Association Mining

Dr. Yan Liu

Department of Biomedical, Industrial and Human Factors Engineering

Wright State University

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Introduction

What is Association Mining Discovering frequent patterns, associations, correlations, or causal structures

among sets of items or objects in transaction databases, relational databases, or other information repositories

Frequent patterns Patterns (such as itemsets, subsequences, or substructures) that occur frequently

Motivation of Association Mining Discovering regularities in data

What products are often purchased together? — Beer and diapers?! What are the subsequent purchases after buying a PC? What kinds of DNA are sensitive to this new drug? Can we automatically classify web documents?

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Association Rules: Basic Concepts I={I1, …, In} is a set of items D is the task-relevant dataset consisting of a set of transactions where each

transaction T is a set of items such that Association Rule

X Y, where X and Y are antecedent and consequent items, respectively

Support Probability that a transaction contains both X and Y, i.e. P(XY)

IT ⊆

ΦYXTYTX =∩,⊂,⊂

P(X Y) = (# of transactions that contain both X and Y) / (total # of transactions)

Confidence Probability that a transaction that contains Y also contains X, i.e. P(Y|X)

P(Y|X) = P(X Y) / P(X) = support (X Y) / support (X)

Mining Association Rules Finding association rules that satisfy the minimum support and confidence thresholds

Min. support 50%Min. confidence 60%

Transaction-id Items bought

1 A, B, C

2 A, C

3 A, D

4 B, E, F

Frequent Itemset Support

{A} 3/4 =75%

{B} 2/4 = 50%

{C} 2/4 = 50%

{A, C} 2/4 = 50%

I={A, B, C, D, E, F}

A C:

support = 50%

confidence = support(A C)/support(A) = 50% / 75% = 66.6%

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Mining Association Rules

Goal Discover rules with high support and confidence values

Two-Step Process Find all frequent itemsets

Itemsets that occur at least as frequently as the predetermined minimum support Generate strong association rules from the frequent itemsets

Generate rules that satisfy minimum support and minimum confidence If we have all frequent itemsets, we can compute support and

confidence!

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Apriori Algorithm

Overview First proposed by Agrawal and Srikant (1994) for mining Boolean association

rules Use prior knowledge of frequent itemset properties

Any subset of a frequent itemset must be frequent (why?) e.g. if itemset{beer, diaper, nuts} is frequent, so is itemset {beer, diaper}

Apriori pruning principle: If there is an itemset is infrequent, its superset is also infrequent and thus should not be generated

Process of Generating Frequent Itemsets Join step

Generate all candidate k-itemsets, Ck, by self-joining frequent (k-1)-itemsets, Lk-1

e.g. L2={ac, bc, be} self-joining L2 x L2 : C3={abc, ace, abe, bce} Prune step

A scan of the database to determine the count of each candidate in Ck to determine Lk

e.g. Pruning C3 gets L3={bce}. But{abc}, {ace}, {abe} are not frequent itemsets because {ab}, {ae}, and{ce} are not in L2

Transaction Database

1st Scan

C1 L1

L2

C2

2nd scan

C3 L3

3rd scan

Tid Items

10 A, C, D

20 B, C, E

30 A, B, C, E

40 B, E

Itemset Sup. count

{A} 2

{B} 3

{C} 3

{D} 1

{E} 3

Itemset Sup. count

{A} 2

{B} 3

{C} 3

{E} 3

Itemset

{A, B}

{A, C}

{A, E}

{B, C}

{B, E}

{C, E}

Itemset Sup. count

{A, B} 1{A, C} 2{A, E} 1{B, C} 2{B, E} 3{C, E} 2

Itemset Sup. count

{A, C} 2{B, C} 2{B, E} 3{C, E} 2

Itemset

{B, C, E}Itemset Sup. count

{B, C, E} 2

Self-Join

Apriori Algorithm Example

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Self-Join

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Apriori Algorithm (Cont.)

Generating Association Rules from Frequent Itemsets For each frequent k-itemset (k≥2), l, generate all nonempty proper subsets of l For each nonempty subset of l, s, output the rule “s(l- s)” if the confidence of

this rule satisfies the minimum confidence threshold, i.e.

support_count (l) / support_count(s) ≥ minimum confidence

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Apriori Algorithm Example (Cont.)Rules Support Confidence

A C 2/4 = 50% 2/2 = 100%

C A 2/4 = 50% 2/3 = 66.7%

B C 2/4 = 50% 2/3 = 66.7%

C B 2/4 = 50% 2/3 = 66.7%

B E 3/4 =75% 3/3 = 100%

E B 3/4 =75% 3/3 = 100%

C E 2/4 = 50% 2/3 = 66.7%

E C 2/4 = 50% 2/3 = 66.7%

B C+E 2/4 = 50% 2/3 = 66.7%

C B+E 2/4 = 50% 2/3 = 66.7%

E B+C 2/4 = 50% 2/3 = 66.7%

B+C E 2/4 = 50% 2/2 = 100%

B+E C 2/4 = 50% 2/3 = 66.7%

C+E B 2/4 = 50% 2/2 = 100%

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Improve Efficiency of Apriori

Challenge in Mining Frequent Itemsets Multiple scans of transaction database are costly Huge number of candidates

e.g. To find frequent itemset {i1,i2…,i100 }: # of scans: 100, # of Candidates:

Transaction Reduction Reduce the number of transactions scanned in future iterations A transaction that does not contain any frequent k-itemsets cannot contain any

frequent (k+1)-itemsets and thus do not need to be considered in future scans

30100100100

2100

1100 10×27.1≈1-2=+...++ CCC

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Improve Efficiency of Apriori (Cont.)

Partitioning Need only two database scans to mine frequent itemsets Scan 1: Divide database into non-overlapping partitions and find local frequent

itemsets for each partition Scan 2: Assess actual support of local frequent itemsets to determine global

frequent patterns Sampling

Randomly select a sample of the database and search for frequent itemset in the sample

Trade off accuracy against efficiency

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Improve Efficiency of Apriori (Cont.)

Dynamic Itemset Counting (DIC) Database is divided into blocks marked by start points New candidates can be added at any start point once all of their subsets are

estimated to be frequent In Apriori, new candidates are added only after a complete database scan

Transactions

1-itemsets2-itemsets

…

1-itemsets

2-itemsets3-itemsetsDIC

Apriori

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Frequency-Pattern (FP) Growth

Purpose Find frequent itemsets without candidate generation

General Idea Compress the database representing frequent items into a FP-tree which

retains the itemset association information Mine the FP-tree to find frequent itemsets

Construct FP-Tree 1st scan of the database: derive the set of frequent items and their support

counts; sort the frequent items in the order of descending support count (the resulting list is denoted L)

Create the root of the tree, labeled “null” 2nd scan of the database: the items in each transaction are processed in L order ,

and a branch is created for each transaction Braches that with share a common prefix are combined To facilitate tree traversal, an item header table is built so that each item points

to its occurrences in the tree via a chain of node-links

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{f, c, a, m, p}{f, c, a, b, m}{f, b}{c, b, p}{f, c, a, m, p}

(ordered) frequent items

1 {f, a, c, d, g, i, m, p} 2 {a, b, c, f, l, m, o} 3 {b, f, h, j, o, w}4 {b, c, k, s, p}5 {a, f, c, e, l, p, m, n}

TID items bought

Minimum support count is 2. L: {{f: 4}, {c: 4}, {a: 3}, {b: 3}, {m:3}, {p: 3}}

{}

f:1

c:1

a:1

m:1

p:1

(root)T1

f:1

c:1

a:1

b:1

m:1

T2

{}

f:2

c:2

a:2

m:1

p:1

b:1

m:1

T1+T2

f:1

b:1

T3

{}

f:3

c:2

a:2

m:1

p:1

b:1

m:1

T1+T2+T3

b:1

c:1

T4

b:1

p:1

f:1

c:1

a:1

m:1

p:1

T5

{}

f:4

c:3

a:3

m:2

p:2

b:1

m:1

b:1

c:1

b:1

p:1

FP-Tree Growth Example

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{}

f:4 c:1

b:1

p:1

b:1c:3

a:3

b:1m:2

p:2 m:1

Header Table

Item Frequency f 4c 4a 3b 3m 3p 3

FP-Tree Registers Compressed Frequent Pattern Information

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Frequency-Pattern (FP) Growth (Cont.)

Mine Frequent Itemsets from FP-Tree Starting from the last item in the header table, for each frequent item, construct

its conditional pattern-base, and then its conditional FP-tree Conditional pattern-base of an item consists of the set of its prefix paths in the FP-

tree co-occurring with the suffix pattern Repeat the process on each newly created conditional FP-tree Until the resulting conditional 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

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{}

f:4 c:1

b:1

p:1

b:1c:3

a:3

b:1m:2

p:2 m:1 Traverse the FP-tree by following the link of each

frequent item p Accumulate all of transformed prefix paths of item p

to form p’s conditional pattern base Construct conditional FP-tree by eliminating non-

frequent items Concatenate items in conditional FP-tree with p to

generate frequent itemsets with p

<f, c, a, m, p: 2>, <c, b, p: 1>Considering p as suffix

<f, c, a, m: 2>, <c, b: 1> Conditional Pattern Base

<f: 2, c: 2, a: 2, m: 2>, <c: 1> Conditional FP-Tree

{f,p:2}, {c,p:3}, {a,p:2}, {m,p:2}{f,c,p:2}, {f,a,p:2}, {f,m,p:2}, {c,a,p:2}, {c,m,p:2}, {a,m,p:2}{f,c,a,p:2}, {c,a,m,p:2}{f,c,a,m,p:2}

Frequent itemsets

FP-Tree Growth Example (Cont.)

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Item Conditional Pattern Base Conditional FP-Tree Frequent Patterns Generated

p <f, c, a, m: 2>, <c, b: 1> <f: 2, c: 2, a: 2, m: 2>, <c: 1> {f,p:2}, {c,p:3}, {a,p:2}, {m,p:2}, {f,c,p:2}, {f,a,p:2}, {f,m,p:2}, {c,a,p:2}, {c,m,p:2}, {a,m,p:2}, {f,c,a,p:2}, {c,a,m,p:2}, {f,c,a,m,p:2}

m <f, c, a: 2>, <f, c, a, b: 1> <f: 3, c: 3, a: 3> {f, m: 3}, {c, m: 3}, {a, m: 3}, {f, c, m: 3}, {f, a, m: 3}, {c, a, m: 3}, {f, c, a, m: 3}

b <f: 1>, <f, c, a : 1>, <c: 1> <f: 2>, <c: 2> {f, b: 2}, {c, b: 2}

a <f, c: 3> <f: 3>, <c: 3> {f, a: 3}, {c, a: 3}

c <f: 3> <f: 3> <f, c: 3>

f  N/A N/A   N/A

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Advantages of FP-Growth over Apriori

Divide-and-Conquer Decompose both the mining task and database according to the frequent

patterns obtained so far Leads to focused search of smaller dataset

Other Factors No candidate generation, no candidate test Compressed database: FP-tree structure No repeated scan of entire database Basic operation — counting local frequent items and building sub FP-tree, no

pattern search and matching

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Mining Various Kinds of Rules or Regularities

Multi-Level Association Rules Involve concepts at different levels of abstraction

Multi-Dimensional Association Rules Involve more than one antecedent

Quantitative Association Rules Involve numeric attributes that have an implicit ordering among values

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Mining Multi-Level Association Rules

Mining Multi-Level Hierarchy Top-down strategy

Starting from the top level in the hierarchy and working downward in the hierarchy toward the more specific concept levels

For each level frequent itemsets and association rules are mined

Variations of Support Threshold Uniform minimum support threshold for all levels

The same minimum support threshold is used for all levels Reduced minimum support threshold at lower levels

Lower-level items usually have lower support Group-based minimum support threshold

Users or experts set up user-specific item- or group-based minimum support threshold

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Uniform support

Milk[support = 10%]

2% Milk [support = 6%]

Skim Milk [support = 4%]

Level 1min_sup = 5%

Level 2min_sup = 5%

Level 1min_sup = 5%

Level 2min_sup = 3%

Reduced support

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Mining Multi-Level Association Rules (Cont.)

Rule Redundancy Some rules may be redundant due to “ancestor” relationships between items A rule is redundant if its support is close to the “expected” value, based on the

rule’s ancestor

e.g. milk is the “ancestor” of “2% milk” Suppose Rule 1: milk wheat bread [support = 8%, confidence = 70%]and we know that about ¼ of the milk is 2% milkIf Rule 2: 2% milk wheat bread [support = 2%, confidence = 72%], then rule 2 is redundant

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Mining Multi-Dimensional Association Rules

Single-Dimensional Rules e.g. buys(X, “milk”) buys(X, “bread”)

Multi-Dimensional Rules: 2 antecedents Inter-dimension assoc. rules (no variable appear in both antecedent and

consequent)e.g. age(X, “19-25”) occupation(X,“student”) buys(X,“coke”)

Hybrid-dimension assoc. rules (variables can appear in both antecedent and consequent)

e.g. age(X,”19-25”) buys(X, “popcorn”) buys(X, “coke”)

Categorical Attributes Finite number of possible values, no ordering among values

Quantitative Attributes Numeric, implicit ordering among values

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Mining Multi-Dimensional Association Rules (Cont.)

Static Discretization Quantitative attributes are discretized using predefined concept hierarchies

(static discretization) e.g. Values of attribute age can be discretized into intervals “0… 20K”,

“21K… 30K”, “31K… 40K”, … Dynamic Discretization

Quantitative attributes are discretized or clustered into “bins” based on data distribution

Treats numeric attribute values as quantities rather than predefined ranges or categories

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Static Discretization of Quantitative Attributes

Quantitative attributes are discretized prior to mining using predefined concept

hierarchies

Numeric values are replaced by intervals

Data cube is well suited for mining multi-dimensional association rules The cells of an k-dimensional cuboid correspond to the itemsets

Store aggregates (such as support counts) in multi-dimensional space

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(income)(age) (buys)

(age,income)(age,buys)

(income,buys)

(age,income,buys)

00-D cuboid

1-D cuboids

2-D cuboids

3-D cuboid

3D Data Cube (each cuboid representing an item or itemset)

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Quantitative Association Rules

Numeric attributes are dynamically discretized to satisfy some mining criteria Such that maximizing the confidence or compactness of the rules mined

2-D Quantitative Association Rules Aquan1 Aquan2 Acat

Aquan1 and Aquan2 are two quantitative predicate attribute intervals (determined dynamically)

Acat is a categorical attribute e.g. age(X, “30…39”) income(X, “42K…48K”) buys(X, “HDTV”)

Association Rule Clustering System Map “adjacent” association rules to form general rules using a 2-D grid Search the grid for clusters of points from which the association rules are

generated

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Quantitative Association Rules

Numeric attributes are dynamically discretized to satisfy some mining criteria Such that maximizing the confidence or compactness of the rules mined

2-D Quantitative Association Rules Aquan1 Aquan2 Acat

Aquan1 and Aquan2 are two quantitative predicate attribute intervals (determined dynamically)

Acat is a categorical attribute e.g. age(X, “30…39”) income(X, “42K…48K”) buys(X, “HDTV”)

Association Rule Clustering System Map “adjacent” association rules to form general rules using a 2-D grid Search the grid for clusters of points from which the association rules are

generated

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Association Rule Clustering System

Step 1: Binning Partition the ranges of quantitative attributes into intervals Equal-width binning

The interval size of each bin is the same Equal-frequency binning

Each bin has approximately the same number of records Clustering-based binning

Clustering is performed on the quantitative attribute to group neighboring points in to the same bin

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Price($)Equi-Width(width $10)

Equi-Depth(depth 2) Clustering-Based

7 [0,10] [7,20] [7,7]20 [11,20] [22,50] [20,22]22 [21,30] [51,53] [50,53]50 [31,40]51 [41,50]53 [51,60]

Illustration of Thee Methods of Binning

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Association Rule Clustering System (Cont.)

Step 2: Finding Frequent Predicate Sets Once the 2-D array containing the count distribution for each category is set up,

it can be scanned to find the frequent predicate sets (i.e. those satisfying minimum support) that also satisfy minimum confidence

Use the rule algorithm generation algorithm (such as Apriori) discussed before Step 3: Clustering Association Rules

Strong association rules obtained in the previous step are mapped to a 2-D grid

• age(X, “34”) income(X, “30K-40K”) buys(X, “HDTV”)• age(X, “34”) income(X, “40K-50K”) buys(X, “HDTV”)• age(X, “35”) income(X, “30K-40K”) buys(X, “HDTV”)• age(X, “35”) income(X, “40K-50K”) buys(X, “HDTV”)

Combined into age(X, “34-35”) income(X, “30K-50K”) buys(X, “HDTV”)

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Basketball Not Basketball Sum (row)

Cereal 2000 1750 3750

Not Cereal 1000 250 1250

Sum (col) 3000 2000 5000

play basketball eat cereal, support = ? Confidence = ?

Support = 2000/5000 = 40%Confidence = 2000/3000 = 66.7%

The overall percentage of students eating cereal (regardless basketball play) is 3750/5000 = 75% > 66.7%, so rule play basketball eat cereal is misleading

play basketball not eat cereal, support = ? Confidence = ?

Support = 1000/5000 = 20%Confidence = 1000/3000 = 33.3%The overall percentage of students not eating cereal (regardless basketball play) is 1250/5000 = 25% < 33.3%, so rule play basketball not eat cereal is more accurate than play basketball eat cereal

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Correlation AnalysisBasketball Not Basketball Sum (row)

Cereal 2000 1750 3750

Not Cereal 1000 250 1250

Sum (col) 3000 2000 5000

play basketball eat cereal, support = ? Confidence = ?

Support = 2000/5000 = 40%Confidence = 2000/3750 = 66.7%

The overall percentage of students eating cereal (regardless basketball play) is 3750/5000 = 75% > 66.7%, so rule play basketball eat cereal is misleading

play basketball not eat cereal, support = ? Confidence = ?

Support = 1000/5000 = 20%Confidence = 1000/3000 = 33.3%The overall percentage of students not eating cereal (regardless basketball play) is 1250/5000 = 25% < 33.3%, so rule play basketball not eat cereal is more accurate than play basketball eat cereal

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Correlation Analysis (Cont.) Why Correlation Analysis

Support and confidence measures can be insufficient in filtering out uninteresting association rules

Correlation measures can augment the support-confidence framework for association rules

Lift χ2 analysis All_confidence Cosine

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Lift If occurrence of A is independent of occurrence of B if P(A and B) = P(A)P(B)

))P(P()and(

=),( BABAP

BAlift

If lift(A, B) < 1, then occurrence of A is negatively correlated with the occurrence of B If lift(A, B) > 1, then occurrence of A is positively correlated with the occurrence of B If lift(A, B) = 1, then occurrences of A and B are independent

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Basketball Not Basketball Sum (row)

Cereal 2000 1750 3750

Not Cereal 1000 250 1250

Sum (col) 3000 2000 5000

play basketball eat cereal, lift= ?

play basketball not eat cereal, lift= ?

P(play basketball and eat cereal) = 2000/5000 = 40%P(play basketball) = 3000/5000 = 60%P(eat cereal) = 3750/5000 = 75%lift(play basketball , eat cereal) = 40%/(60%*75%) = 0.889

P(play basketball and not eat cereal) = 1000/5000 = 20%P(not eat cereal) = 1250/5000 = 25%lift(play basketball , eat cereal) = 20%/(60%*25%) = 1.33

In conclusion, playing basketball and eating cereal are negatively correlated!

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χ2 Analysis

∑ expected)expected-observed(2

2

=χ

Basketball Not Basketball Sum (row)

Cereal 2000 ( 2250) 1750 (1500) 3750

Not Cereal 1000 (750) 250 (500) 1250

Sum (col) 3000 2000 5000

500)500-502(

750)750-0001(

1500)1500-1750(

2250)2250-0002(2

2222

+++=)1(χ = 277.78 >> χ2 0.05(1) = 3.84

• playing basketball and eating cereal are NOT independent• Observed the value of (basketball, Cereal) is less than the expected value of (basketball, Cereal), so playing basketball and eating cereal are negatively correlated

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All_Confidence

Given an itemset X={i1, i2, …, ik}, the all_confidence of X is defined as

}∈∀)|max{sup()sup(

=)(all_conf XiiX

jjX

where }∈∀)|max{sup( Xii jj is the maximum single item support for all the items in X all_confidence of X is the minimal confidence among the set of rules ij → X- ij, where Xi j ∈

X = {basketball, cereal}

sup(X) = 2000/5000 = 40%max{sup(ij)} = max{3000/5000, 3750/5000} = 3750/5000 = 75%all_conf (X) = 40%/75% = 53.3%

• if X={A, B}, when all_conf (X) > 0.5, A and B are positively correlated; when all_conf (X) = 0.5, A and B are independent; when all_conf (X) < 0.5, A and B are negatively correlated

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cosine Measure

Given two itemsets A and B, the cosine measure of A and B is defined as

)sup(×)sup(

)andsup(

)(×)(

)and(== BA

BA

BPAP

BAPcosine (A, B)

• cosine (A, B) > 0.5, A and B are positively correlated; cosine (A, B) = 0.5, A and B are independent; cosine (A, B) < 0.5, A and B are negatively correlated•cosine measure can be viewed as a harmonized lift measure: the square root is taken on P(A) x P(B), so that the cosine value is only influenced by sup(A) and sup(B), not by the number of transactions

A = {basketball}, B = {cereal}

sup(A) = 3000/5000 , sup(B) = 3750/5000, sup(A and B) = 2000/5000cosine(A, B) = 2000/(√3000*3750) = 59.6%

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Comparison of Four Correlation Measures  milk no milkcoffee mc ~mcno cofee m~c ~m~c

Dataset mc ~mc m~c ~m~c all_conf cosine lift χ2

A1 1,000(12) 100 100 100,000 0.91 0.91 83.64 83,452.6A2 1,000(108) 100 100 10,000 0.91 0.91 9.26 9,055.7A3 1,000(550) 100 100 1,000 0.91 0.91 1.82 1,472.7A4 1,000(1008) 100 100 0 0.91 0.91 0.99 9.9B1 1,000(1000) 1,000 1,000 1,000 0.50 0.50 1.00 0.0C1 100(12) 1,000 1,000 100,000 0.09 0.09 8.44 670.0C2 1,000(109) 100 10,000 100,000 0.09 0.29 9.18 8,712.8C3 1(0) 1 100 10,000 0.01 0.07 50.00 48.5

• lift and χ2 are poor indicators because they are greatly affected by the null transaction • all_conf and cosine are better indicators because they are not affected by the null transaction

• cosine is better when ~mc and m~c are unbalanced• Null-invariance (free of the influence of null transactions) is an important property for measuring correlations in large transaction databases

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Comparison of Four Correlation Measures (Cont.)

Dataset gv ~gv g~v ~g~v all_conf cosine lift χ2

D0 4,000(4737) 3,500 2,000 0 0.53 0.60 0.84 1,477.8D1 4,000(4500) 3,500 2,000 500 0.53 0.60 0.89 555.6D2 4,000(2307) 3,500 2,000 10,000 0.53 0.60 1.73 2,913.0

• lift and χ2 show correlation between g and v changes from being rather positive to rather negative• all_conf and cosine cannot precisely assert positive/negative correlations when they are around 0.50

Rule of Thumb: in large transaction databases, perform the all_conf or cosine analysis first, and when the result shows that they are weakly positively/negatively correlated, lift or χ2 can be used to assist analysis

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Constraint-Based Data Mining Problems of Automatic Data Mining

The derived patterns can be too many but not focused Users lack understanding of the derived patterns Users’ domain knowledge cannot be taken advantage of

Interactive Data Mining Users direct data mining process through queries or graphical user interfaces

Constraint-Based Mining Users specify constraints on what “kinds” of patterns to be mined Knowledge type constraints

Specify the type of knowledge to be mined (e.g. association, classification rules) Data constraints

Specify the set of task-relevant data Dimension/level constraints

Specify the desired dimensions (or attributes) of the data, or levels of the concept hierarchies, to be used in mining

Interestingness constraints Specify thresholds on statistical measures of interestingness of patterns (e.g. support,

confidence, correlation of association rules) Rule constraints

Specify the forms of rules to be mined

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Metarule-Guided Association Rules Mining Metarules

Specify the syntactic form of rules that users are interested in mining Rule forms are used as constraints to help improve efficiency of the mining

process

e.g. You are interested in finding associations between customer traits and the items they purchase. However, rather than finding all the association rules that reflect these relationships, you are particularly interested in determining which pairs of customer traits promote the sale of office software.

Metarule:P1(X, Y) P2(X, W) buys(X, “office software”)

P1, P2: predicate variables that instantiated to some attributes from the database during miningX: a variable representing a customer

Y, W: values of attributes assigned to P1 and P2, respectively

age(X, “30..39”) income(X, “41K..60K”) buys(X, “office software”)