1 Fast Algorithms for Mining Association Rules Rakesh Agrawal Ramakrishnan Srikant

©Ofer Pasternak Data Mining Seminar Outline Introduction Formal statement Apriori Algorithm AprioriTid Algorithm Comparison AprioriHybrid Algorithm Conclusions

©Ofer Pasternak Data Mining Seminar Introduction Bar-Code technology Mining Association Rules over basket data (93) Tires ^ accessories  automotive service Cross market, Attached mail. Very large databases.

©Ofer Pasternak Data Mining Seminar Notation Items – I = {i 1,i 2, …,i m } Transaction – set of items – Items are sorted lexicographically TID – unique identifier for each transaction

©Ofer Pasternak Data Mining Seminar Notation Association Rule – X  Y

©Ofer Pasternak Data Mining Seminar Confidence and Support Association rule X  Y has confidence c, c% of transactions in D that contain X also contain Y. Association rule X  Y has support s, s% of transactions in D contain X and Y.

©Ofer Pasternak Data Mining Seminar Notice X  A doesn’t mean X+Y  A – May not have minimum support X  A and A  Z doesn’t mean X  Z – May not have minimum confidence

©Ofer Pasternak Data Mining Seminar Define the Problem Given a set of transactions D, generate all association rules that have support and confidence greater than the user-specified minimum support and minimum confidence.

©Ofer Pasternak Data Mining Seminar Previous Algorithms AIS SETM Knowledge Discovery Induction of Classification Rules Discovery of causal rules Fitting of function to data KID3 – machine learning

©Ofer Pasternak Data Mining Seminar Discovering all Association Rules Find all Large itemsets – itemsets with support above minimum support. Use Large itemsets to generate the rules.

©Ofer Pasternak Data Mining Seminar General idea Say ABCD and AB are large itemsets Compute conf = support(ABCD) / support(AB) If conf >= minconf AB  CD holds.

©Ofer Pasternak Data Mining Seminar Discovering Large Itemsets Multiple passes over the data First pass – count the support of individual items. Subsequent pass – Generate Candidates using previous pass ’ s large itemset. – Go over the data and check the actual support of the candidates. Stop when no new large itemsets are found.

©Ofer Pasternak Data Mining Seminar The Trick Any subset of large itemset is large. Therefore To find large k-itemset – Create candidates by combining large k-1 itemsets. – Delete those that contain any subset that is not large.

©Ofer Pasternak Data Mining Seminar Algorithm Apriori Count item occurrences Generate new k-itemsets candidates Find the support of all the candidates Take only those with support over minsup

©Ofer Pasternak Data Mining Seminar Candidate generation Join step Prune step P and q are 2 k-1 large itemsets identical in all k-2 first items. Join by adding the last item of q to p Check all the subsets, remove a candidate with “ small ” subset

©Ofer Pasternak Data Mining Seminar Example L 3 = { {1 2 3}, {1 2 4}, {1 3 4}, {1 3 5}, {2 3 4} } After joining { { }, { } } After pruning { } {1 4 5} and {3 4 5} Are not in L 3

©Ofer Pasternak Data Mining Seminar Correctness Show that Join is equivalent to extending L k-1 with all items and removing those whose (k-1) subsets are not in L k-1 Prevents duplications Any subset of large itemset must also be large

©Ofer Pasternak Data Mining Seminar Subset Function Candidate itemsets - C k are stored in a hash-tree Finds in O(k) time whether a candidate itemset of size k is contained in transaction t. Total time O(max(k,size(t))

©Ofer Pasternak Data Mining Seminar Problem? Every pass goes over the whole data.

©Ofer Pasternak Data Mining Seminar Algorithm AprioriTid Uses the database only once. Builds a storage set C^ k – Members has the form X k are potentially large k-items in transaction TID. For k=1, C^ 1 is the database. Uses C^ k in pass k+1. Each item is replaced by an itemset of size 1

©Ofer Pasternak Data Mining Seminar Advantage C^ k could be smaller than the database. – If a transaction does not contain k- itemset candidates, than it will be excluded from C^ k. For large k, each entry may be smaller than the transaction – The transaction might contain only few candidates.

©Ofer Pasternak Data Mining Seminar Disadvantage For small k, each entry may be larger than the corresponding transaction. – An entry includes all k-itemsets contained in the transaction.

©Ofer Pasternak Data Mining Seminar Algorithm AprioriTid Count item occurrences Generate new k-itemsets candidates Find the support of all the candidates Take only those with support over minsup The storage set is initialized with the database Build a new storage set Determine candidate itemsets which are containted in transaction TID Remove empty entries

©Ofer Pasternak Data Mining Seminar ItemsTID Set-of- itemsets TID { {1},{3},{4} }100 { {2},{3},{5} }200 { {1},{2},{3},{5} }300 { {2},{5} }400 SupportItemset 2{1} 3{2} 3{3} 3{5} itemset {1 2} {1 3} {1 5} {2 3} {2 5} {3 5} Set-of-itemsetsTID { {1 3} }100 { {2 3},{2 5} {3 5} }200 { {1 2},{1 3},{1 5}, {2 3}, {2 5}, {3 5} } 300 { {2 5} }400 SupportItemset 2{1 3} 3{2 3} 3{2 5} 2{3 5} itemset {2 3 5} Set-of-itemsetsTID { {2 3 5} }200 { {2 3 5} }300 SupportItemset 2{2 3 5} Database C^ 1 L2L2 C2C2 C^ 2 C^ 3 L1L1 L3L3 C3C3

©Ofer Pasternak Data Mining Seminar Correctness Show that C t generated in the kth pass is the same as set of candidate k- itemsets in C k contained in transaction with t.TID

©Ofer Pasternak Data Mining Seminar Correctness Lemma 1  k >1, if C^ k-1 is correct and complete, and L k-1 is correct, Then the set C t generated at the kth pass is the same as the set of candidate k-itemsets in C k contained in transaction with t.TID  t of C^ k t.set-of-itemsets includes all large k-itemsets contained in transaction with t.TID  t of C^ k t.set-of-itemsets doesn’t include any k-itemsets not contained in transaction with t.TID Same as the set of all large k- itemsets

©Ofer Pasternak Data Mining Seminar Proof Suppose a candidate itemset c = c[1]c[2]…c[k] is in transaction t.TID  c 1 = (c-c[k]) and c 2 =(c-c[k-1]) were in transaction t.TID  c 1 and c 2 must be large  c 1 and c 2 were members of t.set-of-items  c will be a member of C t C k was built using apriori-gen(L k-1 )  all subsets of c of C k must be large C^ k-1 is complete

©Ofer Pasternak Data Mining Seminar Proof Suppose c 1 (c 2 ) is not in transaction t.TID  c 1 (c 2 ) is not in t.set-of-itemsets  c of C k is not in transaction t.TID  c will not be a member of C t C^ k-1 is correct

©Ofer Pasternak Data Mining Seminar Correctness Lemma 2  k >1, if L k-1 is correct and the set C t generated in the kth step is the same as the set of candidate k-itemsets in C k in transaction t.TID, then the set C^ k is correct and complete.

©Ofer Pasternak Data Mining Seminar Proof Apriori-gen guarantees  C t includes all large k-itemsets in t.TID, which are added to C^ k  C^ k is complete. C t includes only itemsets in t.TID, only items in C t are added to C^ k  C^ k is correct.

©Ofer Pasternak Data Mining Seminar Correctness Theorem 1  k >1, the set C t generated in the kth pass is the same as the set of candidate k- itemsets in C k contained in transaction t.TID Show: C^ k is correct and complete and L k is correct for all k>=1.

©Ofer Pasternak Data Mining Seminar Proof (by induction on k) K=1 – C^1 is the database. Assume it holds for k=n. – C t generated in pass n+1 consists of exactly those itemsets in C n+1 contained in transaction t.TID. – Apriori-gen guarantees and C t is correct  L n+1 is correct  C^ n+1 will be correct and complete  C^ k is correct and complete for all k>=1  The theorem holds Lemma 2 Lemma 1

©Ofer Pasternak Data Mining Seminar General idea (reminder) Say ABCD and AB are large itemsets Compute conf = support(ABCD) / support(AB) If conf >= minconf AB  CD holds.

©Ofer Pasternak Data Mining Seminar Discovering Rules For every large itemset l – Find all non-empty subsets of l. – For every subset a Produce rule a  (l-a) Accept if support(l) / support(a) >= minconf

©Ofer Pasternak Data Mining Seminar Checking the subsets For efficiency, generate subsets using recursive DFS. If a subset ‘a’ doesn’t produce a rule, we don’t need to check for subsets of ‘a’. Example Given itemset :ABCD If ABC  D doesn’t have enough confidence then surely AB  CD won’t hold

©Ofer Pasternak Data Mining Seminar Why? For any subset a^ of a: Support(a^) >= support(a)  Confidence ( a^  (l-a^) ) = support(l) / support(a^) <= support(l) / support(a) = confidence ( a  (l-a) )

©Ofer Pasternak Data Mining Seminar Simple Algorithm Check all the subsets Check all the large itemsets Output the rule Continue the DFS over the subsets. If there is no confidence the DFS branch cuts here Check confidence of new rule

©Ofer Pasternak Data Mining Seminar Faster Algorithm Idea: If (l-c)  c holds than all the rules (l-c^)  c^ must hold Example: If AB  CD holds, then so do ABC  D and ABD  C C^ is a non empty subset of c

©Ofer Pasternak Data Mining Seminar Faster Algorithm From a large itemset l, – Generate all rules with one item in it’s consequent. Use those consequents and Apriori-gen to generate all possible 2 item consequents. Etc. The candidate set of the faster algorithm is a subset of the candidate set of the simple algorithm.

©Ofer Pasternak Data Mining Seminar Faster algorithm Find all 1 item consequents (using 1 pass of the simple algorithm) Generate new (m+1)- consequents Check the support of the new rule Continue for bigger consequents If a consq. Doesn ’ t hold, don ’ t look for bigger.

©Ofer Pasternak Data Mining Seminar Advantage Example Large itemset : ABCDE One item conseq. : ACDE  B ABCE  D Simple algorithm will check: ABC  DE, ABE  CD, BCE  AD and ACE  BD. Faster algorithm will check: ACE  BD which is also the only rule that holds.

©Ofer Pasternak Data Mining Seminar ABCDE ACDE  B ABCE  D ACD  BE ADE  BC CDE  AB ACE  BD BCE  AD ACE  BD ABE  CD ABC  ED Large itemset Rules with minsup Simple algorithm: Fast algorithm: ACE  BD ABCDE ACDE  B ABCE  D Example

©Ofer Pasternak Data Mining Seminar Results Compare Apriori, and AprioriTid performances to each other, and to previous known algorithms: – AIS – SETM The algorithms differ in the method of generating all large itemsets. Both methods generate candidates “ on-the-fly ” Designed for use over SQL

©Ofer Pasternak Data Mining Seminar Method Check the algorithms on the same databases – Synthetic data – Real data

©Ofer Pasternak Data Mining Seminar Synthetic Data Choose the parameters to be compared. – Transaction sizes, and large itemsets sizes are each clustered around a mean. – Parameters for data generation D – Number of transactions T – Average size of the transaction I – Average size of the maximal potentially large itemsets L – Number of maximal potentially large itemsets N – Number of Items.

©Ofer Pasternak Data Mining Seminar Synthetic Data Expriment values: – N = 1000 – L = 2000 T5.I2.D100k T10.I2.D100k T10.I4.D100k T20.I2.D100k T20.I4.D100k T20.I6.D100k D – Number of transactions T – Average size of the transaction I – Average size of the maximal potentially large itemsets L – Number of maximal potentially large itemsets N – Number of Items. T=5, I=2, D=100,000

©Ofer Pasternak Data Mining Seminar SETM values are too big to fit the graphs. Apriori always beats AIS D – Number of transactions T – Average size of the transaction I – Average size of the maximal potentially large itemsets Apriori is better than AprioriTid in large problems

©Ofer Pasternak Data Mining Seminar Explaining the Results AprioriTid uses C^ k instead of the database. If C^ k fits in memory AprioriTid is faster than Apriori. When C^ k is too big it cannot sit in memory, and the computation time is much longer. Thus Apriori is faster than AprioriTid.

©Ofer Pasternak Data Mining Seminar Reality Check Retail sales – 63 departments – transactions (avg. size 2.47) Small database, C^ k fits in memory.

©Ofer Pasternak Data Mining Seminar Reality Check Mail Order items 2.9 million transactions (avg size 2.62) Mail Customer items transactions (avg size 31)

©Ofer Pasternak Data Mining Seminar So who is better? Look At the Passes. At final stages, C^ k is small enough to fit in memory

©Ofer Pasternak Data Mining Seminar Algorithm AprioriHybrid Use Apriori in initial passes Estimate the size of C^ k Switch to AprioriTid when C^ k is expected to fit in memory The switch takes time, but it is still better in most cases.

©Ofer Pasternak Data Mining Seminar

©Ofer Pasternak Data Mining Seminar Scale up experiment

©Ofer Pasternak Data Mining Seminar Conclusions The Apriori algorithms are better than the previous algorithms. – For small problems by factors – For large problems by orders of magnitudes. The algorithms are best combined. The algorithm shows good results in scale-up experiments.

©Ofer Pasternak Data Mining Seminar Summary Association rules are an important tool in analyzing databases. We’ve seen an algorithm which finds all association rules in a database. The algorithm has better time results then previous algorithms. The algorithm maintains it’s performances for large databases.

©Ofer Pasternak Data Mining Seminar End