Association rules The goal of mining association rules is to generate all possible rules that exceed some minimum user-specified support and confidence thresholds. The problem is thus decomposed into two subproblems:

Association rules 1. Generate all items sets that have a support that exceeds the support threshold. These sets are called large or frequent itemsets, because they have large support (not because of their cardinality). 2. For each large itemset, all the rules that have a minimum confidence are generated: for a large itemset X and Y  X, let Z = X – Y. If support(X)/support(Z) > Confidence threshold, then Z  Y. Z  Y

Association rules Generating rules by using large (frequent) itemsets is straightforward. However, if the cardinality of the set of items is very high, the process becomes very computation-intensive: for m items, the number of distinct itemsets is 2m (power set). Basic algorithms for finding association rules try to reduce the combinatorial search space

Association rules: a Basic algorithm 1. Test the support for itemsets of size 1, called 1-itemsets. Discard those that do not meet Minimum Required Support (MRS). 2. Extend the 1-itemsets by appending one item each time, to generate all candidate 2-itemsets, test MRS and discard those that do not meet it. 3. Continue until no itemsets can be found. 4. Use itemsets found to generate rules (check confidence).

Association rules The naive version of this algorithm is a combinatorial nightmare! Many versions and variants of this algorithm They use different strategies Their resulting sets of rules are all the same: Any algorithm for association rules should find the same set of rules although their computational efficiencies and memory requirements may be different. We will see the Apriori Algorithm

Association rules: The Apriori Algorithm The key idea: The apriori property: any subsets of a frequent itemset are also frequent itemsets x x x ABC ABD ACD BCD x x x AB AC AD BC BD CD Begin with itemsets of size 1 A B C D x

Stage 1. The Apriori Algorithm: Generating large itemsets Find large (frequent, i.e., that meet MRS) itemsets of size 1: F1 From k = 2 Ck = candidates of size k: those itemsets of size k that could be frequent, given Fk-1 Fk = those itemsets that are actually large, Fk  Ck. They meet MRS

Example Minimum support allowed = 50% itemset:count TID Items 1 a, c, d 2 b, c, e 3 a, b, c, e 4 b, e Minimum support allowed = 50% itemset:count 1. scan T  C1: {a}:2, {b}:3, {c}:3, {d}:1, {e}:3  F1: {a}:2, {b}:3, {c}:3, {e}:3  C2: {a,b}, {a,c}, {a,e}, {b,c}, {b,e}, {c,e} 2. scan T  C2: {a,b}:1, {a,c}:2, {a,e}:1, {b,c}:2, {b,e}:3, {c,e}:2  F2: {a,c}:2, {b,c}:2, {b,e}:3, {c,e}:2  C3: {b,c,e} 3. scan T  C3: {b, c, e}:2  F3: {b, c, e}: 2 Sets with d are discarded It is the only candidate because all its subsets are large!

Stage 1. The Apriori Algorithm: Generating large (frequent) itemsets Algorithm Apriori(T, minsup) //T Set of n transactions //minsup = minimum support allowed C1  getC1(T); //Get 1-itemsets and their counts F1  {f | f  C1, f.count / n  minsup}; //Check support FOR (k = 2; Fk-1  ; k++) DO Ck  candidate_gen(Fk-1); //Get k-itemsets using Fk-1 FOR each transaction t  T DO FOR each candidate c  Ck DO IF c is contained in t THEN c.count++; END FOR Fk  {c  Ck | c.count / n  minsup} //Check support RETURN F  k Fk; //Union of all Fk (k >1 for rules) /*Get counts of the k-itemsets*/

Stage 2: The Apriori Algorithm: generating rules from large itemsets Note that frequent itemsets  association rules One more step is needed to generate association rules: For each frequent itemset X: - For each subset A  X, A  . Let B = X – A - A  B is an association rule if confidence(A  B) ≥ minconf, i.e., if (support(A  B)/support(A)) ≥ minconf. Minconf = Minimum confidence

Example Suppose {b,c,e} is a large (frequent) itemset with support  50% Consider subsets of {b,c,e} and their support: {b,c} = 50% {b,e} = 75%, {c,e} = 50%, {b} = 75%, {c} = 75%, {e} = 75%

Example This subset generates the following association rules: {b,c}  {e} confidence = 50/50 = 100% {b,e}  {c} confidence = 50/75 = 66.6% {c,e}  {b} confidence = 50/50 = 100% {b}  {c,e} confidence = 50/75 = 66.6% {c}  {b,e} confidence = 50/75 = 66.6% {e}  {b,c} confidence = 50/75 = 66.6% All rules have support  50% = support({b,c,e})

Hash-based improvement to A-Priori During pass 1 of A-Priori (Get 1-itemsets), most memory is idle Use that memory to keep counts of buckets into which pairs of items are hashed Gives extra condition that candidate pairs must satisfy when getting 2-itemsets (pass 2)

Hash-based improvement to A-Priori The memory is divided like this: Space to count each item (Get 1-itemsets) Use the rest of the space for the described hashing process PCY algorithm (Park, Chen, and Yu)

PCY Algorithm Pass 1: FOR each transaction t in T DO FOR each item in t DO add 1 to item’s count END FOR FOR each pair of items in t DO Hash the pair to a bucket and add 1 to the count for that bucket getC1 Hashing process

PCY Algorithm Pass 2: Count all pairs {i, j} that satisfy the conditions: Both i and j (taken individually!) are frequent items The pair {i, j} hashes to a bucket whose count  the support s (i.e., a frequent bucket) These two conditions are necessary for the pair to have a chance of being frequent.

PCY Example Support s = 3 Items: milk (1), Coke (2), bread (3), Pepsi (4), juice (5). Transactions are t1 = {1, 2, 3}  milk, Coke, bread t2 = {1, 4, 5} t3 = {1, 3} t4 = {2, 5} t5 = {1, 3, 4} t6 = {1, 2, 3, 5} t7 = {2, 3, 5} t8 = {2, 3}

PCY Example Hashing a pair {i, j} to a bucket k, where k = hash(i, j) = (i + j) mode 5. That is, for pairs: (1, 4) and (2, 3)  k = 0 (1, 5) and (2, 4)  k = 1 (2, 5) and (3, 4)  k = 2 (1, 2) and (3, 5)  k = 3 (1, 3) and (4, 5)  k = 4

PCY Example Pass 1: Item’s count: Note that Item 4 does not exceed the support. Item Count 1 5 2 3 6 4

PCY Example For each pair in each transaction: Total: 21 pairs

PCY Example The hash table is Bucket 1 does not exceed the support, i.e., (1, 5) and (2, 4) are not frequent. Bucket Count 6 1 2 4 3 5

PCY Example Pass 2: Frequent items are {1, 2, 3, 5} From the frequent items the candidate pairs are (1,2) (1,3) (1,5) (2,3) (2,5) (3,5) Candidate (1,5) is discarded because bucket 1 is not frequent!  Discarded by the PCY!

PCY Example Counts of the “surviving” pairs Note that pairs (1,2) and (3,5) does not exceed the support. Frequent itemsets are {1} {2} {3} {5} {1,3} {2,3} {2,5} Pair Count (1,2) 2 (1,3) 4 (2,3) (2,5) 3 (3,5)

Association rules Association rules among hierarchies: it is possible to divide items among disjoint hierarchies, e.g., foods in a supermarket. It could be interesting to find associations rules across hierarchies. They may occur among item groupings at different levels. Consider the following example. Levels of a dimension

Hierarchy 1: Beverages Hierarchy 2: Desserts

Association rules Associations such as {Frozen Yoghurt}  {Bottled Water} {Rich Cream}  {Wine Coolers} May produce enough confidence and support to be valid rules of interest.

Association rules: Negative associations Negative associations: a negative association is of the following type “80% of customers who buy W do not buy Z” The problem of discovering negative associations is complex: there are millions of item combinations that do not appear in the database, the problem is to find only interesting negative rules.

Association rules: Negative associations One approach is to use hierarchies. Consider the following example: Positive association Soft drinks Chips Joke Wakeup Topsy Days Nightos Parties

Association rules: Negative associations Suppose a strong positive association between chips and soft drinks If we find a large support for the fact that when customers buy Days chips they predominantly buy Topsy drinks (and not Joke, and not Wakeup), that would be interesting The discover of negative associations remains a challenge

Association rules Additional considerations for associations rules: For very large datasets, efficiency for discovering rules can be improved by sampling  Danger: discovering some false rules Transactions show variability according to geographical location and seasons Quality of data is usually also variable