Lift, leverage, and conviction are three of the best commonly known interest measures for crisp association rules. All of them are based on a comparison of observed support and the support that is expected if the antecedent and consequent part of the rule were stochastically independent. The aim of this paper is to provide a correct definition of lift, leverage, and conviction measures for fuzzy association rules and to study some of their interesting mathematical properties.

Searching for association rules is a broadly discussed, developed, and accepted data mining technique [

Naturally, analysts are interested only in such rules that are somehow interesting, unusual, or exceptional. To assess rule interestingness objectively, there have been developed many measures of rule interest or intensity. Among the most essential,

Searching for association rules fits particularly well on binary or categorical data and many have been written on that topic [

The use of fuzzy logic in connection with association rules has been motivated by many authors (see e.g., [

In this paper, we focus on three measures of rule intensity that are all based on comparison between the observed support and the support that is expected under the assumption of independence of the rule’s antecedent and consequent. These measures are

On the other hand,

As quite many have been written about these measures for crisp association rules, not so much has been done with respect to fuzzy rules. Unfortunately, simplicity of definitions for crisp rules sometimes led to oversimplified definitions for fuzzy rules. Some authors believe the generalization of those crisp measures for fuzzy data is as trivial as substituting crisp terms with analogous fuzzy terminology inside of crisp-case definitions; see, for example, [

In Section

Let

Moreover, let us define a negated predicate

An association rule is a formula

The

If

If

Hence, lift

If

For fuzzy association rules, domain of each fuzzy attribute

product

minimum

Łukasiewicz

Negation of fuzzy predicate

Let

Throughout this text, we assume

For the sake of simplicity, we will sometimes express a fuzzy attribute as a vector of membership degrees. For instance, suppose

Let

A naive approach for introducing lift, leverage, and conviction into the fuzzy association rules framework is to use simply their definitions (

Two vectors

As discussed in Section

Comparison of lift, leverage, and conviction computed with different

Naive lift | Naive leverage | Naive conviction | |
---|---|---|---|

Łukasiewicz |
0.675 | −0.081 | 1.544 |

Product |
1.012 | 0.003 | 1.012 |

Minimum |
1.353 | 0.088 | 0.755 |

The values of naive lift and naive leverage wrongly indicate positive (resp., negative) relationship, if the minimum (resp., Łukasiewicz)

As indicated in Experiment

Given sets

For the sake of simplicity, let us assume

If

Two random variables

By definition,

Similarly, an expected value

In real setting,

Assuming

Since

Now, we are ready to define notions of

Let

Let

if

(1) If

(2) For any

Note that the reverse direction of the first implication of Proposition

Let

If

Let

Let

Let

Let

The second inequality follows from

Now, we are ready to provide a correct definition of lift, leverage, and conviction in the framework of fuzzy association rules.

Let

Let us now study some interesting properties of the newly defined notions.

Let

if

(1) and (2) directly follow from the definitions and from the fact that

(3) Since the membership degrees are defined on interval

(4) If

Let

(1) and (2) directly follow from the definitions and from the fact that

(3) Let

Let

Everything directly follows from Definition

Let

Everything directly follows from Definitions

Corollary

Let

Everything directly follows from Definitions

Let

Everything directly follows from Definitions

The same data as in Experiment

Comparison of lift, leverage, and conviction computed with different

Naive lift | Naive leverage | Naive conviction | |
---|---|---|---|

Łukasiewicz |
1.022 | 0.004 | 1.021 |

Product |
1.012 | 0.003 | 1.012 |

Minimum |
1.010 | 0.003 | 1.011 |

Lift is a ratio of observed support (resp., confidence) to the support (resp., confidence) that is expected under the assumption of independence. Leverage is similar to lift, since it is a difference between observed and expected support. Conviction is often treated as an alternative to confidence; nevertheless, it is defined on the basis of observed and expected support too.

In this paper, a correct definition of lift, leverage, and conviction for fuzzy data was provided. It should be stressed here that there already exist some research papers that use incorrect (a.k.a. “naive”) definition of fuzzy lift (e.g., [

The naive definition does not preserve interpretation of positive/negative relationship. For crisp lift and conviction (resp., leverage), the stochastically independent attributes

All the lift, leverage, and conviction definitions are similar to their “crisp” alternatives (i.e., definitions for binary data) if the

In [

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was supported by the European Regional Development Fund in the Project of IT4Innovations Centre of Excellence (CZ.1.05/1.1.00/02.0070, VP6).