Association Rules

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Association Rules

Apriori Algorithm

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Computation Model Typically, data is kept in a flat file

rather than a database system. Stored on disk. Stored basket-by-basket.

The true cost of mining disk-resident data is usually the number of disk I/O’s.

In practice, association-rule algorithms read the data in passes – all baskets read in turn.

Thus, we measure the cost by the number of passes an algorithm takes.

Item

Item

Item

Item

Item

Item

Item

Item

Item

Item

Item

Item

Basket 1

Basket 2

Basket 3

Etc.

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Main-Memory Bottleneck

For many frequent-itemset algorithms, main memory is the critical resource. As we read baskets, we need to count

something, e.g., occurrences of pairs. The number of different things we can

count is limited by main memory. Swapping counts in/out is a disaster.

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Finding Frequent Pairs

The hardest problem often turns out to be finding the frequent pairs.

We’ll concentrate on how to do that, then discuss extensions to finding frequent triples, etc.

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Naïve Algorithm

Read file once, counting in main memory the occurrences of each pair. From each basket of n items, generate

its n (n -1)/2 pairs by two nested loops.

Fails if (#items)2 exceeds main memory. Remember: #items can be 100K (Wal-

Mart) or 10B (Web pages).

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Details of Main-Memory Counting

Two approaches:1. Count all pairs, using a triangular

matrix.2. Keep a table of triples [i, j, c] = the

count of the pair of items {i,j } is c. (1) requires only 4 bytes/pair.

Note: assume integers are 4 bytes. (2) requires 12 bytes, but only for

those pairs with count > 0.

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4 per pair

Method (1) Method (2)

12 peroccurring pair

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Triangular-Matrix Approach – (1)

Number items 1, 2, …,n Requires table of size O(n). Count {i, j } only if i < j. Keep pairs in the order

{1,2}, {1,3},…, {1,n}, {2,3}, {2,4},…,{2,n}, {3,4},…, {3, n}, … {n -1,n}.

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Triangular-Matrix Approach – (2)

Let n be the number of items. Find pair {i, j } at the position

(i-1)n-i(i+1)/2+j

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

Total number of pairs n (n –1)/2; total bytes about 2n 2.

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Details of Approach #2

Total bytes used is about 12p, where p is the number of pairs that actually occur. Beats triangular matrix if at most 1/3 of

possible pairs actually occur. May require extra space for retrieval

structure, e.g., a hash table.

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Apriori Algorithm for pairs– (1)

A two-pass approach called a-priori limits the need for main memory.

Key idea: monotonicity : if a set of items appears at least s times, so does every subset. Contrapositive for pairs: if item i does

not appear in s baskets, then no pair including i can appear in s baskets.

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Apriori Algorithm for pairs– (2)

Pass 1: Read baskets and count in main memory the occurrences of each item. Requires only memory

proportional to #items.

Pass 2: Read baskets again and count in main memory only those pairs whose both elements were found in Pass 1 to be frequent. Requires memory

proportional to square of frequent items only.

Item counts

Pass 1 Pass 2

Frequent items

Counts ofcandidate pairs

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Detail for A-Priori

You can use the triangular matrix method with n = number of frequent items. Saves space compared with storing

triples.

Trick: number frequent items 1,2,… and keep a table relating new numbers to original item numbers.

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Frequent Triples, Etc. For each k, we construct two sets of k –tuples:

Ck = candidate k - tuples = those that might be frequent sets (support > s ) based on information from the pass for k –1.

Fk = the set of truly frequent k –tuples.

C1 F1 C2 F2 C3Filter Filter ConstructConstruct

Firstpass Second

pass

Allitems

All pairsof itemsfrom F1

Countthe pairs

To beexplained

Countthe items

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Lattice of Itemsetsnull

AB AC AD AE BC BD BE CD CE DE

A B C D E

ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE

ABCD ABCE ABDE ACDE BCDE

ABCDE

Given d items, there are 2d possible candidate itemsets

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Found to be Infrequent

null


A B C D E



ABCDE

Illustrating Apriori Principlenull


A B C D E



ABCDEPruned supersets

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Full Apriori Algorithm Let k=1 Generate frequent itemsets of length 1 Repeat until no new frequent itemsets are found

k=k+11. Generate length-k candidate itemsets from length-k-1

frequent itemsets2. Prune candidate itemsets containing subsets of length k-1 that

are infrequent 3. Count the support of each candidate by scanning the DB and

eliminate candidates that are infrequent, leaving only those that are frequent

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Candidate generationAn effective candidate generation procedure: 1. Should avoid generating too many

unnecessary candidates.

2. Must ensure that the candidate set is complete,

3. Should not generate the same candidate itemset more than once.

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Data Set Example

TID Items

1 Bread, Milk

2 Bread, Diaper, Beer, Eggs

3 Milk, Diaper, Beer, Coke

4 Bread, Milk, Diaper, Beer

5 Bread, Milk, Diaper, Coke

s=3

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Fk-1F1 Method Extend each frequent (k - 1)

itemset with a frequent 1-itemset.

Is it complete? Yes, because every frequent k

itemset is composed of a frequent (k-1) itemset and a frequent 1 itemset.

However, it doesn’t prevent the same candidate itemset from being generated more than once. E.g., {Bread, Diapers, Milk} can

be generated by merging {Bread, Diapers} with {Milk}, {Bread, Milk} with {Diapers}, or {Diapers, Milk} with {Bread}.

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Lexicographic Order Avoid generating duplicate candidates by ensuring

that the items in each frequent itemset are kept sorted in their lexicographic order.

Each frequent (k-1) itemset X is then extended with frequent items that are lexicographically larger than the items in X.

For example, the itemset {Bread, Diapers} can be augmented with {Milk} since Milk is lexicographically larger than Bread and Diapers.

However, we don’t augment {Diapers, Milk} with {Bread} nor {Bread, Milk} with {Diapers} because they violate the lexicographic ordering condition.

Why is it complete?

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Prunning E.g. merging {Beer, Diapers} with {Milk} is

unnecessary because one of its subsets, {Beer, Milk}, is infrequent.

Solution: Prune! How?

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Fk-1F1 Example

{Beer,Diapers,Bread} and {Bread,Milk,Beer} aren't in fact generated if lexicographical ord. is considered.

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Fk-1Fk-1 Method Merge a pair of frequent (k-1) itemsets only if their

first k-2 items are identical.

E.g. frequent itemsets {Bread, Diapers} and {Bread, Milk} are merged to form a candidate 3 itemset {Bread, Diapers, Milk}.

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Fk-1Fk-1 Method We don’t merge {Beer, Diapers} with {Diapers, Milk}

because the first item in both itemsets is different.

But, is this "don't merge" decision Ok? Indeed, if {Beer, Diapers, Milk} is a viable candidate, it

would have been obtained by merging {Beer, Diapers} with {Beer, Milk} instead.

Pruning However, because each candidate is obtained by

merging a pair of frequent (k-1) itemsets, an additional candidate pruning step is needed to ensure that the remaining k-2 subsets of k-1 elements are frequent.

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Fk-1Fk-1 Example

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Another Example

TID List of item ID’s

T1 I1, I2, I5

T2 I2, I4

T3 I2, I3

T4 I1, I2, I4

T5 I1, I3

T6 I2, I3

T7 I1, I3

T8 I1, I2, I3, I5

T9 I1, I2, I3

Min_sup_count = 2

Itemset

{I5}

{I4}

{I3}

{I2}

{I1}

C1

Sup. count

Itemset

2{I5}

2{I4}

6{I3}

7{I2}

6{I1}

F1

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Generate C2 from F1F1


T1 I1, I2, I5

T2 I2, I4

T3 I2, I3

T4 I1, I2, I4

T5 I1, I3

T6 I2, I3

T7 I1, I3

T8 I1, I2, I3, I5

T9 I1, I2, I3

Min_sup_count = 2

Itemset Sup. count

{I1} 6

{I2} 7

{I3} 6

{I4} 2

{I5} 2

F1

{I4,I5}

{I3,I5}

{I3,I4}

{I2,I5}

{I2,I4}

Itemset

{I2,I3}

{I1,I5}

{I1,I4}

{I1,I3}

{I1,I2}

C2

Itemset Sup. C

{I1,I2} 4

{I1,I3} 4

{I1,I4} 1

{I1,I5} 2

{I2,I3} 4

{I2,I4} 2

{I2,I5} 2

{I3,I4} 0

{I3,I5} 1

{I4,I5} 0

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T1 I1, I2, I5

T2 I2, I4

T3 I2, I3

T4 I1, I2, I4

T5 I1, I3

T6 I2, I3

T7 I1, I3

T8 I1, I2, I3, I5

T9 I1, I2, I3

Min_sup_count = 2

Itemset Sup. C

{I1,I2} 4

{I1,I3} 4

{I1,I5} 2

{I2,I3} 4

{I2,I4} 2

{I2,I5} 2

F2

Itemset

{I1,I2,I3}

{I1,I2,I5}

{I1,I3,I5}

{I2,I3,I4}

{I2,I3,I5}

{I2,I4,I5}

Prune

Itemset

{I1,I2,I3}

{I1,I2,I5}

{I1,I3,I5}

{I2,I3,I4}

{I2,I3,I5}

{I2,I4,I5}

Itemset Sup. C

{I1,I2,I3} 2

{I1,I2,I5} 2

F3

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T1 I1, I2, I5

T2 I2, I4

T3 I2, I3

T4 I1, I2, I4

T5 I1, I3

T6 I2, I3

T7 I1, I3

T8 I1, I2, I3, I5

T9 I1, I2, I3

Min_sup_count = 2

Itemset

{I1,I2,I3,I5}

C4

{I1,I2,I3,I5} is pruned because {I2,I3,I5} is infrequent

Itemset Sup. C

{I1,I2,I3} 2

{I1,I2,I5} 2

F3

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Association Rules