What Is The Downward Closure Principle?

In association rule mining, the Apriori algorithm makes association analysis more efficient by identifying frequent itemsets. The downward closure principle can be applied to speed up the search for frequent itemsets.

The principle states that all subsets of a frequent itemset must also be frequent.

By applying the principle, it would make data mining more efficient as we would automatically eliminate all non-frequent itemsets, and we would not have to evaluate the support percentage.

For example, if an item set consists of items {A, B, C, D} is a frequent itemset that meets the minimum user specified threshold, then its subsets of {A}, {B}, {C}, {D}, {AB}, {AC}, {AD}, {BC}, {BD} ,{CD}, {ABC}, {ABD}, {ACD} and {BCD} must also be frequent itemsets that meet the support threshold.

As such, one can reason that if either {A}, {B}, {C}, {D}, {AB}, {AC}, {AD}, {BC}, {BD} and {CD} is not a frequent item set, then {A, B, C, D} itself cannot be a frequent item set.

In another example, if the item set {Milk, Bread, Eggs} is a frequent itemset, we can say that itemsets {Milk, Bread}, {Milk, Eggs} and {Bread, Eggs} are also frequent. Or that every transaction that involves the item set {Milk, Bread, Eggs} must also include its…