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... [44] is a knowledge discovery system with a plug# 2 SPATIAL DATA MINING 5 value 0 1 Low Medium High Figure 1: Fuzzy values for numerical attribute in architecture. The classi\Thetacation system C4.5 =-=[38]-=- is one of the pluggins. In this integrated system, Kepler is used to classify data and Descartes is used to visualize and analyse source data and results of the classi\Thetacation on the map. Fuzzy S...

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...have to compare nodes according to the coordinate number (level(nod) mod k). This structure has one disadvantage: it is sensitive to the order in which the objects are inserted. 3.3 R#tree The R#tree =-=[19, 34]-=- is the modi\Thetacation of B-tree for spatial data. This tree is balanced and splits the space into the rectangles which can overlap. Each node except root contains from m to M children, where 2smsM=...

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...k#d#tree The quad tree is not balanced and its balance depends on the data distribution and the order of inserting the points. 3.2 k#d#tree This method uses a binary tree to split k#dimensional space =-=[3, 15, 34]-=-. This tree splits the space into two subspaces according to one of the coordinates of the splitting point (see Figure 3). Let level(nod) be the length of the path from the root to the node nod and su...

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...ity, so the algorithms for approximate solution are used. The R-tree is one of the most cited spatial data structures and it is very often used for comparison with new structures. 3.4 R*–tree R*-tre=-=e [2, 34] is a -=-modification of R–tree which uses different heuristic for operation INSERT. R–tree tries to minimize the area of all nodes of the tree. R*–tree combines more criteria: the area covered by a boun...

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...k#d#tree The quad tree is not balanced and its balance depends on the data distribution and the order of inserting the points. 3.2 k#d#tree This method uses a binary tree to split k#dimensional space =-=[3, 15, 34]-=-. This tree splits the space into two subspaces according to one of the coordinates of the splitting point (see Figure 3). Let level(nod) be the length of the path from the root to the node nod and su...

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...ce the number of distance computations during the search operation. The construction of the tree requires O(nlogmn) distance computations.s4 DATA STRUCTURES FOR METRIC SPACES 14 4.4 M–tree The M–t=-=ree [6]-=- is designed to partition a metric space with a distance function d (but it works for vector spaces, too). This function has the following properties: 1. symmetry: d(x, y) =d(y, x) 2. non negativity: ...

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...nalysis or \Thetaltering of raster data), measuring functions (e.g. distance, direction between objects), statistic analyses or terrain model analysis (e. g. visibility analysis). Spatial data mining =-=[11, 24]-=- is a special kind of data mining [12]. The main difference between data mining and spatial data mining is that in spatial data mining tasks we use not only non-spatial attributes (as it is usual in d...

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...RIC SPACES 11 1 2 5 6 3 A B C A B C 5 4 5 2 1 3 4 6 Figure 5: R+#tree 5 7 1 6 2 3 4 1 3 2 6 4 5 7 Figure 6: vp#tree 4 Data structures for metric spaces 4.1 Vp#tree The vp#tree (or vantage point tree) =-=[5]-=- partitions the data space around selected points and forms a hierarchical tree structure (see Figure 6). These selected points are called vantage points. Each internal node of the tree is of the form...

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...tion, we will describe several existing systems which can be used for spatial data mining and information sources concerning spatial data mining. 2.2 Spatial data mining systems GeoMiner The GeoMiner =-=[17, 21]-=- is a system for knowledge discovery in large spatial databases. It was developed at Simon Fraser University in Canada. The GeoMiner is an extension of and developed from DBMiner [9, 20]. The DBMiner ...

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...nalysis or \Thetaltering of raster data), measuring functions (e.g. distance, direction between objects), statistic analyses or terrain model analysis (e. g. visibility analysis). Spatial data mining =-=[11, 24]-=- is a special kind of data mining [12]. The main difference between data mining and spatial data mining is that in spatial data mining tasks we use not only non-spatial attributes (as it is usual in d...

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...ows to visualize one attribute with area coloration and another attribute (or attributes) with charts. In [1] the integration of Descartes with Kepler is used to classify spatial related data. Kepler =-=[44]-=- is a knowledge discovery system with a plug# 2 SPATIAL DATA MINING 5 value 0 1 Low Medium High Figure 1: Fuzzy values for numerical attribute in architecture. The classi\Thetacation system C4.5 [38] ...

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...n of the overlap between rectangles decreases the number of paths that must be searched. The implementation of this method is harder, but R*#trees are more effective than R-trees. 3.5 R+#tree R+#tree =-=[40]-=- is an extension of the R#tree. In contrast to R#tree bounding rectangles of the nodes at one level don't overlap in this structure. This feature decreases the number of searched branches of the tree ...

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...es is used to visualize and analyse source data and results of the classi\Thetacation on the map. Fuzzy Spatial OQL for Fuzzy KDD In [4], a fuzzy spatial object query language and fuzzy decision tree =-=[30]-=- are introduced. This language is designed to select, process and mine data from Spatial Object#Oriented Databases and it is based on a fuzzy set theory. In the classical theory, given a set S and an ...

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...artes offers various types of charts (bar, pie, etc.). The combination of both the methods allows to visualize one attribute with area coloration and another attribute (or attributes) with charts. In =-=[1]-=- the integration of Descartes with Kepler is used to classify spatial related data. Kepler [44] is a knowledge discovery system with a plug# 2 SPATIAL DATA MINING 5 value 0 1 Low Medium High Figure 1:...

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...The GeoMiner [17, 21] is a system for knowledge discovery in large spatial databases. It was developed at Simon Fraser University in Canada. The GeoMiner is an extension of and developed from DBMiner =-=[9, 20]-=-. The DBMiner is a relational data mining system, which uses Microsoft SQL Server 7.0 to store data. It contains the \Thetave following data mining modules:sAssociation, Classi\Thetacation, Clustering...

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...fy images not onlys5 DATA STRUCTURES IN SPATIAL DATA MINING 18 according to values of the channels of multispectral image, but we can also use some other information such as terrain model or altitude =-=[23]-=-. The classification of pixel (i. e. land cover type of pixel) depends not only on its own characteristics, but also on the characteristics of neighboring pixels. For example, if the classification of...

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...g spatial structures for both vector and metric spaces are described and structures used in some spatial data mining systems are presented. 1 Spatial data types in GIS A geographic information system =-=[43] i-=-s a special kind of information system, which allows manipulate, analyse, summarize, query, edit and visualize geographically related data. Geographically related data are composed of: • spatial att...

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...rimarily for metric spaces are described. And \Thetanally, structures used in some spatial data mining systems are presented in section 5. 3 Spatial data structures in GIS 3.1 Quad tree The quad tree =-=[13, 34]-=- is used to index 2D space. Each internal node of the tree splits the space into four disjunct subspaces (called NW, NE, SW, SE) according to the axes. Each of these subspaces is split recursively unt...

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...scussed and an inductive query language is proposed. Then, a description of mining system GWiM is given. GWiM is a system for knowledge discovery in spatial data. It is built upon the WiM system. WiM =-=[14]-=- uses inductive logic programming to synthesize closed Horn clauses.s2 SPATIAL DATA MINING 6 Query language of GWiM contains three types of queries. Two of them, characteristic and discriminant rules,...

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...this integrated system, Kepler is used to classify data and Descartes is used to visualize and analyse source data and results of the classi\Thetacation on the map. Fuzzy Spatial OQL for Fuzzy KDD In =-=[4]-=-, a fuzzy spatial object query language and fuzzy decision tree [30] are introduced. This language is designed to select, process and mine data from Spatial Object#Oriented Databases and it is based o...

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...ed primarily for metric spaces are described. And finally, structures used in some spatial data mining systems are presented in section 5. 3 Spatial data structures in GIS 3.1 Quad tree The quad tree =-=[13, 34]-=- is used to index 2D space. Each internal node of the tree splits the space into four disjunct subspaces (called NW, NE, SW, SE) according to the axes. Each of these subspaces is split recursively unt...

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...ity, so the algorithms for approximate solution are used. The R-tree is one of the most cited spatial data structures and it is very often used for comparison with new structures. 3.4 R*#tree R*-tree =-=[2, 34]-=- is a modi\Thetacation of R#tree which uses different heuristic for operation INSERT. R#tree tries to minimize the area of all nodes of the tree. R*#tree combines more criteria: the area covered by a ...

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...e the number of distance computations during the search operation. The construction of the tree requires O(nlog mn) distance computations. 4 DATA STRUCTURES FOR METRIC SPACES 14 4.4 M#tree The M#tree =-=[6]-=- is designed to partition a metric space with a distance functionsd (but it works for vector spaces, too). This function has the following properties: 1. symmetry: d(x; y) = d(y; x) 2. non negativity:...

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...scussed and an inductive query language is proposed. Then, a description of mining system GWiM is given. GWiM is a system for knowledge discovery in spatial data. It is built upon the WiM system. WiM =-=[14]-=- uses inductive logic programming to synthesize closed Horn clauses. 2 SPATIAL DATA MINING 6 Query language of GWiM contains three types of queries. Two of them, characteristic and discriminant rules,...

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... are detected. This step is based on the information that treetops have a greater reectance than their neighborhood. In the \Thetanal steps, duplicate identi\Thetacations of tops are \Thetaltered. In =-=[22]-=- the analysis of traf\Thetac accessibility with respect to public transport accessibility is described. This analysis requires the information about road infrastructure and time schedules of public tr...

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...fy images not only 5 DATA STRUCTURES IN SPATIAL DATA MINING 18 according to values of the channels of multispectral image, but we can also use some other information such as terrain model or altitude =-=[23]-=-. The classi\Thetacation of pixel (i. e. land cover type of pixel) depends not only on its own characteristics, but also on the characteristics of neighboring pixels. For example, if the classi\Thetac...

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...s a list of predicates or hierarchy of predicates. The answer to these queries is a formula of \Thetarst#order logic which characterizes the subset of the database speci\Thetaed by the rule. GeoKD In =-=[26, 27]-=-, a language for knowledge discovery in spatial data is proposed. Interpreter of this language is called GeoKD. This language contains three kinds of rules (classi\Thetacation, characteristic and disc...

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...s a list of predicates or hierarchy of predicates. The answer to these queries is a formula of \Thetarst#order logic which characterizes the subset of the database speci\Thetaed by the rule. GeoKD In =-=[26, 27]-=-, a language for knowledge discovery in spatial data is proposed. Interpreter of this language is called GeoKD. This language contains three kinds of rules (classi\Thetacation, characteristic and disc...

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... 15 3 1 7 4 1 4 7 2 3 5 6 1 2 5 7 4 6 Figure 9: m#tree 5 Data structures in spatial data mining 5.1 Neighborhood graphs Neighborhood graphs and neighborhood paths De\Thetanition: Neighborhood graph G =-=[11, 28]-=- for spatial relation neighbor=2 is a graph G(U,H) where U is a set of nodes and H is a set of edges. Each node represents an object and two nodes N1, N2 are connected by edge iff the objects correspo...

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...s of traf\Thetac accessibility with respect to public transport accessibility is described. This analysis requires the information about road infrastructure and time schedules of public transport. In =-=[29]-=- the distance analysis, network analysis and location#allocation REFERENCES 19 analysis are used in the analysis of Automatic Teller Machines (ATMs) distribution in Bratislava city. Goal of this appli...

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... in database and is used to create neighborhood paths of different lengths. Objects in these paths are used for classi\Thetacation of central pixel (\Thetarst pixel in the path). 5.5 Other methods In =-=[32]-=- the application of classi\Thetacation and trend detection in satellite raster images is described. The result of this application is an identi\Thetacation of the trends in the land use changes in the...

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...rimarily for metric spaces are described. And \Thetanally, structures used in some spatial data mining systems are presented in section 5. 3 Spatial data structures in GIS 3.1 Quad tree The quad tree =-=[13, 34]-=- is used to index 2D space. Each internal node of the tree splits the space into four disjunct subspaces (called NW, NE, SW, SE) according to the axes. Each of these subspaces is split recursively unt...

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...g spatial structures for both vector and metric spaces are described and structures used in some spatial data mining systems are presented. 1 Spatial data types in GIS A geographic information system =-=[43]-=- is a special kind of information system, which allows manipulate, analyse, summarize, query, edit and visualize geographically related data. Geographically related data are composed of: ffl spatial a...

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...e interpretation is an important method of GIS analysis. Given a multispectral satellite image, the goal is to classify all pixels according to their land cover types (e. g. water, field, forest). In =-=[10]-=- the algorithm C4.5 [38], which generates decision trees, is used for image interpretation. This algorithm allows to classify images not onlys5 DATA STRUCTURES IN SPATIAL DATA MINING 18 according to v...

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..., treetops are detected. This step is based on the information that treetops have a greater reflectance than their neighborhood. In the final steps, duplicate identifications of tops are filtered. In =-=[22]-=- the analysis of traffic accessibility with respect to public transport accessibility is described. This analysis requires the information about road infrastructure and time schedules of public transp...

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...ledge> is a list of predicates or hierarchy of predicates. The answer to these queries is a formula of first–order logic which characterizes the subset of the database specified by the rule. GeoKD I=-=n [26, 27]-=-, a language for knowledge discovery in spatial data is proposed. Interpreter of this language is called GeoKD. This language contains three kinds of rules (classification, characteristic and discrimi...

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...ledge> is a list of predicates or hierarchy of predicates. The answer to these queries is a formula of first–order logic which characterizes the subset of the database specified by the rule. GeoKD I=-=n [26, 27]-=-, a language for knowledge discovery in spatial data is proposed. Interpreter of this language is called GeoKD. This language contains three kinds of rules (classification, characteristic and discrimi...

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...NING 15 1 2 4 3 5 6 7 Figure 9: m–tree 1 4 7 1 2 3 4 5 6 7 5 Data structures in spatial data mining 5.1 Neighborhood graphs Neighborhood graphs and neighborhood paths Definition: Neighborhood graph =-=G [11, 28]-=- for spatial relation neighbor/2 is a graph G(U,H) where U is a set of nodes and H is a set of edges. Each node represents an object and two nodes N1, N2 are connected by edge iff the objects correspo...

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...lysis of traffic accessibility with respect to public transport accessibility is described. This analysis requires the information about road infrastructure and time schedules of public transport. In =-=[29] t-=-he distance analysis, network analysis and location–allocationsREFERENCES 19 analysis are used in the analysis of Automatic Teller Machines (ATMs) distribution in Bratislava city. Goal of this appli...

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...s stored in database and is used to create neighborhood paths of different lengths. Objects in these paths are used for classification of central pixel (first pixel in the path). 5.5 Other methods In =-=[32]-=- the application of classification and trend detection in satellite raster images is described. The result of this application is an identification of the trends in the land use changes in the dynamic...