The following questions examine how the support and confidence of an asso- ciation rule may vary in the presence of a concept hierarchy.
(a) Consider an item x in a given concept hierarchy. Let x1, x2, ..., xk
denote the k children of x in the concept hierarchy. Show that s(x) ?
k
i=1 s(xi), where s(·) is the support of an item. Under what conditions
will the inequality become an equality?
(b) Let p and q denote a pair of items, while ˆp and ˆq are their corresponding
parents in the concept hierarchy. If s({p, q}) > minsup, which of the
following itemsets are guaranteed to be frequent? (i) s({p, q ˆ }), (ii)
s({p, qˆ}), and (iii) s({p, ˆ qˆ}).
(c) Consider the association rule {p} ?? {q}. Suppose the confidence of
the rule exceeds minconf. Which of the following rules are guaranteed
to have confidence higher than minconf? (i) {p} ?? {qˆ}, (ii) {pˆ} ??
{q}, and (iii) {pˆ} ?? {qˆ}.
(a) If no transaction contains more than one child of x, then s(x) = k
i=1 s(xi).
(b) All three itemsets are guaranteed to be frequent.
(c) Only {p} ?? {qˆ} is guaranteed to have confidence higher than minconf.
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