association rules mining with sql

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Association Rules Mining with SQL

Kirsten NelsonDeepen Manek

November 24, 2003

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Organization of Presentation Overview – Data Mining and RDBMS Loosely-coupled data and programs Tightly-coupled data and programs Architectural approaches Methods of writing efficient SQL

Candidate generation, pruning, support counting K-way join, SubQuery, GatherJoin, Vertical, Hybrid

Integrating taxonomies Mining sequential patterns

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Early data mining applications Most early mining systems were

developed largely on file systems, with specialized data structures and buffer management strategies devised for each

All data was read into memory before beginning computation

This limits the amount of data that can be mined

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Advantage of SQL and RDBMS Make use of database indexing and

query processing capabilities More than a decade spent on making

these systems robust, portable, scalable, and concurrent

Exploit underlying SQL parallelization For long-running algorithms, use

checkpointing and space management

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Use of Database in Data Mining

“Loose coupling” of application and data How would you write an Apriori program? Use SQL statements in an application Use a cursor interface to read through

records sequentially for each pass Still two major performance problems:

Copying of record from database to memory Process context switching for each record

retrieved

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Tightly-coupled applications Push computations into the database

system to avoid performance degradation

Take advantage of user-defined functions (UDFs)

Does not require changes to database software

Two types of UDFs we will use: Ones that are executed only a few times,

regardless of the number of rows Ones that are executed once for each

selected row

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Tight-coupling using UDFs

Procedure TightlyCoupledApriori():begin

exec sql connect to database;exec sql select allocSpace() into :blob from onerecord;

exec sql select * from sales where GenL1(:blob, TID, ITEMID) = 1;

notDone := true;

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while notDone do {exec sql select aprioriGen(:blob)

into :blob from onerecord;exec sql select *

from sales where itemCount(:blob, TID,

ITEMID)=1;exec sql select GenLk(:blob) into :notDone from onerecord

}

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exec sql select getResult(:blob) into :resultBlob from onerecord;

exec sql select deallocSpace(:blob) from onerecord;

compute Answer using resultBlob;end

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Methodology Comparison done with Association Rules against IBM

DB2 Only consider generation of frequent itemsets using

Apriori algorithm Five alternatives considered:

Loose-coupling through SQL cursor interface – as described earlier

UDF tight-coupling – as described earlier Stored-procedure to encapsulate mining algorithm Cache-mine – caching data and mining on the fly SQL implementations to force processing in the database

Consider two classes of implementations SQL-92 – four different implementations SQL-OR (with object relational extensions) – six implementations

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Architectural Options Stored procedure

Apriori algorithm encapsulated as a stored procedure Implication: runs in the same address space as the DBMS Mined results stored back into the DBMS.

Cache-mine Variation of stored-procedure Read entire data once from DBMS, temporarily cache

data in a lookaside buffer on a local disk Cached data is discarded when execution completes Disadvantage – requires additional disk space for caching Use Intelligent Miner’s “space” option

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Terminology Use the following terminology

T: table of items {tid,item} pairs Data is normally sorted by transaction id

Ck: candidate k-itemsets Obtained from joining and pruning

frequent itemsets from previous iteration Fk: frequent items sets of length k

Obtained from Ck and T

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Candidate Generation in SQL – join step

Generate Ck from Fk-1 by joining Fk-1 with itself

insert into Ck select I1.item1,…,I1.itemk-1,I2.itemk-1

from Fk-1 I1,Fk-1 I2where I1.item1 = I2.item1 and

…I1.itemk-2 = I2.itemk-2 and

I1.itemk-1 < I2.itemk-1

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Candidate Generation Example

item1 item2 item3

1 2 3

1 2 4

1 3 4

1 3 5

2 3 4

item1 item2 item3

1 2 3

1 2 4

1 3 4

1 3 5

2 3 4

F3 is {{1,2,3},{1,2,4},{1,3,4},{1,3,5},{2,3,4}} C4 is {{1,2,3,4},{1,3,4,5}}Table F3 (I1) Table F3 (I2)

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Pruning Modify candidate generation algorithm to ensure all k

subsets of Ck of length (k-1) are in Fk-1

Do a k-way join, skipping itemn-2 when joining with the nth table (2<n≤k)

Create primary index (item1, …, itemk-1) on Fk-1 to efficiently process k-way join

For k=4, this becomesinsert into C4 select I1.item1, I1.item2, I1.item3,I2.item3 from F3 I1,F3

I2,

F3 I3, F3 I4 where I1.item1 = I2.item1 … and I1.item3 < I2.item3 and

I1.item2 = I3.item1 and I1.item3 = I3.item2 and I2.item3 = I3.item3 and

I1.item1 = I4.item1 and I1.item3 = I4.item2 and I2.item3 = I4.item3

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Pruning Example

Evaluate join with I3 using previous example C4 is {1,2,3,4}

item1 item2

item3

1 2 3

1 2 4

1 3 4

1 3 5

2 3 4

item1 item2

item3

1 2 3

1 2 4

1 3 4

1 3 5

2 3 4

item1 item2

item3

1 2 3

1 2 4

1 3 4

1 3 5

2 3 4

Table F3 (I1) Table F3 (I2) Table F3 (I3)

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Support counting using SQL Two different approaches

Use the SQL-92 standard Use ‘standard’ SQL syntax such as joins and

subqueries to find support of itemsets

Use object-relational extensions of SQL (SQL-OR)

User Defined Functions (UDFs) & table functions Binary Large Objects (BLOBs)

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Support Counting using SQL-92

4 different methods, two of which detailed in the papers K-way Joins SubQuery

Other methods not discussed because of unacceptable performance 3-way join 2 Group-Bys

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SQL-92: K-way join Obtain Fk by joining Ck with table T of (tid,item) Perform group by on the itemset

insert into Fk select item1,…,itemk,count(*)

from Ck, T t1, …, T tk,

where t1.item = Ck.item1, … , and

tk.item = Ck.itemk and

t1.tid = t2.tid … and

tk-1.tid = tk.tid

group by item1,…,itemk

having count(*) > :minsup

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K-way join example C3={B,C,E} and minimum support required is 2 Insert into F3 {B,C,E,2}

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K-way join: Pass-2 optimization

When calculating C2, no pruning is required after we join F1 with itself

Don’t calculate and materialize C2- replace C2 in 2-way join algorithm with join of F1 with itself

insert into F2 select I1.item1, I2.item1,count(*)from F1 I1, F1 I2, T t1, T t2

where I1.item1 < I2.item1 andt1.item = I1.item1 and t2.item = I2.item1 andt1.tid = t2.tidgroup by I1.item1,I2.item1


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SQL-92: SubQuery based Split support counting into cascade of k subqueries nth subquery Qn finds all tids that match the

distinct itemsets formed by the first n items of Ck insert into Fk select item1, …, itemk, count(*) from (Subquery Qk) tGroup by item1, item2 … , itemk having count(*) > :minsup

Subquery Qn (for any n between 1 and k): select item1, …, itemn, tidfrom T tn, (Subquery Qn-1) as rn-1

(select distinct item1, …, itemn from CK) as dn

where rn-1.item1 = dn.item1 and … and rn-1.itemn-1 = dn.itemn

and rn-1.tid = tn.tid and tn.item = dn.itemn

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Example of SubQuery based Using previous example from class

C3 = {B,C,E}, minimum support = 2

Q0: No subquery Q0

Q1 in this case becomesselect item1, tid

From T t1,

(select distinct item1from C3) as d1

where t1.item = d1.item1

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Example of SubQuery based cnt’d

Q2 becomes

select item1, item2, tid from T t2, (Subquery Q1) as r1,

(select distinct item1, item2 from C3) as d2 where r1.item1 = d2.item1 and r1.tid = t2.tid and t2.item = d2.item2

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Q3 becomes

select item1,item2,item3, tid from T t3, (Subquery Q2) as r2,

(select distinct item1,item2,item3 from C3) as d3

where r2.item1 = d3.item1 and r2.item2 = d3.item2 and

r2.tid = t3.tid and t3.item = d3.item3

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Output of Q3 is

Insert statement becomesinsert into F3 select item1, item2, item3, count(*) from (Subquery Q3) t group by item1, item2 ,item3 having count(*) > :minsup

Insert the row {B,C,E,2}

For Q2, pass-2 optimization can be used

Item1 Item2 Item3 Tid

B C E 20B C E 30

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Performance Comparisons of SQL-92 approaches

Used Version 5 of DB2 UDB and RS/6000 Model 140

200 Mhz CPU, 256 MB main memory, 9 GB of disk space, Transfer rate of 8 MB/sec

Used 4 different item sets based on real-world data

Built the following indexes, which are not included in any cost calculations

Composite index (item1, …, itemk) on Ck

k different indices on each of the k items in Ck

(item,tid) and (tid,item) indexes on the data table T

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Performance Comparisons of SQL-92 approaches

Best performance obtained by SubQuery approach SubQuery was only comparable to loose-coupling

in some cases, failing to complete in other cases DataSet C, for support of 2%, SubQuery outperforms

loose-coupling but decreasing support to 1%, SubQuery takes 10 times as long to complete

Lower support will increase the size of Ck and Fk at each step, causing the join to process more rows

Datasets # records (millions)

# Transactions (millions)

# Items (thousands)

Avg # Items

Dataset-A 2.5 0.57 85 4.4Dataset-B 7.5 2.5 15.8 2.62Dataset-C 6.6 0.21 15.8 31Dataset-D 14 1.44 480 9.62

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Support Counting using SQL with object-relational extensions

6 different methods, four of which detailed in the papers

GatherJoin GatherCount GatherPrune Vertical

Other methods not discussed because of unacceptable performance

Horizontal SBF

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SQL Object-Relational Extension: GatherJoin

Generates all possible k-item combinations of items contained in a transaction and joins them with Ck

An index is created on all items of Ck

Uses the following table functions Gather: Outputs records {tid,item-list}, with item-list

being a BLOB or VARCHAR containing all items associated with the tid

Comb-K: returns all k-item combinations from the transaction

Output has k attributes T_itm1, …, T_itmk

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GatherJoin

insert into Fk select item1,…, itemk, count(*)

from Ck,

(select t2.T_itm1,…,t2.itmk from T,

table(Gather(T.tid,T.item)) as t1,

table(Comb-K(t1.tid,t1.item-list)) as t2)

where t2.T_itm1 = Ck.item1 and … and

t2.T_itmk = Ck.itemk

group by Ck.item1,…,Ck.itemk


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Example of GatherJoin

t1 (output from Gather) looks like:

t2 (generated by Comb-K from t1) will be joined with C3 to obtain F3

1 row from Tid 10 1 row from Tid 20 4 rows from Tid 30

Insert {B,C,E,2}

t1

Tid Item-List

10 A,C,D20 B,C,E30 A,B,C,E40 B,E

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GatherJoin: Pass 2 optimization

When calculating C2, no pruning is required after we join F1 with itself

Don’t calculate and materialize C2 - replace C2 with a join to F1 before the table function

Gather is only passed frequent 1-itemset rows

insert into F2 select I1.item1, I2.item1, count(*) from F1 I1,

(select t2.T_itm1,t2.T_itm2 from T, table(Gather(T.tid,T.item)) as t1,

table(Comb-K(t1.tid,t1.item-list)) as t2 where T.item = I1.item1)

group by t2.T_itm1,t2.T_itm2


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Variations of GatherJoin - GatherCount Perform the GROUP BY inside the table

function Comb-K for pass 2 optimization Output of the table function Comb-K

Not the candidate frequent itemsets (Ck) But the actual frequent itemsets (Fk) along

with the corresponding support Use a 2-dimensional array to store

possible frequent itemsets in Comb-K May lead to excessive memory use

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Variations of GatherJoin - GatherPrune

Push the join with Ck into the table function Comb-K

Ck is converted into a BLOB and passed as an argument to the table function. Will have to pass the BLOB for each

invocation of Comb-K - # of rows in table T

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SQL Object-Relational Extension: Vertical For each item, create a BLOB containing the

tids the item belongs to Use function Gather to generate {item,tid-

list} pairs, storing results in table TidTable Tid-list are all in the same sorted order

Use function Intersect to compare two different tid-lists and extract common values

Pass-2 optimization can be used for Vertical Similar to K-way join method Upcoming example does not show optimization

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Verticalinsert into Fk select item1, …, itemk, count(tid-list) as cntfrom (Subquery Qk) t where cnt > :minsup

Subquery Qn (for any n between 2 and k)Select item1, …, itemn, Intersect(rn-1.tid-list, tn.tid-list) as tid-listfrom TidTable tn, (Subquery Qn-1) as rn-1

(select distinct item1, …, itemn from CK) as dn

where rn-1.item1 = dn.item1 and … andrn-1.itemn-1 = dn.itemn-1 andtn.item = dn.itemn

Subquery Q1: (select * from TidTable)

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Example of Vertical Using previous example from class

C3 = {B,C,E}, minimum support = 2

Q1 is TidTable Item Tid-List

A 10,30B 20,30,40C 10,20,30D 10E 20,30,40

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Example of Vertical cnt’d Q2 becomes

Select item1, item2, Intersect(r1.tid-list, t2.tid-list) as tid-list

from TidTable t2, (Subquery Q1) as r1

(select distinct item1, item2 from C3) as d2

where r1.item1 = d2.item1 and t2.item = d2.item2

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Example of Vertical cnt’d Q3 becomes

select item1, item2, item3, intersect(r2.tid-list, t3.tid-list) as tid-list

from TidTable t3, (Subquery Q2) as r2

(select distinct item1, item2, item3 from C3) as d3

where r2.item1 = d3.item1 and r2.item2 = d3.item2 and

t3.item = d3.item3

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Performance Comparisons using SQL-OR

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# Items (thousands)

Avg # Items

Dataset-A 2.5 0.57 85 4.4Dataset-B 7.5 2.5 15.8 2.62

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Support 0.10% 0.03% 0.01%

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Legend:PrepPass 1Pass 2Pass 3Pass 4

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Performance Comparisons using SQL-OR

Legend:PrepPass 1Pass 2Pass 3Pass 4



# Items (thousands)

Avg # Items

Dataset-C 6.6 0.21 15.8 31Dataset-D 14 1.44 480 9.62

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Performance comparison of SQL object-relational approaches Vertical has best overall performance,

sometimes an order of magnitude better than other 3 approaches

Majority of time is transforming the data in {item,tid-list} pairs

Vertical spends too much time on the second pass Pass-2 optimization has huge impact on

performance of GatherJoin For Dataset-B with support of 0.1 %, running time for

Pass 2 went from 5.2 hours to 10 minutes Comb-K in GatherJoin generates large number of

potential frequent itemsets we must work with

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Hybrid approach Previous charts and algorithm analysis show

Vertical spends too much time on pass 2 compared to other algorithms, especially when the support is decreased

GatherJoin degrades when the # of frequent items per transaction increases

To improve performance, use a hybrid algorithm Use Vertical for most cases When size of candidate itemset is too large, GatherJoin

is a good option if number of frequent items per transaction (Nf) is not too large

When Nf is large, GatherCount may be the only good option

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Architecture Comparisons Compare five alternatives

Loose-Coupling, Stored-procedure Basically the same except for address space

program is being run in Because of limited difference in performance,

focus solely on stored procedure in following charts

Cache-Mine UDF tight-coupling Best SQL approach (Hybrid)

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Performance Comparisons of Architectures



# Items (thousands)

Avg # Items

Dataset-A 2.5 0.57 85 4.4Dataset-B 7.5 2.5 15.8 2.62

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Data Set B

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Performance Comparisons of Architectures cnt’d



# Items (thousands)

Avg # Items

Dataset-C 6.6 0.21 15.8 31Dataset-D 14 1.44 480 9.62

Legend:Pass 1Pass 2Pass 3Pass 4

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Performance Comparisons of Architectures cnt’d

Cache-Mine is the best or close to the best performance in all cases

Factor of 0.8 to 2 times faster than SQL approach

Stored procedure is the worst Difference between Cache-Mine directly related to

the number of passes through the data Passes increase when the support goes down May need to make multiple passes if all candidates

cannot fit in memory

UDF time per pass decreases 30-50% compared to stored procedure because of tighter coupling with DB

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Performance Comparisons of Architectures cnt’d SQL approach comes in second in performance

to Cache-Mine Somewhat better than Cache-Mine for high support

values 1.8 – 3 times better than Stored-procedure/loose-

coupling approach, getting better when support value decreases

Cost of converting to Vertical format is less than cost of converting to binary format in Cache-Mine

For second pass through data, SQL approach takes much more time than Cache-Mine, particularly when we decrease the support

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Taxonomies - example

Beverages Snacks

Soft Drinks Alcoholic Drinks

Pretzels Chocolate Bar

Pepsi Coke Beer

Parent Child

Beverages Soft Drinks

Beverages Alcoholic Drinks

Soft Drinks Pepsi

Soft Drinks Coke

Alcoholic Drinks Beer

Snacks Pretzels

Snacks Chocolate Bar

Example rule:

Soft Drinks Pretzels with 30% confidence, 2% support

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Taxonomy augmentation

Algorithms similar to previous slides Requires two additions to algorithm

Pruning itemsets containing an item and its ancestor

Pre-computing the ancestors for each item

Will also consider support counting

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Pruning items and ancestors

In the second pass we will join F1 with F1 to give C2

This will give, for example:beverages,pepsi

snacks,coke

pretzels,chocolate bar But beverages,pepsi is redundant!

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Pruning items and ancestors The following modification to the SQL

statement eliminates such redundant combinations from being selected:

insert into C2 (select I1.item1, I2.item1 from F1 I1, F1 I2

where I1.item1 < I2.item1) except(select ancestor, descendant from Ancestor union

select descendant, ancestor from Ancestor)

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Pre-computing ancestors An ancestor table is created

Format (ancestor, descendant) Use the transitive closure operation

insert into Ancestor with R-Tax (ancestor, descendant) as

(select parent, child from Tax union all select p.ancestor, c.child from R-Tax

p, Tax c where p.descendant = c.parent)select ancestor, descendant from R-Tax

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Support Counting

Extensions to handle taxonomies Straightforward, but Non-trivial

Need an extended transaction table For example, if we have {coke, pretzels} We add also {soft drinks, pretzels},

{beverages, pretzels}, {coke, snacks}, {soft drinks, snacks}, {beverages, snacks}

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Extended transaction table Can be obtained by the following SQLQuery to generate T*select item, tid from T unionselect distinct A.ancestor as item, T.tidfrom T, Ancestor Awhere A.descendant = T.item The “select distinct” clause gets rid of items

with common ancestor – e.g. don’t want {beverages, beverages} being added twice from {pepsi, coke}

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Pipelining of Query

No need to actually build T* Make following modification to

SQL:insert into Fk with T*(tid, item) as (Query for T*)select item1,…,itemk,count(*)from Ck, T* t1, …, T* tk,where t1.item = Ck.item1, … , andtk.item = Ck.itemk andt1.tid = t2.tid … and tk-1.tid = tk.tidgroup by item1,…,itemk


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Sequential patterns Similar to papers covered on Nov 17 Input is sequences of transactions

E.g. ((computer,modem),(printer)) Similar to association rules, but dealing

with sequences as opposed to sets Can also specify maximum and

minimum time gaps, as well as sliding time windows Max-gap, min-gap, window-size

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Input and output formats Input has three columns:

Sequence identifier (sid) Transaction time (time) Idem identifier (item)

Output format is a collection of frequent sequences, in a fixed-width table (item1, eno1,…,itemk, enok, len) For smaller lengths, extra column values are

set to NULL

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GSP algorithm Similar to algorithms shown earlier Each Ck has transactions and times, but no

length – has fixed length of k Candidates are generated in two steps

Join – join Fk-1 with itself Sequence s1 joins with s2 if the subsequence obtained

by dropping the first item of s1 is the same as the one obtained by dropping the last item of s2

When generating C2, we need to generate sequences where both of the items appear as a single element as well as two separate elements

Prune All candidate sequences that have a non-frequent

contiguous (k-1) subsequence are deleted

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GSP – Join SQL

insert into Ck

select I1.item1, I1.eno1, ... , I1.itemk-1, I1.enok-1, I2.itemkk-1, I1.enok-1 + I2.enok-1 – I2.enok-2

from Fk-1 I1, Fk-1 I2

where I1.item2 = I2.item1 and ... and I1.itemk-1 = I2.itemk-2 and

I1.eno3-I1.eno2 = I2.eno2 – I2.eno1 and ... and

I1.enok-1 – I1.enok-2 = I2.enok-2 – I2.enok-3

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GSP – Prune SQL Write as a k-way join, similar to before There are at most k contiguous

subsequences of length (k-1) for which Fk-1 needs to be checked for membership

Note that all (k-1) subsequences may not be contiguous because of the max-gap constraint between consecutive elements.

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GSP – Support Counting In each pass, we use the candidate table

Ck and the input data-sequences table D to count the support

K-way join We use select distinct before the group by to

ensure that only distinct data-sequences are counted

We have additional predicates between sequence numbers to handle the special time elements

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GSP – Support Counting SQL

(Ck.enoj = Ck.enoi and abs(dj.time – di.time)≤ window-size) or (Ck.enoj = Ck.enoi + 1 and dj.time – di.time max-gap and dj.time – di.time > min-gap) or (Ck.enoj > Ck.enoi + 1)

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References1. Developing Tightly-Coupled Data Mining

Applications on a Relational Database System Rakesh Agrawal, Kyuseok Shim, 1996

2. Integrating Association Rule Mining with Relational Database Systems: Alternatives and Implications Sunita Sarawagi, Shiby Thomas, Rakesh Agrawal, 1998 Refers to 1) above

3. Mining Generalized Association Rules and Sequential Patterns Using SQL Queries Shiby Thomas, Sunita Sarawagi, 1998 Refers to 1) and 2) above

association rules mining with sql

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