The goal of our visualization is to highlight particularly “discriminatory” items on the one hand and to show relationships among the items and rules on the other. We therefore define new, higher-level measures of interestingness based on the discriminatory measures (D-measure) on individual rules in discrimination- aware data mining. A D-measure can be defined as Extended Lift [138], among other similar measures.
Let conf(X → Y ) be the confidence of the association rule X → Y , and A, B → C be an association rule such that conf(B → C) > 0. The Extended Lift of the rule is with respect to B is defined as:
conf(A, B → C)
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A is the PD itemset, B is the PND itemset that is called the context, and B → C is call ed the base-rule.
The larger the D-measure of a rule is, the more certain we are of this rule being discriminatory against people in disadvantaged groups, such as foreign workers, single mothers, etc. Selection lift (slift) is chosen as the D-measure for assessing direct discrimination: the confidence of the rule with PD item(s) A, divided by the confidence of the rule whose premise contains ¬A. Extended-lift-lower-bound (elb) is chosen as D-measure for assessing indirect discrimination, which adapts
elift, a variation of slift, with the confidence information from the background
knowledge [142].
We combine this with established distribution-based measures of item(set) interestingness: Based on the D-measure, we define the item-focused AD-
measure of the accumulated degree of discrimination of an individual item, and
Mutual Information as a measure of correlation between pairs of items, based on the discriminatory rules these items are involved in. We also combine the rule-focused D-measure with a similarity measure on rule pairs to relate several rules to one another in their degree of discrimination.
Item Focusing: The goal of item focusing is to determine the degree of
discrimination in individual items and to be able to relate these to one another in order to gain a meta-level view.
A simple measure is the supp (support) of an item or items in the mined rule set. Let an item be characterized as the equality of an attribute q to a value (range) v, let N be the number of rules, and N(x) the number of rules satisfying the argument x. Let Q = V denote an itemset: q1= v1, ..., qm= vmfor m ≥ 1.
Then
supp(Q = V ) = N(Q = V )
N (8.2)
This measure does not take into account how discriminatory the PD rules are; therefore we define the AD-measure as a form of averaged support, which is weighted by the D-measure of these rules. Let Ri be the left-hand-side of rule i
and D-measurei its D-measure, then:
AD − measure(Q = V ) = PN i=1D-measurei· b(Q = V, Ri) supp(Q = V ) (8.3) with b(Q = V, Ri) = { 1 if Ricontains Q= V 0 else
The AD-measure’s range is [0, ∞). In the visualizations, we only show the AD-measure for single items.
Other item-focusing interestingness measures have been proposed such as the magnitude-based and association-based interestingness functions in [22] and the summary-based interestingness functions in [94], which are attribute-oriented – the probability distribution of all the items within one attribute is taken into account. However, the attribute-oriented measures are limited in the sense that although the associations between two or more attributes (e.g.
f oreign_worker and purpose) can be well captured and ranked, the association
between two or more items is ill-measured (e.g. foreign_worker = yes and
purpose= new_car). When an attribute only contains one value in all records,
as in most of the rules extracted by DCUBE, the attribute-oriented measures would take it as a non-interesting item (see section 2.2.3 in [22]). In such a case, the items need to be treated individually, in other words, each item is taken as a non-dividable entity without the context of attributes, i.e. item-focusing. The support measure is in accord with this criterion, and so is Mutual Information
(MI) [164] on pairs of items in the rules. We can use MI to characterize the
interdependency between two items:
M I(q1=v1, q2=v2) = log
p(q1=v1, q2= v2)
p(q1=v1) · p(q2=v2) (8.4) with p(Q = V ) = N(Q = V )/M
and M being the total number of items in all rules (including overlaps). Interestingness measures can be categorized into objective and subjective measures. Objective measures such as supp and MI focus on the probability distribution of data. Subjective measures take a user’s judgement and experience as a part of the measurements, such as exceptions [75] and expectations [116], which involve a user’s predefinition of a specific set of items. The “Discrimination Discovery in databases” approach also lets a user subjectively select PD items. Thus, AD-measure is a hybrid objective/subjective measure.2
Rule Focusing: The support-confidence framework can be insufficient to
evaluate the quality of a rule [14]. Many rule-focusing measures of interestingness have been proposed, including objective measures such as rule-interest, lift, conviction, Loevinger index, implication intensity, coverage, strength, performance, Sebag and Schoenauer index and IPEE, and subjective measures such as simplicity, unexpectedness and actionability[116, 24]. The D-measures in the DCUBE system also contain subjectiveness in that they take the user’s
2Like DCUBE, we disregard accuracy-based measures of rule interestingness. Integrating
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prior knowledge of the data domain into account. We will use the D-measures (slift and elb) to characterize each rule’s interestingness.
To characterize the similarity/distance between two rules, we concentrate on all the items in the rule (for association rules) or all items on the left-hand-side of the rule (for classification rules) and regard them as an itemset or vector in the space spanned by all attributes. We can then apply a number of measures such as the Hamming distance, Jaccard distance, Dice’s coefficient, or Cosine similarity and its binary version, the Tanimoto coefficient. Another measure of the association between two (or more) rules is through fuzzy meta-association rule extraction [187].
We will use the Jaccard distance to indicate the degree of dissimilarity between two rules. In contrast to other measures, it is applicable in a straightforward way to the sets with different items as well as readily implemented. Let S(R) denote the itemset of R. Then the Jaccard distance between R1and R2is
J(R1, R2) = 1 −S(R1) ∩ S(R2) S(R1) ∪ S(R2)
. (8.5)