Association Rule Mining

The Task Two ways of defining the task General Input: A collection of instances Output: rules to predict the values of any attribute(s) (not just the class attribute) from values of other attributes E.g. if temperature = cool then humidity =normal If the right hand side of a rule has only the class attribute, then the rule is a classification rule Distinction: Classification rules are applied together as sets of rules Specific - Market-basket analysis Input: a collection of transactions Output: rules to predict the occurrence of any item(s) from the occurrence of other items in a transaction E.g. {Milk, Diaper} -> {Beer} General rule structure: Antecedents -> Consequents

Association Rule Mining Given a set of transactions, find rules that will predict the occurrence of an item based on the occurrences of other items in the transaction Market-Basket transactions Example of Association Rules {Diaper}  {Beer}, {Milk, Bread}  {Eggs,Coke}, {Beer, Bread}  {Milk}, Implication means co-occurrence, not causality!

Definition: Frequent Itemset A collection of one or more items Example: {Milk, Bread, Diaper} k-itemset An itemset that contains k items Support count () Frequency of occurrence of an itemset E.g. ({Milk, Bread,Diaper}) = 2 Support Fraction of transactions that contain an itemset E.g. s({Milk, Bread, Diaper}) = 2/5 Frequent Itemset An itemset whose support is greater than or equal to a minsup threshold

Definition: Association Rule An implication expression of the form X  Y, where X and Y are itemsets Example: {Milk, Diaper}  {Beer} 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 Example:

Association Rule Mining Task Given a set of transactions T, the goal of association rule mining is to find all rules having support ≥ minsup threshold confidence ≥ minconf threshold Brute-force approach: List all possible association rules Compute the support and confidence for each rule Prune rules that fail the minsup and minconf thresholds  Computationally prohibitive!

Mining Association Rules 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

Mining Association Rules Two-step approach: Frequent Itemset Generation Generate all itemsets whose support  minsup Rule Generation Generate high confidence rules from each frequent itemset, where each rule is a binary partitioning of a frequent itemset Frequent itemset generation is still computationally expensive

Power set Given a set S, power set, P is the set of all subsets of S Known property of power sets If S has n number of elements, P will have N = 2n number of elements. Examples: For S = {}, P={{}}, N = 20 = 1 For S = {Milk}, P={{}, {Milk}}, N=21=2 For S = {Milk, Diaper} P={{},{Milk}, {Diaper}, {Milk, Diaper}}, N=22=4 For S = {Milk, Diaper, Beer}, P={{},{Milk}, {Diaper}, {Beer}, {Milk, Diaper}, {Diaper, Beer}, {Beer, Milk}, {Milk, Diaper, Beer}}, N=23=8

Brute Force approach to Frequent Itemset Generation For an itemset with 3 elements, we have 8 subsets Each subset is a candidate frequent itemset which needs to be matched against each transaction 1-itemsets Itemset Count {Milk} 4 {Diaper} {Beer} 3 2-itemsets Itemset Count {Milk, Diaper} 3 {Diaper, Beer} {Beer, Milk} 2 3-itemsets Itemset Count {Milk, Diaper, Beer} 2 Important Observation: Counts of subsets can’t be smaller than the count of an itemset!

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

Illustrating Apriori Principle Items (1-itemsets) Pairs (2-itemsets) (No need to generate candidates involving Coke or Eggs) Minimum Support = 3 Triplets (3-itemsets) If every subset is considered, 6C1 + 6C2 + 6C3 = 41 With support-based pruning, 6 + 6 + 1 = 13 Write all possible 3-itemsets and prune the list based on infrequent 2-itemsets

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 Prune candidate itemsets containing subsets of length k that are infrequent Count the support of each candidate by scanning the DB Eliminate candidates that are infrequent, leaving only those that are frequent Note: This algorithm makes several passes over the transaction list

Rule Generation 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 2k – 2 candidate association rules (ignoring L   and   L) Because, rules are generated from frequent itemsets, they automatically satisfy the minimum support threshold Rule generation should ensure production of rules that satisfy only the minimum confidence threshold

Rule Generation 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 Example: Consider the following two rules {Milk}  {Diaper, Beer} has cm=0.5 {Milk, Diaper}  {Beer} has cmd=0.67 > cm

Computing Confidence for Rules Itemset Support {Milk} 4/5=0.8 {Diaper} {Bread} Itemset Support {Milk, Diaper} 3/5=0.6 {Diaper, Bread} {Bread, Milk} Unlike computing support, computing confidence does not require several passes over the transaction list Supports computed from frequent itemset generation can be reused Tables on the side show support values for all (except null) the subsets of itemset {Bread, Milk, Diaper} Confidence values are shown for the (23-2) = 6 rules generated from this frequent itemset Itemset Support {Bread, Milk, Diaper} 3/5=0.6 Example of Rules: {Bread, Milk}  {Diaper} (s=0.6, c=1.0) {Milk, Diaper}  {Bread} (s=0.6, c=1.0) {Diaper, Bread}  {Milk} (s=0.6, c=1.0) {Bread}  {Milk, Diaper} (s=0.6, c=0.75) {Diaper}  {Milk, Bread} (s=0.6, c=0.75) {Milk}  {Diaper, Bread} (s=0.6, c=0.75)

Rule generation in Apriori Algorithm For each frequent k-itemset where k > 2 Generate high confidence rules with one item in the consequent Using these rules, iteratively generate high confidence rules with more than one items in the consequent if any rule has low confidence then all the other rules containing the consequents can be pruned (not generated) {Bread, Milk, Diaper} 1-item rules {Bread, Milk}  {Diaper} {Milk, Diaper}  {Bread} {Diaper, Bread}  {Milk} 2-item rules {Bread}  {Milk, Diaper} {Diaper}  {Milk, Bread} {Milk}  {Diaper, Bread}

Evaluation Support and confidence used by Apriori allow a lot of rules which are not necessarily interesting Two options to extract interesting rules Using subjective knowledge Using objective measures (measures better than confidence) Subjective approaches Visualization – users allowed to interactively verify the discovered rules Template-based approach – filter out rules that do not fit the user specified templates Subjective interestingness measure – filter out rules that are obvious (bread -> butter) and that are non-actionable (do not lead to profits)

Drawback of Confidence Coffee Tea 15 5 20 75 80 90 10 100 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

Objective Measures Confidence estimates rule quality in terms of the antecedent support but not consequent support As seen on the previous slide, support for consequent (P(coffee)) is higher than the rule confidence (P(coffee/Tea)) Weka uses other objective measures Lift (A->B) = confidence(A->B)/support(B) = support(A->B)/(support(A)*support(B)) Leverage (A->B) = support(A->B) – support(A)*support(B) Conviction(A->B) = support(A)*support(not B)/support(A->B) conviction inverts the lift ratio and also computes support for RHS not being true