A join-less approach for co-location pattern mining A summar

A Join-less Approach for Co-location Pattern Mining:A Summary of Results

Jin Soung Yoo,Shashi Shekhar,Mete Celik

Computer Science Department,University of Minnesota,Minneapolis,MN,USA

jyoo,shekhar,mcelik@a335bf124431b90d6c85c737

Abstract

Spatial co-location patterns represent the subsets of fea-tures whose instances are frequently located together in geographic space.Co-location pattern discovery presents challenges since the instances of spatial features are em-bedded in a continuous space and share a variety of spatial relationships.A large fraction of the computation time is devoted to identifying the instances of co-location patterns. We propose a novel join-less approach for co-location pat-tern mining,which materializes spatial neighbor relation-ships with no loss of co-location instances and reduces the computational cost of identifying the instances.The join-less co-location mining algorithm is ef?cient since it uses an instance-lookup scheme instead of an expensive spatial or instance join operation for identifying co-location in-stances.We prove the join-less algorithm is correct and complete in?nding co-location rules.The experimental evaluations using synthetic datasets and real world datasets show the join-less algorithm performs more ef?ciently than a current join-based algorithm and is scalable in dense spa-tial datasets.

1Introduction

The explosive growth of spatial data and widespread use of spatial databases emphasize the need for the auto-mated discovery of spatial knowledge.Spatial data min-ing[7,8]is the process of discovering interesting and pre-viously unknown,but potentially useful patterns from spa-tial databases.Extracting interesting patterns from spatial datasets is more dif?cult than extracting the corresponding patterns from traditional numeric and categorical data due to the complexity of spatial data types,spatial relationships and spatial autocorrelation[10].

A spatial co-location pattern represents a subset of spa-

This work was partially supported by NSF grant0431141and Oak Ridge National Laboratory grant.The content of this work does not nec-essarily re?ect the position or policy of the government and no of?cial endorsement should be inferred.tial features whose instances are frequently located in a spa-tial neighborhood.For example,ecologists have found that Nile Crocodiles and Egyptian Plover birds are frequently co-located.The co-location rule,i.e.,Nile Crocodile Egyptian Plover,predicts the presence of Egyptian Plover birds in areas with Nile Crocodiles.Spatial co-location pat-terns may yield important insights for many applications. For example,a mobile service provider may be interested in mobile service patterns frequently requested by geographi-cally neighboring users.The frequent neighboring request sets can be used for providing attractive location-sensitive advertisements,etc.Other application domains include Earth science,public health,biology,transportation,etc.

Co-location rule discovery presents challenges due to the following reasons:First,it is dif?cult to?nd co-location in-stances since the instances of spatial features are embedded in a continuous space and share neighbor relationships.A large fraction of the computation time is devoted to iden-tifying the co-location instances.Second,it is non-trivial to reuse association rule mining algorithms[3,5]for co-location pattern mining since there are no pre-de?ned trans-actions in many spatial datasets.Thus,a current co-location mining algorithm[9]uses a join-based approach to?nd co-location instances.Its computational performance suffers, however,due to the large number of joins required as the number of features and their instances increases.

In this paper,we propose a method to materialize the neighbor relationships of a spatial dataset with no duplica-tion of the neighbor relationships and no loss of co-location instances,and present a novel join-less approach for co-location pattern mining.The join-less co-location min-ing algorithm reduces the computational cost of identify-ing the instances of co-location patterns using an instance-lookup scheme,and also has a coarse pruning step which can?lter candidate co-locations without?nding exact co-location instances.We analytically prove our join-less algo-rithm is correct and complete,i.e.,there are no false drop-pings or false admissions in?nding co-location rules.The experimental evaluations using synthetic datasets and real world datasets show the join-less co-location mining algo-rithm outperforms the join-based algorithm and is scalable

in dense spatial datasets.

The remainder of the paper is organized as follows.Sec-tion2gives an overview of the basic concepts of co-location pattern mining and the problem de?nition,and discusses related works.In Section3,we presents our join-less ap-proach for co-location pattern mining.In Section4,the analytical analysis of the join-less co-location mining al-gorithm is given.Section5presents the experimental eval-uation.The conclusion and future work are discussed in Section6.

2Co-location Pattern Mining

In this section,we describe the basic concepts of co-location pattern mining and the problem de?nition,and dis-cuss the related works.

2.1Basic Concepts

Given a set of spatial features,a set of their instances ,and a neighbor relationship over,a co-location is a subset of spatial features whose instances form a clique using a neighbor relationship.A co-location rule is of the form:,where

,is the prevalence measure,and is the conditional probability.For example,when a spatial neighbor relationship is a Euclidean distance metric and its threshold value,two spatial objects are neighbors if they satisfy the neighbor relationship,e.g.,(A.1,B.1) ((A.1,B.1)).Figure1(a)shows an example dataset with three spatial features,A,B and C.Each object is represented by its feature type and the unique instance id of each feature type,e.g.,A.1.Identi?ed neighbor objects are connected by solid lines.The instance of a co-location is a set of objects which includes an object of each feature type in the co-location and forms a clique relationship among them.For example,in Figure1(a), A.2,B.4,C.2is an instance of co-location A,B,C since(A.2)=A,

(B.4)=B and(C.2)=C,and(A.2,B.4), (A.2,C.2)and(B.4,C.2).

The interest of a co-location pattern can be measured by its prevalence and conditional probability[9].The conditional probability of a co-location rule is the fraction of instances of in the neighborhood of instances of,i.e.,

. The participation index is used as a co-location preva-lence measure.First,the participation ratio

of feature in a co-location is the fraction of objects of features in the neighborhood of instances of co-location,i.e.,

.The partic-ipation index of a co-location is de?ned as.A high participation index value indicates that the spatial features in a co-location pattern likely show up together.For example,in the dataset of Figure1(a),feature A has four instances,feature B has?ve instances,and feature C has three instances.Consider the prevalence values of co-location=A,B,C.The instances of co-location are A.2,B.4,C.2and A.3,B.3,C.1as shown in Figure1 (c).The participation ratio of feature A in the co-location ,(,A)is since only A.2and A.3among four feature

A objects are involved in the co-location instances.(,

B)is and(,C)is.Thus the participation index of co-location,,is(,A),(,B),(,C) =.

Lemma1The participation ratio and the participation in-dex are monotonically non increasing with increases in the size of the co-location.

For example,the participation index value of a size3 co-location is not greater than the participation index value of any size2co-location,e.g.,=

=in Figure1(c).Please refer to[9]for the proof of Lemma1.

2.2Problem De?nition

The formal problem de?nition for the co-location pat-tern mining is as follows.We focus on?nding a correct and complete set of co-location rules with reducing the compu-tation cost.

Given:

1)A set of spatial features and a set of their instances where

is a set of instances of feature and each instance

is a vector feature type,instance id,location,where location a spatial framework

2)A neighbor relationship over locations

3)A minimum prevalence threshold()and a min-imum conditional probability threshold() Find:

A set of co-location rules with participation index

and conditional probability. Objective:

1)Find a correct and complete set of co-location rules.

2)Reduce the computation cost.

Constraints:

1)is a distance metric based neighbor relationship and has a symmetric property.

2)The spatial dataset is a point dataset.

2.3Related Work

The problem of mining association rules based on spatial relationships was?rst discussed in[5].The work discovers

.

.

(a)

.

.

(b)

.

(c)(d)

Figure1.Different approaches for?nding co-location instances(a)Example dataset(b)Space parti-tion(c)Instance join(d)Clique partition and partial join

the subsets of spatial features frequently associated with a

speci?c feature,e.g.,cancer.Directly applying this method

to a co-location problem may not capture our co-location

meaning with no speci?c reference feature.

Previous works on spatial co-location mining have pre-

sented different approaches for identifying co-location in-

stances.[6]uses space partitioning for identifying neigh-

boring objects for a frequent neighboring feature set.Fig-

ure1(b)shows the space partition method to?nd the neigh-

boring objects of a subset of features,A,C.First,it

decides the partition center points with base objects,e.g.,

feature A objects,A.1,A.2,A.3and A.4,and decomposes

the space from the partitioning points using a geometric ap-

proach,i.e.,V oronoi diagram,and then?nds feature C ob-

jects within a distance threshold from the partitioning point

in each partition area.In this example,the identi?ed neigh-

boring objects of A,C are A.3,C.1and A.2,C.2.

However,note that A.1,C.1and A.4,C.1are also

neighboring objects of A,C but they are not found by

the disjoint space partitions.Thus the distinct space parti-

tioning approach may miss co-location instances across par-

tition areas and generate incorrect results.

[9]proposes an instance join-based co-location mining

algorithm similar to[3].First,after?nding

all neighbor pair objects(size2co-location instances)us-

ing a geometric method,the method?nds the instances of

size()co-locations by joining the instances of its size

subset co-locations where the?rst objects are

common.Figure1(c)shows the procedure to generate the

instances of co-location A,B,C.The instances of co-

location A,B and the instances of co-location A,C

are joined with the?rst objects,and then the neighbor rela-

tionships between the second objects are checked.This ap-

proach?nds correct and complete co-location instance sets.

However,the join-based approach is computationally ex-

pensive with the increase of co-location patterns and their

instances.[11]proposes a partial join approach.It transac-

tionizes a continuous spatial data into a set of disjoint clique

neighborhoods while keeping track of the spatial neighbor

relations not modeled by the transactionization as shown

in Figure1(d).This approach reduces the number of ex-

pensive join operations dramatically in?nding co-location

instances.However,the performance depends on the dis-

tribution of the spatial dataset,exactly the number of cut

neighbor relations.

3A Join-less Approach for Co-location Pat-

tern Mining

In this section,we discuss a join-less approach for min-

ing co-location patterns.First,we describe our method to

materialize spatial neighbor relationships,and then present

the join-less co-location algorithm.

3.1Neighborhood Materialization

The ideal neighborhood materialization for co-location

mining is to?nd all maximal clique relationships from an

input dataset.However,it is computationally expensive.We

propose to materialize disjoint star neighbor relationships as

a framework for ef?cient co-location mining.

De?nition1Given a spatial object whose feature

type is,the star neighborhood of is de?ned as a

set of spatial objects

,where is the feature type of and is

a neighbor relationship.

We de?ne the star neighborhood of an object is a set of

the center object and objects in its neighborhood whose fea-

Figure2.Neighborhood materialization

ture types are greater than the feature type of the center ob-ject in a lexical order.Figure2illustrates the method to materialize neighbor relationships of a spatial dataset.The neighborhood areas of objects A.1,A.3,and B.4are rep-resented by dotted circles whose radii are a user speci?c neighbor distance.The black solid lines in each circle rep-resent a star neighbor relationship with the center object.

A.1has two neighboring objects,

B.1and

C.1.The star neighborhood of A.1is A.1,B.1,C.1including the cen-ter object A.1.In the case of A.3,three neighboring objects are present,A.4,B.3and C.1.However,A.4is not included in the star neighborhood set of A.3since we focus on co-location relationships among different feature types.Next consider the neighborhood of B.4. B.4has two neighbor objects,A.2and C.2.However,A.2is not included in the star neighborhood set of B.4since the neighbor relationship between A.2and B.4is already re?ected in the star neigh-borhood set of A.2.A set of all star neighborhoods of the spatial dataset is listed in Figure2.

De?nition2Let be a set of spatial objects whose feature types are different.If all objects in are neighbors to the?rst object,is called a star instance of co-location=.

In Figure2,a subset of the A.1star neighborhood in-cluding A.1, A.1,B.1,C.1is a star instance of A,B, C.

3.2Join-less Co-location Mining Algorithm

The join-less co-location mining algorithm has three phases.The?rst phase converts an input spatial dataset into a set of disjoint star neighborhoods.The second phase gathers the star instances of candidate co-locations from the star neighborhood set,and coarsely?lters candidate co-locations by the prevalence value of the star instances. The third phase?lters co-location instances from the star instances,and?nds prevalent co-locations and generates co-location rules.Figure3illustrates a join-less algorithm trace.Algorithm1shows the pseudo code.

Figure3.Join-less algorithm trace Convert a spatial dataset to a set of disjoint star neigh-

borhoods(Step1):Given an input dataset and a neigh-

bor relationship,?rst?nd all neighboring object pairs using

a geometric method such as plane sweep[4],or a spatial

query method using quaternary tree or R-tree[8].The star

neighborhoods are generated by grouping the neighboring

objects per each object.Figure3shows the star neighbor-

hoods sorted by the feature type of the center objects.

Generate candidate co-locations(Step4):First,we ini-

tialize all features to size1prevalent co-locations by the

de?nition of the participation index measure.The number

of instances per each feature can be known during the scan

of the input spatial dataset for materializing the neighbor re-

lationships.Size()candidate co-locations are gener-

ated from prevalent size co-locations.Here,we have a

feature level?ltering of co-locations.If any subset of a can-

didate co-location is not prevalent,the candidate co-location

is pruned.

Filter the star instances of candidate co-locations from

the star neighborhood set(Step6):The star instances of

a candidate co-location are gathered from the star neigh-

borhoods whose center object feature type is the same as

the?rst feature of the candidate co-location.For example,

the instances of a candidate co-location B,C are gathered

from the feature B star neighborhoods,and the instances of A,B,C are gathered from the feature A star neighbor-hoods.Notice that the number of candidate co-locations

examined in each star neighborhood is much smaller than

the number of actual candidate co-locations.

Select coarse prevalent co-locations using their star in-

stances(Step9):The size2star instances are clique in-

stances since our neighbor relationship is symmetric(step

8).Thus,we go to step12to?nd prevalent co-locations.

For size3or more,we need to check if the star instance is

Algorithm1Join-less co-location mining algorithm

Inputs

:a set of spatial feature types

:a spatial dataset,:a neighbor relationship ,

Output

A set of all prevalent co-location rules with participation index and conditional probability

Variables

=:a set of star neighborhoods

:a set of size candidate co-locations

:star instances of size candi co-locations

:clique instances of size candi co-locations :a set of size prevalent co-locations

:a set of size co-location rules

Method

1)=gen star neighborhoods(,,);

2)=;=2;

3)while(not empty)do

4)=gen candidate co-locations();

5)for do

6)=filter star instances();

7)end do

8)if=2then

9)else do=select coarse prev co-location

()

10)=filter clique instances();

11)end do

12)=select prev co-location();

13)=gen co-location rules();

14)=+1;

15)end do

16)return;

a clique instance.Before this procedure,we have a coarse ?ltering step of co-locations.We?lter the candidate co-locations using the participation index from their star in-stances.For example,in Figure3,the participation index of candidate co-location A,B,C from the star instances is. If it is less than a user speci?ed minimum prevalent thresh-old,the candidate co-location A,B,C is pruned without examining exact co-location instances.

Filter co-location instances(Step10):From the star in-stances of a candidate co-location,we?lter its co-location instances by looking up all the instances of the co-location of features except the?rst feature of the candidate co-location.For example,to check the cliqueness of a star instance A.1,B.1,C.1of co-location A,B,C,we ex-amine if a subinstance B.1,C.1except A.1is in the set of clique instances of co-location B,C.This instance look up operation can be performed ef?ciently by an instance key which is composed of the ids of objects in the instance. As shown in Figure3, A.1,B.1,C.1is not a co-location instance,but A.2,B.4,C.2and A.3,B.3,C.1are co-location instances.

Select prevalent co-location patterns(Step12):The re?nement?ltering of co-locations is done by the partic-ipation index values calculated from their co-location in-stances.Prevalent co-locations satisfying the minimum prevalence threshold are selected.

Generate co-location rules(Step13):All co-location rules satisfying a given minimum conditional probability are generated from a set of prevalent co-locations.Steps3-15are repeated as the size of co-location patterns increases. 4Analytical Analysis

We analyze our join-less co-location mining algorithm for completeness and a335bf124431b90d6c85c737pleteness means the join-less algorithm?nds all co-location rules whose partici-pation index and conditional probability satisfy a user spec-i?ed minimum prevalence threshold and conditional prob-ability threshold.Correctness means that all co-location rules generated by the join-less algorithm have a partici-pation index and a conditional probability above a user-speci?ed minimum prevalence threshold and conditional probability.First we provide related lemmas.

Lemma2The star partition model does not miss any neighbor relationship of an input spatial data.

Proof The disjoint star partition model includes all neigh-bor relations of each object and excludes only duplicate neighbor relations which are already included in a star neighborhood by De?nition1.

Lemma3Let be a size co-location and be a set of star instances of.The participation index of from is not less than the true participation index of.

Proof The participation ratio of from is the maxi-mum possible probability that the objects of feature of have clique relationships with the objects of the other features in since only objects of feature in the star instances can be included in a clique co-location instance of.The participation ratio of() from is also the maximum possible probability that the objects of feature have clique relationships with the ob-jects of features in since our neighbor relationship is symmetric.Thus the participation index of calculated from the star instances is not less than the true partici-pation index of,

.

Lemma4Let an instance be a star in-stance of a co-location.If the subin-stance except is a clique,the instance is a clique.

Proof In a star instance,the?rst object has neighbor relationships to the other objects,

by De?nition2.Object()has a neighbor rela-tionship to since the neighbor relationship is symmetric and also has neighbor relationships to all the other objects where and since is a clique.Thus each object()has neighbor rela-tionships to all other objects in.is a clique.

0 50 100 150 200 250 300

350

400 10

15

20 25 30 35 40 45 50E x e c u t i o n t i m e (s e c )

Number of total points(K)

join-less(dense)join-based(dense)join-less(sparse)join-based(sparse)

(a) 0

50 100 150 200

250 20 25 30

35 40 45 50 55 60E x e c u t i o n t i m e (s e c )

Number of features

join-less(dense)join-based(dense)join-less(sparse)join-based(sparse)

(b) 0 100 200 300 400 500 10

20 30

40 50 60 70 80 90 100

E x e c u t i o n t i m e (s e c )

Neighbor distance

join-less join-based

(c)

Figure 4.Scalibity of the join-less algorithm:(a)by number of points,(b)by number of features,(c)by neighbor distance

Theorem 1The join-less co-location mining algorithm is complete.

Proof The completeness of the join-less algorithm can be shown by the following two parts.The ?rst is that the method to materialize the neighbor relationships of an input spatial data (step 1),the method to gather star instances(step 6),and the method to ?lter clique instances(step 10)are cor-rect.The star partition model does not miss any neighbor relationship of a input spatial dataset by Lemma 2.The star instances of co-locations gathered from the star neigh-borhoods whose center object feature type is the same as the ?rst feature of the co-location,have correct star neigh-bor relationships.Any potential co-location instance is not missed since the star instances are a super set of the clique instances.The method to ?lter co-location instances from the star instances does not drop a true clique instance by Lemma 4.Next,we show that the ?ltering steps of co-locations do not drop true co-locations.The feature level ?l-tering by prevalent subsets(step 4)is complete by Lemma 1.The coarse ?ltering of co-locations(step 10)does not elim-inate any true prevalent co-locations by Lemma 3.The re-?nement ?ltering(step 12)prunes only co-locations whose true participation index is less than the threshold.Step 13ensures that no co-location rules satisfying a user speci?c conditional probability are missed.

Theorem 2The join-less co-location mining algorithm is correct.

Proof The correctness of the join-less algorithm can be guaranteed by steps 12and 13.Step 12selects only co-locations whose participation indexes satisfy a user speci?c prevalence threshold.The generated co-location rules by step 13also satisfy a user speci?c conditional probability.

5Experimental Evaluation

We evaluate the join-less co-location algorithm with the join-based co-location algorithm [9]using synthetic and real datasets.Synthetic datasets were generated using a spatial data generator similar to the data generator used in [9].The number of features is 20,the average size of co-locations is 5,the neighbor distance is 10and the preva-lence threshold is 0.3.All the experiments were performed on a Sun SunBlade 1500with 1.0GB main memory and 177MHz CPU.

5.1Evaluation with Synthetic Datasets

We examined the scalability of the join-less algorithm in the number of point objects,the number of feature types and distance neighbor threshold.

1)Effect of the number of point objects:First,we com-pared the effect of the number of points.We used two dif-ferent spatial frames,and .In the ?rst frame,even if the number of points is increased from 10K to 50K,the two algorithms showed similar exe-cution time since the datasets are still sparse.In the second frame,with the increase of number of points,the join-based algorithm execution time is dramatically increased due to the increase of data density.As shown in Figure 4(a),the join-less algorithm shows scalibility to large dense datasets.2)Effect of the number of features:In the second exper-iment,we compared the performance of the algorithms as a function of the number of features.We also used two differ-ent dense datasets of 15K points.Figure 4(b)shows the re-sults.In the sparse dataset,the algorithms show similar ex-ecution time even if the number of features increases.In the dense dataset,overall execution time is decreased with the increase of features.The reason is that under the same num-

0 50 100 150 200 250 300 350

400 0.1

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5E x e c u t i o n t i m e (s e c )

Prevalence threshold join-less join-based

(a) 0 50 100 150 200 250 300 350 100

120 140 160 180 200 220 240

E x e c u t i o n t i m e (s e c )

Neighbor distance(m)

join-less join-based

(b)

Figure 5.Real datasets:(a)A climate dataset,(b)A chimpanzee behavior dataset

ber of points,the increase of features causes the number of points per each feature to be decreased,which in turn may lead to a decrease in the number of co-location instances.Overall the join-less algorithm shows better performance.3)Effect of neighbor distance:The third experiment examined the effect of different neighbor distances 10,20,40,80and 100.As shown in Figure 4(c),the join-less al-gorithm shows less increase in the execution time with the increase of distance threshold.The join-based algorithm shows a rapid increase since the neighbor distance increase makes the neighborhood areas larger and increases the num-ber of co-location instances.

5.2Evaluation with Real Datasets

We used two different types of real world datasets.One was an Earth dataset relating climate to vegetation growth from [1].Another dataset was an Ecology animal behavior dataset.It contained female chimpanzee behavior observa-tion data from 1999to 2001from [2].

1)Earth climate data:The earth climate dataset in-cludes monthly measurements of variables such as global plant growth, e.g.,Net Primary Production(NPP),and climate variables, e.g.,precipitation(PREC)on latitude-longitude spherical grids.For example,(NPP-Hi,PREC-Low)is one of the co-location patterns discovered,where Hi(Low)denotes an unusually high(low)value of the mea-surements.The total number of event features was 18.The total number of feature instances was 15,515.We used 4as a neighborhood distance which means 4cells (each grid cell is 1degree 1degree).Figure 5(a)presents the execution time of the three algorithms as a function of the prevalence threshold.The join-less method shows much better perfor-mance at the lower threshold values.The performance dif-ference between the partial join method and the join-based

method is relatively small because the cut relation ratio was almost 0.8.

2)Ecology animal behavior data:The animal behavior dataset has 24chimpanzee features.We assigned a unique instance id to different location points per chimpanzee id.The total number of point instances was 698.Figure 5(b)presents the execution time of the algorithms by differ-ent neighbor distances.The execution time of the join-less method increases more slowly than the other methods with the increase of distance and shows better performance.

6Conclusion and Future Work

In this paper,we propose a join-less co-location mining algorithm with a neighborhood materialization.The algo-rithm is ef?cient since it does not require expensive spa-tial joins or instance joins for identifying co-location in-stances.The experimental evaluation shows the join-less method outperforms the join-based method and is scalable to dense datasets.As future work,we plan to explore meth-ods to answer temporal questions such as how a co-location changes over time as well as methods to identify objects showing similar moving patterns.

References

[1]Discovery of patterns in the global climate system using data

mining.a335bf124431b90d6c85c737/nasa-umn/.

[2]Nsf-sei project:Spatio-temporal data analysis for ecology

behavior.a335bf124431b90d6c85c737/research/chimps/.

[3]R.Agarwal and R.Srikant.Fast algorithms for Mining as-sociation rules.In Proc.of Int’l Conference on Very Large Databases(VLDB),1994.

[4]M.Berg,M.Kreveld,O.M,and a335bf124431b90d6c85c737pu-tational Geometry .Springer,2000.

[5]K.Koperski and J.Han.Discovery of Spatial Association

Rules in Geographic Information Databases.In Proc.of Int’l Symposium on Large Spatial Data bases,Maine.47-66,1995.

[6]Y.Morimoto.Mining Frequent Neighboring Class Sets in

Spatial Databases.In Proc.ACM SIGKDD Int’l Conference on Knowledge Discovery and Data Mining,2001.

[7]J.Roddick and M.Spiliopoulou.A Bibliography of Tempo-

ral,Spatial and Spatio-Temporal Data Mining Research.In Proc.SIGKDD Explorations1(1):34-38,1999.

[8]S.Shekhar and S.Chawla.Spatial Databases:A Tour.Pren-

tice Hall,2003.

[9]S.Shekhar and Y.Huang.Co-location Rules Mining:A

Summary of Results.In Proc.of Int’l Symposium on Spatio and Temporal Database(SSTD),2001.

[10]S.Shekhar,P.Zhang,Y.Huang,and R.Vatsavai.Trends

in Spatial Data Mining,as a book chapter in Data Min-ing:Next Generation Challenges and Future Directions.H.

Kargupta,A.Joshi,K.Sivakumar and Y.Yesha(editors), AAAI/MIT Press,2004.

[11]J.Yoo and S.Shekhar.A Partial Join Approach for Mining

Co-location Patterns.In Proc.of ACM Int’l Symposium on Advances in Geographic Information Systems(ACM-GIS), 2004.

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