association rule mining

Association Rule Mining

Debapriyo Majumdar

Data Mining – Fall 2014

Indian Statistical Institute Kolkata

August 4 and 7, 2014

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Transaction id

Items

1 Bread, Ham, Juice, Cheese, Salami, Lettuce

2 Rice, Dal, Coconut, Curry leaves, Coffee, Milk, Pickle

3 Milk, Biscuit, Bread, Salami, Fruit jam, Egg

4 Tea, Bread, Salami, Bacon, Ham, Sausage, Tomato

5 Rice, Egg, Pickle, Curry leaves, Coconut, Red chilly

Market Basket Analysis

Scenario: customers shopping at a supermarket

What can we infer from the above data? An association rule: {Bread, Salami} {Ham}, with

confidence ~ 2/3

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Applications Information driven marketing Catalog design Store layout Customer segmentation based on buying patterns

Several papers by Rakesh Agrawal and others in the 1990s

Rakesh Agrawal and Ramakrishnan Srikant

Fast Algorithms for Mining Association Rules

The VLDB 1994

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The Market-Basket Model A (large) set of binary attributes, called items

I = {i1, …, in}

e.g. milk, bread, the items sold at the market

A transaction T consists of a (small) subset of I

e.g. the list of items (bill) bought by one customer at once

The database D is a (large) set of transactions

D = {T1, …, TN}

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The Market-Basket Model Goal: mining associations between the items– The transactions or customers also may have associations,

but here we are interested in such relations

Approach: finding subset of items that are present together in transactions frequently

An itemset: any subset X of I

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Support of an Itemset Let X be an itemset Support count σ(X) = # of transactions containing all items of X support(X) = fraction of transactions containing all items of X

Makes sense (statistically significant) only when – support count is at least a few hundreds– in a database of several thousand transactions

support({Bread, Salami})

support({Rice, Pickle, Coconut})

= 0.6

= 0.4

T-ID Items






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Association Rule Association rule: an implication of the form X Y

where X, Y I, and X Y = ϕ.

support(XY) =

– Transactions containing all items of both X and Y

confidence(XY) =

UUI

T-ID Items






σ(X U Y)

| D |

σ(X U Y)

σ(X)

R : {Bread, Salami} {Ham}

support(R) =

confidence(R) =

2

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3

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Association Rule Mining Task Given a set of items I, a set of transactions D, a

minimum support thresholds minsup and a minimum confidence threshold minconf

Find all rules R such that

support(R) ≥ minsup

confidence(R) ≥ minconf

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One Approach Observe:

support(X Y) = == support(Z)

where Z = X U Y If Z = W U V, support(X Y) = support(W V)

– Each binary partition of Z represents an association rule– With same support– However, the confidences may be different

Approach: frequent itemset generation1. Find all itemsets Z with support(Z) ≥ minsup. Call such itemsets

frequent itemsets.

2. From each Z, generate rules with confidence(Z) ≥ minconf

σ(X U Y)

| D |σ(Z)

| D |

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Finding Frequent Itemsets If | I | = n, then number of possible itemsets = 2n

For each itemset, compute the support by scanning the lists of items of each transaction– O(N × w), where w is the average length of transactions

Overall complexity: O(2n× N × w) Computationally very expensive!!

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Anti-monotone Property of Support If an itemset is frequent, all its subsets are also

frequent– Because if X ⊆ Y, then support(X) ≥ support(Y)– For all transactions T such that Y ⊆ T, we have X ⊆ T

T-ID Items






Support({Bread, Salami}) ≥ Support({Bread, Ham, Salami})

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The A-Priori AlgorithmNotation:

Lk = The set of frequent (large) itemsets of size k

Ck= The candidate set of frequent (large) itemsets of size.

Algorithm:

L1 = {Frequent 1-itemsets};

for ( k = 2; Lk – 1 ≠ 0; k++ ) do begin

Ck = apriori_gen(Lk-1); /* Generating new candidates */

for all transactions T in D do begin

CT = subset(Ck,T) /* Keeping only the valid candidates */

for all candidates c in CT do

c.count++;

end

Lk = {c in Ck | c.count ≥ minsup}

end

Output = Union of all Lk for k = 1, 2, … , n

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Generating candidate itemsets Lk

A join of Lk-1 with itself

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

What does it do?L3 L3

{1, 2, 3} {1, 2, 3}

{1, 2, 4} {1, 2, 4}

{1, 3, 4} {1, 3, 4}

{1, 3, 5} {1, 3, 5}

{2, 3, 4} {2, 3, 4}

C4 = { {1, 2, 3, 4}, {1, 3, 4, 5} }

A prune step:

{1, 3, 4, 5} will be pruned because {1, 4, 5} ∉ L3

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Checking Support for candidates One approach:

for each candidate itemset c ∈ Ck

for each transactions T ∈ D do begin

check if c ⊆ T

end

end

Complexity?

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Using a Hash TreeLet us have 12 candidate itemsets of size 3

{1 2 5}, {1 2 7}, {1 3 9}, {2 4 5}, {2 8 9}, {3 5 7},

{4 5 9}, {4 7 8}, {5 6 7}, {5 7 9}, {6 7 8}, {6 7 9}

Hash function

1, 4, 7

2, 5, 8

3, 6, 9

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The Hash Tree{1 2 5}, {1 2 7}, {1 3 9}, {2 4 5}, {2 8 9}, {3 5 7}, {4 5 9}, {4 7 8}, {5 6 7}, {5 7 9}, {6 7 8}, {6 7 9}

Hash Function

1, 4, 7

2, 5, 8

3, 6, 9

Root

1,4,7+ 2,5,8+ 3,6,9+

{1 2 5}{1 2 7} {1 3 9} {2 4 5} {2 8 9} {3 5 7}

{4 5 9}

{4 7 8}

{5 6 7}

{5 7 9}

{6 7 8} {6 7 9}

Subsets of the transaction

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{1 6 7}{1 6 8}

{1 2 6}{1 2 7}{1 2 8}

{1 2 6 7 8}

{6 7 8}{1 2 6 7 8} {2 6 7 8}

{1 2 6 7 8} {1 6 7 8} {1 7 8} {2 6 7 8} {2 7 8}

{2 6 7}{2 6 8}

All subsets of size 3 for a transaction{1 2 6 7 8}, ordered by the item id

Subsets starting with 1

Subsets starting with 12

Hashing in the same style

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The Subset Operation using Hash TreeTransaction: {1 2 5 6 8}, ordered by item id Hash Function

1, 4, 7

2, 5, 8

3, 6, 9

Root

1,4,7+ 2,5,8+ 3,6,9+

{1 2 5}{1 2 7} {1 3 9} {2 4 5} {2 8 9} {3 5 7}

{4 5 9}

{4 7 8}

{5 6 7}

{5 7 9}

{6 7 8} {6 7 9}

{1 2 5 6 8} {2 5 6 8} {5 6 8}

{1 2 5 6 8}

{1 2 5}

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Where are we now? Computed frequent itemsets, i.e. the itemsets with required

support minsup Each frequent k-itemset X gives rise to several association

rules Ignoring X ϕ and ϕ X, 2k – 2 rules Rules generated from different itemsets are also different The rules need to be checked for minimum confidence All these rules already satisfy the support condition

How many?

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Rules Generated from the Same Itemset Let X ⊂ Y, for non empty itemsets X, and Y Then X Y - X is an association rule Theorem: If X’ ⊂ X, then c(X Y – X) ≥ c(X’ Y – X’)

– Example: c({1 2 3} {4 5}) ≥ c({1 2} {3 4 5})

Proof. Observe: c(X Y – X) = σ(Y)/σ(X)

c(X’ Y – X’) = σ(Y)/σ(X’)since X’ ⊂ X, σ(X’) ≥ σ(X)

so c(X Y – X) ≥ c(X’ Y – X’)

Corollary: If X Y – X is not a high-confidence association rule, then X’ Y – X’ is also not a high confidence rule.

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Level-wise Approach for Rule Generation

Frequent itemset: {1 2 3 4}

{1 3 4} {2} {2 3 4} {1}{1 2 4} {3}

{1} {2 3 4}

{1 2} {3 4}

{1 2 3} {4}

{1 3} {2 4} {1 4} {2 3} {2 3} {1 4} {2 4} {1 3} {3 4} {1 2}

{2} {1 3 4} {3} {1 2 4} {4} {1 2 3}

{1 2 3 4} {}

Suppose {1 2 4} {3} fails the confidence bar The whole tree under {1 2 4} {3} can be discarded

Maximal Frequent itemsetsMaximal frequent itemset: an itemset, for which none of its immediate supersets are frequent

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{3} {4}{2}

{1 2 3}

{1 2}

{1}

{1 3} {1 4} {2 3} {2 4} {3 4}

{1 2 4} {1 3 4} {2 3 4}

{}

{1 2 3 4}


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{3} {4}{2}

{1 2 3}

{1 2}

{1}

{1 3} {1 4} {2 3} {2 4} {3 4}

{1 2 4} {1 3 4} {2 3 4}

{}

{1 2 3 4}

Not frequent


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{3} {4}{2}

{1 2 3}

{1 2}

{1}

{1 3} {1 4} {2 3} {2 4} {3 4}

{1 2 4} {1 3 4} {2 3 4}

{}

{1 2 3 4}

Not frequent

Maximal frequent

Maximal Frequent itemsetsAll frequent itemsets are subsets of one of the maximal frequent itemsets.

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{3} {4}{2}

{1 2 3}

{1 2}

{1}

{1 3} {1 4} {2 3} {2 4} {3 4}

{1 2 4} {1 3 4} {2 3 4}

{}

{1 2 3 4}

Not frequent

Maximal frequent

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Maximal Frequent Itemsets Valuable compact representation of the frequent

itemsets

But

Do not contain the support information of the subsets– Says all supersets have lesser support, but does not say if

any subset also has the same support

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Closed Frequent Itemsets Closed itemset: an itemset X for

which none of its immediate supersets has exactly the same support count as X – If X is not closed, at least one of

its immediate supersets have the same support as the support of X

Closed frequent itemset: an itemset which is closed and frequent (support ≥ minsup)

Support for non-closed frequent itemsets can be determined from the support information of the closed frequent itemsets

Frequent itemsets

Closed frequent itemsets

Maximal frequent itemsets

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Evaluation of Association Rules Even from a small dataset a very large number of

rules can be generated– For example, as support and confidence conditions are

relaxed, number of rules explode

Interestingness measure for patterns / rules is required

Objective interestingness measure: a measure that uses statistics derived from the data– Support, confidence, correlation, … – Domain independent– Requires minimal human involvement

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Subjective Measure of Interestingness The rule {Salami} {Bread} is not so interesting because it is

obvious! Rules such as{Salami} {Dish washer detergent}, {Salami}

{Diper}, etc are less obvious Subjectively more interesting for marketing experts

– Non-trivial cross sell

Methods for subjective measurement– Visualization aided: human in the loop– Template-based: constrains are provided for rules– Filter obvious and non-actionable rules

?

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Contingency TableCoffee Coffee

Tea 150 50 200Tea 650 150 800

800 200 1000

B B’A f11 f10 f1+

A’ f01 f00 f0+

f+1 f+0

Frequency tabulated for a pair of binary variables Used as a useful evaluation and illustration tool Generally:

A’ (or B’) denotes the transactions in which A (or B) is absentf1+ = support count of A

f+1 = support count of B

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Limitations of Support & Confidence Tuning the support threshold is tricky Low threshold – Too many rules generated! High threshold – Potentially interesting patterns may

fall below the support threshold

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Limitation of Confidence

But: Overall 80% people have coffee– i.e., the rule{} {Coffee} has confidence 80%. – Among tea takers, the percentage actually drops to 75%!!

Where does it go wrong? Confidence measure ignores the support of Y for a

rule X Y

Coffee CoffeeTea 150 50 200Tea 650 150 800

800 200 1000

Consider the rule:{Tea} {Coffee}Support = 15%Confidence = 75%

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Interest factor

Lift: Lift(X Y) =

For binary variables, lift is equivalent to interest factor

Interest factor: I(X,Y) = =

Similar to baseline frequency comparison under statistical independence assumption– If X and Y are statistically independent, their baseline frequency

(expected frequency of X and Y both occurring) is

f11 =

c(X Y)

σ(Y)

s(X UY)

s(X) s(Y)

N f11

f1+ . f+1

f1+ . f+1

N

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Interest factor Intuitively

I(X,Y) = 1, if X and Y are independent

> 1, if X and Y have a positive correlation

< 1, if X and Y have a negative correlation

Verify for the tea – coffee example

I(Tea, Coffee)

= 0.15 / (0.2 × 0.8)

= 0.94

Coffee CoffeeTea 150 50 200Tea 650 150 800

800 200 1000

I =N f11

f1+ . f+1

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Limitation of Interest Factor

Observe: I(Text, Analysis) = 1.02, I(Graph, Mining) = 4.08 Text and Analysis are more related than Graph and Mining Confidence measure:

c(Text Analysis) = 94.6%

c(Graph Mining) = 28.6% What goes wrong here?

Text TextAnalysis 880 50 930Analysis 50 20 70

930 70 1000

Mining MiningGraph 20 50 70Graph 50 880 930

70 930 1000

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More Measures Correlation coefficient for binary variables:

IS Measure: I and S measures combined

Mathematically equivalent to cosine measure of binary variables

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Properties of Objective MeasuresB B’

A f11 f10 f1+

A’ f01 f00 f0+

f+1 f+0

Inversion property: Invariant under inversion operation– Exchange f11 with f00 and f01 with f10

– The value of the measure remains the same

Null addition property: Invariant under addition of counts for other variables, i.e. the value of the measure remains the same if f00 is increased

Which measures have which properties?

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References Rakesh Agrawal and Ramakrishnan Srikant

Fast Algorithms for Mining Association Rules

VLDB 1994 Introduction to Data Mining, by Tan, Steinbach, Kumar

– The webpage: http://www-users.cs.umn.edu/~kumar/dmbook/index.php

– Chapter 6 is available online: http://www-users.cs.umn.edu/~kumar/dmbook/ch6.pdf

http://www-users.cs.umn.edu/~kumar/dmbook/index.php



http://www-users.cs.umn.edu/~kumar/dmbook/ch6.pdf



association rule mining

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