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Spatial Congeries Pattern Mining
Presented by: Iris Zhang
Supervisor: Dr. David Cheung
24 October 2003
Outline Introduction Motivation Related work Formal definition Algorithms Experiments Conclusion
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]
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
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}
Motivation (Cont’) Another case
[HSX02]
Related work Neighboring Class Sets Mining Co-location Pattern Mining
Neighboring Class Sets Access records of mobile services
ID Position Services …
xxx (14975,27020) Weather …
xxx (16723,24301) Timetable …
xxx (15521,26441) Ticket …
xxx … … …
xxx (14737,26752) Timetable …
Neighboring Class Sets Neighboring class sets
((timetable,ticket),4),
((timetable,weather)3),
((ticket,weather),2),
((timetable,ticket,weather),2)
[Mor01]
Neighboring Class Sets Grouping of points
[Mor01]
Neighboring Class Sets Grouping of points
[Mor01]
Neighboring Class Sets Grouping of points
[Mor01]
Neighboring Class Sets Apriori generation of valid instances
[Mor01]
Problems Undercount the number of instances Depend on the order of classes to generate
instances for k-neighboring class set (k>2) Provide an absolute number to be support
threshold
Co-location Patterns Mining Co-location: a subset of Boolean features
E.g.: (drought, EL Nino, substantial increase in vegetation, extremely high precipitation)
Co-location Patterns Mining Row instance I ={i1,i2,…,ik} of a co-
location C={f1,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}
Co-location Patterns Mining Participant ratio for feature fi:
Pr({A,B},A}=3/4=75%, Pr({A,B},B}=2/5=40%
Participant index of a co-location C:
Pi({A,B})=min(0.75,0.4)=0.4
Co-location Pattern Mining Co-location rule: C1C2(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:
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
Co-location Patterns Mining Candidate generation
Join
Prune
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(k>2)
co-location
Co-location Patterns Mining Example
Problems The participant index measure may overate
some co-location
The features are binary
Pr({A,B},A)=2/8=25%
Pr({A,B},B)=6/6=100%
Pi({A,B})=min(25%,100%)=25%
{B}{A}(25%, 100%)
{A}{B}(25%, 25%)
Probability({A,B})=7/(8*6)15%
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
Spatial Congeries Patterns Mining Example of datasets
*Attribute values can be translated to categorical values
** {VD:10 WD:shallow DOP: near NL:existent} can be a pattern
ID Attribute Type Description
D1 Vegetation durability Ordinal Ordinate scale from 10 to 100
D2 Water depth Numeric In centimeters
D3 Distance to open water Numeric In meters
D4 Nest location Binary Existence or absence of bird nest
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.: VD:10 WD:shallow DOP: near
N:existent E.g.: VD:100 WD:depth DOP:far N:absent
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.
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
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/(8*6)15%
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.
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, v
2,…, v
k-1} and size k itemset Ik =
{v1, v
2,…, v
k-1,v
k}
*mq is the number of instances of Iq
**nq is the number of instances of item vq.
121
1
121
1
21 .........
k
k
kk
kk
k
kk nnn
m
nnnn
nm
nnn
mf
121
11 ...
k
kk nnn
mf
kk ff 1
Algorithm-1 Step 1: generate complete set of size 2
patterns by Rtree-join on complete Rtrees
Algorithm-1 Step 1: generate complete set of size 2
patterns by Rtree-join on complete Rtrees
Algorithm-1 Step 1: generate complete set of size 2
patterns by Rtree-join on complete Rtrees
Algorithm-1 Step 1: generate complete set of size 2
patterns by Rtree-join on complete Rtrees
Algorithm-1 Step 2:generate size k (k>2) patterns level
by level Generate size k (k>2) candidates
Join two size k-1 patterns Prune those candidates which have subsets that are
not frequent
Generate size k (k>2) instances
Sample
Square: a1Triangle: a2Circle: b1Diamond: c1
a2Y5X5
a1Y4X4
a1Y3X3
a2Y2X2a1Y1X1
b1Y8X8
b1Y7X7
b1Y6X6
c1Y9X9
Datasets A
Datasets B
Datasets C
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/(3*3) 56%, fa2b1 = 3/(2*3) = 50%,
fa1c1 = 2/(3*1) 67%, fa2c1 = 0
fb1c1 = 0
Process of Algorithm-1 Sort-merge-join to find the instances of
size k (k>2) 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)
Algorithm-2 Step 1:generate 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(k>2) patterns.
Step 2: generate all patterns for another combination until there is no combination E.g.: The second combination is a2b1c1.
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
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
Experiment-1
*No. of Datasets: 3*No. of Attribute Values: 5*Distance threshold : 100*Frequency threshold: 0.01
Effects of No. of Points
0
500
1000
1500
2000
2500
1000 2000 3000 4000 5000 6000 7000 8000 9000
No. of Points
CP
U T
ime
(s)
cpu time-C
cpu time-P
Experiment-1
*No. of Datasets: 3*No. of Attribute Values: 5*Distance threshold : 100*Frequency threshold: 0.01
Effects of No. of Points
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
1000 2000 3000 4000 5000 6000 7000 8000 9000
No. of Points
IO-T
ime(
s)
IO time-C
IO time-P
Experiment-1
*No. of Datasets: 3*No. of Attribute Values: 5*Distance threshold : 100*Frequency threshold: 0.01
Effects of No. of Points
0
500
1000
1500
2000
2500
1000 2000 3000 4000 5000 6000 7000 8000 9000
No. of Points
Tim
e(s)
total-C
total-P
Experiment-2
*No. of Points in each datasets: 1000*No. of Attribute Values: 5*Distance threshold : 100*Frequency threshold: 0.01
Effects of No. of Datasets
0
200
400
600
800
1000
1200
1400
1600
1800
3 4 5 6 7 8 9
No. of Datasets
To
tal T
ime(
Clo
cks)
total-C
total-P
Experiment-3
*No. of Datasets: 5*No. of Points in each datasets: 1000*No. of Attribute Values: 5*Distance threshold: 100
Effects of Frequency Threshold
0
100
200
300
400
500
600
700
800
900
1000
0.05 0.03 0.005 0.004 0.003 0.002 0.001 0.0005
Frequency Threshold
Tim
e(s)
total-C
total-P
Experiment-4
*No. of Datasets: 3*No. of Points in each datasets: 1000*No. of Attribute Values: 5*Frequency threshold: 0.01
Effects of Distance Threshold
0
50
100
150
200
250
300
50 100 150 200 250 300
Distance Threshold
Tim
e(s
)
total-C
total-P
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
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.
Q&A