1. Mining Association Rules

2. Data Mining Overview Data Mining
Data warehouses and OLAP (On Line Analytical Processing.)
Association Rules Mining
Clustering: Hierarchical and Partitional approaches
Classification: Decision Trees and Bayesian classifiers
Sequential Patterns Mining
Advanced topics: outlier detection, web mining

3. Association Rules: Background
Given: (1) database of transactions, (2) each transaction is a list of items (purchased by a customer in a visit)
Find: all association rules that satisfy user-specified minimum support and minimum confidence interval
Example: 30% of transactions that contain beer also contain diapers; 5% of transactions contain these items
30%: confidence of the rule
5%: support of the rule
We are interested in finding all rules rather than verifying if a rule holds

4. Rule Measures: Support and Confidence
Customer
buys both
Customer
buys diaper
Find all the rules X & Y  Z with minimum confidence and support
support, s, probability that a transaction contains {X  Y  Z}
confidence, c, conditional probability that a transaction having {X  Y} also contains Z
Customer
buys beer
Let minimum support 50%, and minimum confidence 50%, we have
A  C (50%, 66.6%)
C  A (50%, 100%)

5. Application Examples Market Basket Analysis
*  Maintenance Agreement (What the store should do to boost Maintenance Agreement sales?)
Home Electronics  * (What other products should the store stocks up on if the store has a sale on Home Electronics?)
Attached mailing in direct marketing
Detecting “ping-pong”ing of patients
Transaction: patient
Item: doctor/clinic visited by patient
Support of the rule: number of common patients
HIC Australia “success story”

6. Problem Statement I = {i1, i2, …, im}: a set of literals, called items
Transaction T: a set of items s.t. T I
Database D: a set of transactions
A transaction contains X, a set of items in I, if X T
An association rule is an implication of the form X  Y,
where X,Y I
The rule X  Y holds in the transaction set D with confidence c if c% of transactions in D that contain X also contain Y
The rule X  Y has support s in the transaction set D if s% of transactions in D contain X Y
Find all rules that have support and confidence greater than user-specified min support and min confidence

7. Association Rule Mining: A Road Map
Boolean vs. quantitative associations (Based on the types of values handled)
buys(x, “SQLServer”) ^ buys(x, “DMBook”) ® buys(x, “DBMiner”) [0.2%, 60%]
age(x, “30..39”) ^ income(x, “42..48K”) ® buys(x, “PC”) [1%, 75%]
Single dimension vs. multiple dimensional associations (see ex. Above)
Single level vs. multiple-level analysis
What brands of beers are associated with what brands of diapers?
Various extensions
Correlation, causality analysis
Association does not necessarily imply correlation or causality
Constraints enforced
E.g., small sales (sum < 100) trigger big buys (sum > 1,000)?

8. Problem Decomposition
1. Find all sets of items that have minimum support (frequent itemsets)
2. Use the frequent itemsets to generate the desired rules

9. Problem Decomposition – Example
For min support = 50% = 2 trans,
and min confidence = 50%
For the rule Shoes  Jacket
Support = Sup({Shoes,Jacket)}=50%
Confidence = =66.6%
Jacket  Shoes has 50% support and 100% confidence

10. Discovering Rules Naïve Algorithm for each frequent itemset l do
for each subset c of l do
if (support(l ) / support(l - c) >= minconf) then
output the rule (l – c )  c,
with confidence = support(l ) / support (l - c )
and support = support(l )

11. Discovering Rules (2)
Lemma. If consequent c generates a valid rule, so do all subsets of c. (e.g. X  YZ, then XY  Z and XZ  Y)
Example: Consider a frequent itemset ABCDE
If ACDE  B and ABCE  D are the only one-consequent rules with minimum support confidence, then
ACE  BD is the only other rule that needs to be tested

12. Mining Frequent Itemsets: the Key Step
Find the frequent itemsets: the sets of items that have minimum support
A subset of a frequent itemset must also be a frequent itemset
i.e., if {AB} is a frequent itemset, both {A} and {B} should be a frequent itemset
Iteratively find frequent itemsets with cardinality from 1 to k (k-itemset)
Use the frequent itemsets to generate association rules.

13. The Apriori Algorithm
Lk: Set of frequent itemsets of size k (those with min support)
Ck: Set of candidate itemset of size k (potentially frequent itemsets)
L1 = {frequent items};
for (k = 1; Lk !=; k++) do begin
Ck+1 = candidates generated from Lk;
for each transaction t in database do
increment the count of all candidates in Ck that are contained in t
Lk+1 = candidates in Ck+1 with min_support
end
return k Lk;

14. The Apriori Algorithm — Example
Min support =50% = 2 trans
Database D L1 C1
Scan D C2 C2 L2
Scan D C3 L3
Scan D

15. How to Generate Candidates?
Suppose the items in Lk-1 are listed in order
Step 1: self-joining Lk-1
insert into Ck
select p.item1, p.item2, …, p.itemk-1, q.itemk-1
from Lk-1 p, Lk-1 q
where p.item1=q.item1, …, p.itemk-2=q.itemk-2, p.itemk-1 < q.itemk-1
Step 2: pruning
forall itemsets c in Ck do
forall (k-1)-subsets s of c do
if (s is not in Lk-1) then delete c from Ck

16. Example of Generating Candidates
L3={abc, abd, acd, ace, bcd}
Self-joining: L3*L3
abcd from abc and abd
acde from acd and ace
Pruning:
acde is removed because ade is not in L3
C4={abcd}

17. How to Count Supports of Candidates?
Why counting supports of candidates a problem?
The total number of candidates can be very huge
One transaction may contain many candidates
Method:
Candidate itemsets are stored in a hash-tree
Leaf node of hash-tree contains a list of itemsets and counts
Interior node contains a hash table
Subset function: finds all the candidates contained in a transaction

18. Hash-tree:search
Given a transaction T and a set Ck find all members of its members contained in T
Assume an ordering on the items
Start from the root, use every item in T to go to the next node
If you are at an interior node and you just used item i, then use each item that comes after i in T
If you are at a leaf node check the itemsets

19. Methods to Improve Apriori’s Efficiency
Transaction reduction: A transaction that does not contain any frequent k-itemset is useless in subsequent scans
Partitioning: Any itemset that is potentially frequent in DB must be frequent in at least one of the partitions of DB
Sampling: mining on a subset of given data, lower support threshold + a method to determine the completeness
Dynamic itemset counting: add new candidate itemsets only when all of their subsets are estimated to be frequent

20. Is Apriori Fast Enough? — Performance Bottlenecks
The core of the Apriori algorithm:
Use frequent (k – 1)-itemsets to generate candidate frequent k-itemsets
Use database scan and pattern matching to collect counts for the candidate itemsets
The bottleneck of Apriori: candidate generation
Huge candidate sets:
104 frequent 1-itemset will generate 107 candidate 2-itemsets
To discover a frequent pattern of size 100, e.g., {a1, a2, …, a100}, one needs to generate 2100  1030 candidates.
Multiple scans of database:
Needs (n +1 ) scans, n is the length of the longest pattern

21. Max-Miner
Max-miner finds long patterns efficiently: the maximal frequent patterns
Instead of checking all subsets of a long pattern try to detect long patterns early
Scales linearly to the size of the patterns

22. Max-Miner: the idea f Pruning: (1) set infrequency
Set enumeration tree of
an ordered set f 1 2 3 4
Pruning: (1) set infrequency
(2) Superset frequency
1,2
1,3
1,4
2,3
2,4
3,4
1,2,3
1,2,4
1,3,4
2,3,4
Each node is a candidate group g
h(g) is the head: the itemset of the node
t(g) tail: an ordered set that contains all items that can appear in the subnodes
1,2,3,4
Example: h({1}) = {1} and t({1}) = {2,3,4}

23. Max-miner pruning
When we count the support of a candidate group g, we compute also the support for h(g), h(g) t(g) and h(g) {i} for each i in t(g)
If h(g) t(g) is frequent, then stop expanding the node g and report the union as frequent itemset
If h(g) {i} is infrequent, then remove I from all subnodes (just remove i from any tail of a group after g)
Expand the node g by one and do the same

24. The algorithm Max-Miner Set candidate groups C {}
Set of Itemsets F {Gen-Initial-Groups(T,C)}
while C not empty do
scan T to count the support of all candidate groups in C
for each g in C s.t. h(g) U t(g) is frequent do
F  F U {h(g) U t(g)}
Set candidate groups Cnew{ }
for each g in C such that h(g) U t(g) is infrequent do
F F U {Gen-sub-nodes(g, Cnew)}
C 
remove from F any itemset with a proper superset in F
remove from C any group g s.t. h(g) U t(g) has a superset in F
return F

25. The algorithm (2) Gen-Initial-Groups(T, C)
scan T to obtain F1, the set of frequent 1-itemsets
impose an ordering on items in F1
for each item i in F1 other than the greatest itemset do
let g be a new candidate with h(g) = {i}
and t(g) = {j | j follows i in the ordering}
C C U {g}
return the itemset F1 (an the C of course)
Gen-sub-nodes(g, C) /* generation of new itemsets at the next level*/
remove any item i from t(g) if h(g) U {i} is infrequent
reorder the items in t(g)
for each i in t(g) other than the greatest do
let g’ be a new candidate with h(g’) = h(g) U {i} and t(g’) = {j | j in t(g)
and j is after i in t(g)}
C  C U {g’}
return h(g) U {m} where m is the greatest item in t(g) or h(g) if t(g) is empty

26. Item Ordering
Re-ordering items we try to increase the effectiveness of frequency-pruning
Very frequent items have higher probability to be contained in long patterns
Put these item at the end of the ordering, so they appear in many tails