advanced topics in data mining: association rules

Advanced Topics in Data Mining:

Association Rules

What Is Association Mining?• Association Rule Mining

– Finding frequent patterns, associations, correlations, or causal structures among item sets in transaction databases, relational databases, and other information repositories

• Applications– Market basket analysis (marketing strategy: items to put

on sale at reduced prices), cross-marketing, catalog design, shelf space layout design, etc

• Examples– Rule form: Body ead [Support, Confidence].– buys(x, “Computer”) buys(x, “Software”) [2%, 60%]– major(x, “CS”) ^ takes(x, “DB”) grade(x, “A”) [1%,

75%]

Market Basket Analysis

Typically, association rules are considered interesting if they satisfy both a minimum support threshold and a minimum confidence threshold.

Rule Measures: Support and Confidence

• Let minimum support 50%, and minimum confidence 50%, we have– A C [50%, 66.6%]

– C A [50%, 100%]

Transaction ID Items Bought1000 A,B,C2000 A,C3000 A,D4000 B,E,F

Support & Confidence

Association Rule: Basic Concepts

• Given– (1) database of transactions, – (2) each transaction is a list of items

(purchased by a customer in a visit)

• Find all rules that correlate the presence of one set of items with that of another set of items

• Find all the rules A B with minimum confidence and support– support, s, P(A B)– confidence, c, P(B|A)

Association Rule Mining:A Road Map

• Boolean vs. quantitative associations (Based on the types of values handled in the rule set)– buys(x, “SQLServer”) ^ buys(x, “DM Book”) 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

• Single level vs. multiple-level analysis (Based on the levels of abstractions involved in the rule set)

Terminologies• Item

– I1, I2, I3, …– A, B, C, …

• Itemset– {I1}, {I1, I7}, {I2, I3, I5}, …– {A}, {A, G}, {B, C, E}, …

• 1-Itemset– {I1}, {I2}, {A}, …

• 2-Itemset– {I1, I7}, {I3, I5}, {A, G}, …

Terminologies

• K-Itemset– If the length of the itemset is K

• Frequent K-Itemset– If the length of the itemset is K and the itemset

satisfies a minimum support threshold.

• Association Rule– If a rule satisfies both a minimum support

threshold and a minimum confidence threshold

Analysis• The number of itemsets of a given cardinality

tends to grow exponentially

Mining Association Rules: Apriori Principle

• For rule A C:– support = support({A C}) = 50%– confidence = support({A C})/support({A}) = 66.6%

• The Apriori principle:– Any subset of a frequent itemset must be frequent

Transaction ID Items Bought1000 A,B,C2000 A,C3000 A,D4000 B,E,F

Frequent Itemset Support{A} 75%{B} 50%{C} 50%

{A,C} 50%

Min. support 50%Min. confidence 50%

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

Example

Apriori Algorithm

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}

Another Example 1Database D1 3 42 3 51 2 3 52 5

scan D

count C1

C1 count1 22 33 34 15 3

generate L1

L1

1 2 3 5

scan D

count C2

C2 count12 113 215 123 225 335 2

generate L2

L2

13232535

C2

121315232535

generate C2

scan D

count C3

C3 count235 2

generate L3L3

235C3

235generate C3

Another Example 2

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 to collect counts for the candidate itemsets

• The bottleneck of Apriori:– 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

Demo-IBM Intelligent Minner

Demo Database

Methods to Improve Apriori’s Efficiency

• Hash-based itemset counting: A k-itemset whose

corresponding hashing bucket count is below the threshold

cannot be frequent

• 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

Partitioning

Hash-Based Itemset Counting

= *+

Compare Apriori & DHP (Direct Hash & Pruning)

Apriori

Compare Apriori & DHP

DHP

DHP: Database Trimming

DHP (Direct Hash & Pruning)

• A database has four transactions• Let min_sup = 50%

Example: Apriori

Example: DHP

Mining Frequent Patterns Without Candidate Generation

• Compress a large database into a compact, Frequent-Pattern tree (FP-tree) structure– highly condensed, but complete for frequent pattern mining

– avoid costly database scans

• Develop an efficient, FP-tree-based frequent pattern mining method– A divide-and-conquer methodology: decompose mining

tasks into smaller ones

– Avoid candidate generation & sub-database test only!

Construct FP-tree from a Transaction DB

Construction Steps

• Scan DB once, find frequent 1-itemset (single item pattern)

• Order frequent items in frequency descending order

• Sorting DB according to the frequency descending order

• Scan DB again, construct FP-tree

Benefits of the FP-Tree Structure

• Completeness– never breaks a long pattern of any transaction– preserves complete information for frequent pattern

mining

• Compactness– reduce irrelevant information—infrequent items are

gone– frequency descending ordering: more frequent items

are more likely to be shared– never be larger than the original database (if not count

node-links and counts)– Compression ratio could be over 100

Frequent Pattern Growth

• For I5– {I1, I5}, {I2, I5}, {I2, I1, I5}

• For I4– {I2, I4}

• For I3– {I1, I3}, {I2, I3}, {I2, I1, I3}

• For I1– {I2, I1}

Order frequent items in frequency descending order

Frequent Pattern Growth• For I5

– {I1, I5}, {I2, I5}, {I2, I1, I5}

• For I4– {I2, I4}

• For I3– {I1, I3}, {I2, I3}, {I2, I1, I3}

• For I1– {I2, I1}

SubDB

SubDB

SubDB

SubDB

TrimmingDatabases

FP-tree

Conditional FP-tree from Conditional Pattern-Base

Conditional FP-tree

Conditional FP-tree fromConditional Pattern-Basefor I3

Mining Results Using FP-tree

• For I5 (不產生 NULL)– Conditional Pattern Base

• {(I2I1:1), (I2I1I3:1)}

– Conditional FP-tree

– Generate Frequent Itemsets• I2:2 Rule: I2I5:2

• I1:2 Rule: I1I5:2

• I2I1:2 Rule: I2I1I5:2

Item ID

Support

count

Node

Link

I2 2

I1 2

◎ “I2”: 2

◎ “I1”: 2

◎ NULL: 2


• For I4

– Conditional Pattern Base• {(I2I1:1), (I2:1)}



Item ID

Support

count

Node

Link

I2 2 ◎ “I2”: 2

◎ NULL: 2


• For I3

– Conditional Pattern Base• {(I2I1:2), (I2:2), (I1:2)}


Item ID

Support

count

Node

Link

I2 4

I1 4

◎ “I2”: 4

◎ “I1”: 2

◎ NULL: 4

◎ “I1”: 2


• For I1/I3

– Conditional Pattern Base• {(NULL:2), (I2:2)}


Item ID

Support

count

Node

Link

I2 2 ◎ “I2”: 2

◎ NULL: 4

– Generate Frequent Itemsets• Null:4 Rule: I1I3:4• I2:2 Rule:

I2I1I3:2


• For I2/I3

– Conditional Pattern Base• {(NULL:4)}


◎ NULL: 4

– Generate Frequent Itemsets• Null Rule: I2I3:4


• For I1

– Conditional Pattern Base• {(NULL:2), (I2:4)}



Item ID

Support

count

Node

Link

I2 4 ◎ “I2”: 4

◎ NULL: 6

Mining Frequent PatternsUsing FP-tree

• General idea (divide-and-conquer)– Recursively grow frequent pattern path using the FP-tree

• Method – For each item, construct its conditional pattern-base, and

then its conditional FP-tree

– Repeat the process on each newly created conditional FP-tree

– Until the resulting FP-tree is empty, or it contains only one path (single path will generate all the combinations of its sub-paths, each of which is a frequent pattern)

Major Steps to Mine FP-tree

• Construct conditional pattern base for each

item in the FP-tree

• Construct conditional FP-tree from each

conditional pattern-base

• Recursively mine conditional FP-trees and

grow frequent patterns obtained so far If the conditional FP-tree contains a single path,

simply enumerate all the patterns

Virtual Items in Association Mining• Different region exhibit different selling patterns. Thus,

including as virtual item the information on the location or the type of stores (existing or new) where the purchase was made will enable the comparisons between locations or types within a single chain

• Virtual item may include information on whether the purchase was made with cash, a credit card or check. The inclusion of such virtual item allows to analyze the association between the payment method and items purchased.

• Virtual item may include information on the day of the week or the time of the day the transaction occurred. The inclusion of such virtual item allows to analyze the association between the transaction time and items purchased

Virtual Items: An Example

Dissociation Rules• A dissociation rule is similar to an association rule

except that it can have “not item-name” in the condition or the result of the rule– A and not B C – A and D not E

• Dissociation rules can be generated by a simple adaptation of the association rule analysis

Discussions• The size of a typical transaction grows because it now

includes inverted items• The total number of items used in the analysis doubles

– Since the amount of computation grows exponentially with the number of items, doubling the number of items seriously degrades performance

• The frequency of the inverted items tends to be much larger than the frequency of the original items. So, it tends to produce rules in which all items are inverted. These rules are less likely to be actionable. – not A and not B not C

• It is useful to invert only the most frequent items in the set used for analysis. It is also useful to invert some items whose inverses are of interest.

Interestingness Measurements

• Subjective Measures – A rule (pattern) is interesting if

• it is unexpected (surprising to the user)

• actionable (the user can do something with it)

• Objective Measures– Two popular measurements:

• Support

• confidence

Criticism to Support and Confidence

• Example 1– Among 5000 students

• 3000 play basketball• 3750 eat cereal• 2000 both play basket ball and eat cereal

– play basketball eat cereal [40%, 66.7%] is misleading because the overall percentage of students eating cereal is 75% which is higher than 66.7%.

basketball not basketball sum (row)cereal 2000 1750 3750

not cereal 1000 250 1250sum (col.) 3000 2000 5000


• Example 2– X and Y: positively correlated,– X and Z, negatively related– support and confidence of X=>Z dominates

• We need a measure of dependent or correlated events

X 1 1 1 1 0 0 0 0Y 1 1 0 0 0 0 0 0Z 0 1 1 1 1 1 1 1

nt)(Improveme )()(

)(, BPAP

BAPcorr BA

Rule Support ConfidenceX=>Y 25% 50%X=>Z 37.50% 75%


• Improvement (Correlation)– Taking both P(A) and P(B) in consideration

– P(A^B)=P(B)*P(A), if A and B are independent events

– A and B negatively correlated, if the value is less than 1

otherwise A and B positively correlated

– When improvement is less than 1, negating the result produces a better

rule

• X => NOT Z

X 1 1 1 1 0 0 0 0Y 1 1 0 0 0 0 0 0Z 0 1 1 1 1 1 1 1

)()(

)(, BPAP

BAPcorr BA

Itemset Support ImprovementX,Y 25% 2X,Z 37.50% 0.9Y,Z 12.50% 0.57

Multiple-Level Association Rules

• Items often form hierarchy• Items at the lower level are expected to

have lower support• Rules regarding itemsets at appropriate

levels could be quite useful• Transaction database can be encoded based

on dimensions and levels• We can explore multi-level mining

Transaction Database

Concept Hierarchy

Mining Multi-Level Associations

• A top_down, progressive deepening approach– First find high-level strong rules

milk bread [20%, 60%]

– Then find their lower-level “weaker” rules2% milk wheat bread [6%, 50%]

• Variations at mining multiple-level association rules.– Cross-level association rules (Generalized Asso. Rules)

2% milk Wonder wheat bread

– Association rules with multiple, alternative hierarchies2% milk Wonder bread

Multi-level Association: Uniform Support vs. Reduced Support

• Uniform Support: the same minimum support for all levels– One minimum support threshold is needed.– Lower level items do not occur as frequently. If support threshold

• too high miss low level associations• too low generate too many high level associations

• Reduced Support: reduced minimum support at lower levels– There are 4 search strategies:

• Level-by-level independent• Level-cross filtering by k-itemset• Level-cross filtering by single item• Controlled level-cross filtering by single item

Uniform Support

• Optimization Technique– The search avoids examining itemsets containing

any item whose ancestors do not have minimum support.

Uniform Support : An Example

L1: 2,3,4,5,6L2: 23,24,25,26,34,45,46,56L3: 234,245,246,256,456L4: 2456

L1: 2,3,6,8,9L2: 23,68,69,89L3: 689

L1: 2,3,4L2: 23,24,34L3: 234

L1: 9

min_sup=50%

min_sup=60%

min_sup=50%

min_sup=60%


Crab

King’s Crab Sunset Milk Dairyland Milk Dairyland Cheese Best Cheese

Milk Cheese

All

Best Bread Wonder Bread

Bread

Goldenfarm Apple Westcoast Bread

Apple Pie

Tasty Pie

1 2 3

4 5 6

1 2 6 3 5

4 9 7 10 8


L1: 2,3,4,5,6L2: 23,24,25,26,34,45,46,56L3: 234,245,246,256,456L4: 2456

L2: 23,68,69,89L3: 689

min_sup=50%

min_sup=50%

L1: 2,3,6,8,9min_sup=50%

C2: 23,36,29,69,28,68,39,89C3: 239,369,289,689

Scan DB

Scan DB

(1) (2)

(3)

Apriori/DHPFP Growth


Crab


Milk Cheese

All


Bread


Apple Pie

Tasty Pie

1 2 3

4 5 6

1 2 6 3 5

4 9 7 10 8

Reduced Support

• Each level of abstraction has its own minimum support threshold.

Search Strategies forReduced Support

• There are 4 search strategies:– Level-by-level independent

• Full-Breadth Search

• No pruning– No background knowledge of frequent itemsets is used for pruning

– Level-cross filtering by single item• An item at the ith level is examined if and only if its parent node

at the (i-1)th level is frequent.

– Level-cross filtering by k-itemset• An k-itemset at the ith level is examined if and only if its corresp

onding parent k-itemset at the (i-1)th level is frequent.

– Controlled level-cross filtering by single item

Level-Cross Filtering by Single Item

Reduced Support : An Example

L2: 23,69

min_sup=60%

min_sup=50%

L1: 2,3,6,9min_sup=50%

Scan DB


(1)

(2)

(3)

(1)

L1: 2,3,4L2: 23,24,34L3: 234Apriori/DHP

FP Growth


Crab


Milk Cheese

All


Bread


Apple Pie

Tasty Pie

1 2 3

4 5 6

1 2 6 3 5

4 9 7 10 8

Level-Cross Filtering by K-Itemset


L2: 23,69

min_sup=60%

min_sup=50%

L1: 2,3,6,9min_sup=50%

C2: 23,36,29,69,39C3: 239,369

Scan DB

Scan DB

(1) (2)

(3)


L1: 2,3,4L2: 23,24,34L3: 234


Crab


Milk Cheese

All


Bread


Apple Pie

Tasty Pie

1 2 3

4 5 6

1 2 6 3 5

4 9 7 10 8

Reduced Support• Level-by-level independent

– It is very relaxed in that it may lead to examining numerous infrequent items at low levels, finding associations between items of little importance.

• Level-cross filtering by k-itemset– It allows the mining system to examine only the children of frequent

k-itemsets.– This restriction is very strong in that there usually are not many k-

itemsets.– Many valuable patterns may be filtered out.

• Level-cross filtering by single item– A compromise between the above two approaches– This method may miss associations between low level items that are

frequent based on a reduced minimum support, but whose ancestors do not satisfy minimum support.

Controlled Level-Cross Filtering by Single Item


min_sup=60%

min_sup=50%

L1: 2,3,6,8,9min_sup=50%

Scan DB


(1)

(2)

(3)

(1)

L1: 2,3,4L2: 23,24,34L3: 234Apriori/DHP

FP Growth

level_passage_sup=50%L1: 2,3,4,5,6

L2: 23,68,69,89L3: 689


Crab


Milk Cheese

All


Bread


Apple Pie

Tasty Pie

1 2 3

4 5 6

1 2 6 3 5

4 9 7 10 8

Multi-Dimensional Association

• Single-Dimensional (Intra-Dimension) Rules: Single Dimension (Predicate) with Multiple Occurrences.– buys(X, “milk”) buys(X, “bread”)

• Multi-Dimensional Rules: 2 Dimensions– Inter-dimension association rules (no repeated

predicates)• age(X,”19-25”) occupation(X,“student”) buys(X,“coke”)

– hybrid-dimension association rules (repeated predicates)

• age(X,”19-25”) buys(X, “popcorn”) buys(X, “coke”)

Summary• Association rule mining

– Probably the most significant contribution from the database community in KDD

– A large number of papers have been published

• Some Important Issues– Generalized Association Rules

– Multiple-Level Association Rules

– Association Analysis in Other Types of Data• Spatial Data, Multimedia Data, Time Series Data, etc.

– Weighted Association Rules

– Quantitative Association Rules

Weighted Association Rules

• Why Weighted Association Analysis? – In previous work, all items in a transactional database

are treated uniformly

– Items are given weights to reflect their importance to the user

– The weights may correspond to special promotions on some products, or the profitability of different items

• Some products may be under promotion and hence are more interesting, or some products are more profitable and hence rules concerning them are of greater values

Weighted Association Rules

• A simple attempt to solve this problem is to eliminate the items with small weights– However, a rule for a heavy weighted item may also

consist of low weighted items

• Is Apriori algorithm feasible?– Apriori algorithm depends on the downward closure

property which governs that subsets of a frequent itemset are also frequent

– However, it is not true for the weighted case

Weighted Association Rules:An Example

• Total Benefits: 500– Benefits for the First Transaction: (40+30+30+20+20+10+ 10)=160

– Benefits for the Second Transaction: (40+30+20+ 20+10+10)=130

– Benefits for the Third Transaction: (40+30+20+10+10)= 110

– Benefits for the Fourth Transaction: (30+30+20+10+10)=100

• Suppose Weighted_Min_Sup = 40%– Minimum Benefits = 500 * 40% = 200

An Example

• Minimum Benefits = 500 * 40% = 200

• Itemset {3,5,6,7}– Benefits: 70

– Support Count (Frequency): 3

– 70 * 3 = 210 >= 200 {3,5,6,7}is a Frequent Itemset

• Itemset {3,5,6}– Benefits: 60

– Support Count ( Frequency): 3

– 60 * 3 = 180 < 200 {3,5,6} is not a Frequent Itemset

Apriori Principle can not be applied!

K-Support Bound• If Y is a frequent q-itemset

– Support_Count(Y) (Weighted_Min_Sup * Total_Benefits) / Benefits(Y)

• Example– {3,5,6,7} is a Frequent 4-Itemset

• Support_Count({3,5,6,7}) = 3 (40% * 500) / Benefits({3,5,6,7}) = (40% * 500) / 70 = 2.857

• If X is a frequent k-itemset containing q-itemset Y– Minimum_Support_Count(X)

(Weighted_Min_Sup * Total_Benefits) / (Benefits(Y) + (k-q) Maximum Remaining Weights)

• Example– X is a Frequent 5-Itemset containing {3,5,6,7}

• Minimum_Support_Count(X) (40% * 500) / (70 + 40) = 1.81

K-Support Bound

K-Support Bound• Itemset {1,2}

– Benefits: 70– Support_Count({1,2}) = 1 <

(40% * 500) / Benefits({1,2}) = (40% * 500) / 70 = 2.857– {1,2} is not a Frequent Itemset

• If X is a frequent 3-itemset containing {1,2}– Minimum_Support_Count(X) (40% * 500) / (70 + 30) = 2– But, Maximum_Support_Count(X) = 1– No frequent 3-itemsets containing {1,2}

• If X is a frequent 4-itemset containing {1,2}– Minimum_Support_Count(X) (40% * 500) / (70 + 30 + 20) = 1.667– But, Maximum_Support_Count(X) = 1– No frequent 4-itemsets containing {1,2}

• Similarly, no frequent 5, 6, 7-itemsets containing {1,2}• The algorithm is designed based on this k-support bound

MINWAL Algorithm

Step by Step

• Input: Product & Transactional Databases

Total Profits: 1380

Weighted_Min_Sup = 50%

Step 2, 7

• Search(D)– This subroutine finds out the maximum transaction size in

that transactional database D• Size = 4 in this case

• Counting(D, w)– This subroutine cumulates the support counts of the 1-ite

msets– The k-support bounds of each 1-itemset will be calculated,

and the 1-itemsets with support counts greater than any of the k-support bounds will be kept in C1

Step 7

K-Support Bound (50% * 1380) / (10 + 90) = 6.9

Step 11

• Join(Ck-1)

– The Join step generates Ck from Ck-1 as Apriori Algorithm

• If we have {1, 2, 3}, {1, 2, 4} in Ck-1 {1, 2, 3, 4} will be generated in Ck

– In this case,• C1 = {1 (4), 2 (5), 4 (6), 5 (7)}

• C2 = Join(C1) = {12, 14, 15, 24, 25, 45}

Support_Count

Step 12• Prune(Ck)

– The itemset will be pruned in either of the following cases• A subset of the candidate itemset in Ck does not exist in Ck-1

• Estimate an upper bound on the support count (SC) of the joined itemset X, which is the minimum support count among the k different (k-1)-subsets of X in Ck-1. If the estimated upper bound on the support count shows that the itemset X cannot be a subset of any large itemset in the coming passes (from the calculation of k-support bounds for all itemsets), that itemset will be pruned

– In this case,• C2 = Prune(C2) = {12 (4), 14 (4), 15 (4), 24 (5), 25 (5), 45 (6)}

• Using K-Support-Bound

Estimated_Support_Count

Step 12• Prune(Ck)

– Using K-Support-Bound (No one is pruned)

Step 13• Checking(Ck, D)

– Scan DB & Generate Lk

L2

C2

Step 11, 12

• Join(C2)– C2 = {15 (4), 24 (5), 25 (5), 45 (6)}

– C3 = Join(C2) = {245}

• Prune(C3)– C3 = Prune(C3) = {245 (5)}

– Using K-Support-Bound (No one is pruned)

Step 13

• Checking(C3, D): Scan DB

• C3 = L3 = {245}

• Finally, L = {45, 245}

Step 15

• Generate Rules for L = {45, 245}– 4 5 (confidence = 100%)

– 5 4 (confidence = 85.7%)

– 24 5 (confidence = 100%)

– 25 4 (confidence = 100%)

– 45 2 (confidence = 83.3%)

– 2 45 (confidence = 100%)

– 4 25 (confidence = 83.3%)

– 5 24 (confidence = 71.4%)

Min_Conf=90%

Generalized Association Rules

Quantitative Association Rules

– {A(1..2), B(3)} 50%– {A(1..2), B(3..5)} 60%– {A(1..2), B(1..5)} 60%– {B(1..5), D(1..2)} 60%– {B(3..5), D(1..3)} 60%– {B(1..5), D(1..3)} 70%– {B(1..3), D(1..2)} 50%– {B(3..5), D(1..2)} 50%– {A(1..2), B(3..5), D(1..2)} 50%

• Let min_sup = 50%, we have– {A, B} 60%

– {B, D} 70%

– {A, B, D} 50%

advanced topics in data mining: association rules

Documents

support support

support confidence slide

minimum support threshold

pba slide

association rule mining

mining frequent itemsets

association mining

apriori algorithm slide