1. 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:

2. 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

3. 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

4. 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).

5. 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

6. Association rules: The Apriori Algorithm
The key idea:
The apriori property: any subsets of a frequent itemset are also frequent itemsets x x x
ABC ABD ACD BCD x x x
AB AC AD BC BD CD
Begin with
itemsets of size 1
A B C D x

7. 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

8. Example Minimum support allowed = 50% itemset:count
TID
Items 1
a, c, d 2
b, c, e 3
a, b, c, e 4
b, e
Minimum support allowed = 50%
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
Sets with d are discarded
It is the only candidate because all its subsets are large!

9. Stage 1. The Apriori Algorithm: Generating large (frequent) itemsets
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 support
FOR (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 FOR
Fk  {c  Ck | c.count / n  minsup} //Check support
RETURN F  k Fk; //Union of all Fk (k >1 for rules)
/*Get counts of the k-itemsets*/

10. Stage 2: The Apriori Algorithm: generating rules from large 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.
Minconf = Minimum confidence

11. 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%

12. 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})

13. 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)

14. 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)

15. 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
getC1
Hashing process

16. 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.

17. 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}

18. 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

19. PCY Example Pass 1: Item’s count:
Note that Item 4 does not exceed the support.
Item
Count 1 5 2 3 6 4

20. PCY Example For each pair in each transaction:
Total: 21 pairs

21. PCY Example The hash table is
Bucket 1 does not exceed the support, i.e., (1, 5) and (2, 4) are not frequent.
Bucket
Count 6 1 2 4 3 5

22. 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!

23. PCY Example
Counts of the
“surviving” pairs
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)
(2,5) 3
(3,5)

24. 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

25. Hierarchy 1: Beverages
Hierarchy 2: Desserts

26. 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.

27. 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.

28. Association rules: Negative associations
One approach is to use hierarchies. Consider the following example:
Positive association
Soft drinks
Chips
Joke
Wakeup
Topsy
Days
Nightos
Parties

29. 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

30. 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