2. Supermarket shelf management – Market-basket model:  Goal: Identify items that are bought together by sufficiently many customers  Approach: Process the sales data collected with barcode scanners to find dependencies among items  A classic rule:  If someone buys diaper and milk, then he/she is likely to buy beer  Don’t be surprised if you find six-packs next to diapers! J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org2

3.  A large set of items  e.g., things sold in a supermarket  A large set of baskets  Each basket is a small subset of items  e.g., the things one customer buys on one day  Want to discover association rules  People who bought {x,y,z} tend to buy {v,w}  Amazon! J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org3 Rules Discovered: {Milk} --> {Coke} {Diaper, Milk} --> {Beer} Rules Discovered: {Milk} --> {Coke} {Diaper, Milk} --> {Beer} Input: Output:

4.  Items = products; Baskets = sets of products someone bought in one trip to the store  Real market baskets: Chain stores keep TBs of data about what customers buy together  Tells how typical customers navigate stores, lets them position tempting items  Suggests tie-in “tricks”, e.g., run sale on diapers and raise the price of beer  Need the rule to occur frequently, or no $$’s  Amazon’s people who bought X also bought Y J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org4

5.  Baskets = sentences; Items = documents containing those sentences  Items that appear together too often could represent plagiarism  Notice items do not have to be “in” baskets  Baskets = patients; Items = drugs & side-effects  Has been used to detect combinations of drugs that result in particular side-effects  But requires extension: Absence of an item needs to be observed as well as presence J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org5

6.  A general many-to-many mapping (association) between two kinds of things  But we ask about connections among “items”, not “baskets”  For example:  Finding communities in graphs (e.g., Twitter) J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org6

7.  Finding communities in graphs (e.g., Twitter)  Baskets = nodes; Items = outgoing neighbors  Searching for complete bipartite subgraphs K s,t of a big graph J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org7  How?  View each node i as a basket B i of nodes i it points to  K s,t = a set Y of size t that occurs in s buckets B i  Looking for K s,t  set of support s and look at layer t – all frequent sets of size t … … … A dense 2-layer graph s nodes t nodes

8.  Simplest question: Find sets of items that appear together “frequently” in baskets  Support for itemset I : Number of baskets containing all items in I  (Often expressed as a fraction of the total number of baskets)  Given a support threshold s, then sets of items that appear in at least s baskets are called frequent itemsets J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org8 Support of {Beer, Bread} = 2

9.  Items = {milk, coke, pepsi, beer, juice}  Support threshold = 3 baskets B 1 = {m, c, b}B 2 = {m, p, j} B 3 = {m, b}B 4 = {c, j} B 5 = {m, p, b}B 6 = {m, c, b, j} B 7 = {c, b, j}B 8 = {b, c}  Frequent itemsets: {m}, {c}, {b}, {j}, J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org9, {b,c}, {c,j}. {m,b}

10. 10  Association Rules: If-then rules about the contents of baskets  {i 1, i 2,…,i k } → j means: “if a basket contains all of i 1,…,i k then it is likely to contain j ”  In practice there are many rules, want to find significant/interesting ones!  Confidence of this association rule is the probability of j given I = {i 1,…,i k } J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

11.  Not all high-confidence rules are interesting  The rule X → milk may have high confidence for many itemsets X, because milk is just purchased very often (independent of X ) and the confidence will be high  Interest of an association rule I → j : difference between its confidence and the fraction of baskets that contain j  Interesting rules are those with high positive or negative interest values (usually above 0.5) J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org11

12. B 1 = {m, c, b}B 2 = {m, p, j} B 3 = {m, b}B 4 = {c, j} B 5 = {m, p, b}B 6 = {m, c, b, j} B 7 = {c, b, j}B 8 = {b, c}  Association rule: {m, b} → c  Confidence = 2/4 = 0.5  Interest = |0.5 – 5/8| = 1/8  Item c appears in 5/8 of the baskets  Rule is not very interesting! J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org12

13. Example: l Association Rule – An implication expression of the form X  Y, where X and Y are itemsets – Example: {Milk, Diaper}  {Beer} l Rule Evaluation Metrics – Support (s)  Fraction of transactions that contain both X and Y – Confidence (c)  Measures how often items in Y appear in transactions that contain X

14.  Problem: Find all association rules with support ≥ s and confidence ≥ c  Note: Support of an association rule is the support of the set of items on the left side  Hard part: Finding the frequent itemsets!  If {i 1, i 2,…, i k } → j has high support and confidence, then both {i 1, i 2,…, i k } and {i 1, i 2,…,i k, j} will be “frequent” J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org14

15.  Step 1: Find all frequent itemsets I  (we will explain this next)  Step 2: Rule generation  For every subset A of I, generate a rule A → I \ A  Since I is frequent, A is also frequent  Variant 1: Single pass to compute the rule confidence  confidence(A,B→C,D) = support(A,B,C,D) / support(A,B)  Variant 2:  Observation: If A,B,C → D is below confidence, so is A,B → C,D  Can generate “bigger” rules from smaller ones!  Output the rules above the confidence threshold J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org15

16. B 1 = {m, c, b}B 2 = {m, p, j} B 3 = {m, c, b, n}B 4 = {c, j} B 5 = {m, p, b}B 6 = {m, c, b, j} B 7 = {c, b, j}B 8 = {b, c}  Support threshold s = 3, confidence c = 0.75  1) Frequent itemsets:  {b,m} {b,c} {c,m} {c,j} {m,c,b}  2) Generate rules:  b → m: c =4/6 b → c: c =5/6 b,c → m: c =3/5  m → b: c =4/5 … b,m → c: c =3/4  b → c,m: c =3/6 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org16

17. Example of Rules: {Milk,Diaper}  {Beer} (s=0.4, c=0.67) {Milk,Beer}  {Diaper} (s=0.4, c=1.0) {Diaper,Beer}  {Milk} (s=0.4, c=0.67) {Beer}  {Milk,Diaper} (s=0.4, c=0.67) {Diaper}  {Milk,Beer} (s=0.4, c=0.5) {Milk}  {Diaper,Beer} (s=0.4, c=0.5) Observations: All the above rules are binary partitions of the same itemset: {Milk, Diaper, Beer} Rules originating from the same itemset have identical support but can have different confidence Thus, we may decouple the support and confidence requirements

18.  Two-step approach: 1.Frequent Itemset Generation – Generate all itemsets whose support  minsup 2.Rule Generation – Generate high confidence rules from each frequent itemset, where each rule is a binary partitioning of a frequent itemset

19.  Given a frequent itemset L, find all non-empty subsets f  L such that f  L – f satisfies the minimum confidence requirement  If {A,B,C,D} is a frequent itemset, candidate rules: ABC  D, ABD  C, ACD  B, BCD  A, A  BCD,B  ACD,C  ABD, D  ABC AB  CD,AC  BD, AD  BC, BC  AD, BD  AC, CD  AB,  If |L| = k, then there are 2 k – 2 candidate association rules (ignoring L   and   L)

20. J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org20

21.  How to efficiently generate rules from frequent itemsets?  In general, confidence does not have an anti- monotone property c(ABC  D) can be larger or smaller than c(AB  D)  But confidence of rules generated from the same itemset has an anti-monotone property  e.g., L = {A,B,C,D}: c(ABC  D)  c(AB  CD)  c(A  BCD)  Confidence is anti-monotone w.r.t. number of items on the RHS of the rule

22. Lattice of rules Pruned Rules Low Confidence Rule

23.  Association rule algorithms tend to produce too many rules  many of them are uninteresting or redundant  Redundant if {A,B,C}  {D} and {A,B}  {D} have same support & confidence  Interestingness measures can be used to prune/rank the derived patterns  In the original formulation of association rules, support & confidence are the only measures used

24. Interestingness Measures

25.  Given a rule X  Y, information needed to compute rule interestingness can be obtained from a contingency table YY Xf 11 f 10 f 1+ Xf 01 f 00 f o+ f +1 f +0 |T| Contingency table for X  Y f 11 : support of X and Y f 10 : support of X and Y f 01 : support of X and Y f 00 : support of X and Y Used to define various measures u support, confidence, lift, Gini, J-measure, etc.

26. Coffee Tea15520 Tea75580 9010100 Association Rule: Tea  Coffee Confidence= P(Coffee|Tea) = 0.75 but P(Coffee) = 0.9  Although confidence is high, rule is misleading  P(Coffee|Tea) = 0.9375  Knowing a person is a Tea drinker actually lessons prob. that they are a coffee drinker

27.  Population of 1000 students  600 students know how to swim (S)  700 students know how to bike (B)  420 students know how to swim and bike (S,B)  P(S  B) = 420/1000 = 0.42  P(S)  P(B) = 0.6  0.7 = 0.42  P(S  B) = P(S)  P(B) => Statistical independence  P(S  B) > P(S)  P(B) => Positively correlated  P(S  B) Negatively correlated

28.  Measures that take into account statistical dependence Lift & Interest are equivalent in the case of binary variables

29. Coffee Tea15520 Tea75580 9010100 Association Rule: Tea  Coffee Confidence= P(Coffee|Tea) = 0.75 but P(Coffee) = 0.9  Lift = 0.75/0.9= 0.8333 (< 1, therefore is negatively associated)

30. YY X100 X090 1090100 YY X900 X010 9010100 Statistical independence: If P(X,Y)=P(X)P(Y) => Lift/Interest = 1 Suggests positive correlation But x & y don’t appear together

31. There are lots of measures proposed in the literature Some measures are good for certain applications, but not for others What criteria should we use to determine whether a measure is good or bad? What about Apriori- style support based pruning? How does it affect these measures?