07. Closet - An Efficient Algorithm for Mining Frequent

8/14/2019 07. Closet - An Efficient Algorithm for Mining Frequent

1/9

VIGNAN INSTOFINFORMATION TECHNOLOGY

DUVVADA, VISAKHAPATNAM.

TITLE: CLOSET- An Efficient Algorithm for Mining Frequent Closed Itemsets

PRESENTED BY :

P.BHAKTHAVATHSALANAIDU K.K.S.SWAROOP

[email protected]

[email protected]

CLOSET: An Efficient Algorithm for Mining Frequent Closed

Itemsets

ABSTRACT

Association mining may often derive an undesirably large set of frequent itemsets and

association rules. Recent studies have proposed an interesting alternative: mining frequent closed

itemsets and their corresponding rules, which has the same power as association mining but

substantially reduces the number of rules to be presented.

An efficient algorithm CLOSET, for mining closed itemsets, with the development of

three techniques:

(1) Applying a compressed frequent pattern tree FP-tree structure for mining closed itemsets

without candidate generation,

(2) Developing a single prefix path compression technique to identify frequent closed

itemsets quickly, and

(3) Exploring a partition bases projection mechanism for scalable mining in large databases.

CLOSET is efficient and scalable over lager databases and is faster than the previously

proposed methods.

INTRODUCTION:

It has been well recognized that frequent pattern mining plays an essential role in many

important data mining tasks, e.g. associations, sequential patterns , episodes, partial periodicity,

etc. However, it is also well known that frequent pattern mining often generates a very large
mailto:[email protected]:[email protected]:[email protected]:[email protected]


2/9

number of frequent itemsets and rules, which reduces not only efficiency but also effectiveness

of mining since users have to sift through a large number of mined rules to find useful ones.

There is an interesting alternative: instead of mining the complete set of frequent itemsets

and their associations, association mining only their corresponding rules. An important

implication is that mining frequent closed itemset has the same power as mining the complete set

of frequent itemsets, but it will substantially reduce redundant rules to be generated and increase

both efficiency and effectiveness of mining.

Definition: Association rule mining searches for interesting relationships among items in a given

data set.

Interestingness Measures:

Certainty: Each discovered pattern should have a measure of certainty associated with it that

assesses the validity or trustworthiness of the pattern. A certainty measure for association rules

of the form AB, where A and B are sets of items, is Confidence. Given a set task-relevant

data tuples, the confidence of is defined as AB is defined as

Confidence (AB) = # tuples containing both A and B

# tuples containing A

Utility : The potential usefulness of a pattern is a factor defining its interestingness. It can be

estimated by a utility function, such as support. The support of an association pattern refers to the

percentage of task-relevant data tuples for which the pattern is true. For association rules of the

form AB where A and B are sets of items, it is defined as

Support(AB) = # tuples containing both A and B

total # of tuples

Association rules that satisfy both a user specified minimum confidence and user specified

minimum support threshold are referred to as Strong Association Rules.

Association Rule Mining is a two step process:

1. Find all frequent itemsets: each of these itemsets will occur at least as frequently as a pre-

determined minimum support count.

2. Generate strong association rules from the frequent itemsets: These rules must satisfy

minimum support and minimum confidence.

The Apriori Algorithm: Finding Frequent Itemsets Using Candidate Generation:

Apriori is an influential algorithm for mining frequent itemsets for Boolean association rules.

The names of the algorithm are based on the fact that the algorithm uses prior knowledge of


3/9


4/9

2) Suppose that the minimum transactions support count required is 2. The set of frequent 1-

itemsets, L1, can then be determined. It consists of the candidate 1-itemsets satisfying

minimum support.

3) To discover the set of frequent 2-itemsets, L2 the algorithm uses L1 1X1 L1 to generate a

candidate set of 2-itemsets C2.

In this way we will find candidate sets until a candidate set is null.

Generating Association Rules from Frequent Itemsets

Once the frequent itemsets from transactions in a database D have been found, it is

straightforward to generate strong association rules from them. This can be done using the

following equation for confidence. Where the conditional probability is expressed in terms of

itemset support count.

Confidence (AB) = support count (A U B)

Support count (A)

where support count (AUB) is the number of transactions containing the itemsets AUB, and

support count (A) is the number of transactions containing the itemset A. Based on this equation,

association rules can be generated as follows:

for each frequent itemset l, generate all nonempty subsets of l.


5/9

for every nonempty subset of l, output the rule s(l-s) if support count(l) min_conf,

support count(s)

where min_conf is the minimum confidence threshold.

Since the rules are generated from frequent itemsets, each one automatically satisfies minimum

support. Frequent itemsets can be stored ahead of time in hash tables along with their counts so

that they can be accessed quickly.

Eg:- Suppose the data contain the frequent itemset l= {I1,I2,I5}. What are the association rules

that can be generated from l?

The nonempty subsets of l are {I1, I2}, {I1, I5}, {I2, I5}, {I1}, {I2} and {I5}. The resulting

association rules are as shown below, each listed with its confidence.

I1I2I5, confidence = 2/4 = 50%






If the minimum confidence threshold is, say, 70% then only the second, third and last rules

above are output, since these are the only ones generated that are strong.

Mining Frequent Itemsets without Candidate GenerationThe Apriori algorithm suffers from two non-trivial costs:

1) It may need to generate a huge number of candidate sets.

2) It may need to repeatedly scan the database and checks large set of candidates by pattern

matching

An interesting method that mines the complete set of frequent itemsets without candidate

generation is called frequent-pattern growth of simply FP-growth, which adopts a divide-and

conquer strategy as follows: compress the database representing frequent items into a set of

conditional databases, each associated with one frequent item, and mine each such database

separately.

Reexamine the mining of transaction database, using the frequent-pattern growth approach.

The first scan of the database is the same as Apriori, which derives the set of frequent items (1-

items) and their support counts. Let the minimum count be 2. The set of frequent items is sorted


6/9

in the order of descending support count. This resulting set or list is denoted L. Thus, we have

L=[I2: 7,I1: 6,I3: 6,I4: 2,I5: 2].

An FP-tree is then constructed as follows. First, create the root of the tree, labeled

with null. Scan database D a second time. The items in each transaction are processed in L

order and a branch is created for each transaction. For example, the scan of the first transaction,

T100:I1, I2, I5, which contains three items (I2, I1, I5) in L order, leads to the construction of

the first branch of the tree with three nodes:, where I2 is linked as child of

the root, I1 is linked to I2 and I5 is linked to I2. The second transaction, T200, contains the items

I2 and I4 in L order, which would result in a branch where I2 is linked to the root and I4 is linked

to I2. However this branch would share a common prefix, (I2:2), with the existing path for T100.

Therefore, we instead increment the count of the I2 node when considering the branch to be

added for a transaction, the count of each node along a common prefix is incremented by 1, and

nodes for the item following the prefix are created and linked accordingly.

To facilitate tree traversal, an item header table is built so that each item points to its

occurrence in the tree via a chain of node-links. The tree obtained after scanning all of the

transactions is

Item Conditional pattern base Conditional FP-tree Frequent patterns generated

I5 {(I2 I1: 1) , (I2 I1 I3: 1)} I2 I5: 2, I1 I5: 2, I2 I1 I5: 2

I4 {(I2 I1: 1) , (I2: 1)} I2 I4: 2

The mining of the FP-tree proceeds as follows. Start from each frequent length-1 pattern,

construct its conditional pattern base (a sub database which consists of the set of prefix paths in

the FP-tree co-occurring with the suffix pattern), then construct its (conditional) FP-tree and


7/9

perform mining recursively on such a tree. The pattern growth is achieved by the concatenation

tree.

Lets first consider I5 which is the last item in L, rather than the first. The reasoning

behind this will become apparent as we explain the FP-tree mining process. I5 occurs in two

branches of the FP-tree. The paths formed by these branches are < (I2, I2, I5:1)> and < (I2, I1,

I3, I5:1)>. Therefore considering I5 as a suffix, its corresponding two prefix paths are and < (I2, I1, I3:1) >, which form its conditional pattern base. Its conditional FP-tree

contains only a single path, (I2:2, I1:2); I3 is not included because its support count of 1 is less

than the minimum support count. The single path generates all the combinations of frequent

patterns: I2 I5:2, I1 I5:2, I2 I1 I5:2.

In the same way find the frequent itemsets for all other Items. The FP-growth method

transforms the problem of finding long frequent patterns to looking for shorter ones recursively

and then concatenating the suffix. It uses the least frequent items as suffix, offering good

selectivity. The method substantially reduces the search costs.

CLOSET: An Efficient Algorithm for Mining Frequent Closed Itemsets

PROBLEM DEFINITION:

An itemset X is contained in transaction if X Y. Given a transaction database

TDB, the support of an itemset X, denoted as sup(X), is the number of transactions in TDB

which contain X. An association rule R: XY is an implication between two itemsets X and Y

where X, YI and XY =. The support of the rule, denoted as sup(XY), is defined as sup

(XUY). The confidence of the rule, denoted as conf(XY), is defined as sup (XUY)/sup(X).

The requirement of mining the complete set of association rules leads to two problems:

1) There may exist a large number of frequent itemsets in a transaction database, especially

when the support threshold is low.

2) There may exist a huge number of association rules. It is hard for users to comprehend

and manipulate a huge number of rules.

An interesting alternative to this problem is the mining offrequent closed itemsets and their

corresponding association rules.

Frequent closed itemset: An itemset X is a closed itemset if there exist no itemset X such that

(1) X is a proper superset of X and (2) every transaction containing X also contains X. A closed

itemset X is frequent if its support passes the given support threshold.


8/9

How to find the complete set of frequent closed itemsets efficiently from large database,

which is called the frequent closed itemset mining problem

For the transaction database in table1 with min_sup = 2, the divide and conquer method for

mining frequent closed itemset.

1) Find frequent items. Scan TDB to find the set of frequent items and derive a global

frequent item list, called f_list, and f_list = {c:4, e:4, f:4, a:3, d:2}, where the items are

sorted in support descending order any infrequent item, such as b are omitted..

2) Divide search space. All the frequent closed itemsets can be divided into 5 non-overlap

subsets based on the f_list: (1) the ones containing items d,(2) the ones containing item a

but no d, (3) the ones containing item f but no a not d, (4) the ones containing e but no f,

a nor d, and (5) the one containing only c. once all subsets are found, the complete set of

frequent closed itemsets is done.

3) Find subsets of frequent closed itemsets. The subsets of frequent closed itemsets can be

mined by constructing corresponding conditional database and mine each recursively.


9/9

Find frequent closed itemsets containing d. Only transaction containing d are needed. The d-

conditional database, denoted as TDB|d, contains all the transactions having d, which is {cefa,

cfa}. Notice that item d is omitted in each transaction since it appears in every transaction in the

d-conditional database.

The support of d is 2. Items c, f and a appear twice respectively in TDB|d. Therefore, cfad: 2 is a

frequent closed itemset. Since this itemset covers every frequent items in TDB|d finishes.

In the same way find the frequent closed itemsets for a, f, e, and c.

4) The set of frequent closed itemsets fund is {acdf :2, a :3, ae :2, cf :4, cef :3, e :4}

Optimization 1: Compress transactional and conditional databases using FP-tree structures. FP-

tree compresses databases for frequent itemset mining. Conditional databases can be derived

from FP-tree efficiently.

Optimization 2: Extract items appearing in every transaction of conditional databases.

Optimization 3: Directly extract frequent closed itemsets from FP-tree.

Optimization 4: Prune search branches.

PERFORMANCE STUDY

Comparison of A-close, CHARM, and CLOSET, CLOSET out performs both CHARM

and A-close. CLOSET is efficient and scalable in mining frequent closed itemsets in large

databases. It is much faster than A-close, and also faster than CHARM.

CONCLUSION

CLOSET leads to less and more interesting associations rules then the other previously

proposed methods.

07. Closet - An Efficient Algorithm for Mining Frequent

Documents

Transcript of 07. Closet - An Efficient Algorithm for Mining Frequent