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Reducing Number of Candidates

Reducing Number of Candidates. Apriori principle : If an itemset is frequent, then all of its subsets must also be frequent Apriori principle holds due to the following property of the support measure: Support of an itemset never exceeds the support of its subsets

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Reducing Number of Candidates

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  1. Reducing Number of Candidates • Apriori principle: • If an itemset is frequent, then all of its subsets must also be frequent • Apriori principle holds due to the following property of the support measure: • Support of an itemset never exceeds the support of its subsets • This is known as the anti-monotone property of support

  2. Apriori Algorithm • Method: • Let k=1 • Generate frequent itemsets of length 1 • Repeat until no new frequent itemsets are identified • Generate length (k+1) candidate itemsets from length k frequent itemsets • Prunecandidate itemsets containing subsets of length k that are infrequent • Count the support of each candidate by scanning the DB • Eliminatecandidates that are infrequent, leaving only those that are frequent

  3. L1 Large 2-itemset Generation Candidate Generation C2 “Large” Itemset Generation L2 Large 3-itemset Generation Candidate Generation C3 “Large” Itemset Generation L3 … Apriori • Join Step • Prune Step Disadvantage 1: It is costly to handle a large number of candidate sets Disadvantage 2: It is tedious to repeatedly scan the database and check the candidate patterns Counting Step

  4. Max-patterns • Max-pattern: frequent patterns without proper frequent super pattern • BCDE, ACD are max-patterns • BCD is not a max-pattern Min_sup=2 Frequent-pattern mining methods

  5. Maximal Frequent Itemset An itemset is maximal frequent if none of its immediate supersets is frequent Maximal Itemsets Infrequent Itemsets Border Frequent-pattern mining methods

  6. A Simple Example Database TDB Here is a database that has 5 transactions: A, B, C, D, E. Let the min support = 2. Find all frequent max itemsets using Apriori algorithm. 1stscan: count support Generate frequent itemsets of length 1 C1 L1 Eliminate candidates that are infrequent

  7. A Simple Example C2 L1 Generate length 2 candidate itemsets from length 1 frequent itemsets 2nd scan: Count support C2 L2 generate frequent itemsets of length 2

  8. A Simple Example L2 Generate length 3 candidate itemsets from length 2 frequent itemsets C3 3rd scan: Count support C3 L3 generate frequent itemsets of length 3 The algorithm stops here

  9. A Simple Example Supmin = 2 Database TDB L1 C1 1st scan C2 C2 L2 2nd scan L3 C3 3rd scan

  10. Draw a graph to illustrate the result Null Green and Yellow: Frequent Itemsets Border A B C D E AB AC AD AE BC BD BE CD CE DE ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ABDE ACDE ABCD ABCE BCDE ABCDE Maximum Frequent Itemsets

  11. FP-growth Algorithm • Scan the database once to store all essential information in a data structure called FP-tree(Frequent Pattern Tree) • The FP-tree is concise and is used in directly generating large itemsets • Once an FP-tree has been constructed, it uses a recursive divide-and-conquer approach to mine the frequent itemsets

  12. FP-growth Algorithm • Step 1: Deduce the ordered frequent items. For items with the same frequency, the order is given by the alphabetical order. • Step 2: Construct the FP-tree from the above data • Step 3: From the FP-tree above, construct the FP-conditional tree for each item (or itemset). • Step 4: Determine the frequent patterns.

  13. A Simple Example of FP-tree Problem: Find all frequent itemsets with support >= 2)

  14. FP-growth Algorithm • Step 1: Deduce the ordered frequent items. For items with the same frequency, the order is given by the alphabetical order. • Step 2: Construct the FP-tree from the above data • Step 3: From the FP-tree above, construct the FP-conditional tree for each item (or itemset). • Step 4: Determine the frequent patterns.

  15. A Simple Example: Step 1 Threshold = 2

  16. A Simple Example: Step 1

  17. FP-growth Algorithm • Step 1: Deduce the ordered frequent items. For items with the same frequency, the order is given by the alphabetical order. • Step 2: Construct the FP-tree from the above data • Step 3: From the FP-tree above, construct the FP-conditional tree for each item (or itemset). • Step 4: Determine the frequent patterns.

  18. Step 2: Construct the FP-tree from the above data Root A: 1 B: 1

  19. Step 2: Construct the FP-tree from the above data Root A: 1 B: 1 B: 1 C: 1 D: 1

  20. Step 2: Construct the FP-tree from the above data Root A: 2 B: 1 C: 1 B: 1 C: 1 D: 1 D: 1 E: 1

  21. Step 2: Construct the FP-tree from the above data Root A: 3 B: 1 B: 1 C: 1 C: 1 D: 1 D: 1 D: 1 E: 1 E: 1

  22. Step 2: Construct the FP-tree from the above data Root A: 4 B: 1 B: 2 C: 1 C: 1 D: 1 C: 1 D: 1 D: 1 E: 1 E: 1

  23. FP-growth Algorithm • Step 1: Deduce the ordered frequent items. For items with the same frequency, the order is given by the alphabetical order. • Step 2: Construct the FP-tree from the above data • Step 3: From the FP-tree above, construct the FP-conditional tree for each item (or itemset). • Step 4: Determine the frequent patterns.

  24. Step 3:From the FP-tree above, construct the FP-conditional tree for each item (or itemset). Root A: 4 B: 1 B: 2 C: 1 C: 1 D: 1 C: 1 Cond. FP-tree on “E” D: 1 D: 1 { } (A:1, C:1, D:1, E:1), E: 1 E: 1

  25. Step 3:From the FP-tree above, construct the FP-conditional tree for each item (or itemset). Root Threshold = 2 A: 4 B: 1 B: 2 C: 1 C: 1 D: 1 C: 1 Cond. FP-tree on “E” D: 1 D: 1 { } (A:1, C:1, D:1, E:1), (A:1, D:1, E:1), E: 1 E: 1

  26. Step 3:From the FP-tree above, construct the FP-conditional tree for each item (or itemset). Root Threshold = 2 A: 4 B: 1 B: 2 C: 1 C: 1 D: 1 C: 1 2 Cond. FP-tree on “E” D: 1 D: 1 { } (A:1, C:1, D:1, E:1), (A:1, D:1, E:1), E: 1 E: 1

  27. Step 3:From the FP-tree above, construct the FP-conditional tree for each item (or itemset). Threshold = 2 conditional pattern base of “E” Cond. FP-tree on “E” (A:1, C:1, D:1, E:1), { } { } (A:1, D:1, E:1), (A:1, D:1, E:1), Root (A:1, D:1, E:1), A:2 D:2

  28. Step 3:From the FP-tree above, construct the FP-conditional tree for each item (or itemset). Root Threshold = 2 A: 4 B: 1 B: 2 C: 1 C: 1 D: 1 C: 1 Cond. FP-tree on “D” D: 1 D: 1 { } (A:1, C:1, D:1), E: 1 E: 1

  29. Step 3:From the FP-tree above, construct the FP-conditional tree for each item (or itemset). Root Threshold = 2 A: 4 B: 1 B: 2 C: 1 C: 1 D: 1 C: 1 Cond. FP-tree on “D” D: 1 D: 1 { } (A:1, C:1, D:1), (A:1, D:1), E: 1 E: 1

  30. Step 3:From the FP-tree above, construct the FP-conditional tree for each item (or itemset). Root Threshold = 2 A: 4 B: 1 B: 2 C: 1 C: 1 D: 1 C: 1 Cond. FP-tree on “D” D: 1 D: 1 { } (A:1, C:1, D:1), (A:1, D:1), E: 1 (B:1, C:1, D:1), E: 1

  31. Step 3:From the FP-tree above, construct the FP-conditional tree for each item (or itemset). Root Threshold = 2 A: 4 B: 1 B: 2 C: 1 C: 1 D: 1 C: 1 3 Cond. FP-tree on “D” D: 1 D: 1 { } (A:1, C:1, D:1), (A:1, D:1), E: 1 (B:1, C:1, D:1), E: 1

  32. Step 3:From the FP-tree above, construct the FP-conditional tree for each item (or itemset). Threshold = 2 Cond. FP-tree on “D” { } (A:1, C:1, D:1), { } (A:1, D:1), (A:1, D:1), (C:1, D:1), (B:1, C:1, D:1), Root A:2 C:2

  33. Step 3:From the FP-tree above, construct the FP-conditional tree for each item (or itemset). Root Threshold = 2 A: 4 B: 1 B: 2 C: 1 C: 1 D: 1 C: 1 3 Cond. FP-tree on “C” D: 1 D: 1 { } (A:1, B:1, C:1), (A:1, C:1), E: 1 (B:1, C:1), E: 1

  34. Step 3:From the FP-tree above, construct the FP-conditional tree for each item (or itemset). Threshold = 2 Cond. FP-tree on “C” { } (A:1, B:1, C:1), { } (A:1, C:1), (A:1, C:1), (B:1, C:1), (B:1, C:1), Root A:2 B:2

  35. Step 3:From the FP-tree above, construct the FP-conditional tree for each item (or itemset). Root Threshold = 2 A: 4 B: 1 B: 2 C: 1 C: 1 D: 1 C: 1 3 Cond. FP-tree on “B” D: 1 D: 1 { } (A:2, B:2), (B:1), E: 1 E: 1

  36. Step 3:From the FP-tree above, construct the FP-conditional tree for each item (or itemset). Threshold = 2 Cond. FP-tree on “B” { } (A:2, B:2), { } (A:2, B:2), (B:1), (B:1), Root A:2

  37. Step 3:From the FP-tree above, construct the FP-conditional tree for each item (or itemset). Root Threshold = 2 A: 4 B: 1 B: 2 C: 1 C: 1 D: 1 C: 1 4 Cond. FP-tree on “A” D: 1 D: 1 { } (A:4), E: 1 { } (A:4), E: 1 Root

  38. FP-growth Algorithm • Step 1: Deduce the ordered frequent items. For items with the same frequency, the order is given by the alphabetical order. • Step 2: Construct the FP-tree from the above data • Step 3: From the FP-tree above, construct the FP-conditional tree for each item (or itemset). • Step 4: Determine the frequent patterns.

  39. 3 Cond. FP-tree on “C” Root Cond. FP-tree on “E” 2 Root A:2 A:2 B:2 D:2 Cond. FP-tree on “B” 3 Root Root A:2 Cond. FP-tree on “D” 3 A:2 Cond. FP-tree on “A” 4 C:2 Root

  40. 3 Cond. FP-tree on “C” Root Cond. FP-tree on “E” 2 Root 1. Before generating this cond. tree, we generate {C} (support = 3) 1. Before generating this cond. tree, we generate {E} (support = 2) A:2 A:2 2. After generating this cond. tree, we generate {A, C}, {B, C}(support = 2) B:2 2. After generating this cond. tree, we generate {A, D, E}, {A, E}, {D, E} (support = 2) D:2 Cond. FP-tree on “B” 3 Root 1. Before generating this cond. tree, we generate {B} (support = 3) Root A:2 2. After generating this cond. tree, we generate {A, B} (support = 2) Cond. FP-tree on “D” 3 1. Before generating this cond. tree, we generate {D} (support = 3) A:2 C:2 Cond. FP-tree on “A” 4 2. After generating this cond. tree, we generate {A, D}, {C, D} (support = 2) 1. Before generating this cond. tree, we generate {A} (support = 4) Root 2. After generating this cond. tree, we do not generate any itemset.

  41. Step 4: Determine the frequent patterns Answer: According to the above step 4, we can find all frequent itemsets with support >= 2): {E}, {A,D,E}, {A,E}, {D,E}, {D}, {A,D}, {C,D}, {C}, {A,C}, {B,C}, {B}, {A,B} {A}

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