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Data Mining Techniques for Query Relaxation

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Title: Data Mining Techniques for Query Relaxation


1
Data Mining Techniques for Query Relaxation
2
Query Relaxation via Abstraction
Abstraction is context dependent 69 guard ?
big guard 69 forward ? medium forward 69
center ? small center
  • Abstraction must be automated for
  • Large domains
  • Unfamiliar domains

Heights of guards
A conceptual query Find me a big guard
lt 6
lt 64
gt 64
small
medium
large
3
Related Work
  • Maximum Entropy (ME) method
  • Maximization of entropy (- S p log p)
  • Only considers frequency distribution
  • Conceptual clustering systems
  • Only allows non-numerical values (COBWEB)
  • Assume a certain distribution (CLASSIT)

4
Supervised vs. Unsupervised Learning
  • Supervised Learning
  • Given instances with known class information,
    generate rules/decision tree that can be used to
    infer class of future instances
  • Examples ID3, Statistical Pattern Recognition
  • Unsupervised Learning
  • Given instances with unknown class information,
    generate concept tree that cluster instances into
    similar classes
  • Examples COBWEB, TAH Generation (DISC, PBKI)

5
Automatic Construction of TAHs
  • Necessary for Scaling up CoBase
  • Sources of Knowledge
  • Database Instance
  • Attribute Value Distributions
  • Inter-Attribute Relationships
  • Query and Answer Statistics
  • Domain Expert
  • Approach
  • Generate Initial TAH
  • With Minimal Expert Effort
  • Edit the Hierarchy to Suit
  • Application Context
  • User Profile

6
For Clustering Attribute Instances with
Non-Numerical Values
7
Pattern-Based Knowledge Induction (PKI)
  • Rule-Based
  • Cluster attribute values into TAH based on other
    attributes in the relation
  • Provides Attribute Correlation value

8
Definitions
  • The cardinality of a pattern P, denoted P, is
    the number of distinct objects that match P.
  • The confidence of a rule A ? B, denoted by x(A ?
    B), is
  • x (A ? B) A I B / A
  • Let A ? B be a rule that applies to a relation
    R. The support of the rule over R is defined as
  • h(A ? B) A / R

9
Knowledge Inference A Three-Step Process
  • Step 1 Infer Rules
  • Consider all rules of basic form A ? B.
  • Calculate Confidence and Support.
  • Confidence measures how well a rule applies to
    the database.
  • A ? B has a confidence of .75 means that if A
    holds, B has a 75 chance of holding as well.
  • Support measures how often a rule applies to the
    database.
  • A ? B has a support of 10 means that it applies
    to 10 tuples in the database (A holds for 10
    tuples).

10
Knowledge Inference (contd)
  • Step 2 Combine Rules
  • If two rules share a consequence and have the
    same attribute as a premise (with different
    values), then those values are candidates for
    clustering.
  • Color red ? style sport (x1)
  • Color black ? style sport (x2)
  • Suggests red and black should be clustered.
  • Correlation is product of the confidences of the
    two rules
  • g x1 x x2

11
Clustering
  • Algorithm Binary Cluster (Greedy Algorithm)
  • repeat
  • INDUCE RULES and determine g
  • sort g in descending order
  • for each g(ai, aj)
  • if ai and aj are unclustered
  • replace ai and aj in DB with joint value Ji,j
    until fully clustered
  • Approximate n-ary g using binary g
  • cluster a set of n values if the g between all
    pairs is above threshold
  • Decrease threshold and repeat

12
Knowledge Inference (contd)
  • Step 3 Combine Correlations
  • Clustering Correlation between two values is the
    weighted sum of their correlations.
  • Combines all the evidence that two values should
    be clustered together into a single number (g(a1,
    a2)).
  • g(a1, a2) S i 1 wi x x(A a1 ? Bi bi) x
    x(A a2 ? Bi bi) / (m-1)
  • Where a1, a2 are values of attribute A, and there
    are m attributes B1, …, Bm in the relation with
    corresponding weights w1, …, wm

m
13
Pattern-Based Knowledge Induction (Example)
A B C a1 b1 c1 a1 b2 c1 a2 b1 c1 a3 b2 c1
1st iteration
Rules A a1 ? B b1 confidence 0.5 A a2
? B b1 confidence 1.0 A a1 ? C
c1 confidence 1.0 A a2 ? C c1 confidence
1.0 correlation (a1, a2) 0.5x1.01.0x1.0/ 2
0.75 correlation (a1, a3) 0.75 correlation
(a2, a3) 0.5
14
Pattern-Based Knowledge Induction (contd)
2nd iteration
0.67
A B C a12 b1 c1 a12 b2 c1 a12 b1 c1 a3 b2 c1
0.75
a3
a1
a2
15
Example for Non-Numerical Attribute Value The
PEOPLE Relation
16
TAH for People
17
  • Cor(a12, a3) is computed as follows
  • Attribute origin Same (Holland)
  • contributes 1.0
  • Attribute hair Same
  • contributes 1.0
  • Attribute eye Different
  • contributes 0.0
  • Attribute height Overlap on MEDIUM
  • 5/10 of a12 and 2/2 of a3
  • contributes 5/10 2/2 0.5
  • cor(a12, a3) 1/4 (1100.5) 0.63

18
Correlation Computation
  • Compute correlation between European and Asian.
  • Attributes ORIGIN and HAIR COLOR
  • No overlap between Europe and Asia, no
    contributions to correlation
  • Attribute EYE COLOR
  • BROWN is the only attribute that has overlap
  • 1 out of 24 Europeans have BROWN
  • 12 out of 12 Asians have BROWN
  • Attribute BROWN contributes 1/24 12/12 0.0416
  • Attribute Height
  • SHORT 5/24 Europeans and 8/12 of Asians
  • Medium 11/24 and 3/12
  • Tall 8/24 and 1/12
  • Attribute HEIGHT contributes
  • 5/24 8/12 11/24 3/12 8/12 1/12
    0.2812
  • Total Contribution 0.0416 0.2812 0.3228

19
Extensions
  • Pre-clustering
  • For non-discrete domains
  • Reduces computational complexity
  • Expert Direction
  • Identify complex rules
  • Eliminate unrelated attributes
  • Eliminating Low-Popularity Rules
  • Set Popularity Threshold q
  • Do not keep rules below q
  • Saves Time and Space
  • Loses Knowledge about Uncommon Data
  • In the Transportation Example, q 2 improves
    efficiency by nearly 80.
  • Statistical sampling for very large domains.

20
Clustering of Attribute Instances with Numerical
Values
21
Conventional Clustering Methods I. Maximum
Entropy (ME)
  • Maximization of entropy (- S p log p)
  • Only considers frequency distribution
  • Example 1,1,2,99,99,100 and
  • 1,1,2,3,100,100
  • have the same entropy (2/6,1/6,2/6,1/6)
  • ME cannot distinguish between
  • (1) 1,1,2,99,99,100 good partition
  • (2) 1,1,2,3,100,100 bad partition
  • Me does not consider value distribution.
  • Clusters have no semantic meaning.

22
Conventional Clustering Methods II. Biggest Gap
(BG)
  • Consider only value distribution
  • Find cuts at biggest gaps
  • 1,1,1,10,10,20 is partitioned to
  • 1,1,1,10,10 and 20 ? bad
  • A good partition
  • 1,1,1 and 10,10,20

23
New Notion of Goodness of Clusters Relaxation
Error
24
Relaxation Error of a Cluster
25
Relaxation Error of a Partition
26
Distribution Sensitive Clustering (DISC) Example
27
  • Relaxation Error
  • RE(B) average pair-wise difference
  • 3 2 3 8
  • 9 9 9 9
  • RE(C) 0.5
  • RE(A) 2.08
  • correlation (B) 1 - RE(B) 1 - 0.89 0.57
  • RE(A) 2.08
  • correlation (C) 1- 0.5 0.76
  • 2.08
  • correlation (A) 1- 2.08 0
  • 2.08

28
Examples
  • Example 1 1,1,2,3,100,100
  • ME 1,1,2,3,100,100
  • RE(1,1,2) (010111)/9 0.44
  • RE(3,100,100) 388/9 43.11
  • RE(1,1,2,3,100,100) 0.443/6 43.113/6
    21.78
  • Ours RE(1,1,2,3,100,100) 0.58
  • Example 2 1,1,1,10,10,20
  • BG 1,1,1,10,10,20
  • RE(1,1,1,10,10,20) 3.6
  • Ours RE(1,1,1,10,10,20) 2.22

29
An Example
  • Example
  • The table SHIPS has 153 tuples and the attribute
    LENGTH has 33 distinct values ranging from 273 to
    947. DISC and ME are used to cluster LENGTH into
    three sub-concepts SHORT, MEDIUM, and LONG.

30
An Example (contd)
  • Cuts by DISC
  • between 636,652 and 756,791
  • average gap 25.5
  • Cuts by ME
  • between 540,560 and 681,685 (a bad cut)
  • average gap 12
  • Optimal cuts by exhaustive search
  • between 605,635 and 756,791
  • average gap 32.5
  • DISC is more effective than ME in discovering
    relevant concepts in the data.

31
An Example
Clustering of SHIP.LENGTH by DISC and ME Cuts by
DISC - - - Cuts by ME - . - .
32
Quality of Approximate Answers
33
DISC
  • For numeric domains
  • Uses intra-attribute knowledge
  • Sensitive to both frequency and value
    distributions of data.
  • RE average difference between exact and
    approximate answers in a cluster.
  • Quality of approximate answers are measured by
    relaxation error (RE) the smaller the RE, the
    better the approximate answer.
  • DISC (Distribution Sensitive Clustering)
    generates AAHs based on minimization of RE.

34
DISC
  • Goal automatic generation of TAH for a numerical
    attribute
  • Task given a numerical attribute and a number s,
    find the optimal s-1 cuts that partition the
    attribute into s sub-clusters
  • Need a measure for optimality of clustering.

35
Quality of Partitions
If RE(C) is too big, we could partition C into
smaller clusters. The goodness measure for
partitioning C into m sub-clusters C1, …, Cm is
given by the relaxation error reduction per
cluster (category utility CU)
For efficiency, use binary partitions to obtain
m-ary partitions.
36
The Algorithms DISC and BinaryCut
  • Algorithm DISC(C)
  • if the number of distinct values e C lt T, return
    / T is a threshold /
  • let cut the best cut returned by BinaryCut(C)
  • partition values in C based on cut
  • let the resultant sub-clusters be C1 and C2
  • call DISC(C1) and DISC(C2)
  • Algorithm BinaryCut(C)
  • / input cluster C x1, …, xn /
  • for h 1 to n 1 / evaluate each cut /
  • Let P be the partition with clusters C1 x1,
    …, xh and
  • C2 xh1, …, xn
  • computer category utility CU for P
  • if CU lt MinCU then
  • MinCU CU, cut h / the best cut /
  • Return cut as the best cut

37
The N-ary Partition Algorithm
  • Algorithm N ary Partition(C)
  • let C1 and C2 by the two sub-clusters of C
  • compute CU for the partition C1, C2
  • for N 2 to n 1
  • let Ci by the sub-cluster of C with maximum
    relaxation error
  • call BinaryCut to find the best sub-clusters Ci1
    and Ci2 of Ci
  • compute and store CU for the partition C1, …,
    Ci-1, Ci1, Ci2, Ci1, …, CN
  • if current CU is less than the previous CU
  • stop
  • else
  • replace Ci by Ci1 and Ci2
  • / the result is an N ary partition of C /

38
Using TAHs for Approximate Query Answering
  • select CARGO-ID
  • from CARGOS
  • where SQUARE-FEET 300
  • and WEIGHT 740
  • no answers
  • The query is relaxed according to TAHs.

39
Approximate Query Answering
  • select CARGO-ID
  • from CARGOS
  • where 294 lt SQUARE-FEET lt 300
  • and 737 lt WEIGHT lt 741
  • CARGO-ID SQUARE-FEET WEIGHT

10 296 740
Relaxation error (4/11.950)/2 0.168 Further
Relaxation select CARGO-ID from CARGOS where 294
lt SQUARE-FEET lt 306 and 737 lt WEIGHT lt
749 CARGO-ID SQUARE-FEET WEIGHT
10 296 740 21 301 737
30 304 746 44 306 745
Relaxation error (3.75/11.953.5/9.88)/2 0.334
40
Performance of DISC
  • Theorem Let D and M be the optimal binary cuts
    by DISC and ME respectively. If the data
    distribution is symmetrical with respect to the
    median, then D M (i.e., the cuts determined by
    DISC and ME are the same).
  • For skewed distributions, clusters discovered by
    DISC have less relaxation error than those by the
    ME method.
  • The more skewed the data, the greater the
    performance difference between DISC and ME.

41
Multi-Attribute TAH (MTAH)
In many applications, concepts need to be
characterized by multiple attributes, e.g.,
near-ness of geographical locations.
  • As MTAH
  • As a guidance for query modification
  • As a semantic index

42
Multi-Attribute TAH (MTAH)
43
Multi-Attribute DISC (M-DISC) Algorithm
  • Algorithm M-DISC(C)
  • if the number of objects in C lt T, return / T
    is a threshold /
  • for each attribute a 1 to m
  • for each possible binary cut h
  • compute CU for h
  • if CU gt MaxCU then / remember the best cut /
  • MaxCU CU, BestAttribute a, cut h
  • partition C based on cut of the attribute
    BestAttribute
  • let the resultant sub-clusters be C1 and C2
  • call M-DISC(C1) and M-DISC(C2)

44
Greedy M-DISC Algorithm gM-DISC
  • Algorithm gM-DISC(C)
  • if the number of objects in C lt T, return / T
    is a threshold /
  • for each attribute a 1 to m
  • for each possible binary cut h
  • compute REa for h
  • if REa gt Max RE then / remember the
    best cut /
  • Max RE REa, BestAttribute a, cut
    h
  • partition C based on cut of the attribute
    BestAttribute
  • let the resultant sub-clusters be C1 and C2
  • call gM-DISC(C1) and gM-DISC(C2)

45
MTAH of RECTANGLES (Height, Width)
46
The Database Table AIRCRAFT
How to find similar aircrafts?
47
MTAH for AIRCRAFT
48
Example for Numerical Attribute Value
Motor Data from PartNet (http//PartNet)
49
TAH for Motor Capability
50
TAH for Motor Size and Weight
51
TAHs for Motor
  • The Motor table was adapted from Housed Torque
    from Part Net. After inputting the data, two
    TAHs were generated automatically from the DISC
    algorithm.
  • One TAH was based on peak torque, peak torque
    power, and motor constant. The other was based
    on outer diameter, length, and weight. The leaf
    nodes represent part number. THE intermediate
    nodes are classes. The relaxation error (average
    pair-wise distance between the parts) of each
    node are also given.

52
Application of TAHs
  • The TAHs can be used jointly to satisfy
    attributes in both TAHs. For example, find part
    similar to T-0716 in terms of peak torque, peak
    torque power, motor constant, outer diameter,
    length, and weight. By examining both TAHs, we
    know that QT-0701 is similar to T-0716 with an
    expected relaxation error of (0.06 0.1)/2 0.08

53
Performance of TAH
  • Performance measures
  • accuracy
  • efficiency
  • where all relevant answers are the best n
    answers determined by exhaustive search.
  • Compare an MTAH with a traditional 2-d index tree
    (based on frequency distribution).

retrieved relevant answers
all relevant answers
retrieved relevant answers
all retrieved answers
54
Performance of MTAHs
  • Based on attributes Longitudes and Latitudes of
    972 geographical locations from a transportation
    database.
  • 500 queries with the form
  • find the n locations nearest to (long,lat)
  • where n is randomly selected from 1 to 20, and
    long and lat are generated based on the
    distributions of the geographical locations.

MTAH is more accurate than 2-d-tree. MTAH is more
efficient than Exhaustive Search.
55
Generation of Evolutionary TAH
  • Approximate query answering for temporal data
    (given as a set of time sequences)
  • Find time sequences that are similar to a given
    template sequence.
  • A time sequence S of n stages is defined as an
    n-tuple S s1, …, sn where si is a numerical
    value.
  • Issues
  • Needs a similarity measure for sequences
  • Use clustering for efficient retrieval
  • Evaluation of work

56
Automatic Constructions of TAHs
  • Necessary for scaling up CoBase
  • Sources of Knowledge
  • Database Instance
  • Attribute Value Distributions
  • Inter-Attribute Relationships
  • Query and Answer Statistics
  • Domain Exert
  • Approach
  • Generate Initial TAH
  • With Minimal Expert Effort
  • Edit the Hierarchy to Suit
  • Application Context
  • User Profile

57
The CoBase Knowledge-Base Editor
  • Tool for Type Abstraction Hierarchies
  • Display available TAHs
  • Visualize TAHs as graphs
  • Edit TAHs
  • Add/Delete/Move nodes and sub-trees
  • Assign names to nodes
  • Interface to Knowledge Discovery Tools
  • Cooperative Operators
  • Specify parameter values
  • Approximate
  • Near-To, Similar-To

58
An Example of Using the KB Editor
59
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