1 Association Graphs Selim Mimaroglu University of Massachusetts Boston

2 This Presentation is Built On:  A Graph-Based Approach for Discovering Various Types of Association Rules (MAIN) by Show-Jane Yen and A.L.P. Chen at IEEE Transactions on Data and Knowledge Engineering 2001  Mining Multiple-Level Association Rules from Large Databases by Jiawei Han and Yongjian Fu at VLDB ’95  Mining Multilevel Association Rules from Transaction Databases section 6.3 ISBN  Mining Generalized Association Rules by R. Srikant, and R. Agrawal at VLDB ’95  Introduction to Data Mining by Tan, Steinbach, Kumar ISBN:

3 Organization  Primitive Association Rules Definition Association Graphs for finding large (frequent) item sets  Multiple-Level Association Rules Definition Why it’s important? Association Graphs for finding large item sets Interest Measure  Generalized Association Rules Definition Association Graphs for finding large item sets

4 Association Rules  Discovering patterns from a large database (generally data warehouse) is computationally expensive  To find all the rules that satisfy minsupport and minconf  X → Y  Transactions in the database contain the items in X tend also contain the items in Y.

5 Definitions  Let I={i1, i2, …, id} be set of all items in a market basked data  Let T={t1, t2,..tN} be the set of all transactions  A collection of items is termed itemset TIDi1i2…id t1101 t2100 …001 tN111

6 Definitions Let X and Y be two disjoint itemsets  Support Count of X  Support of X → Y  Confidence of X → Y

7 Taxonomy (or Concept Hierarchy)  This is created by the domain expert, e.g. store manager.  There may be more than one taxonomy for the same items  is-a relation  “Folgers Coffee Classic Roast Singles - 19 ct” (barcode ) is-a “Regular Ground Coffee”  “Regular Ground Coffee” is-a “Coffee”  “Coffee” is-a “Beverage”  “Beverage” is-a “General Grocery Item”

8 Taxonomy Example ……………………………………………………

9 Primitive Association Rules  Deals with the lowest level items of the taxonomy (most widely used)  Very well studied Algorithms using Apriori principle FP-Growth Algorithms  Steps Large (frequent) itemset generation Rule Generation

10 Found to be Infrequent Frequent Itemset Generation (Apriori) Pruned supersets

11 Motivation  Many database scans for checking if the candidate itemsets are qualified (i.e. ≥ minsupport )  Reducing the database scans FP-Growth: 2 scans Association Graphs: 1 scan ?

12 Mining Primitive Association Rules Create Bit Vectors TIDABCDE Number the Attributes A:1D:4 B:2E:5 C:3 BV BV BV BV BV

13 Bit Vectors (Column Store)  This is a column store (as opposed to a row store)  Read Efficient  Row stores are write efficient  For an item or itemset a 64 bit processor can count the support count of 64 rows in one instruction only  Logical AND, OR minsupport = 50% (2 transactions) Is item 1 (column A) frequent ? Yes BV Is the itemset {1, 3} frequent? Yes BV BV {1,3} =

14 Association Graph Construction Property 1: The support for the itemset {i 1, i 2, …, i k } is the number of 1s in BVi 1 ∧ BVi 2 … ∧ BVi k, where the notation “ ∧ ” is a logical AND operator ACG: Association Graph Construction: For every two large items i and j ( i < j), if the number of 1s in BV i ∧ BV j achieves the user-specified minimum support, a directed edge from item i to item j is created (The Association Graph is a Directed Acyclic Graph. Proven in Appendix A). Also, itemset (i, j) is a large 2 - itemset

15 Example 1: BV BV BV BV BV

16 Lemma 1: If an itemset is not a large itemset, then any itemset which contains this itemset cannot be a large itemset (Proof: Apriori principle) Lemma 2: For a large itemset (i 1, i 2, …, i k ), if there is no directed edge from i k to an item v, then itemset (i 1, i 2, …, i k, v) cannot be a large itemset Proof of Lemma 2: If there is not an edge from i k to v then (i k, v) is not a large itemset. If (i k, v) is not a large itemset then none of the supersets of (i k, v) can be a large itemset (e.g. (i 1, i 2, …, i k, v) isn’t a large itemset )

17 Finding All Large Itemsets by Using an Association Graph  Suppose (i 1, i 2, …, i k ) is a large k-itemset. If there is no directed edge from i k to v then the itemset need not be extended into k+1-itemset because (i 1, i 2, …, i k, v) must not be a large itemset according to Lemma 2  If there is a directed edge from i k to u, then the itemset (i 1, i 2, …, i k ) is extended to k+1-itemset (i 1, i 2, …, i k, u). The itemset (i 1, i 2, …, i k, u) is a large k+1 itemset if the number of 1s in BVi 1 ∧ BVi 2 … ∧ BVi k ∧ BVi u achieves the minsupport Fig 1: Association Graph

18 Example 2: Finding all large itemsets by using the Association Graph of Example1 Large 2-itemsets: (1, 3) (2, 3) (2, 5) (3,5) minsupport : 50% (2 rows) Fig 1: Association Graph

19 Large 3-itemset Generation  (1, 3, 5)  (2, 3, 5) BV BV BV (1, 3, 5) = BV BV = (2, 3, 5) BV

20 Is it really 1 (one) scan?  Creating an Association Graph may be computationally expensive  This really is not 1 scan only.  Bit Vectors make reading extremely efficient.  This is almost as good as it gets Fig 2: An itemset lattice (at level 2) Fig 1: An Association Graph

21 Multilevel Association Rules all computer desktop IBMDell laptop SonyToshiba software educational Microsoft… financial management … printer color HP… b/w Sony… computer accessory wrist pad Ergoway mouse Logitech

22 Why Multilevel Association Rules?  Support at the lower concept levels is low  You can miss the association rules at higher levels (bigger picture)  Support: Uniform support for all levels Reduced minimum support at lower levels Level by level independent Other approaches

23 Mining Multiple level Association Rules  Replace items/concepts at level k with the concepts at level k-1  Apply an Association Rules Algorithm at the items at level k-1 computer desktoplaptop IBMDellSonyToshiba

24 Example 3 for Multilevel Associations  For level 4  Generate Association Graphs on this table  Upgrade the items at Table 1 to level 3 (by looking at the taxonomy tree) TIDItems Purchased T1IBM desktop computer, Sony b/w printer T2Microsoft educational software, Microsoft financial software T3Logitech mouse, Microsoft financial software T4IBM desktop computer, Microsoft financial software T5IBM desktop computer Table 1: Market basket data (at lowest level taxonomy: level 4) TIDItems Purchased T1desktop computer, b/w printer T2educational software, financial software T3mouse, financial software T4desktop computer, financial software T5desktop computer Table 2: Market basket data (at level 3)

25 Redundant Multilevel Rules desktop computer → b/w printer [support=8%, confidence = 70%](R1) IBM desktop computer → b/w printer [support=2%, confidence=72%] (R2)  A rule is interesting if: it doesn’t have a parent rule it can’t be deduced from its parent rule Suppose that one quarter of all “desktop computers” are “IBM desktop computers”, so we expect following Confidence of R2 shall be around 70% Support of R2 shall be around 8% * ¼ = 2% R2 is redundant (uninteresting) because it does not convey any additional information All redundant rules shall be pruned

26 Mining Generalized Association Rules all computersoftwareprinter desktoplaptop IBMDell computer accessory SonyToshiba educational financial management Microsoft…… colorb/w HP…Sony… wrist pad Ergoway mouse Logitech

27 Rule Generation  To generate generalized association patterns one can add all ancestors of each item and then apply the basic algorithm  {IBM desktop computer, Sony b/w printer}  {IBM desktop computer, desktop computer, computer, Sony b/w printer, b/w printer, printer}  This way works, but inefficient If {IBM desktop computer} is a large itemset then {IBM desktop computer, desktop computer}, {IBM desktop computer, computer} and {desktop computer, computer} are large itemsets but redundant

28 Definition  A generalized association rule is an implication of the form X→Y, where X ⊂ I, Y ⊂ I, X∩Y= ∅, and no item in Y is an ancestor of any item in X.  X →ancestor(X) is trivially true with 100% confidence and hence redundant

29 Support for an item and its ancestor Lemma 3: The support for an itemset X that contains an item x i and its ancestor y i will be the same as the support for the itemset X- y i Proof of Lemma 3: X={x 1,..., x i, y i,…, x n } The support for the itemset X is: BV x 1 ∧ … ∧ BV x i ∧ BV y i … ∧ BV x n Note that: BV x i = BV x i ∧ BV y i The support of X becomes as the support of X- y i BV x 1 ∧ … ∧ BV x i ∧ … ∧ BV x n

30 Post Order Numbering  Number the items using the Post Order Numbering method (PON)  Lemma 4: For every two items i and j (i<j), item v is an ancestor of item i, but not an ancestor of item j, then v<j An example of concept hierarchy

31 Support for an ancestor  Lemma 5: Suppose items i 1, i 2, …, and i m are all specific descendants of the generalized item i n. The bit vector BV i n associated with item in is BV i 1 ∨ BV i 2 ∨ …… ∨ BV i m and the number of 1s in this bit vector is the support for item in where ∨ stands for logical OR operation.  Lemma 6: If an itemset X is large itemset, then any itemset generated by replacing an item in itemset X with its ancestor is also a large itemset

32 Creating the Generalized Association Graph  Lemma 7: if (the number of 1s in BV i ∧ BV j )≥ minsupport, then for each ancestor u of item i and for each ancestor v of item j, (the number of 1s in BV u ∧ BV j )≥minsupport and (the number of 1s in BV i ∧ BV v )≥minsupport From Lemma 7, if an edge from item i to item j is created, the edges from item i to the ancestors of item j, which are not ancestors of item i, are also created From Lemma 4, the ancestors of item i, which are not ancestors of item j, are all less than j. Hence if an edge from item i to item j is created, the edges from the ancestors of item i, which are not ancestors of j, to item j are also created (A. Graph Construction Pr.)

33 Example of Generalized Association Rule with minsupport 40% (2 transactions) The generalized Association Graph

34 Finally Theorem 1: Any itemset generated by traversing the Generalized Association Graph (GAG) will not contain both an item and its ancestor Proof of Theorem 1: Basis of Induction: In GAG there will be no edge between an item and its ancestor. Therefore all 2-itemsets are free of ancestors Inductive Hypothesis: We assume that any large k-itemset (i 1, i 2, …, i k ) does not contain both an item and its ancestor Inductive Step: Large k-itemset (i 1, i 2, …, i k ) is extended to k+1-itemset (i 1, i 2, …, i k,, w). Suppose v 1, v 2, …, v k-1 are ancestors of i 1, i 2, …, i k-1 respectively, but none are ancestors of item i k. Because items are numbered by PON method, i k > v j (1≤ j ≤ k-1). Hence, there are no edges from item i k to the ancestors of items i 1, i 2, …, i k. So, item w cannot be an ancestor of item i 1, i 2, …, i k