A Join-less Approach for Mining Spatial Co-location Patterns

Transcription

1 IEEE TRNSTIONS ON KNOWLEDGE ND DT ENGINEERING DRFT 1 Join-less pproach for Mining Spatial o-location Patterns Jin Soung Yoo, Student Member, IEEE, and Shashi Shekhar, Fellow, IEEE bstract Spatial co-locations represent the subsets of features which are frequently located together in geographic space o-location pattern discovery presents challenges since spatial objects are embedded in a continuous space, whereas classical data is often discrete large fraction of the computation time is devoted to identifying the instances of co-location patterns We propose a novel join-less approach for efficient co-location pattern mining The join-less co-location mining algorithm uses an instance-lookup scheme instead of an expensive spatial or apriori join operation for identifying co-location instances We prove the join-less algorithm is correct and complete in finding co-location rules We also describe a partial join approach for a spatial dataset often clustered in neighborhood areas We provide the algebraic cost models to characterize the performance dominance zones of the join-less method and the partial join method with a current join-based co-location mining method, and compare their computational complexities In the experimental evaluation using synthetic and real world datasets, our methods performed more efficiently than the join-based method, and show more scalability in dense data Index Terms Spatial data mining, ssociation rule, olocation pattern, Spatial neighbor relationship I INTRODUTION The explosive growth of spatial data and widespread use of spatial databases emphasize the need for the automated discovery of spatial knowledge Spatial data mining[1], [2] is the process of discovering interesting and previously unknown, but potentially useful patterns from spatial databases Extracting interesting patterns from spatial datasets is more difficult than extracting the corresponding patterns from traditional numeric and categorical data due to the complexity of spatial data types, spatial relationships and spatial autocorrelation [3] spatial co-location represents a subset of spatial features whose instances are frequently located together in spatial neighborhoods For example, epidemiologists have found that West Nile disease and stagnant water sources are frequently co-located The co-location rule, ie, stagnant water source West Nile disease, predicts the presence of West Nile disease in areas with stagnant water sources Spatial co-location patterns may yield important insights for many applications For example, a mobile service provider may be interested in service patterns frequently requested by geographically Manuscript received ugust 3, 25; revised May 15, 26 Jin Soung YOO(orresponding uthor) is with the omputer Science Department, University of Minnesota, 4-192, EE/S ldg, 2 Union Street SE, Minneapolis, MN, 55455, US (telephone: , jyoo@csumnedu) Shashi SHEKHR is with the omputer Science Department, University of Minnesota, 4-192, EE/S ldg, 2 Union Street SE, Minneapolis, MN, 55455, US (telephone: , shekhar@csumnedu) neighboring users The frequent neighboring request sets may be used for providing attractive location-sensitive advertisements, recommendations, etc In another example, ecologists are interested in animal behaviors that frequently co-occur with their nearby environmental or social factors, eg, food abundance, alpha attack [4] Other application domains include Earth science, biology, public health, transportation, etc o-location rule discovery is a process to identify colocation patterns from a spatial dataset o-location rule mining presents challenges due to the following reasons: First, it is difficult to find co-location instances since spatial objects are embedded in a continuous space and share neighbor relationships large fraction of the computation time is devoted to identifying the instances of co-location patterns Second, it is non-trivial to reuse association rule mining algorithms [5], [6] for co-location pattern mining since, unlike market-basket data, spatial datasets often have no predefined transactions Thus, a current co-location mining algorithm [7] uses a joinbased approach to find co-location instances Its computational performance suffers, however, due to the large number of joins required as the number of co-location patterns and their instances increases In this paper, we propose a novel join-less approach for co-location pattern mining which 1) materializes spatial neighbor relationships without any loss of co-location instances, and 2) reduces the computational cost of identifying co-location instances using an instance-lookup scheme Related Work The problem of mining association rules based on spatial relationships (eg, proximity, adjacency) was first discussed in [6] The work discovers the subsets of spatial features frequently associated with a specific feature, eg, cancer Fig 1 (a) shows an example dataset with three spatial features,, and Each object is represented by its feature type and unique instance id, eg, 1 Identified neighbor objects are connected by solid lines Fig 1 (b) shows the neighbor objects near the objects of a specific reference feature type, set of neighboring objects of each reference object is converted to a transaction ll rules generated from the transactions are related to the specific feature However, directly applying this approach to the co-location mining may not capture our colocation meaning with no specific reference feature Previous works on co-location pattern mining have presented different approaches for identifying co-location instances and choosing the interest measures of co-location patterns [8] discovers frequent neighboring class (eg, colocated features) sets using a support count measure This

2 IEEE TRNSTIONS ON KNOWLEDGE ND DT ENGINEERING DRFT 2 Fi : instance i of feature type F :identified neighbor relation <= d: neighbor distance * specific reference feature : 4 neighborhood transactions graph I 5 2 graph IV d graph II 1 1 lose 4 3 not count 1 graph III not count 3 3 (a) (b) (c) Fig 1 (a) n example spatial dataset (b) reference feature based spatial association rule mining (c) Subgraph mining approach uses a space partitioning and non overlap grouping scheme for identifying neighboring objects However, the explicit space partitioning approach may miss co-location instances across partitions like 4, 1 in Fig 2 (a) [7] proposed statistically meaningful interest measures for co-location patterns and a join-based co-location mining algorithm The instance join operation for generating co-location instances is similar to ÔÖ ÓÖ Ò[5] First, after finding all neighbor object pairs (size 2 co-location instances), the approach finds size ( ¾) co-location instances by joining the instances of its size ½subset co-locations where the first ¾objects are common, and checking the neighbor relationship between ½th objects Fig 2 (b) shows generating the instances of co-location,, This approach finds correct and complete co-location instance sets However, the apriori-join operation is computationally expensive with the increase of colocation instances In the spatial database literature, multi-way join techniques using R-trees for multiple spatial features [9], [1] can be used as alternatives to the apriori-join However, we assume our spatial dataset has no spatial index, and thus it is difficult to apply these techniques to our co-location mining directly In our previous work [11], we proposed a partial join algorithm to reduce the number of expensive join operations in finding co-location instances This approach transactionizes objects using clique neighbor relationships, eg, posing simple grids in Fig 2(c), and uses the apriorijoin only for the residual instances not modeled by the explicit transactionization This method reduces the number of join operations significantly However, its performance depends on the number of cut instances by explicit partitioning o-location pattern mining may appear to be similar with subgraph mining [12], [13], which finds frequent subgraphs in a large graph database The prevalent interest measure used is support, the ratio of graphs which include a subgraph pattern For example, in Fig 1(c), the support of subgraph is¾ Subgraph mining is a different problem from our colocation mining, which finds all frequent subsets(ie, clique subgraphs) from a spatial dataset (ie, conceptually a single graph which represents input spatial objects and their neighbor relationships) spatial dataset can be represented to a set of disjoint graphs, and a subgraph mining technique might be applied to it However, some neighbor relationships can be lost by the distinct partition, eg, 1, 1 in Fig 1(c), and the support measure may count one instance in a graph, eg, 3, 1, but not other instances, eg, 4, 1 Our ontributions We took the first step of introducing a join-less algorithm for co-location pattern discovery in [14] We continue that work here and make the following contributions First, we propose to materialize the neighbor relationships of spatial data for efficient co-location pattern mining We present two novel neighborhood partition models, star neighborhood partitioning and clique neighborhood partitioning, and compare the models We also discuss our design decision for co-location instance filtering and prevalent co-location filtering Second, we develop a join-less co-location mining algorithm based on the star neighborhood partition model The algorithm is efficient since it uses an instance-lookup scheme instead of an expensive spatial or instance join operation for filtering co-location instances It also has a coarse pruning step which can filter candidate colocations without finding exact co-location instances We analytically prove our join-less algorithm is correct and complete, ie, there are no false droppings nor false admissions in finding co-location rules Third, we apply the clique partition model to spatial data especially clustered in neighborhood areas and describe the partial join co-location mining algorithm with a new coarse filtering scheme Fourth, we provide algebraic cost models to analyze the performance dominance zones of the join-less method, the partial join method and a current join-based method [7] We compare the computational complexity of the different methods Finally, we experimentally evaluate our algorithms using synthetic datasets and real world datasets The join-less method outperforms the join-based method in overall parameter settings and shows scalability to dense data Scope and Outline This paper addresses the co-location pattern mining of spatial point data We focus on the analytical and experimental

3 IEEE TRNSTIONS ON KNOWLEDGE ND DT ENGINEERING DRFT Vornoi partition 1 1 lose size size 2 join size 3 check neighbor relationships d/sqer(2) a clique transaction d/sqer(2) Trace cut instances 3 (a) (b) (c) Fig 2 Different approaches for finding co-location instances (a) Space partition (b) Instance join (c) lique partition and residual instances comparisons with a current join-based algorithm [7] to examine the effects of our algorithmic design decisions The discussion of optimized partitioning methods for the clique partition model and the discussion of the significance of colocation interest measures are beyond the scope of this paper The remainder of the paper is organized as follows Section II gives an overview of the basic concepts of co-location pattern mining and the problem definition In Section III, we describe our algorithmic design decision for co-location pattern mining In Section IV, we present our join-less colocation mining algorithm, and the completeness and correctness of the algorithm In Section V, we describe the partial join method briefly The analytical analysis of our algorithms and a current join based co-location algorithm is given in Section VI Section VII presents the experimental evaluation The conclusion and future work are discussed in Section VIII II O-LOTION PTTERN MINING In this section, we describe the basic concepts of co-location pattern mining and the problem definition asic oncepts Given a set of spatial features, a set of their instance objectsë, and a neighbor relationshipêoverë, a co-location is a subset of spatial features,, whose instances form a clique using a neighbor relationshipê When the neighbor relationshipêis a Euclidean distance metric with its threshold value,, two spatial objects are neighbors if they satisfy the neighbor relationship, ie,ê(1, 1) ( Ø Ò (1, 1) ) co-location instanceáis a set of objects,á Ë, which includes the objects of all features in the co-location and forms a clique relationship In Fig 1 (a), 2, 4, 2 is an instance of co-location,, since the feature type of 2 is, the feature type of 4 is and the feature type of 2 is, andê(2, 4),Ê(2, 2) andê(4, 2) In this paper, we use the terms instance or clique instance interchangeably to refer to a co-location instance co-location rule is of the form: ½ ¾ Ô Ôµ, where ½, ¾ and ½ ¾ The interest of a colocation rule can be measured by its prevalence(ô) and conditional probability(ô) We use the co-location interest measures defined by [7] First, the participation ratio ÈÖ µof ÆÙÑ ÖÓ Ø ÒØÓ Ø Ó Ò Ò Ø Ò Ó feature in ÆÙÑ ÖÓ Ó Ø Ó a co-location ½,½, is the fraction of objects of feature in the neighborhood of instances of co-location asèö µ It is defined The participation indexè µof a co-location ½ is defined asè µ Ñ Ò ¾ ÈÖ µ high participation index of a co-location indicates that the spatial features of the co-location likely show up together For example, in the dataset of Fig 1 (a), feature has four instances, feature has five instances, and feature has three instances onsider the prevalence values of co-location=,, The instances of co-locationare 2, 4, 2 and 3, 3, 1 The participation ratio of feature in the co-location,èö(, ) is¾ since only 2 and 3 among four feature objects are involved in the co-location instancesèö(, ) is¾ and ÈÖ(, ) is¾ Thus the participation index of co-location,è µ, isñ Ò ÈÖ(, ),ÈÖ(, ),ÈÖ(, ) =¾ The conditional probabilityè ½ ¾µof a co-location rule ½ ÆÙÑ ÖÓ Ø ÒØ Ò Ø Ò Ó ½ Ò Ò Ø Ò Ó ½ ¾ ÆÙÑ ÖÓ Ò Ø Ò Ó ½ asè ½ ¾µ ¾is the fraction of instances of ¾in the neighborhood of instances of ½ Formally, it is estimated In the example of Fig 1 (a), the conditional probability of colocation rule,è µis¾ Lemma 1: The participation ratio and the participation index are monotonically non increasing with increases in the size of the co-location For example, the participation index of a size 3 co-location is not greater than the participation index value of any size 2 colocation, eg,è µ=¾ È µ= in Fig 1 (a) Please refer to [7] for the proof of Lemma 1 Problem Definition We focus on finding a correct and complete set of colocation rules while reducing the computation cost The formal problem definition is objectsë ˽ ËÒwhereË ½ types ½ Ò as follows Given: 1) set of spatial feature 2) set of spatial Òµis a set of instances of feature type Each objectó ¾Ë has a vector information of feature type, instance id and locationü Ý where½ Ë

4 IEEE TRNSTIONS ON KNOWLEDGE ND DT ENGINEERING DRFT 4 Spatial dataset Neighbor relationship Prevalence threshold onditional probability threshold Materialize neighbor relationships Generate candidate co locations Filter co location instances Filter prevalent co locations Generate co location rules Fig 3 asic procedure of co-location pattern mining 3) spatial neighbor relationshipê 4) minimum prevalence threshold (Ñ ÒÔÖ Ú) index Ñ ÒÔÖ Ú and a minimum conditional probability threshold (Ñ ÒÓÒ ÔÖÓ ) Find: set of co-location rules with participation and conditional probability Ñ ÒÓÒ ÔÖÓ Objective: Find a correct and complete set of co-location rules while reducing the computation cost onstraints: Êis a distance metric based neighbor relationship and has a symmetric property III DESIGN DEISIONS FOR O-LOTION PTTERN MINING LGORITHM In this section, we present our algorithmic design decisions for the co-location pattern mining asic lgorithm Given a spatial dataset, a neighbor relationship, and interest measure thresholds, the basic co-location pattern mining procedure involves four steps as shown in Fig 3 First, candidate co-location sets are generated and their co-location instances are gathered from the spatial dataset Prevalent co-location sets satisfying the given prevalence threshold are filtered Finally, co-location rules satisfying the conditional probability threshold are generated Most of the computational time of co-location pattern mining is devoted to finding co-location instances with clique neighbor relationships Thus we propose to add a step for materializing the spatial neighbor relationships to increase the efficiency of co-location mining In the following subsections, we discuss our design decisions in three areas: neighborhood materialization, co-location instance filtering and prevalent co-location filtering Neighborhood Materialization Given a neighbor relationship, a spatial dataset can be represented as a neighbor graph in which a node is a spatial object and an edge between two nodes represents the neighbor relationship Fig 4 (a) depicts a spatial dataset as a neighbor graph The materialization of neighbor relationships for co-location mining should satisfy two criteria No neighbor relation should be missed during the materialization process and the computation cost has to be cheap In addition, it is preferable to minimize duplicate neighbor relations We propose two neighborhood materialization models, star neighborhood partitioning and clique neighborhood partitioning, and then compare them 1 Star neighborhood partition model The star neighborhood partition model (simply, the star partition model) partitions neighbor relationships using a star neighbor relationship For example, Fig 4 (b) shows the possible neighborhood areas of star relationships with objects 1, 3 and 4, which are represented by dotted circles whose radii are a user specific neighbor distance The black solid lines in each circle represent a star neighbor relationship with the center object For example, 1 has a star neighbor relationship with 1 and 1 The star partition model can be divided into two types, overlap star partitioning and disjoint star partitioning Overlap star partitioning results in duplicate neighbor relations, eg, a neighbor relationship between 1 and 1 is duplicated in the star relationship of 1 and in the star relationship of 1 In contrast, disjoint star partitioning divides the whole neighbor relationships to a set of disjoint star relations The disjoint star partition model is preferred since it can lead to no duplication nor any loss of neighbor relationships The star neighborhood of an object identified by the disjoint star objectsëæ Ó ¾Ë Ó Ó partition model is defined as follows Definition 1: Given a spatial objectó ¾Ëwhose feature type is ¾, the star neighborhood ofó is defined as a set of spatial Ê Ó Ó µµ, where ¾ is the feature type ofó andêis a neighbor relationship The star neighborhood of an object is a set of the center object and objects in its neighborhood whose feature types are greater than the feature type of the center object in a lexical order In Fig 4 (b), 1 has two neighboring objects, 1 and 1 The star neighborhood of 1 is 1, 1, 1 including the center object 1 In the case of 3, three neighboring objects, 4, 3 and 1, are present However, 4 is not included in the star neighborhood of 3 since we focus on co-location patterns among different feature types Next, 4 has two neighbor objects, 2 and 2 However, 2 is not included in the star neighborhood of 4 since the neighbor relationship between 2 and 4 is already reflected in the star neighborhood of 2 ll star neighborhoods of the spatial dataset are listed in Fig 4 (b) 2 lique neighborhood partition model The clique neighborhood partition model (simply, the clique partition model) partitions a spatial dataset into sets of objects having clique relationships For example, in Fig 4 (c), 2, 4 and 2 have a clique relationship, and become one partition The clique partition model can also be divided

5 IEEE TRNSTIONS ON KNOWLEDGE ND DT ENGINEERING DRFT 5 y ircle : star neigborhood area : star neighbor relationship of 1, 3, 4 Disjoint star neighborhoods y Fi : instance i of feature type F : identified neighbor relationship <= d : neighbor distance d enter feature object neighborhood area of 1 3 x (b) Star neighborhood partition model Neighbor objects 1, 1, 1 2, 4, 2 3, 3, 1 4, , 1, x (a) Spatial dataset (as neighbor graph) y ircle : clique neigborhood area : cut neighbor relationship d Disjoint clique neighborhoods lique No Neighor objests , 5 1, 1 3, 4, 1 3, 3 2, 4, 2 ut neighbor relations {1, 1} {3, 3} {3, 1} x (c) lique neighborhood partition model Fig 4 Neighborhood materialization into two types, overlap clique partitioning and disjoint clique partitioning The overlap clique partitioning generates a set of maximal cliques In contrast, the disjoint clique partitioning divides a spatial dataset into a set of disjoint cliques in which each clique is not always maximal The disjoint clique partition model is preferred since overlap clique partitioning requires finding all maximal cliques, which is computationally expensive(np-omplete) The following is the definition of a clique neighborhood by the disjoint clique partition model Definition 2: clique neighborhood is a set of spatial objects Æ Ëthat forms a clique under a neighbor relationshipê spatial datasetëis a union of disjoint clique neighborhoods, ƽ ÆÑ where Æ Æ, and ƽ ÆÑ Ë Each spatial object belongs to one clique neighborhood exactly Fig 4 (c) illustrates the clique partition model dashed circle represents a neighborhood area whose radius is /2 The objects within the circle are neighbors of each other For example, 1 and 1 form a clique neighborhood However, note that this disjoint clique partitioning can lose neighbor relationships across the partitions, eg,ê(1, 1) Fig 4 (c) presents cut neighbor relationships as dotted lines 1, 1, 3, 3 and 3, 1 are the cut neighbor relations Our clique partition model also keeps the cut neighbor relations for completeness Fig 4 (c) shows a set of the disjoint clique neighborhoods and cut neighbor relations 3 Model omparison We compare the star partition model and the clique partition model in terms of their characteristics, the materialized data size, and advantages and disadvantages in co-location pattern mining First, if we consider the two models from the perspective of a graph, the star partition model partitions edges disjointly using a star neighbor relationship In contrast, the clique partition model partitions nodes disjointly using clique neighbor relationships The star partition model hasòstar neighborhoods, whereòis the total number of objects in a spatial dataset The number of clique neighborhoods depends on the data distribution In a declustered spatial dataset where each object is one clique neighborhood, the number of clique neighborhoods isò In general, the number of clique neighborhoods is much smaller than the number of star neighborhoods Next, we compare the advantages and disadvantages of the models in co-location pattern mining Star neighborhoods are easily generated from a neighbor graph However, a star neighborhood does not model a clique relationship directly and can contain false positives of co-location instances In contrast, the cost to generate clique neighborhoods depends on the partitioning method, eg, from a simple grid partition to a fine partition considering data distribution The ideal case is to maximize the number of objects in a partition and to minimize the number of cut relationships The benefit of the clique partition model is that it models clique relationships directly However, this model requires a procedure to trace false dismissals split by the disjoint object partition, ie, colocation instances related to cut relationships o-location Instance Filtering The algorithmic design for filtering co-location instances from a spatial dataset depends on the neighborhood mate-

6 IEEE TRNSTIONS ON KNOWLEDGE ND DT ENGINEERING DRFT 6 rialization model We present two schemes for co-location instance filtering, instance-lookup and instance-join We define the following terms to distinguish the instances of co-locations from different neighborhood materialization models Definition 3: LetÁ Ó½ Ó Ëbe a set of spatial objects whose feature types ½ are different If all objects ináare neighbors to the first objectó½,áis called a star instance of co-location = ½ For example, in Fig 4 (b), a subset of 1 star neighborhood, 1, 1 is a star instance of co-location, Next, we define terms related to the clique neighborhoods an (clique) instance of co-location If all objects inábelong to a same clique neighborhood,áis called an intra neighborhood Definition 4: LetÁ Ó½ Ó Ëbe instance (simply, an intra instance) of co-location For example, in Fig 4 (c), 1, 1 is an intra instance of co-location, Definition 5: neighbor relationship between two objects, Ó½ Ó¾¾Ë Ó½ Ó¾is called a cut neighbor relationship if Ó½andÓ¾are neighbors of each other but belong to different clique neighborhoods Definition 6: LetÁ Ó½ Ó Ëbe an (clique) instance of a co-location If all objects ináhave at least one cut neighbor relationship,áis called an inter neighborhood instance (simply, an inter instance) of co-location For example, in Fig 4 (c), 1, 1 is an inter instance of co-location, because 1 has a cut neighbor relationship with 1, and 1 also has cut neighbor relationships with 1 and 3 The objects in a inter instance are located across one more clique neighborhoods 1 Instance-lookup ½ instance Ó½ Ó We propose an instance-lookup scheme to filter co-location instances from star instances star of co-location can be gathered from star neighborhoods whose center object type is ½by Definition 3 Note that the star instance is not a co-location instance since it is not guaranteed that it has a clique relationship However, if all objects in it except the first object(the center object of the star relationship) form a clique, the star instance is a co-location instance The cliqueness of the subset can be checked by searching other star instances, eg, the second objectó¾ s star instance, Ó¾ Ó, the star instance, Ó Ó, third objectó s etc, or by looking up the clique instances of co-location ¾ except ½ 2 Instance-join Intra neighborhood instances (ie, instances within neighborhoods) can be gathered by scanning the clique neighborhood set n intra instance is a co-location instance since a subset of a clique neighborhood is also a clique In contrast, inter neighborhood instances (ie, instances over different neighborhoods) are not found from the materialized set directly We adopt an instance join method [7] for tracking the inter instances The inter instances of a size co-location can be generated by joining the inter instances of its size ½co-locations For example, in Fig 4 (c), an inter instance 3, 3, 1 can be generated by joining size 2 inter instances, 3, 3 and 3, 1, and checking the neighbor relationship between 3 and 1 D o-location Filtering We use three filtering steps to find prevalent co-locations: feature-level filtering, coarse filtering and refinement filtering Feature-level filtering is possible due to the anti-monotonic property of participation index by Lemma 1 candidate colocation can be pruned without examining its instances if any subset of it is not prevalent oarse filtering is done after finding all star instances or after finding all intra instances of co-locations, which depends on the materialization model This step can reduce the expensive filtering operation for colocation instances If the prevalence value of star instances of a candidate co-location does not satisfy a given threshold, the candidate is pruned without the cliqueness check of its star instances In the clique partition model, the intra instances of a candidate co-location do not give enough information for the coarse pruning Thus we use cardinal objects from the inter instances of its subsets as well as its intra instances If the prevalence value from them does not satisfy the threshold, the co-location is pruned without finding its inter instances In the last step, the refinement filtering decides prevalent co-location sets after filtering all co-location instances E Design Decision The choice of the neighborhood materialization models can depend on the distribution of input spatial data For general spatial datasets, we choose the star neighborhood partition model and use the instance lookup scheme for filtering colocation instances We present the join-less co-location mining algorithm in the next section In contrast, the clique partition model is appropriate for spatial data mostly clustered in neighborhood areas having fewer neighbor relationships with their nearby neighborhoods This is because clustered data can be modeled to clique neighborhoods and fewer neighbor interactions with nearby neighborhoods can reduce the number of cut relationships The partial join approach is briefly discussed in Section V IV JOIN-LESS PPROH In this section, we present a join-less co-location mining algorithm based on the star neighborhood partition model We show the algorithm is correct and complete in mining colocation patterns Join-less o-location Mining lgorithm The join-less co-location mining algorithm has three phases The first 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 materialized set, and coarsely filters the candidates using the prevalence values of their star instances The third phase filters their colocation instances from the star instances, and finds prevalent co-locations and generates co-location rules The second and third phases are repeated with the increase of co-location size Fig 5 illustrates a trace example of the join-less algorithm lgorithm 1 shows the pseudo code The detailed explanation is given as follows

7 IEEE TRNSTIONS ON KNOWLEDGE ND DT ENGINEERING DRFT 7 PHSE I Feature star neighborhoods 1, 1, 1 2, 4, 2 3, 3, 1 4, 1 Feature star neighborhoods 1 2 3, 1, Feature star neighborhoods Fig Level 1 PHSE II PHSE III 1, 1 2, 4 3, 3 1, 1 2, 2 3, 1 4, 1 single instance count 1, 1 2, 4 3, 3 3/4 3/5 Level 2 1, 1 2, 2 3, 1 4, 1 4/4 2/3 3, 1 3, 3 4, 2 3, 1 3, 3 4, 2 2/5 3/3 1, 1, 1 2, 4, 2 3, 3, 1 3/4 3/5 2/3 1, 1, 1 2, 4, 2 3, 3, 1 2/5 2/3 Level 3 participation index on star instances if < threshold, it s pruned 2/4 heck if it is a clique true participation index n illustration of the join-less co-location mining algorithm star instances clique instances 1) onvert a spatial dataset to a set of disjoint star neighborhoods (Step 1): Given an input spatial dataset and a neighbor relationship, first find all neighboring object pairs using a geometric method such as plane sweep [15] The star neighborhoods can be generated from the neighbor pairs by grouping the neighbor objects per each object Fig 5 shows the same star neighborhood set in Fig 4 (b) 2) Generate candidate co-locations(step 4): First, we initialize all features to size 1 prevalent co-locations by the definition of participation index The number of instances per each feature can be known during the neighborhood materialization procedure Size ( ½) candidate co-locations are generated from prevalent size ½co-locations(step 4) Here, we have the feature level filtering of the co-locations If any subset of a candidate co-location is not prevalent, the candidate colocation is pruned 3) Filter the star instances of candidate co-locations from the star neighborhood set(step 5): The star instances of a candidate co-location are gathered from the star neighborhoods whose center object s feature typ is the same as the first feature of the co-location For example, the instances of co-location, are gathered from the feature star neighborhoods, and the instances of co-location,, are gathered from the feature star neighborhoods Notice that the number of candidate co-locations examined in each star neighborhood is much smaller than the number of actual candidates 4) Select coarsely prevalent co-locations using their star instances(step 1): The size 2 star instances are clique instances since our neighbor relationship is symmetric Thus, we go to step 13 to find prevalent co-locations For size 3 or more, we need to check if the star instances are clique instances efore this procedure, we have a coarse filtering of the co-locations We filter a candidate co-location using the participation index of its star instances For example, in Fig 5, the participation index of candidate co-location,, lgorithm 1 Join-less co-location mining algorithm Inputs ½ Ò :a set of spatial feature types Ë:a set of spatial objects Ê:a spatial neighbor relationship Ñ ÒÔÖ Ú:prevalence Ñ ÒÓÒ ÔÖÓ threshold, Ñ ÒÓÒ ÔÖÓ :conditional probability threshold Output set of all co-location rules with participation index Ñ ÒÔÖ Úand conditional probability ËÆ= ËÆ ½ ËÆ Ò Variables :a set of feature star neighborhoods :co-location size :a set of size candidate co-locations ËÁ :a set of star instances of size candidates Á :a set of clique instances of size candidates È :a set of size prevalent co-locations Ê :a set of size co-location rules Method 1)ËÆ=gen star neighborhoods(,ë,ê); 2)Ƚ= ; =2; 3) while (not emptyè ½) do 4) =gen candidate co-locations(è ½); the first feature of ½ µ 7) ËÁ =filter star instances( Ø); 8) end do 5) if =2then Á ËÁ for in 1 to n do 6) forø¾ëæ where ½, ½is 9) 1) else do =select coarse prevalent co-locations( ËÁ Ñ ÒÔÖ Ú) 11) Á =filter clique instances( ËÁ ); 12) end do 13) È =select prevalent co-locations ( Á Ñ ÒÔÖ Ú); 15) = +1; 16) end do 17) returnë ʾ Ê µ; 14) Ê =gen co-location rules(è Ñ ÒÓÒ ÔÖÓ ); from its star instances is If it is less than a user specified minimum prevalent threshold, the candidate is pruned without examining exact co-location instances 5) Filter co-location instances(step 11): From the star instances of a candidate co-location, we filter its co-location instances using the instance look-up scheme For example, in Fig 5, to check the cliqueness of a star instance 1, 1, 1 of co-location,,, we examine if a sub instance 1, 1 except 1 is in the set of clique instances of co-location, This instance look up operation can be performed efficiently by an instance key which is composed of the ids of objects in the instance s shown in Fig 5, 1, 1, 1 is not a co-location instance, but 2, 4, 2 and 3, 3, 1 are co-location instances 6) Select prevalent co-location sets(step 13): The refinement filtering of co-locations is done by the true participation index values calculated from their co-location instances Prevalent co-locations satisfying a given threshold are selected 7) Generate co-location rules(step 14): Finally we generate all co-location rules satisfying a given minimum conditional probability from a set of prevalent co-locations Steps 3-16 are repeated as the co-location size increases ompleteness and orrectness ompleteness means the join-less algorithm finds all colocation rules whose participation index and conditional prob-

8 IEEE TRNSTIONS ON KNOWLEDGE ND DT ENGINEERING DRFT 8 ability satisfy a user specified minimum prevalence threshold and conditional probability threshold orrectness means that all co-location rules generated by the join-less algorithm have participation index and conditional probability values above the minimum thresholds First we provide related lemmas Lemma 2: The star partition model does not miss any neighbor relationship of an input spatial dataset Proof: The disjoint star partition model includes all neighbor relations of each object and excludes only duplicate neighbor relations which are already included in a star neighborhood by Definition 1 Lemma 3: Let ½ be a size co-location andëábe the star instances of The participation index of fromëáis not less than the true participation index of Proof: The participation ratio of ½fromËÁis the maximum possible probability that the objects of feature ½ have clique relationships with the objects of the other features ¾ in since only objects of feature ½in the star instances can be included in the clique co-location instances of The participation ratio fromëá of (¾ ) is also the maximum possible probability that the objects of feature have clique relationships with the objects of features ½since our neighbor relationship is symmetric Thus the participation index of calculated from the star instances is not less than the true participation index of, ie, Ñ Ò ¾ ÔÓ Ð Ñ ÜÈÖ µ Ñ Ò ¾ ÈÖ µ Lemma 4: Let an instanceá Ó½ Ó be a star instance of co-location ½ If the sub instance Ó¾ Ó exceptó½is a clique,áis a co-location instance of Proof: In a star instanceá Ó½ Ó, the first object Ó½has neighbor relationships to the other objects,ó¾ Ó, by Definition 3 If a subset Ó¾ Ó is a clique, object Ó (¾ ) has neighbor relationships with all objects in the subset as well asó½ ThusÁis a clique co-location instance Theorem 1: The join-less co-location mining algorithm is complete Proof: The completeness of the join-less algorithm can be shown in two ways The first is that the method to materialize the neighbor relationships of an input spatial dataset (step 1), the method to gather star instances(step 7), and the method to filter clique instances(step 11) are correct The star partition model does not miss any neighbor relationship of the input data by Lemma 2 The star instances gathered from the star neighborhoods where the feature type of the center object is the same with the first feature of the co-location have correct star neighbor relationships by Definition 1 and 3 ny potential co-location instance is not missed since the star instances are a super set of the clique instances The method to filter co-location instances from the star instances does not drop a true instance by Lemma 4 Next, we show that the filtering steps of co-locations do not miss true colocations The feature level filtering by prevalent subsets(step 4) is complete by Lemma 1 The coarse filtering(step 1) does not eliminate any true prevalent co-locations by Lemma 3 The refinement filtering(step 13) prunes only co-locations whose lique Neighborhood lique No Neighoring objests Fig 6 PHSE I 2, 5 1, 1 3, 4, 1 3, 3 2, 4, 2 ut neighbor relations {1, 1} {3, 3} {3, 1} PHSE II PHSE III PHSE IV Level 1 Level single instance count join intra instances inter instances /4 3/5 4 1 participation ratios 4/4 2/ cardinal objects from size 2 inter instances /5 3/3 Level 3 intra instances estimated inter instance objects /4 2/5 2/3 if < threshold, pruned inter instances /4 2/5 2/3 participation index n illustration of the partial join co-location mining algorithm true participation index is less than the threshold Step 14 ensures that no co-location rules satisfying a given conditional probability are missed Theorem 2: The join-less co-location mining algorithm is correct Proof: The correctness of the join-less algorithm can be guaranteed by steps 13 and 14 Step 13 selects only colocations whose participation indexes satisfy a user specific prevalence threshold The generated co-location rules by step 14 also satisfy a user specific conditional probability V PRTIL JOIN PPROH In this section, we briefly describe a partial join approach for spatial datasets that are mostly clustered in neighborhood areas with fewer neighbor relationships with their nearby neighborhoods The partial join co-location mining algorithm was proposed in our previous work [11] Here we added a coarse filtering scheme to further reduce the number of join operations The partial join approach uses the clique partition model for materializing clustered data to clique neighborhoods directly, and keeps track of co-location instances split by the disjoint object partitioning Fig 6 (b) shows a trace example of the partial join co-location mining algorithm size Ô¾ Given a spatial dataset and a neighbor relationship, first we need to generate a set of disjoint cliques To do this, we use grid partitioning The rectangular grids of Ô¾, where is a neighbor distance, are posed on a spatial framework The objects in each grid are generated to a clique neighborhood ut neighbor relations can be detected by examining neighbor pairs whose partition areas are different Size 2 inter instances(ie, instances across neighborhoods) are the cut relations and the pairs of objects having at least one cut neighbor relationship in each partition The intra instances(ie, instances within neighborhoods) of a size ( ¾) candidate co-location are enumerated from the clique neighborhood set For example, in Fig 6, 2, 4, 2 is an intra instance of

9 IEEE TRNSTIONS ON KNOWLEDGE ND DT ENGINEERING DRFT 9 co-location,, efore finding the inter instances of a candidate co-location, we conduct a coarse filtering of the colocation using its intra instances and the cardinal objects from the inter instances of its size ½co-locations For example, in Fig 6, the cardinal objects which can be involved in the inter instances of co-location,, are 3 of feature, 3 of feature and 1 of feature If the participation index calculated from them does satisfy a given threshold value, we further go to find the exact inter instances We use an instance join method [7] for generating the inter instances For example, the inter instances of,, are produced by joining an inter instance of,, 3, 3 and an inter instance of,, 3, 1, and by examining the neighbor relationship between 3 and 1 The true participation index of a co-location is calculated from its intra instances and inter instances ll co-location rules whose prevalence value and conditional probability satisfy given thresholds are generated VI NLYTIL OMPRISON In this section, we analytically compare our join-less and partial join co-location algorithms with a pure join-based colocation algorithm [7] First we examine the computational complexities of the different methods and then compare them omputational omplexities LetÌ Ð,ÌÔ andì be the total computation cost of the join-less method, the partial join method, and the join-based ÌÔ =ÌÐ ÕÙ Ò ÓÖ ÓÓ ²ÙØÖ Ð Ø ÓÒ Ëµ+ÌÔ ¾µ Ì Ð=Ì Ø ÖÒ ÓÖ ÓÓ Ëµ+Ì Ð ¾µ+È ¾Ì Ð µ method, respectively The following equations show the total cost +È ¾ÌÔ µ functions Ì =ÌÒ ÓÖÔ Ö Ëµ+Ì ¾µ+È ¾Ì µ Ëdenotes an input spatial datasetì Ð ¾µ,ÌÔ ¾µand Ì ¾µrepresent Ì Ð µ=ì Ò Ò È the cost for finding size¾co-location patterns +Ì ÐØ ÖÓ Ö ÓÐÓ µ+ì ÐØ ÖÐ ÕÙ Ò Ø ¼ µ in each method The following equations are the costs of +Ì ÐØ ÖÔÖ ÚÓÐÓ ¼ µ+ì ÒÖÙÐ È µ ½µ+Ì ÐØ Ö Ø Ö Ò Ø µ finding size ( 2) co-location patterns ÌÔ µ=ì Ò Ò È Ì ÐØ Ö Ø Ö Ò Ø µ+ì ÐØ ÖÐ ÕÙ Ò Ø ¼ µ +Ì ÐØ ÖÓ Ö ÓÐÓ µ+ì ÐØ Ö ÒØ Ö Ò Ø ¼¼ +Ì ÐØ ÖÔÖ ÚÓÐÓ ¼¼ ½µ+Ì ÐØ Ö ÒØÖ Ò Ø µ Ì ÐØ Ö ÒØÖ Ò Ø µ+ì ÐØ Ö ÒØ Ö Ò Ø ¼¼ µ+ì ÒÖÙÐ È µ µ µ Ì µ=ì Ò Ò È +Ì ÐØ ÖÔÖ ÚÓÐÓ µ+ì ÒÖÙÐ È µ Ì ÐØ Ö Ò Ø µ ½µ+Ì ÐØ Ö Ò Ø µ È ½is a size ½prevalent co-location set, is a size candidate co-location set, and each ¼ and ¼¼ is a size candidate co-location set filtered by the coarse filtering in the join-less algorithm and the partial join algorithm In these equations,ì ÒÖÙÐ is the same cost in all three methods Ì Ò Ò,Ì ÐØ ÖÓ Ö ÓÐÓandÌ ÐØ ÖÔÖ ÚÓÐÓcan be ignored when compared with other computation factors as shown in the experiment results of the next section omparison of omputational omplexities We compare the computational complexities of the different methods in two parts One is the computation cost of materializing neighborhoods and generating size 2 co-location patterns and, the other is the computational cost of generating size ( 2) co-location patterns 1) Neighborhood materialization & size 2 co-locations : The computational costs of materializing neighborhoods and ÌÐ ÕÙ Ò ÓÖ ÓÓ ²ÙØÖ Ð Ø ÓÒ Ëµ ÌÔ ¾µ generating size 2 co-location patterns in the different methods are compared by Ì Ø ÖÒ ÓÖ ÓÓ Ëµ Ì Ð ¾µ the following equation ÌÒ ÓÖÔ Ö Ëµ Ì ¾µ First, each method needs to find all neighboring pairs The cost is the same in the three methods The join-based method generates size 2 co-locations by gathering the neighbor pairs per candidate co-location, and calculating their prevalence measures In both the join-less method and the partial join method, the size 2 co-locations can be found also from the neighbor pairs or by scanning their materialized neighborhood set If we use the former, the cost is the same in all three methods However, the join-less method has an additional cost to materialize the star neighborhoods from the neighbor pairs In the partial join method, we generate the disjoint clique neighborhoods using a simple grid partitioning The cut relations are also easily found during the neighbor pair search, but an additional cost is required to find all size 2 inter instances with cut relationships The overall cost is expected to be a litte bigger than the cost to generate the star neighborhood set 2) Join-less vs Join-based with size ( 2) co-locations: We compare the join-less algorithm with the join-based algorithm in generating size ( 2) co-location patterns Ì µ Ì ÐØ Ö Ø Ö Ò Ø µ Ì ÐØ ÖÐ ÕÙ Ò Ø ¼ µ Ì Ð µ Equation (1) shows the computation Ì ÐØ Ö Ò Ø µ ratio Ø Ò Ô Ð ØÐÓÓ ÙÔ Ø Ò Ô Ð ØÐÓÓ ÙÔ Ø Ó Ò Ø Ó Ò ½µ Ø Òis the average cost to collect the star instances of a candidate co-location by scanning the materialized neighborhood setô Ðis the coarse filtering ratioøðóó ÙÔis the average cost isô Ð ØÐÓÓ ÙÔ to check the cliqueness Ø Ó Ò of its star instancesø Ó Òis the average cost to generate co-location instances per co-location using the instance join If the input dataset is dense,ø Ò Ø Ó Òand the ratio We assume that the number of instances involved in the join operation and the number of instances involved in the lookup operation per co-location is similar Even if the coarse pruning is not effective(ô Ð=1), the join-less algorithm is expected to take less computation time than the join-based algorithm sinceøðóó ÙÔ Ø Ó Ò We examine the efficiency oføðóó ÙÔoverØ Ó Òwith different density datasets later in the experiment With the increase of neighbor density,

10 IEEE TRNSTIONS ON KNOWLEDGE ND DT ENGINEERING DRFT 1 the number of instances involved in the join operation or the lookup operation increases and the computation difference betweenøðóó ÙÔandØ Ó Òis expected to be bigger The overall data density is controlled by spatial frame size, and the coarse filtering ratioô Ðis controlled by an overlap parameter, which is the ratio of the number of points included in different colocation instances over the total number of points 3) Partial-join vs Join-based with size ( ÌÔ µ Ì µ Ì ÐØ Ö ÒØÖ Ò Ø µ Ì ÐØ Ö ÒØ Ö Ò Ø ¼¼ µ 2) colocations Ø Ò ÒØÖ Áµ ÔÔ Ø Ó Ò ÒØ ÖÁµ Ì ÐØ Ö Ò Ø µ : Ø Ò ÒØÖ Áµ ÔÔ Ø Ó Ò ÒØ ÖÁµ Ø Ó Ò Áµ Ø Ó Ò Áµ ¾µ In equation (2), ÒØÖ Áis the intra instances from the clique neighborhoods, ÒØ ÖÁis the inter instances joined in the partial join method andáis the instances joined in the joinbased method In general,ø Ò ÒØÖ Áµ Ø Ó Ò Áµsince the cost of gathering instances by scanning a neighborhood materialization set is much cheaper than the join operation for generating whole instances The number of inter instances in the partial join method is a major factor in this ratio If ÒØ ÖÁ Á, the have less computation cost If ÒØ ÖÁ ³ Á and pruning is not effective(ôô =1),Ø Ó Ò ÒØ ÖÁµ³Ø Ó Ò Áµand Ø Ò ÒØÖ Áµcost is partial join method is expected to the coarse an ÖÓ added to the partial join method The join-based method is expected a little better To control the number of inter instances in the experiment, we use ratio which is ratio of instances across different neighborhood areas ÌÔ µ Ì ÐØ Ö Ø Ö Ò Ø µ Ì ÐØ ÖÐ ÕÙ Ò Ø ¼ µ Ì Ð µ over total instances 4) Join-less Ì ÐØ Ö ÒØÖ Ò Ø µ Ì ÐØ Ö ÒØ Ö Ò Ø ¼¼ vs Partial-join with size ( 2) co-locations: Ô Ð ØÐÓÓ ÙÔ Áµ µ ÔÔ Ø Ó Ò ÒØ ÖÁµ µ ÔÔ Ø Ó Ò ÒØ ÖÁµ Ô Ð ØÐÓÓ ÙÔ Áµ Ì Ø Ö ÒØÖ Ò Ø µ and are expected relatively smaller than other factors If we assume the coarse filter ratios are the same, the comparison of the two methods is the same with the comparisons ofìðóó ÙÔandÌ Ó Ò, and of Á and ÒØ ÖÁ In datasets of dense in neighborhoods with fewer inter instances, the partial join method is expected to show better performance In the experiment presented later, we compare the behavior of the two algorithms in data density in neighborhood area and the ratio of cut instances Each factor is controlled by clumpy degree and across ratio variables In dense data, Ì Ø Ö Ø Ö Ò Ø µ VII EXPERIMENTL EVLUTION We evaluate the join-less and the partial join co-location algorithms with the join-based algorithm [7] using synthetic and real datasets We conducted the following experiments We examined the effects of our algorithm design decisions Specifically, we compared the computation costs of generating up to size 2 co-location patterns with and without a materialization step We examined the efficiency of the instance-lookup scheme, and the effect of coarse filtering In addition, we compared the costs of the computation complexity factors of the three algorithms to support our analytical comparisons in Section VI We examined the scalability of the join-less algorithm in terms of the number of points, the number of features, the distance neighbor threshold, and the prevalence threshold We examined the behavior of the partial join algorithm with different neighborhood densities and cut relation ratios Finally, we evaluated the algorithms with two real datasets, a climate dataset and an Ecology dataset ll algorithms were implemented in /++, and are memory based algorithms ll the experiments were performed on a Sun Sunlade 15 with 1 G memory and 177MHz PU Synthetic Data Generation Synthetic datasets were generated using a spatial data generator similar to [7] Table I describes the parameters used for the data generation First, overall spatial frame size is determined with the frame side length size frame was used The whole frame was divided into regular grids whose side length was a neighborhood distance In order to generate co-location instances located across the grids, we divided each grid into two areas, a non-overlapping area and an overlapping area n object in the non-overlapping area has neighbor relationships only with other objects in the same area y contrast, objects in the overlapping area can have neighbor relationships with objects in the nearby grids The following is the procedure we used to generate general synthetic datasets in which co-location instances were randomly distributed First we generatedèinitial patterns whose average size wasë The feature types of each pattern were randomly chosen from features verageáinstances per each pattern were generated The total number of points wasæ For locating a co-location instance, first we chose a grid cell randomly ll points of the instance were randomly located within the chosen grid cell To generate our specialized datasets, first, overall data density was controlled by spatial frame size Under a fixed total number of points, a smaller frame generates more dense data Second, the data density in neighborhood areas was controlled by aðùñôýdegree When a new grid cell was chosen randomly for locating a co-location instance, a ÐÙÑÔÝnumber of instances was generated in the same grid cell The defaultðùñôývalue was 1 Third, the ÖÓ ratio controlled the number of instances across different grid cells n across instance was generated in the overlapping area of a chosen grid and its nearby grid Finally, the number of points overlapped in different co-location instances was controlled by theóú ÖÐ ÔratioÆ ÓÚ ÖÐ Ôpoints were randomly selected EachÓÚ ÖÐ Ôpoint was involved in four co-location instances The parameter values for the synthetic dataset used in each experiment are described in Table I

11 IEEE TRNSTIONS ON KNOWLEDGE ND DT ENGINEERING DRFT 11 ÈË Experiment no Symbols ÆÁ Meaning verage size ofýco-locations 5 Number ofýco-locations 2 verage number ofýco-location instances 15 ÐÙÑÔÝ Number of features 2 * 2 Number of points 15K * 15K Spatial frame size( ) * 5 1, Neighborhood distance threshold 1 * 1 ÓÚ ÖÐ Ô ÖÓ Prevalence threshold * 3 Number of co-location instances generated in a same neighborhood area 1 * 15 Ratio of instances across different neighborhood areas over total instances 1 * Ratio of points overlapped in different co-location instances over total points * Ý: initial core co-location, * : variable values TLE I EXPERIMENTL PRMETERS ND THEIR VLUES IN EH EXPERIMENT method factor(%) data type Tgen_candi Tfilter_coarse_coloc Tfilter_prev_coloc Tfilter_star_inst Tfilter_clique_inst Tfilter_intra_inst Tfilter_inter_inst Tfiter_inst total execution time(sec) Join less partial join join based sparse dense sparse dense sparse dense * Execution time is sec Other data values are % Materialization and size 2 execution time (sec) join-less(star partition materialization) join-less(size2 co-locations) partial-join(clique partition materialization) partial-join(size2 co-locations) join-based(no materialization + size 2 co-location) Neighbor degree (a) (b) Fig 7 (a) omputation of computational complexity factors (b) omputation comparison of neighborhood materialization and size 2 co-location generation Experiment Results We present a summary of our experiment results 1 Effects of algorithm design decision 1) omparison of computational complexity factors: We examined the costs of the computational complexity factors discussed in Section VI We used two datasets of different density, ie, a sparse dataset with neighbor degree 5 and a dense dataset with neighbor degree 15 In Fig 7 (a), data values except total execution time represent percent values Overall, we can notice thatì Ò Ò,Ì ÐØ ÖÓ Ö ÓÐÓand Ì ÐØ ÖÔÖ ÚÓÐÓtake much less portion than the other costs In addition,ì ÐØ Ö Ø Ö Ò ØandÌ ÐØ Ö ÒØÖ Ò Øof gathering instances from each materialization set are relatively smaller thanì ÐØ Ö ÒØ Ö Ò ØandÌ ÐØ Ö Ò Øusing instance join operations The difference is greater in the dense dataset 2) omparison of neighborhood materialization methods: We compared the computation cost of generating up to size 2 co-locations, including the costs of neighborhood materialization Size 2 co-locations were generated from neighboring object pairs in all methods Fig 7 (b) shows the execution times of three approaches, with star partition materialization, with clique partition materialization, and with no materialization s can be seen, the join-less approach without materialization shows less execution time In the materialization methods, the star partitioning of join-less approach took less computation time than the clique partitioning of partial join approach 3) omparison of instance filtering methods: We examined the efficiency of our instance look-up scheme over the instance join method in the join-based algorithm with different dense datasets Data density determines the neighbor degree of an object, which in turn affects the number of co-location instances Neighbor degree is the average number of neighbor points per each point Different spatial frame sizes were used to control the overall data density of the same number of points Fig 8 (a) compares the execution time of finding size 3 or more co-location instances of the different schemes We turned off the coarse filtering option to see only the effects of instance filtering schemes In sparse datasets(eg, neighbor degree 1), the two methods show similar execution time However, the execution time increases as the neighbor degree increases, and here the instance-lookup method shows better performance than the instance-join method Moreover, the difference in execution times of the two methods is much greater with the increase of neighbor degree 4) Effect of coarse pruning: We examined the effect of the coarse filtering scheme in the join-less algorithm Fig 8 (b) shows the percent of candidate co-locations pruned in the coarse filtering step over the total candidate co-locations after the feature-level pruning We considered only size 3

12 IEEE TRNSTIONS ON KNOWLEDGE ND DT ENGINEERING DRFT 12 Size 3 or more execution time (sec) join-less(instance-lookup) join-based(instance-join) Neighbor degree Pruning ratio(%) Overlap ratio(%) (a) (b) Fig 8 (a) omparison of instance filtering schemes (b) Effect of coarse pruning in the join-less method Execution time (sec) join-less(dense) partial-join(dense) join-based(dense) join-less(sparse) partial-join(sparse) join-based(sparse) Execution time (sec) join-less(dense) partial-join(dense) join-based(dense) join-less(sparse) partial-join(sparse) join-based(sparse) Number of total points(k) Number of features (a) (b) Execution time (sec) join-less partial-join join-based Execution time (sec) join-less partial-join join-based Neighbor distance Particiatpion index threshold (c) (d) Fig 9 Scalability of the join-less algorithm (a) by number of points (b) by number of features (c) by neighbor distance (d) by prevalence threshold or more candidates since the coarse pruning is effective in filtering them With the increase of overlap ratio, the chance of pruning the candidate co-locations increased since the participation ratios of features of overlap points decrease 2 Scalability of the join-based algorithm We examined the scalability of the join-less algorithm with several workloads, ie, the number of point objects, the number of feature types, distance neighbor threshold, and prevalence threshold We compared the total computation time of finding all prevalent co-locations including the materialization cost sizes,½¼ ¼¼¼ ½¼ ¼¼¼ 1) Effect of the number of point objects: First, we compared the effect of the number of points in the different algorithms We used two different spatial frame and½ ¼¼¼ ½ ¼¼¼ In the first frame, even if the number of points is increased from 1K to 5K, the three algorithms showed similar execution time since the datasets are still sparse However, in the second frame, with the increase of number of points, the join-based algorithm s execution time is dramatically increased due to the increase of data density s shown in Fig 9 (a), the join-less algorithm shows scalability to large dense datasets 2) Effect of the number of features: In the second experiment, we compared the performance of the algorithms as a function of the number of different features We also used two different dense datasets of 15K points Fig 9 (b) shows the results In the sparse dataset, the algorithms show similar execution 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 number 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 instances per co-location Overall the join-less algorithm shows better performance 3) Effect of neighbor distance: The third experiment examined the effect of different neighbor distances s shown in Fig 9 (c), the join-less algorithm shows less increase in the execution time with the increase of distance threshold The

13 IEEE TRNSTIONS ON KNOWLEDGE ND DT ENGINEERING DRFT 13 Execution Time (sec) join-less partial-join join-based Execution time (sec) join-less partial-join join-based lumpy degree cross ratio(%) (a) (b) Fig 1 The effectiveness of the partial join algorithm (a) by data density on a neighborhood area (b) by cut neighbor relations Execution time (sec) join-less partial-join join-based Execution time (sec) join-less partial-join join-based Prevalence threshold Neighbor distance(m) (a) (b) Fig 11 Real datasets (a) climate dataset (b) chimpanzee behavior dataset join-based algorithm shows a rapid increase since the increase of neighbor distance makes the neighborhood areas larger and increases the number of co-location instances 4) Effect of prevalence threshold: In the final experiment of scalability, we examined the performance effect as the prevalence threshold increases Overall the execution time decreases for all three algorithms with the increase of the prevalence threshold, as shown in Fig 9 (d) However, the join-less algorithm reduces the computation time by a magnitude at lower threshold values 3 ehavior of the partial join algorithm We examined the behavior of the partial join algorithm with the density of neighborhood areas and the cut instances by the explicit partition of spatial data 1) Effect of data density in neighborhood areas: The density of neighborhood areas was controlled byðùñôýdegree in Fig 1 (a) The synthetic datasets had around 1% cut neighbor relations by disjoint clique partitioning in the partial join algorithm The partial join algorithm performed well even if the data density increases withðùñôýdegree The reason is that the points are clustered in some neighborhood areas with fixed cut relations y contrast, the join-less algorithm shows an increase in execution time with the increase of density The join-based algorithm always shows higher execution time than the other algorithms 2) Effect of cut relations: In the second experiment, low ÖÓ under a fixed clumpy degree of 15, we increased the number of cut relations The number of cut relations is controlled by the ÖÓ ratio in synthetic data generation t ratios, the partial join algorithm with fewer cut relations shows better performance than the join-less algorithm However, the execution time increases with the increase of the ÖÓ ratio because of the increase of inter neighborhood instances and the increase of instance join operations to trace them Overall, the join-based algorithms increase in execution time with increase of the across ratio due to the increase of neighbor density 4 Evaluation with real datasets We used two different types of real world datasets One was an Earth climate dataset related with vegetation growth [16] nother was an Ecology animal behavior dataset that contained chimpanzee observation data from 1999 to 21 [4] 1) Earth climate data: The earth climate dataset includes monthly measurements of variables such as global plant growth, eg, Net Primary Production(NPP), and climate variables, eg, precipitation(pre) on latitude-longitude spherical grids n example of a discovered co-location pattern was (NPP-Hi, PRE-Low), where Hi(Low) denotes an unusually high(low) value of the measurements The total number of event features was 18 The total number of instances was 15,515 We used 4 as a neighborhood distance, which means 4 cells (each grid cell is 1 degree 1degree) In Fig 11 (a), the join-less method shows much better performance at the lower threshold values The performance difference between the partial join method and the join-based method is relatively small because the cut relation ratio was almost 8 2) Ecology animal behavior data: The animal behavior dataset has 24 chimpanzee features We assigned a unique instance id to different location points per chimpanzee id The total number of point instances was 698 Fig 11 (b) presents

14 IEEE TRNSTIONS ON KNOWLEDGE ND DT ENGINEERING DRFT 14 the execution time of the algorithms by different 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 ccording to domain scientists in the application [4], the neighbor distances between 2 and 3 are considered meaningful distance thresholds In this context, our join-less method showed much better performance compared with the join-based method VIII ONLUSION ND FUTURE WORK In this paper, we propose the materialization of neighborhood information for efficient co-location pattern mining We present two neighborhood materialization models, a star partition model and a clique partition model, with no loss of co-location instances We compare the advantages and disadvantages of the two models in co-location pattern mining, and present the instance filtering schemes and prevalent co-location filtering schemes We develop a join-less co-location mining algorithm on the star partition model The algorithm is efficient since it does not require expensive spatial joins or instance joins but uses an instance-lookup scheme for identifying colocation instances We prove the join-less algorithm is correct and complete in finding co-location rules In addition, we describe a partial join approach for spatial data which are clustered in neighborhood areas We analyze the computation complexity of the join-less algorithm and the partial join algorithm, and provided analytical comparisons with the previous instance-join based algorithm In the experimental evaluation using synthetic and real world datasets, we examined the effects of our algorithm design decision, the scalability of the join-less algorithm and the behavior of the partial join algorithm The join-based method appears to be appropriate for sparse datasets However, the join-less method outperforms the join-based method in overall parameter settings and shows scalability to dense data In future work, many application domains include spatiotemporal features Scientists in these domains are interested in understanding the evolution of co-location patterns among stationary features as well as co-location patterns over moving feature types We plan to explore methods to answer temporal questions such as how a co-location changes over time as well as methods to identify moving object co-location patterns [4] Nsf-sei project: Spatio-temporal analysis for ecology behavior, [5] R garwal and R Srikant, Fast algorithms for Mining association rules, in Proc of the 2th VLD, 1994 [6] K Koperski and J Han, Discovery of Spatial ssociation Rules in Geographic Information Databases, in Proc of Fourth International Symposium on Large Spatial Data bases, Maine 47-66, 1995 [7] S Shekhar and Y Huang, o-location Rules Mining: Summary of Results, in Proc of International Symposium on Spatio and Temporal Database(SSTD), 21 [8] Y Morimoto, Mining Frequent Neighboring lass Sets in Spatial Databases, in Proc M SIGKDD International onference on Knowledge Discovery and Data Mining, 21 [9] D Papadias, N Mamoulis, and Y Theodoridis, Processing and Optimization of Multiway Spatial Joins Using R-Trees, in POPS, 1999 [1] H Park, G ha, and hung, Multi-way Spatial Joins Using R-Trees: Methodology and Performance Evaluation, in SSD, 1999 [11] J Yoo and S Shekhar, Partial Join pproach for Mining o-location Patterns, in Proc of M International Symposium on dvances in Geographic Information Systems(M-GIS), 24 [12] M Kuramochi and G Karypis, Fequent Subgraph Discovery, in IEEE International onference on Data Mining, 21 [13] X Yan and J Han, gspan: Graph-aseed Substructure Pattern Mining, in IEEE International onference on Data Mining, 21 [14] J Yoo and S Shekhar, Join-less pproach for o-location Pattern Mining: Summary of Results, in Proc of IEEE International onference on Data Mining(IDM), 25 [15] L rge, O Procopiuc, S Ramaswamy, T Suel, and J Vitter, Scalable Sweeping-ased Spatial Join, in Proc of the Int l onference on Very Large Databases, 1998 [16] Discovery of patterns in the global climate system using data mining, KNOWLEDGMENT This work was partially supported by NSF grant and Oak Ridge National Laboratory The content of this work does not necessarily reflect the position or policy of the government and no official endorsement should be inferred REFERENES [1] J Roddick and M Spiliopoulou, ibliography of Temporal, Spatial and Spatio-Temporal Data Mining Research, in Proc SIGKDD Explorations 1(1): 34-38, 1999 [2] S Shekhar and S hawla, Spatial Databases: Tour Prentice Hall, ISN , 23 [3] S Shekhar, P Zhang, Y Huang, and R Vatsavai, Trends in Spatial Data Mining, as a book chapter in Data Mining: Next Generation hallenges and Future Directions H Kargupta, Joshi, K Sivakumar and Y Yesha (editors), I/MIT Press, 24