單調性約束與反單調性約束的區別 monotonicity and anti-monotonicity constraint

▚ 01 Monotone Constraints 單調性約束

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1.1 Definitions 定義

【Monotone Constraints | SpringerLink 】

A constraint C is monotone if and only if for all itemsets S and S′:
if S ⊇ S′ and S violates C, then S′ violates C.

1.2 Key Points 要點

   Monotone constraints possess the following property. If an itemset S violates a monotone constraint C, then any of its subsets also violates C. Equivalently, all supersets of an itemset satisfying a monotone constraint C also satisfy C (i.e., C is upward closed). By exploiting this property, monotone constraints can be used for reducing computation in frequent itemset mining with constraints. As frequent itemset mining with constraints aims to find frequent itemsets that satisfy the constraints, if an itemset S satisfies a monotone constraint C, no further constraint checking needs to be applied to any superset of S because all supersets of S are guaranteed to satisfy C. Examples of monotone constraints include min(S. Price) ≤ $30, which expresses that the minimum price of all items in an itemset S is at most $30. Note that, if the minimum price of all items in S is at most $30, adding more items to S would not increase its minimum price (i.e., supersets of S would also satisfy such a monotone constraint).

▚ 02 Anti-monotone Constraints 反單調性約束

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2.1 Definitions 定義

【Anti-monotone Constraints | SpringerLink】

A constraint C is anti-monotone if and only if for all itemsets S and S′:
if S ⊇ S′and S satisfies C, then S′ satisfies C.

2.2 Key Points 要點

   Anti-monotone constraints possess the following nice property. If an itemset S satisfies an anti-monotone constraint C, then all of its subsets also satisfy C (i.e., C is downward closed). Equivalently, any superset of an itemset violating an anti-monotone constraint C also violates C. By exploiting this property, anti-monotone constraints can be used for pruning in frequent itemset mining with constraints. As frequent itemset mining with constraints aims to find itemsets that are frequent and satisfy the constraints, if an itemset violates an anti-monotone constraint C, all its supersets (which would also violate C) can be pruned away and their frequencies do not need to be counted. Examples of anti-monotone constraints include min(S. Price) ≥ $20 (which expresses that the minimum price of all items in an itemset S is at least $20) and the usual frequency constraint support(S)  ≥ minsup (i.e., frequency(S)  ≥ minsup). For the former, if the minimum price of all items in S is less than $20, adding more items to S would not increase its minimum price (i.e., supersets of S would not satisfy such an anti-monotone constraint). For the latter, it is widely used in frequent itemset mining, with or without constraints. It states that (i) all subsets of a frequent itemset are frequent and (ii) any superset of an infrequent itemset is also infrequent. This is also known as the Apriori property.

▚ 03 Explanation 解釋

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假設：我們將S violates C作為事件A，S′ violates C作為事件B；則 S satisfies C為事件not A，then S′ satisfies C為事件not B.

此時，根據Monotone Constraints定義知(A → B)，也即(not B → not A)；
根據Anti-monotonicity Constraints定義知(not A → not B)，也即(B → A)；

因為(A → B)並不一定意味著(B → A)，所以這兩者 (Monotone Constraints & Anti-monotonicity Constraints) 的聲明是不同的。

▚ 04 Example 示例

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For an example. Consider-  
C1 = Sum of elements is greater than 5  
C2 = Sum of elements is at most 5  
U(universe) = Set of non-negative real numbers

In case of C1,
If S violates C1, then S’ obviously violates C1 as well (S being a superset of S’)
Eg. S = {1, 2}, S’ = {2}
Hence C1 is monotonic.

In case of C2,
If S satisfies C2, then S’ obviously satisfies C2 as well (S being a superset of S’)
Eg. S = {1, 2}, S’ = {2}
Hence C2 is anti-monotonic.

▚ 05 數據挖掘的約束限制

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Constraint-Based Mining — A General Picture

參考文章

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database - What does Antimonotonicity and monotonicity constraint mean? - Code Utility - Code Utility

CS6220: Data Mining Techniques (ucla.edu)