srivastava. Selecting the Right Interestingness Measure for Association Patterns. Information Systems, 29(4), June 2004, pp. 293-331. [13] R. J. Hilderman, and H. J. Hamilton. Knowledge Discovery and Interestingness Measures: A Survey, Technical Report, Department of Computer Science, University of Regina, Canada, 1999. [14] R. Meo. Theory of Dependence Values. ACM Transactions on Database Systems, 25(3), September 2000, pp. 380-406. [15] S. Brin, R. Motwani, and C. Silverstein. Beyond Market Baskets: Generalizing Association Rules to Correlations. Proceedings of the 1997 ACM SIGMOD International Conference on Management of Data, May 1997, pp.265276. [16] S. Brin, R. Motwani, J. D. Ullman, and S. Tsur. Dynamic Itemset Counting and Implication Rules for Market Basket Data. Proceedings of the ACM SIGMOD International Conference on Management of Data, May 1997, pp. 255264. [17] S. Jaroszewicz, and D. A. Simovici. A General Measure of Rule Interestingness. Proceedings of the 5th European Conference on Principles of Data Mining and Knowledge Discovery (PKDD 2001), Springer-Verlag, LNAI 2168, Freiburg, September 2001, pp. 253-266. [18] S. Sahar. Interestingness Via What is Not Interesting. Proceedings of the 5th ACM, SIGKDD International Conference on Knowledge Discovery and Data Mining, 1999, pp. 332-336. [19] T. Brijs, G. Swinnen, K. Vanhoof, and G. Wets. Building and Association Rules Framework to Improve Product Assortment Decisions. Data Mining and knowledge Discovery, 8, 2004, pp. 7-23. [20] W. Teng, M. Hsieh, and M. Chen. On the Mining of Substitution Rules for Statistically Dependent Items. Proceedings of IEEE International Conference on Data Mining (ICDM’02), 2002, pp. 442-449.