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...n defined, we can apply clustering algorithm on spatial features. Clustering techniques may be broadly classified into partition based, hierarchical, density-based, grid-based and model-based schemes =-=[8]-=-. Resulting clusters may be globular, such as those found by partitionbased K-means or complete link hierarchical clustering algorithms or arbitrary shaped, such as those found by Proceedings of the 1...

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...ed with an spatial feature f ∈Fas f.O. The problem of finding spatial co-location patterns is to find subset of spatial features whose objects tend to locate together in spatial proximity. Clustering =-=[10]-=- divides a set of objects into several groups where objects in the same group are most similar to each other while objects from different groups are most dissimilar from each other. The first attempt ...

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...1 00 11 Figure 1. Illustration of Spatial Co-location Patterns. Symbols represent different spatial features. Spatial features • and ∆ tend to be located together. Mining spatial co-location patterns =-=[16, 13, 3]-=- is an important spatial data mining task with broad applications. Spatial co-location patterns represent subsets of the spatial features whose objects are often located in close geographic proximity....

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...ature j divided by the density of objects of spatial feature i in S Yes Yes No [0, area(S) πd2 ] Table 1. Proximity Measures for Spatial Features i and j in a Spatial Framework S density-based DBScan =-=[5]-=- or single link hierarchical clustering algorithms. Also clusters may be overlapping, such as those found by the model-based EM algorithm or nonoverlapping, such as those found by most partition based...

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...-negativity: sim(x, y) ≥ 0, ∀x, y ∈ O, (ii)commutativity: sim(x, y) =sim(y, x), ∀x, y ∈ O, and (iii) reflexivity: sim(x, y) =m ⇔ x = y where m is the largest possible value of the similarity function =-=[17]-=-. This means that x is most similar to itself and if two objects are most similar in the object set, they are the same object. Our goal is to use density ratio or some similar measures (will be discus...

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.... Transaction based approaches focus on defining transactions over space so that an Apriori-like algorithm [2] can be used. Transactions over space may be defined by a reference-feature centric model =-=[11]-=-. Under this model, transactions are created around objects of one user-specified spatial feature. The association rules are derived using the Apriori [2] algorithm. The rules found are all related to...

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...tion in the form of interesting and potentially useful patterns that were previously unknown. The automatic discovery of such patterns is being widely investigated with the use of spatial data mining =-=[14, 15]-=- techniques. 01 01 01 00 11 00 11 00 11 00 11 00 11 01 01 00 11 00 00 11 00 11 00 11 00 11 01 00 00 1100 00 11 00 11 00 11 01 01 01 01 00 11 01 00 11 01 01 01 00 00 11 11 00 11 00 11 f3 f1 f2 01 00 11...

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...tion in the form of interesting and potentially useful patterns that were previously unknown. The automatic discovery of such patterns is being widely investigated with the use of spatial data mining =-=[14, 15]-=- techniques. 01 01 01 00 11 00 11 00 11 00 11 00 11 01 01 00 11 00 00 11 00 11 00 11 00 11 01 00 00 1100 00 11 00 11 00 11 01 01 01 01 00 11 01 00 11 01 01 01 00 00 11 11 00 11 00 11 f3 f1 f2 01 00 11...

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...function and investigate its properties. Proximity between a pair of spatial features can also be defined by various spatially meaningful measures based on their objects including participation index =-=[9]-=-, join selectivity, G nearest neighbor function, F nearest neighbor function, J function, Ripley’s K function and its variations [3]. We summarize their properties as a proximity function for co-locat...

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...00 11 cluter all clustering objects analysis Figure 3. Mixed Clustering Approach Colocation Patterns tial features, resulting clusters do not represent spatial colocations. Association pattern mining =-=[2]-=- algorithms may be applied to derive co-location patterns among spatial features. Cluster sizes, overlapping property, and shapes are likely impact the quality of the mined co-location patterns. Vario...

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...1 00 11 Figure 1. Illustration of Spatial Co-location Patterns. Symbols represent different spatial features. Spatial features • and ∆ tend to be located together. Mining spatial co-location patterns =-=[16, 13, 3]-=- is an important spatial data mining task with broad applications. Spatial co-location patterns represent subsets of the spatial features whose objects are often located in close geographic proximity....

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...1 00 11 Figure 1. Illustration of Spatial Co-location Patterns. Symbols represent different spatial features. Spatial features • and ∆ tend to be located together. Mining spatial co-location patterns =-=[16, 13, 3]-=- is an important spatial data mining task with broad applications. Spatial co-location patterns represent subsets of the spatial features whose objects are often located in close geographic proximity....

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...n Section 5. 2 Related Work Due to the close relationships between clustering and colocation mining, finding co-location patterns via clustering, the focus of this paper, has attracted much attention =-=[7, 6]-=-. Clustering have been mainly applied in two ways: layer based clustering and mixed clustering as illustrated in Figure 2 and Figure 3 respectively. f1,f2,...fK 01 01 01 01 01 00 11 01 00 11 01 00 11 ...

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...n Section 5. 2 Related Work Due to the close relationships between clustering and colocation mining, finding co-location patterns via clustering, the focus of this paper, has attracted much attention =-=[7, 6]-=-. Clustering have been mainly applied in two ways: layer based clustering and mixed clustering as illustrated in Figure 2 and Figure 3 respectively. f1,f2,...fK 01 01 01 01 00 11 01 00 11 01 00 11 01 ...

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...co-location is a set of spatial features whose objects tend to locate together in spatial proximity. A spatial framework is a mapped set of geographic regions that supports agency programs or studies =-=[12]-=-. Formally, the spatial co-location mining problem is defined as follows: We are given a database D of spatial objects in a spatial framework S. Let F = {f1,f2,...,fK} be a set of spatial features. Ea...