Association rules

• The goal of mining association rules is to generate all possible rules that exceed some minimum user-specified support and confidence thresholds.

• The problem is thus decomposed into two subproblems:

Association rules

1. Generate all items sets that have a support that exceeds the support threshold. These sets are called large or frequent itemsets, because they have large support (not because of their cardinality).

2. For each large itemset, all the rules that have a minimum confidence are generated: for a large itemset X and Y X, let Z = X – Y. If support(X)/support(Z) > Confidence threshold, then Z Y.

Z Y

Association rules

• Generating rules by using large (frequent) itemsets is straightforward. However, if the cardinality of the set of items is very high, the process becomes very computation-intensive:

for m items, the number of distinct itemsets is 2m (power set).

• Basic algorithms for finding association rules try to reduce the combinatorial search space

Association rules: a Basic algorithm

1. Test the support for itemsets of size 1, called 1-itemsets. Discard those that do not meet Minimum Required Support (MRS).

2. Extend the 1-itemsets by appending one item each time, to generate all candidate 2-itemsets, test MRS and discard those that do not meet it.

3. Continue until no itemsets can be found.

4. Use itemsets found to generate rules (check confidence).

Association rules

• The naive version of this algorithm is a combinatorial nightmare!

• Many versions and variants of this algorithm• They use different strategies• Their resulting sets of rules are all the same:

Any algorithm for association rules should find the same set of rules although their computational efficiencies and memory requirements may be different.

• We will see the Apriori Algorithm

Association rules: The Apriori Algorithm

• The key idea:

The apriori property: any subsets of a frequent itemset are also frequent itemsets

AB AC AD BC BD CD

A B C D

ABC ABD ACD BCD

x

xxx

xxx

Begin with itemsets of size 1

Stage 1. The Apriori Algorithm: Generating large itemsets

• Find large (frequent, i.e., that meet MRS) itemsets of size 1: F1

• From k = 2– Ck = candidates of size k: those itemsets of

size k that could be frequent, given Fk-1

– Fk = those itemsets that are actually large,

Fk Ck. They meet MRS

Example

itemset:count

1. scan T C1: {a}:2, {b}:3, {c}:3, {d}:1, {e}:3

F1: {a}:2, {b}:3, {c}:3, {e}:3

C2: {a,b}, {a,c}, {a,e}, {b,c}, {b,e}, {c,e}

2. scan T C2: {a,b}:1, {a,c}:2, {a,e}:1, {b,c}:2, {b,e}:3, {c,e}:2

F2: {a,c}:2, {b,c}:2, {b,e}:3, {c,e}:2

C3: {b,c,e}

3. scan T C3: {b, c, e}:2

F3: {b, c, e}: 2

TID Items

1 a, c, d

2 b, c, e

3 a, b, c, e

4 b, e

Minimum support allowed = 50%

It is the only candidate because all its subsets are large!

Sets with d are discarded

T

Algorithm Apriori(T, minsup) //T Set of n transactions //minsup = minimum support allowed

C1 getC1(T); //Get 1-itemsets and their counts F1 {f | f C1, f.count / n minsup}; //Check supportFOR (k = 2; Fk-1 ; k++) DO

Ck candidate_gen(Fk-1); //Get k-itemsets using Fk-1 FOR each transaction t T DO FOR each candidate c Ck DO

IF c is contained in t THEN c.count++;

END FOREND FOR

Fk {c Ck | c.count / n minsup} //Check support END FOR

RETURN F k Fk; //Union of all Fk (k >1 for rules)

/*Get counts of the k-itemsets*/

Stage 1. The Apriori Algorithm: Generating large (frequent) itemsets

• Note that frequent itemsets association rules• One more step is needed to generate

association rules:

• For each frequent itemset X: - For each subset A X, A .

Let B = X – A

- A B is an association rule if

confidence(A B) ≥ minconf, i.e.,

if (support(A B)/support(A)) ≥ minconf.

Stage 2: The Apriori Algorithm: generating rules from large itemsets

Minconf = Minimum confidence

Example

• Suppose {b,c,e} is a large (frequent) itemset with support 50%

• Consider subsets of {b,c,e} and their support: {b,c} = 50%{b,e} = 75%, {c,e} = 50%, {b} = 75%, {c} = 75%, {e} = 75%

Example

This subset generates the following association rules:

• {b,c} {e} confidence = 50/50 = 100%• {b,e} {c} confidence = 50/75 = 66.6%• {c,e} {b} confidence = 50/50 = 100%• {b} {c,e} confidence = 50/75 = 66.6%• {c} {b,e} confidence = 50/75 = 66.6%• {e} {b,c} confidence = 50/75 = 66.6%

All rules have support 50% = support({b,c,e})

Hash-based improvement to A-Priori

• During pass 1 of A-Priori (Get 1-itemsets), most memory is idle

• Use that memory to keep counts of buckets into which pairs of items are hashed

• Gives extra condition that candidate pairs must satisfy when getting 2-itemsets (pass 2)

Hash-based improvement to A-Priori

• The memory is divided like this:

– Space to count each item (Get 1-itemsets)– Use the rest of the space for the described

hashing process

• PCY algorithm (Park, Chen, and Yu)

PCY Algorithm Pass 1:FOR each transaction t in T DO

FOR each item in t DO

add 1 to item’s count

END FOR

FOR each pair of items in t DO

Hash the pair to a bucket and add 1 to the

count for that bucket

END FOR

END FOR

getC1

Hashing process

PCY Algorithm Pass 2:

• Count all pairs {i, j} that satisfy the conditions:– Both i and j (taken individually!) are frequent

items– The pair {i, j} hashes to a bucket whose count

the support s (i.e., a frequent bucket)

• These two conditions are necessary for the pair to have a chance of being frequent.

PCY Example

• Support s = 3• Items: milk (1), Coke (2), bread (3), Pepsi (4), juice (5). • Transactions are• t1 = {1, 2, 3} milk, Coke, bread• t2 = {1, 4, 5}• t3 = {1, 3}• t4 = {2, 5}• t5 = {1, 3, 4}• t6 = {1, 2, 3, 5}• t7 = {2, 3, 5}• t8 = {2, 3}

PCY Example

• Hashing a pair {i, j} to a bucket k, where k = hash(i, j) = (i + j) mode 5. That is, for pairs:

• (1, 4) and (2, 3) k = 0

• (1, 5) and (2, 4) k = 1

• (2, 5) and (3, 4) k = 2

• (1, 2) and (3, 5) k = 3

• (1, 3) and (4, 5) k = 4

PCY Example

• Pass 1:• Item’s count:

• Note that Item 4 does not exceed the support.

Item Count1 52 53 64 25 4

PCY Example

• For each pair in each transaction:• t1 = (1,2)3 (2,3)0 (1,3)4 • t2 = (1,4)0 (1,5)1 (4,5)4

• t3 = (1,3)4

• t4 = (2,5)2

• t5 = (1,3)4 (3,4)2 (1,4)0

• t6 = (1,2)3 (1,3)4 (1,5)1 (2,3)0 (2,5)2 (3,5)3

• t7 = (2,3)0 (2,5)2 (3,5)3

• t8 = (2,3)0

Total: 21 pairs

PCY Example

• The hash table is

• Bucket 1 does not exceed the support, i.e., (1, 5) and (2, 4) are not frequent.

Bucket Count0 61 22 43 44 5

PCY Example

• Pass 2:

• Frequent items are {1, 2, 3, 5}

• From the frequent items the candidate pairs are (1,2) (1,3) (1,5) (2,3) (2,5) (3,5)

• Candidate (1,5) is discarded because bucket 1 is not frequent! Discarded by the PCY!

PCY Example

• Note that pairs (1,2) and (3,5) does not exceed the support.

• Frequent itemsets are

{1} {2} {3} {5} {1,3} {2,3} {2,5}

Pair Count(1,2) 2(1,3) 4(2,3) 4(2,5) 3(3,5) 2

Counts of the “surviving” pairs

Association rules

• Association rules among hierarchies: it is possible to divide items among disjoint hierarchies, e.g., foods in a supermarket.

• It could be interesting to find associations rules across hierarchies. They may occur among item groupings at different levels.

• Consider the following example.Levels of a dimension

Hierarchy 1: Beverages

Hierarchy 2: Desserts

Association rules

• Associations such as

{Frozen Yoghurt} {Bottled Water}

{Rich Cream} {Wine Coolers}

May produce enough confidence and

support to be valid rules of interest.

Association rules: Negative associations

• Negative associations: a negative association is of the following type “80% of customers who buy W do not buy Z”

• The problem of discovering negative associations is complex: there are millions of item combinations that do not appear in the database, the problem is to find only interesting negative rules.

Association rules: Negative associations

• One approach is to use hierarchies. Consider the following example:

Soft drinks

Joke Wakeup Topsy

Chips

Days Nightos Parties

Positive association

Association rules: Negative associations

• Suppose a strong positive association between chips and soft drinks

• If we find a large support for the fact that when customers buy Days chips they predominantly buy Topsy drinks (and not Joke, and not Wakeup), that would be interesting

• The discover of negative associations remains a challenge

Association rules

Additional considerations for associations rules:• For very large datasets, efficiency for

discovering rules can be improved by sampling Danger: discovering some false rules

• Transactions show variability according to geographical location and seasons

• Quality of data is usually also variable