Spatial Congeries Pattern Mining

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Spatial Congeries Pattern Mining

• Presented by Iris Zhang
• Supervisor Dr. David Cheung
• 24 October 2003

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Outline

• Introduction
• Motivation
• Related work
• Formal definition
• Algorithms
• Experiments
• Conclusion

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Introduction

• KDD
• Discovery of interesting, implicit, and
  previously unknown knowledge from large databases
  FPM91
• Spatial data mining
• Extraction of implicit knowledge, spatial
  relations, or other patterns not explicitly
  stored in spatial databases KH95

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Feature of Spatial Data Mining

• Spatial autocorrelation
• Everything is related to everything else but
  nearby things are more related than distant
  things (Tobler, 1979)
• Spatial heterogeneity
• The variation in spatial data is a function of
  location

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Motivation

• A famous historical example
• In 1909, the residents of Colorado Springs were
  discovered that they had healthy teeth and the
  local drinking water had high level of fluoride.
  Researchers confirmed the positive role of
  fluoride in controlling tooth decay.
• healthy teeth, high level of fluoride

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Motivation (Cont)

• Another case

HSX02
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Related work

• Neighboring Class Sets Mining
• Co-location Pattern Mining

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Neighboring Class Sets

• Access records of mobile services

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Neighboring Class Sets

• Neighboring class sets
• ((timetable,ticket),4),
• ((timetable,weather)3),
• ((ticket,weather),2),
• ((timetable,ticket,weather),2)

Mor01
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Neighboring Class Sets

• Grouping of points

Mor01
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Neighboring Class Sets

• Grouping of points

Mor01
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Neighboring Class Sets

• Grouping of points

Mor01
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Neighboring Class Sets

• Apriori generation of valid instances

Mor01
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Problems

• Undercount the number of instances
• Depend on the order of classes to generate
  instances for k-neighboring class set (kgt2)
• Provide an absolute number to be support threshold

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Co-location Patterns Mining

• Co-location a subset of Boolean features
• E.g. (drought, EL Nino, substantial increase in
  vegetation, extremely high precipitation)

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Co-location Patterns Mining

• Row instance I i1,i2,,ik of a co-location
  Cf1,f2,,fk
• ij is an instance of fj (j 1,2,k)
• ip and iq are neighbors to each other
• (A.1,B.1) is a row instance of co-location A,B
• Table instance T of C is the set of all row
  instances of C
• (A.1,B.1), (A.2,B.4), (A.3,B.4) is table
  instance of A,B

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Co-location Patterns Mining

• Participant ratio for feature fi
• Pr(A,B,A3/475, Pr(A,B,B2/540
• Participant index of a co-location C
• Pi(A,B)min(0.75,0.4)0.4

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Co-location Pattern Mining

• Co-location rule C1?C2(p,cp)
• C1 and C2 are co-locations
• C1 ? C2 ?
• p participant index, cp conditional probability
• A?B(40, 75)
• Conditional probability of a co-location rule

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Co-location Patterns Mining

• Apriori-property
• Participant index is monotonically non-increasing
  as the size of the co-location increasing
• Apriori-like mining algorithm
• Candidate generation
• Instances generation

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Co-location Patterns Mining

• Candidate generation
• Join
• Prune

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Co-location Patterns Mining

• Instance generation
• Geometric approach
• Rtree join
• Combinatorial approach
• Sort-merge join
• Hybrid approach
• Rtree join to get instances for size 2
  co-location
• Sort-merge join to get instances for size k(kgt2)
  co-location

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Co-location Patterns Mining

• Example

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• The participant index measure may overate some
  co-location
• The features are binary

Pr(A,B,A)2/825 Pr(A,B,B)6/6100 Pi(A,B)
min(25,100)25 B?A(25,
100) A?B(25, 25) Probability(A,B)7/(86)
?15
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Spatial Congeries Patterns Mining

• Input
• D D1,D2,,Dn
• Spatial relation to regulate the relation of
  objects in patterns
• min_fre threshold to determine whether an itemset
  is frequent
• Output
• Complete set of Spatial Congeries patterns

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Spatial Congeries Patterns Mining

• Example of datasets

Attribute values can be translated to
categorical values VD10 WDshallow DOP near
NLexistent can be a pattern
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Formal Definition

• Item an attribute value in a dataset. I is the
  set of all items.
• E.g. water depth shallow
• Itemset subset of I
• E.g. VD10 WDshallow DOP near Nexistent
• E.g. VD100 WDdepth DOPfar Nabsent

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Formal Definition

• Spatial relation rule to regulate the spatial
  relation of objects in patterns
• Instances of an item i points which has
  attribute value i
• Instances of an itemset if instances of all
  items in the itemset satisfy the spatial
  relation, the combination of these instances is
  an instance of the itemset.

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Observation

• The number of instances of itemsets is not
  monotonically non-increasing
• E.g. an instance of triangle, circle can
  construct two instances of triangle, circle,
  rectangle
• Conclusion the number of instances of an itemset
  can be used to be the measure to determine
  whether the itemset is a pattern

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Formal Definition

• Frequency of an itemset
• Number of instances of the itemset over all
  possible combinations of instances of items
• E.g. Frequency(A,B)7/(86)?15

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Formal Definition

• Spatial Congeries pattern
• If the frequency of an itemset is no less than
  frequency threshold min_fre, the itemset is a
  Spatial Congeries pattern.

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Property of Frequency

• Lemma the frequency of an itemset is
  monotonically non-increasing with the size of the
  itemset increasing.
• Proof (simplified)
• For size k-1 itemset Ik-1 v1, v2,, vk-1 and
  size k itemset Ik v1, v2,, vk-1,vk

mq is the number of instances of Iq nq is the
number of instances of item vq.
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Algorithm-1

• Step 1 generate complete set of size 2 patterns
  by Rtree-join on complete Rtrees

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Algorithm-1

• Step 1 generate complete set of size 2 patterns
  by Rtree-join on complete Rtrees

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Algorithm-1

• Step 1 generate complete set of size 2 patterns
  by Rtree-join on complete Rtrees

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Algorithm-1

• Step 1 generate complete set of size 2 patterns
  by Rtree-join on complete Rtrees

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Algorithm-1

• Step 2generate size k (kgt2) patterns level by
  level
• Generate size k (kgt2) candidates
• Join two size k-1 patterns
• Prune those candidates which have subsets that
  are not frequent
• Generate size k (kgt2) instances

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Sample
Square a1 Triangle a2 Circle b1 Diamond c1
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Process of Algorithm-1

• RJ to find the instances of size 2 candidates
• Build Rtree for each dataset A, B and C
• Do RJ find the instances of size 2 candidates
• ma1b1 5, ma2b1 3, ma1c1 2, ma2c1 0, mb1c1
  0
• Get size 2 patterns a1b1, a2b1,a1c1 according to
  the frequency threshold 50
• fa1b1 5/(33) ? 56, fa2b1 3/(23) 50,
• fa1c1 2/(31) ? 67, fa2c1 0
• fb1c1 0

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Process of Algorithm-1

• Sort-merge-join to find the instances of size k
  (kgt2) candidates
• Generate size 3 candidates
• Join size 2 pattern a1b1 and a1c1 to form a1b1c1
• Prune a1b1c1 because b1c1 is not a pattern
• Get size 3 patterns ( there is no size 3
  patterns)

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Algorithm-2

• Step 1generate all patterns for a combination of
  subsets. Each subset corresponds to an item. All
  points in the subset have the item as their
  attribute value.
• E.g. The first combination is a1b1c1. It needs
  to build rtrees for subsets of a1, b1, c1 in
  order to generate size 2 patterns. Then it do
  sort-merge join to generate size k(kgt2) patterns.
• Step 2 generate all patterns for another
  combination until there is no combination
• E.g. The second combination is a2b1c1.

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Process of Algorithm-2

• Generate patterns for combination a1b1c1
• RJ on Rtrees for a1, b1 and c1 to get instances
  of candidates a1b1, a1c1, b1c1
• Suppose a1b1 and a1c1 are patterns, size 3
  candidates is a1b1c1
• Sort-merge-join to get instances of a1b1c1
• Generate patterns for combination a2b1c1
• RJ on Rtrees for a2, b1, c1 to get instances of
  candidates a2b1 and a2c1. Because the instances
  of b1c1 has been generated, there is no need to
  do it again
• Suppose a2b1 is pattern
• There is no size 3 candidate

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Experiment

• Environment
• CPU type Pentium III Xeon 700MHz
• RAM 4096M
• Dataset
• Synthetic dataset with Gauss distribution
• No. of clusters 5
• Map size 800
• E.g. (622, 478, 5) is a point in a dataset

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Experiment-1
No. of Datasets 3 No. of Attribute Values
5 Distance threshold 100 Frequency threshold
0.01
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Experiment-1
No. of Datasets 3 No. of Attribute Values
5 Distance threshold 100 Frequency threshold
0.01
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Experiment-1
No. of Datasets 3 No. of Attribute Values
5 Distance threshold 100 Frequency threshold
0.01
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Experiment-2
No. of Points in each datasets 1000 No. of
Attribute Values 5 Distance threshold
100 Frequency threshold 0.01
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Experiment-3
No. of Datasets 5 No. of Points in each
datasets 1000 No. of Attribute Values
5 Distance threshold 100
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Experiment-4
No. of Datasets 3 No. of Points in each
datasets 1000 No. of Attribute Values
5 Frequency threshold 0.01
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Conclusions

• Neighboring class set mining and co-location
  pattern mining problem are introduced
• Spatial Congeries pattern mining is formulated
  and provided with two Apriori-like mining
  algorithms
• Future work
• More pruning methods should be used to reduce the
  time and space requirement
• The experiments should be done on real datasets

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References

• HSX02 Huang Y., Shekhar S., Xiong H.
  Discovering Co-location Patterns from Spatial
  Datasets A General Approach. Submited to IEEE
  TKED (under second round review)
• HXSP03 Huang Y., Xiong H., Shekar S., Pei J.
  Mining Confident Co-location Rules without A
  Support Threshold. Proc. of 18th ACM Symposium on
  Applied Computing (ACM SAC), 2003
• Mor01 Morimoto Y. Mining Frequent Neighboring
  Class Sets in Spatial Databases. Proc. of ACM
  SIGKDD Int. Conf. on Knowledge Discovery and Data
  Mining, 2001.

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QA
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