panagiotis papapetrou department of computer science boston university constraint-based mining of...

Panagiotis Papapetrou

Department of Computer Science

Boston University

Constraint-based Mining of Frequent Arrangements of

Temporal Intervals

Master Thesis Defense

Introduction and Motivation Sequential pattern mining has received particular attention in the last decade:

Database of sequences: ordered lists of instantaneous events.

Extract frequent sequential patterns.

In many applications events occur over time intervals.

Extracting frequent arrangements of these temporally correlated labeled

intervals may lead to useful observations.

So far, algorithms concentrate on the case where events occur

instantaneously.

Several works on mining temporal patterns of interval-based events.

However, the mining algorithms were apriori-based and in some cases [1] the

extracted patterns were restricted to certain forms.

1. P. Kam and A. W. Fu. “Discovering temporal patterns of Interval-based Events”. In Proc. of the DaWak, pages 317–326, London, UK, 2000. Springer-Verlag.

Applications (1/4)Linguistics ASL Database

Collections of utterances. Utterance:

Associates a segment of video with a detailed transcription. Number of ASL fields occurring over time intervals. Syntactic Structures:

Wh-Question. Negation. Yes/No Question.

Gestural Fields: Head-shake. Eye-brow raise/lower.

Applications (2/4)Linguistics (An example)

> Who drove the car?

(Eye-brow Lower)

(Wh-Question)

(Wh-Word)

time

(Rapid head shake) (Rapid head shake)

Applications (3/4)Networks

Router 1

Router 2IPs IPs

A B

(D, C)(D, B)

(A, B)

D

C

time

Applications (4/4)Biology

Human Gene

Region ofNucleodite A

Region ofNucleodite G

Region ofNucleodite C

(Nucleodite C)

(Nucleodite G)

(Nucleodite A)

Position in the Gene

Main Contributions

Formal definition of the problem of mining frequent temporal arrangements of

intervals in an interval database using temporal and structural constraints.

Development of three algorithms:

BFS-based

DFS-based

Prefix-based

Further improvement of the mining process with the incorporation of

interestingness measures for the extracted arrangement rules.

Extensive experimental evaluation and comparison with a standard sequential

pattern mining method both on real and synthetic datasets.

Outline Preliminaries Problem Formulation Proposed Algorithms

BFS-based DFS-based Prefix-based

Extraction of Arrangement Rules Experimental Evaluation Related Work Conclusions and Future Work

Preliminaries (1/9) There can be many types of relations between two event intervals2. We consider seven of them:

2. J. F. Allen and G. Ferguson. “Actions and events in interval temporal logic”. Technical Report 521, The University of Rochester, July 1994”.

A[tstart, tend] B[tstart, tend]

(a) Meet of A and B

A[tstart, tend]

B[tstart, tend]

(d)

A[tstart, tend] B[tstart, tend]

(g)

Contain of A and B

Follow of A and B

+/- e

A[tstart, tend]

B[tstart, tend]

(e) Left Contain of A and B

A[tstart, tend]

B[tstart, tend]

(f) Right Contain of A and B

+/- e +/- e

A[tstart, tend]

B[tstart, tend]

(c) Overlap of A and B

A[tstart, tend]

B[tstart, tend]

(b) Match of A and B

+/- e+/- e

Preliminaries (2/9)

Let SS = { = {EE11, , EE22, …, , …, EEmm} be an ordered set of event intervals, called } be an ordered set of event intervals, called

event interval sequenceevent interval sequence, or, or e-sequence e-sequence..

Each Each EEii is a triple (eis a triple (eii, t, tiistartstart, t, tii

endend))

eeii: an event label.: an event label.

ttiistart:start:: the event start time.: the event start time.

ttiiend:end:: the event end time.: the event end time.

Note: Note: SS is ordered by t is ordered by tiistartstart..

k-e-sequencek-e-sequence: an e-sequence of size k.

e-sequence databasee-sequence database D: D: a set of e-sequences.

Preliminaries (3/9)

Example of a 5-e-sequence:

SS = { (A,1,7), (B,3,19), (D,4,30), (C,7,15), C,23,42) }= { (A,1,7), (B,3,19), (D,4,30), (C,7,15), C,23,42) }

A

B

CC

31 4 7 15 19 23 30 42

D

Preliminaries (4/9) k-Arrangement:k-Arrangement: a set of kk temporally correlated events in an

e-sequence, denoted as A = {EE , R}, where: E E : the set of labels of the event intervals in the arrangement.

R R : the set of temporal relations between the events in E.

)}E ,(Er ... ),E ,(Er ..., ),E ,(Er ),E ,(Er ..., ),E ,(Er ),E ,(E{r R n1-nn232n13121

}, |, ||, { )E ,(Er ji A

B

C

} }|, |, { C},B,{A, {

E1 Ei Ei+1 EnEi+3Ei+2Ei-1

where is the temporal relation between EEii and EEjj.

Preliminaries (5/9)

Given an e-sequence SS and an arrangement AA = {EE , RR}:

SS contains AA, if all the events in EE appear in SS, with the

relations defined in RR.

Given an e-sequence database DD and a minimum support

threshold min_supmin_sup:

An arrangement AA is frequent, if it is contained in at least

min_supmin_sup e-sequences (i.e. records) of DD.

Preliminaries (6/9)

A

B

C

A

B

CC

31 4 7 15 19 23 30 42

D

SS = {(A,1,7), (B,3,19), (D,4,30), (C,7,15), C,23,42)}= {(A,1,7), (B,3,19), (D,4,30), (C,7,15), C,23,42)}

Example of an arrangement AA, contained in an e-sequence SS:

} }, |, { C},B,{A, {

Preliminaries (7/9)

Arrangement Rule:Arrangement Rule:

AA = {EE , RR} is split into:

AAii = {EEii , RRii}

AAjj = {EEjj , RRjj}

Ei Ej = Ø

jRDi AAr ij

,:

Rij: defines the relations between Ei and Ej.

λ: an interestingness measure.

Preliminaries (8/9)

C

D

B

C

D

B

C

B

} }, , { D},C,{B, { Example of an arrangement rule r, r, given arrangement

AA =

rr : {}}{{D},}}{C},{{B, },{

Preliminaries (9/9) Monotone Interestingness Measures:Monotone Interestingness Measures:

Support (A) = |A|/|D|Support (A) = |A|/|D|

All-Confidence (A) = sup(A)/max{sup(AAll-Confidence (A) = sup(A)/max{sup(Akk)})}

Anti-Monotone Interestingness Measures:Anti-Monotone Interestingness Measures:

Confidence (r) = support (r) / coverage (r)Confidence (r) = support (r) / coverage (r)

Lift (r) = support (r) / cover (A) * cover (B)Lift (r) = support (r) / cover (A) * cover (B)

Leverage (r) = support (r) – cover (A) * cover (B)Leverage (r) = support (r) – cover (A) * cover (B)

Conviction (r) = (1-support(B))/(1-confidence (r))Conviction (r) = (1-support(B))/(1-confidence (r))

Cover (A) = |A|/|D| Coverage (r : A->B) = Cover (A)Cover (A) = |A|/|D| Coverage (r : A->B) = Cover (A)

Problem Formulation

1. Find the complete set of frequent arrangements given:

An e-sequence database D.D.

A minimum support threshold min_sup.min_sup.

2. Find the top K frequent arrangement rules given:

An e-sequence database D.D.

A minimum support threshold min_sup.min_sup.

A set of constraints CC.

An interestingness measure λλ.

An integer KK.

Constraints Regular Expressions RR:

A set of regular expressions that limit the form of the extracted patterns.

Gap Constraint CCgg:

A Follow should be separated by at most Cg units.

Overlap Constraint CCoo = {Col, Cor}:

An Overlap should be between Col% and Cor%.

Contain Constraint CCtt = {Ctl, Ctr}:

A Contain should be between Ctl% and Ctr%.

Duration Constraint CCdd:

Each event interval should have a duration of at least Cd units.

Apply a sequential pattern mining algorithm?

Consider start and end points of an interval as two instantaneous events.

Convert each e-sequence into a regular sequence. Apply an efficient sequential pattern mining algorithm + post-

processing. Basic drawbacks:

k-e-sequence = sequence of 2k events. May produce 22k patterns. Can we reduce it to 2k?

Extracted patterns will carry lots of redundant information.A

B

{Astart, Bstart, Aend, Bend},

but also: {Astart, Bstart},…

Sequential Pattern MiningAlgorithm will produce

Frequent Arrangement Mining Algorithms

Use a logical Tree-like structure to enumerate the

arrangements4.

Traverse the Tree using:

BFS

DFS

Hybrid DFS

BFS for the first two levels.

DFS for the rest of the mining process.

4. R. J. Bayardo. “Efficiently mining long patterns from databases”. In Proc. of ACM SIGMOD, pages 85–93, 1998.

The Arrangement Enumeration Tree

NULL

{A, B} {A, C} {B, A} {B, C} {C, A} {C, B}{A, A} {B, B} {C, C}

A->A A>B A->B AC A|C A||C A>C A->C

{A} {B} {C}

{A, A, A} {A, A, B} {A, B, A} {A, B, B} {A, B, C}{A, A, C}

AB*A|C*B||CAB*AC*B|C A||B*A->C*BC ...

AA A|A A||A A >A A||BAB A|B

Let },,{ CBAE

LEVEL3

LEVEL2

LEVEL1

Intermediate

Intermediate

BFS-based Approach (1/4) Traverse the Tree in BFS order.

2 database scans.

On each step k:

Build candidate k-arrangements based on (k-1)-arrangements.

Find 2-relations by scanning the second level of the Tree.

Determine frequency: min_supmin_sup threshold must be satisfied.

If a node is not frequent, do not expand sub-tree (Apriori Principle)5.

Stop at step k, where no frequent arrangements are found.

5. R. Agrawal and R. Srikant. “Fast algorithms for mining association rules”. In Proc. of VLDB, pages 487-499, 1994.

BFS-based Approach (2/4)An Example

{A, B} {A, C} {B, A} {B, C} {C, A} {C, B}{A, A} {B, B} {C, C}

A->A A>B A->B AC A|C A||C A>C A->C

{A, A, B} {A, B, A} {A, B, B}{A, A, C}

AA, A|C, A||C

AA A|A A||A A >A A||BAB A|B

… …

{A, B, C}{A, A, A}

BFS-based Approach (3/4) Creating a 2-arrangement (Example)

A

esid Intv-List

1

1

2

2

3

3

4

[1, 3]

[6, 12]

[1, 2]

[10, 12]

[4, 7]

[9, 11]

[6, 14]

B

esid

1

1

2

2

3

3

4

[1, 3]

[8, 11]

[2, 6]

[11, 15]

[1, 3]

[11, 12]

[1, 5]

Intv-List

4 [7, 10]

Meet (A, B)

esid

2

3

[1, 2] , [2, 6]

Intv-List

[9, 11] , [11, 12]

Follow (A, B)

esid

1

2

[1, 3] , [8, 11]

Intv-List

[1, 2] , [11, 15]

3 [4, 7] , [11, 12]

{A, B}

Contain (A, B)

esid

1 [6, 12] , [8, 11]

Intv-List

4 [6, 14] , [7, 10]

BFS-based Approach (4/4)Creating a 3-arrangement (Example)

{A, B, C}

Contain (A, B) * Contain (A, C) * Contain (B, C)

esid

1

4

Intv-List

Contain (A, B)

esid

1 [6, 12] , [8, 11]

Intv-List

4 [6, 14] , [7, 10]

Contain (B, C)

esid

1

4

[8, 11] , [9, 10]

Intv-List

[7, 10] , [8, 9]

Contain (A, C)

esid

1

4

[6, 12] , [9, 10]

Intv-List

[6, 14] , [8, 9]

DFS-based Approach Candidate generation in DFS order.

Leads to frequent large arrangements faster.

Skips expansions of nodes that are definitely going to lead to frequent

arrangements.

DFS is inappropriate:

For each node we would have to scan the database multiple times to

detect the 2-relations among the items in the node.

Hybrid-DFS

Generates the first two levels of the Tree using BFS, then uses DFS.

Eliminates multiple database scans, 2-relations are available.

Support Counting{A, B}

A|C

{A, B, C}

AB*A|C*B||C

AB B||C

SID BIT

1

0

1

0

0

1

1

1

2

3

4

5

6

7

SID BIT

1

1

0

0

0

1

1

1

2

3

4

5

6

7

SID BIT

1

0

1

1

1

1

1

1

2

3

4

5

6

7

SID BIT

1

0

0

0

0

1

1

1

2

3

4

5

6

7

Prefix-based Approach (1/8)The Sequential Approach

Prefix and Suffix (Projection) <a>, <aa>, <a(ab)> and <a(abc)> are prefixes

of sequence <a(abc)(ac)d(cf)> Given sequence <a(abc)(ac)d(cf)>

Prefix Suffix (Prefix-Based Projection)

<a> <(abc)(ac)d(cf)>

<aa> <(_bc)(ac)d(cf)>

<ab> <(_c)(ac)d(cf)>

Prefix-based Approach (2/8)Example

Sequence_id Sequence

10 <a(abc)(ac)d(cf)>

20 <(ad)c(bc)(ae)>

30 <(ef)(ab)(df)cb>

40 <eg(af)cbc>

(min_sup=2)

Prefix-based Approach (3/8)The Sequential Approach (continued)

Step1: Find length-1 sequential patterns;

<a>:4, <b>:4, <c>:4, <d>:3, <e>:3, <f>:3

patternsupport

Step2: Divide search space;

six subsets according to the six prefixes;

Step3: Find subsets of sequential patterns;

By constructing corresponding projected databases and mine

each recursively.

Prefix-based Approach (4/8)Example (continued)

Sequence_id OriginalSequences

ProjectedSequences

10 <a(abc)(ac)d(cf)> <a(abc)(ac)d(cf)>

20 <(ad)c(bc)(ae)> <(ad)c(bc)(ae)>

30 <(ef)(ab)(df)cb> <(ef)(ab)(df)cb>

40 <eg(af)cbc> <eg(af)cbc>

New locally frequent items:

a : 2 b : 4 d : 2

c : 4 f : 3

Prefix-based Approach (5/8)Example (continued)

Sequence_id OriginalSequence

ProjectedSequences





Sequence_id OriginalSequences

ProjectedSequences





Prefix-based Approach (6/8)The Interval-based Approach

Use similar definition for the projection of an e-sequence SS with

respect to an arrangement AA to that of the sequential approach.

Problem:

May skip frequent patterns.

Solution:

Find every occurrence of AA in SS and project with respect to each

one of them.

Prefix-based Approach (7/8)An Example of A Projection

A

B

CC

31 4 7 15 19 23 30 42

D

time

A

B

C

C

31 4 7 15 19 23 30 42

D

time

Prefix Arrangement A:

Projection with respect to A

Prefix-based Approach (8/8)An Example That Works And One That Does Not

A

C

time

Prefix Arrangement: ASupport Threshold: 2

A

A

time

C

time

A

C

DetectedArrangements:

A

C

Detected twice(Correct)

Record 1

Record 2 A

C

C

A

time

Support = 1Correct

Support = 1Wrong

Extracting Arrangement Rules Discover top KK arrangement rules that maximize a given

interestingness measure λλ.

How deep can we push λ in the mining process?

Depends on antimonotonicity.

If λ is antimonotone:

Can prune a subset of the candidate arrangement rules.

If λ is non-antimonotone:

Pruning cannot be done.

Non-Antimonotone λ (1/2)

First discover the set of frequent arrangements.

The set of constraints CC is applied during the mining process.

Infer the arrangement rules from the extracted patterns after

the completion of the mining process.

Non-Antimonotone λ (2/2) Given a frequent arrangement AA = {EE , RR}

A is split into:

AAii = {EEii , RRii}

AAjj = {EEjj , RRjj}

Rule is defined.

If rr satisfies λ, add it into the set of valid rules.jRDi AAr ij

,:

Antimonotone λ

If A A is reached and valid no rule is inferred from AA

The sub-tree of AA is pruned.

Otherwise, a set of rules RRAA exists for node AA.

For each new arrangement CC = {EE , RR}

EE is split into EE11 and E E2.2.

If AAii = = {EEii,R,Rii} in the antecedent part of any rule in RRAA

such that

Then EE1 1 cannot be the antecedent part of any rule inferred from C.C.

The Split is skipped.

i1 E E

Experimental Setup (1/4)Real Datasets SignStream Database

Created by the National Center for Sign Language and

Gesture Resources at Boston University.

Collection of 884 utterances.

Some types of event labels: Grammatical or syntactic structures:

Wh-Question.

Negation.

Yes/No Question.

Gestural Fields: Head-shake.

Eye-brow raise/lower.

Experimental Setup (2/4)Real Datasets Network Data

Sampled from flow data.

Two routers with high communication rate:

ATLA: router in Atlanta.

LOSA: router in LA.

Monitored communication for 10 days, between 200 IPs.

An e-sequence is a set of IP connections for every 15 minutes:

An event label is the two IPs (source-destination).

The interval corresponds to the duration of this communication.

Size of dataset: 960 e-sequences.

Experimental Setup (3/4)Synthetic Datasets

Generated considering the following factors: Number of e-sequences in the Database.

Average e-sequence size.

Number of distinct items.

Density of frequent patterns.

Experimental Setup (4/4)Algorithms Compared:

BFS.

Hybrid-DFS.

Prefix-based.

SPAM6, modified as follows:

Considered the start and end points of each interval as two

instantaneous events.

Post-processed the extracted sequential patterns to convert

them into arrangements.

6. J. Ayres, J. Gehrke, T. Yiu, and J. Flannick. Sequential pattern mining using a bitmap representation. In Proc. of ACM SIGKDD, pages 429–435, 2002.

Performance Analysis

BFS outperforms SPAM in large database sizes and

small supports.

Hybrid-DFS outperforms both SPAM and BFS.

In low supports Hybrid-DFS is twice as fast as BFS.

In all cases the Prefix-based algorithm performs

worse.

Sample Results (1/4) SignStream Database

Head: tilt side

Negation

Eye-brow raise

Negation 72 %

87 %

Eye-brow raise

Yes/No Word

68 % Eye-brow lower

Yes/No Word

66 %

Head: tilt side

Negations:

YES/NO Questions:



WH-questions:

For more detailed results visit the following web page:http://cs-people.bu.edu/panagpap/Research/asl_mining.htm

Eye-brow lower

Wh-word

Head: jut forward

Wh-word

Eye-aperture: squint

Wh-word

Head: rapid shake

Wh-word85 %

56 %

51 %

85 %

Sample Results (4/4)Network Dataset

Performance of Different Interestingness MeasuresASL Dataset

Some Arrangement Rules (1/2)ASL Dataset

Some Arrangement Rules (2/2)ASL Dataset

Related Work (1/2) Problem of sequential pattern mining first introduced in:

R. Agrawal and R. Srikant. Fast algorithms for mining association rules. In proc. of VLDB, pages 487-499, 1994.

An extension to episodes (i.e. combinations of events with a partially specified order) was proposed in: H. Mannila, H. Toivonen, and A. Verkamo. Discovering Frequent episodes in

sequences. In Proc. of ACM SIGKDD, pages 210–215, 1995.

The Itemset Enumeration Tree was described in: R. J. Bayardo. Efficiently mining long patterns from

databases. In Proc. of ACM SIGMOD, pages 85–93, 1998.

Some efficient sequential pattern mining algorithms have been proposed in: M. Zaki. Spade: An efficient algorithm for mining sequences. Machine Learning,

40:31–60, 2001. J. Ayres, J. Gehrke, T. Yiu, and J. Flannick. Sequential pattern mining using a bitmap

representation. In Proc. of ACM SIGKDD, pages 429–435, 2002.

Related Work (2/2) Closed sequential pattern mining:

X. Yan, J. Han, and R. Afshar. Clospan: Mining closed sequential patterns in large databases. In Proc. of SDM, 2003.

J.Wang and J. Han. Bide: Efficient mining of frequent closed sequences. In Proc. of IEEE ICDE, pages 79–90, 2004.

Mining association rules in temporal and spatio-temporal databases: T. Abraham and J. F. Roddick. Incremental meta-mining from large temporal data sets.

In ER ’98: Proceedings of the Workshops on Data Warehousing and Data Mining , pages 41–54, 1999.

X. Chen and I. Petrounias. Mining temporal features in association rules. In Proc. of PKDD, pages 295–300, London, UK, 1999. Springer-Verlag.

I. Tsoukatos and D. Gunopulos. Efficient mining of spatiotemporal patterns. In Proc. of the SSTD, pages 425–442, 2001.

Discovering temporal patterns of Interval-based Events: P. Kam and A. W. Fu. Discovering temporal patterns of Interval-based Events. In Proc.

of the DaWak, pages 317–326, London, UK, 2000. Springer-Verlag.

Conclusions The problem of constraint-based mining frequent arrangements of

temporal intervals has been formally defined.

Three efficient methods for solving the problem have been discussed.

An efficient algorithm for applying interestingness measures on the

discovered patterns and extracting interesting arrangement rules has

been proposed.

Both BFS and DFS approaches use an arrangement enumeration tree to

discover the set of frequent arrangements.

The DFS-based approach further improves performance over BFS: Longer arrangements are reached faster.

The need to examine smaller subsets of these arrangements is

eliminated.

The Prefix-based approach performs worse due to projections.

Future Work

Apply our algorithms on biological sequences: DNA.

Proteins.

Consider e-sequences with categorical domains: Series of medical treatments for a disease.

Result (Cure/Death).

EXTRA SLIDES

Apply a closed sequential pattern mining algorithm3?

Noise again…

A

B

A

B

A

B

{Astart, Bstart, Aend, Bend}: 2/3

But also:

{Astart, Aend, Bend}: 3/3

Closed Sequential PatternMining Algorithm

3. J.Wang and J. Han. “Bide: Efficient mining of frequent closed sequences”. In Proc. of IEEE ICDE, pages 79–90, 2004.

The ISIdList Structure (1/2) An ISIdList is defined for every arrangement generated

throughout the mining process. The ISIdList for an arrangement AA = { , R} in an e-

sequence database DD, has the following structure: Head: Arrangement representation using and R. A record for each e-sequence in the database that supports AA. Each record is of type (idid, intv-Listintv-List), where:

idid is the id of the e-sequence in DD. intv-List:intv-List:

set of intervals where AA occurs in the e-sequence A (for | | ≤ 2). set of pointers to records of ISIdLists of the second level (for | | > 2).

E

E

EE

The ISIdList Structure (2/2) (Example)

Database D

id e-sequence

1

2

4

3

A [1, 3], B [1, 3], A [6, 12], B [8, 11], C [ 9, 10]

A [1, 2], B [2, 6], A [10, 12], B [11, 15], C [14, 17]

B [1, 3], A [4, 7], A [9, 11], B [11, 12] , C [12, 14]

B [1, 5], A [6, 14], B [7, 10], C [8, 9]

A

esid Intv-List

1

1

2

2

3

3

4

[1, 3]

[6, 12]

[1, 2]

[10, 12]

[4, 7]

[9, 11]

[6, 14]

B

esid

1

1

2

2

3

3

4

[1, 3]

[8, 11]

[2, 6]

[11, 15]

[1, 3]

[11, 12]

[1, 5]

Intv-List

4 [7, 10]

C

esid

1

2

3

4

[9, 10]

[14, 17]

[12, 14]

[8, 9]

Intv-List

Let DD consist of a set or e-sequences of event intervals with labels A, B, C.

The set of frequent 1 arrangements is {A, B, C}, with the following ISIdLists:

BFS-based Approach

At each Step k: Use Tree to generate candidate arrangements:

Build N(k) from N(k-1). Construct IMk. For every 2-relation, point to the

second level of the Tree. Check support. If it satisfies min_sup, then add to

Fk. Continue with the rest of the Tree in a BFS order. If a node is found not to be frequent, do not expand

its sub-tree (Apriori Principle)3. Stop at step k, where Fk = empty.

3. R. Agrawal and R. Srikant. “Fast algorithms for mining association rules”. In Proc. of VLDB, pages 487-499, 1994.

BFS-based Approach (1/4) DD: an input e-sequence database. FF: the complete set of frequent

arrangements. FFkk: the complete set of frequent k-

arrangements. CCkk: the current set of candidate k-

arrangements. min_supmin_sup: the minimum support threshold. ISIdList (A)ISIdList (A): the ISIdList of arrangement A.

BFS-based Approach (2/4)

BFS:STEP 1: Find F1

Use Tree to generate C1

Build N(1). For each ni

1 in N(1): Build ISIdList (Ai), where Ai is the arrangement that

corresponds to ni1.

If the number of records in ISIdList (Ai) is at least min_sup,min_sup, then A is inserted into F1.


BFS:STEP k: Find Fk

Use Tree to generate Ck

Build N(k) from N(k-1). Construct IMk.

For each node in IMk: Build ISIdList. If the number of records in the ISIdList is at least

min_sup,min_sup, insert arrangement into F1.

Continue with the rest of the Tree in a BFS order.


Continue with the rest of the Tree in a BFS order.

If a node is found not to be frequent, do not expand its sub-tree (Apriori Principle)1.

Stop at step k, where Fk = empty.

1. R. Agrawal and R. Srikant. Fast algorithms for mining association rules. In proc. of VLDB, pages 487-499, 1994.

Hybrid DFS-based Approach

DFS is inappropriate:

For each node we would have to scan the database multiple

times to detect the 2-relations among the items in the node.

Though in BFS these relations are already available.

Generate the first two levels of the Tree using BFS.

Then use DFS.

Eliminates multiple database scans, since now the 2-relations are

available.

Experimental SetupReal Datasets

Dataset 1: Utterances of WH-Questions.

Size: 73 e-sequences.

# of labels: 400.

Dataset 2: SignStream Database.

Size: 884 e-sequences.

# of labels: 400.

panagiotis papapetrou department of computer science boston university constraint-based mining of...

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