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What's Special about Spatial Data Mining?

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  • 1. ' $ What’s Special about Spatial Data Mining? Shashi Shekhar Pusheng Zhang Yan Huang Ranga Raju Vatsavai Department of Computer Science and Engineering University of Minnesota Sea Surface Temperature (SST) in March, 1982 & %
  • 2. ' $ Application Domains Spatial data mining is used in • NASA Earth Observing System (EOS): Earth science data • National Inst. of Justice: crime mapping • Census Bureau, Dept. of Commerce: census data • Dept. of Transportation (DOT): traffic data • National Inst. of Health(NIH): cancer clusters Sample Global Questions from Earth Science • How is the global Earth system changing? • What are the primary forcings of the Earth system? • How does the Earth system respond to natural and human- included changes? • What are the consequences of changes in the Earth system for human civilization? • How well can we predict future changes in the Earth sys- tem & % What’s Special About Spatial Data Mining? 1
  • 3. ' $ Example of Application Domains Sample Local Questions from Epidemiology[TerraSeer] • What’s overall pattern of colorectal cancer? • Is there clustering of high colorectal cancer incidence any- where in the study area? • Where is colorectal cancer risk significantly elevated? • Where are zones of rapid change in colorectal cancer inci- dence? Figure 1: Geographic distribution of male colorectal cancer in Long Island, New York(in courtesy of TerraSeer) & % What’s Special About Spatial Data Mining? 2
  • 4. ' $ Spatial Slicing Spatial heterogeneity • “Second law of geography”[M. Goodchild, UCGIS 2003] • Global model might be inconsistent with regional models – spatial Simpson’s Paradox (a) Global Model (b) Regional Models Spatial Slicing • Slicing inputs can improve the effectiveness of SDM • Slicing output can illustrate support regions of a pattern – e.g., association rule with support map & % What’s Special About Spatial Data Mining? 3
  • 5. ' $ Location As Attribute Location as attribute in spatial data mining What value is location as an explanatory variable? • most events are associated with space and time • space is an important surrogate variable • critical to hypothesis formation about relationships among variables Domain Spatial Observa- Hypothesis Science tions Social Science central places, e.g., power law observed in social cities networks Animal Behavior co-occurrence(pant- chimpanzees use observed in hoot, food-bout) in pant-hoot to Gombe dataset space and time share abundant food sources Physical Science co-location(water in water carries ele- fluoride and den- Colorado Springs, ments related to tal health dental health) dental health Physical Science 1854, London: water carries 1883: germ the- co-location(water cholera agents ory pump, cholera) & % What’s Special About Spatial Data Mining? 4
  • 6. ' $ Spatial Data Mining (SDM) The process of discovering • interesting,useful, non-trivial patterns • from large spatial datasets Spatial patterns • Spatial outlier, discontinuities – bad traffic sensors on highways (DOT) • Location prediction models – model to identify habitat of endangered species • Spatial clusters – crime hot-spots (NIJ), cancer clusters (CDC) • Co-location patterns – predator-prey species, symbiosis – Dental health and fluoride & % What’s Special About Spatial Data Mining? 5
  • 7. ' $ Example Spatial Pattern: Spatial Cluster The 1854 Asiatic Cholera in London & % What’s Special About Spatial Data Mining? 6
  • 8. ' $ Example Spatial Pattern: Spatial Outliers and Predictive Models Spatial Outliers Average Traffic Volume(Time v.s. Station) 180 160 10 140 I35W Station ID(South Bound) 20 120 100 30 80 40 60 40 50 20 60 0 50 100 150 200 250 Time Predictive Models Nest sites for 1995 Darr location 0 10 20 30 40 Marsh land Nest sites 50 60 70 80 0 20 40 60 80 100 120 140 160 nz = 85 & % What’s Special About Spatial Data Mining? 7
  • 9. ' $ Example Spatial Pattern: Co-locations (backup) Given: • A collection of different types of spatial events Illustration Find: Co-located subsets of event types & % What’s Special About Spatial Data Mining? 8
  • 10. ' $ Overview Spatial Data Mining • Find interesting, potentially useful, non-trivial patterns from spatial data Components of Data Mining: • Input: table with many columns, domain(column) • Statistical Foundation • Output: patterns and interest measures – e.g., predictive models, clusters, outliers, associations • Computational process: algorithms & % What’s Special About Spatial Data Mining? 9
  • 11. ' $ General Approaches in SDM Materializing spatial features • e.g., spatial association rule mining[Koperski, Han, 1995] • commercial tools: e.g., Arc/Info family Spatial slicing • e.g., association rule with support map[P. Tan et al] 90 15 14 70 13 50 12 30 11 10 10 9 −10 8 −30 7 −50 6 −70 5 −90 4 −180 −140 −100 −60 −20 20 60 100 140 180 Support Figure 2: Association rule with support map(FPAR-high → NPP-high) • commercial tools: e.g.,Matlab, SAS, R, Splus Customized spatial techniques • e.g., MRF-based Bayesian Classifier • commercial tools – e.g.,Splus spatial/R spatial/terraseer + customized codes & % What’s Special About Spatial Data Mining? 10
  • 12. ' $ Overview ⇒ Input Statistical Foundation Output Computational process & % What’s Special About Spatial Data Mining? 11
  • 13. ' $ Overview of Input Data • Table with many columns(attributes) tid f1 f2 ... fn Table 1: Example of Input Table – e.g., tid: tuple id; fi : attributes • Spatial attribute: geographically referenced • Non-spatial attribute: traditional Relationships among Data • Non-spatial • Spatial & % What’s Special About Spatial Data Mining? 12
  • 14. ' $ Data in Spatial Data Mining Non-spatial Information • Same as data in traditional data mining • Numerical, categorical, ordinal, boolean, etc • e.g., city name, city population Spatial Information • Spatial attribute: geographically referenced – Neighborhood and extent – Location, e.g., longitude, latitude, elevation • Spatial data representations – Raster: gridded space – Vector: point, line, polygon – Graph: node, edge, path Figure 3: Raster and Vector Data for UMN Campus (in courtesy of UMN, MapQuest) & % What’s Special About Spatial Data Mining? 13
  • 15. ' $ Relationships on Data in Spatial Data Mining Relationships on non-spatial data • Explicit • Arithmetic, ranking(ordering), etc. • Object is instance of a class, class is a subclass of another class, object is part of another object, object is a mem- bership of a set Relationships on Spatial Data • Many are implicit • Relationship Categories – Set-oriented: union, intersection, and membership, etc – Topological: meet, within, overlap, etc – Directional: North, NE, left, above, behind, etc – Metric: e.g., Euclidean: distance, area, perimeter – Dynamic: update, create, destroy, etc – Shape-based and visibility • Granularity Granularity Elevation Example Road Example local elevation on road? focal slope adjacent to road? zonal highest elevation in a zone distance to nearest road Table 2: Examples of Granularity & % What’s Special About Spatial Data Mining? 14
  • 16. ' $ Mining Implicit Spatial Relationships Choices: • Materialize spatial info + classical data mining • Customized spatial data mining techniques Relationships Materialization Customized SDM Tech. Topological Neighbor, Inside, Outside Classical Data Mining NEM, co-location Euclidean Distance, can be used K-means density DBSCAN Directional North, Left, Above Clustering on sphere Others Shape, visibility Table 3: Mining Implicit Spatial Relationships What spatial info is to be materialized? • Distance measure: – Point: Euclidean – Extended objects: buffer-based – Graph: shortest path • Transactions: i.e., space partitions – Circles centered at reference features – Gridded cells – Min-cut partitions – Voronoi diagram & % What’s Special About Spatial Data Mining? 15
  • 17. ' $ Overview √ Input ⇒ Statistical Foundation Output Computational process & % What’s Special About Spatial Data Mining? 16
  • 18. ' $ Statistics in Spatial Data Mining Classical Data Mining • Learning samples are independently distributed • Cross-correlation measures, e.g., χ2 , Pearson Spatial Data Mining • Learning sample are not independent • Spatial Autocorrelation – Measures: ∗ distance-based(e.g., K-function) ∗ neighbor-based(e.g., Moran’s I) • Spatial Cross-Correlation – Measures: distance-based, e.g., cross K-function • Spatial Heterogeneity & % What’s Special About Spatial Data Mining? 17
  • 19. ' $ Overview of Statistical Foundation Spatial Statistics[Cressie, 1991] • Geostatistics – Continuous – Variogram: measure how similarity decreases with distance – Spatial prediction: spatial autocorrelation • Lattice-based statistics – Discrete location, neighbor relationship graph – Spatial Gaussian models ∗ Conditionally specified spatial Gaussian model ∗ Simultaneously specified spatial Gaussian model – Markov Random Fields, Spatial Autoregressive Model • Point process – Discrete – Complete spatial randomness (CSR): Poisson process in space – K-function: test of CSR Point Process Lattice Geostatistics √ √ raster √ √ √ vector point √ line √ √ polygon graph Table 4: Data Types and Statistical Models & % What’s Special About Spatial Data Mining? 18
  • 20. ' $ Spatial Autocorrelation(SA) First Law of Geography • ”All things are related, but nearby things are more related than distant things. [Tobler, 1970]” White Noise −No spatial autocorrelation Vegetation distribution across the marshland 0 0 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 0 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160 nz = 5372 nz = 5372 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 10 20 30 40 50 60 70 80 90 (a) Pixel property with independent iden- (b) Vegetation Durability with SA tical distribution Figure 4: Spatial Randomness vs. Autocorrelation Spatial autocorrelation • Nearby things are more similar than distant things • Traditional i.i.d. assumption is not valid • Measures: K-function, Moran’s I, Variogram, ··· & % What’s Special About Spatial Data Mining? 19
  • 21. ' $ Spatial Autocorrelation: Distance-based Measure K-function Definition: • Test against randomness for point pattern • K(h) = λ−1E [number of events within distance h of an arbitrary event] – λ is intensity of event • Model departure from randomness in a wide range of scales Inference • For Poisson complete spatial randomness(csr): K(h) = πh2 • Plot Khat(h) against h, compare to Poisson csr – >: cluster – <: decluster/regularity 1400 Poisson CSR Cluster Process Decluster Process 1200 Envelope 1000 800 K−function 600 400 200 0 −200 0 2 4 6 8 10 12 14 16 18 20 Distance h & % What’s Special About Spatial Data Mining? 20
  • 22. ' $ Spatial Autocorrelation: Topological Measure Moran’s I Measure Definition: zW z t MI = zz t • z = {x1 − x, . . . , xn − x} ¯ ¯ – xi : data values – ¯ x: mean of x – n: number of data • W : the contiguity matrix Ranges between -1 and +1 • higher positive value ⇒ high SA, Cluster, Attract • lower negative value ⇒ interspersed, de-clustered, repel • e.g., spatial randomness ⇒ MI = 0 • e.g., distribution of vegetation durability ⇒ MI = 0.7 • e.g., checker board ⇒ MI = -1 & % What’s Special About Spatial Data Mining? 21
  • 23. ' $ Cross-Correlation Cross K-Function Definition • Kij (h) = λ−1E [number of type j event within distance h j of a randomly chosen type i event] • Cross K-function of some pair of spatial feature types • Example – Which pairs are frequently co-located? – Statistical significance Co−location Patterns − Sample Data 80 70 60 50 40 Y 30 20 10 0 0 10 20 30 40 50 60 70 80 X Figure 5: Example Data (o and * ; x and +) & % What’s Special About Spatial Data Mining? 22
  • 24. ' $ Illustration of Cross-Correlation Illustration of Cross K-Function for Example Data Cross−K function of pairs of spatial features 1000 2 y=pi*h 800 o and * x and + Cross−K function * and x 600 * and + 400 200 0 0 2 4 6 8 10 Distance h Figure 6: Cross K-function for Example Data & % What’s Special About Spatial Data Mining? 23
  • 25. ' $ Overview √ Input √ Statistical Foundation ⇒ Output Computational process & % What’s Special About Spatial Data Mining? 24
  • 26. ' $ Overview of Data Mining Output Supervised Learning: Prediction • Classification • Trend Unsupervised Learning: • Clustering • Outlier Detection • Association Input Data Types vs. Output Patterns Patterns Point Process Lattice Geostatistics √ √ Prediction √ Trend √ √ Clustering √ √ √ Outliers √ √ Associations Table 5: Output Patterns vs. Statistical Models & % What’s Special About Spatial Data Mining? 25
  • 27. ' $ Illustrative Application to Location Prediction (Backup) Nest sites for 1995 Darr location Vegetation distribution across the marshland 0 0 10 10 20 20 30 30 40 40 Marsh land 50 Nest sites 60 50 70 60 80 70 0 20 40 60 80 100 120 140 160 nz = 5372 80 0 20 40 60 80 100 120 140 160 nz = 85 0 10 20 30 40 50 60 70 80 90 (a) Nest Locations (b) Vegetation Water depth variation across marshland Distance to open water 0 0 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 0 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160 nz = 5372 nz = 5372 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 (c) Water Depth (d) Distance to Open Water & % What’s Special About Spatial Data Mining? 26
  • 28. ' $ Prediction and Trend Prediction • Continuous: trend, e.g., regression – Location aware: spatial autoregressive model(SAR) • Discrete: classification, e.g., Bayesian classifier – Location aware: Markov random fields(MRF) Classical Spatial y = Xβ + y = ρW y + Xβ + P r(Ci|X) = P r(X|Ci)P r(Ci) P r(ci|X, CN ) = P r(ciP r(X,C )N |ci) P r(X) )∗P r(X,C N Table 6: Prediction Models • e.g., ROC curve for SAR and regression ROC Curve for learning data(Darr 95) ROC Curve for testing data(Stubble marshland 1995) 1 1 0.9 0.9 0.8 0.8 Truth Positive Rate 0.7 0.7 Truth Positive Rate 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 Classical Regression Classical Regression 0.1 0.1 Spatial Regression Spatial Regression 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 False Positive Rate False Positive Rate (e) ROC curves for learning (f) ROC curves for testing Figure 7: (a) Comparison of the classical regression model with the spatial autoregressive model on the Darr learning data. (b) Comparison of the models on the Stubble testing data. & % What’s Special About Spatial Data Mining? 27
  • 29. ' $ Prediction and Trend Open Problems • Estimate W for SAR • Spatial interest measure: e.g., avg dist(actual, predicted) P Legend A P P A P A = nest location A = actual nest in pixel P P P = predicted nest in pixel A A A A A A (a) (b) (c) (d) Figure 8: An example showing different predictions: (a)The actual sites, (b)Pixels with actual sites, (c)Prediction 1, (d)Prediction 2. Prediction 2 is spatially more accurate than 1. & % What’s Special About Spatial Data Mining? 28
  • 30. ' $ Clustering Clustering: Find groups of tuples Statistical Significance • Complete spatial randomness, cluster, and decluster Figure 9: Inputs: Complete Spatial Random (CSR), Cluster, and Decluster Figure 10: Classical Clustering 2 3 1 3 2 3 Data is of Complete 2 2 3 Data is of 4 1 Spatial Randomness 2 Decluster Pattern 3 3 1: Unusually Dense 2: Desnse 3: Mean Dense 4: Sparse Figure 11: Spatial Clustering & % What’s Special About Spatial Data Mining? 29
  • 31. ' $ Clustering Similarity Measures • Non-spatial: e.g., soundex • Classical clustering: Euclidean, metric, graph-based • Topological: neighborhood EM – Implicitly based on locations • Interest measure: – spatial continuity – cartographic generalization – unusual density – keep nearest neighbors in common cluster & % What’s Special About Spatial Data Mining? 30
  • 32. ' $ Outlier Detection Spatial Outlier Detection • Finding anomalous tuples • Global and spatial outlier • Detection Approaches – Graph-based outlier detection: variogram, moran scatter plot – Quantitative outlier detection: scatter plot, and z-score Location-awareness • All tuples/No tuple: classical • Some tuple: locations for neighborhood and non-spatial attributes for difference test Original Data Points Average Traffic Volume(Time v.s. Station) 8 Data Point 180 G Fitting Curve ←S I35W Station ID(North Bound) 7 160 10 6 140 20 Attribute Values 120 5 P→ 100 4 30 D↑ 80 3 Q→ 40 60 2 40 L 50 1 20 60 0 0 0 2 4 6 8 10 12 14 16 18 20 2 4 6 8 1012 14 16 18 20 22 24 Location Time (a) Outliers in Example Data (b) Outliers in Traffic Data & % What’s Special About Spatial Data Mining? 31
  • 33. ' $ Association Association • Domain(fi) = union { any, domain(fi )} • Finding frequent itemsets from fi • Co-location – Effect of transactionizing: loss of info – Alternative: use spatial join, statistics A1 C1 A1 C1 B1 B1 A2 A2 B2 B2 C2 C2 Figure 12: Different Transactionizing Schemes Location-awareness • All tuples: co-location mining • No tuple: classical association rule mining • Some tuple: future work & % What’s Special About Spatial Data Mining? 32
  • 34. ' $ Output Patterns Output Patterns vs. Input Vector Data SDM Techniques Point Line Polygon Raster Data √ √ √ classification - √ √ association - √ √ clustering - √ √ outlier detection Table 7: Output Patterns vs. Input Output Patterns vs. Interest Measures Traditional Non-spatial Spatial Mixture Predictive Classification accuracy Spatial accuracy, e.g., avg Future Work Model dist(actual site, predicted site) Cluster Low coupling and high cohe- Spatial continuity, unusual Future Work sion in feature space density, cartographic general- ization Outlier Different from population or Geographically distant from Significant at- neighbors in feature space neighbors tribute discon- tinuity in geo- graphic space Association Subset prevalence, Clique prevalence Future Work P r[B ∈ T | A ∈ T ], P r[B ∈ N (L) | A at L] Correlation: e.g., Cross K-Function Table 8: Output Patterns vs. Interest Measures & % What’s Special About Spatial Data Mining? 33
  • 35. ' $ Output Patterns vs. Location Awareness Output Patterns vs. Location Awareness • No awareness: no location info • Total awareness: location info available for all tuples • Partial awareness: location info missing for some tuples No Awareness Total Awareness Partial Awareness Prediction Decision tree, nearest neighbor, kriging, MRF Bayesian classi- future work Bayesian classifier, neural net- fier, self-organizing map, spa- work, regression tial autoregressive model Clustering EM in feature space, k-means, Neighborhood EM future work density-based, graph-based Outliers Neighbor def: feature domain Neighbor def: geographic do- future work main Difference test def: feature do- Difference test def: feature do- main main Association Association rules Co-location future work Table 9: Output vs. Location Awareness & % What’s Special About Spatial Data Mining? 34
  • 36. ' $ Overview √ Input √ Statistical Foundation √ Output ⇒ Computational process & % What’s Special About Spatial Data Mining? 35
  • 37. ' $ Computational Process Most algorithmic strategies are applicable Algorithmic Strategies in Spatial Data Mining: Classical Algorithms Algorithmic Strategies in SDM Comments Divide-and-Conquer Space Partitioning possible info loss Filter-and-Refine Minimum-Bounding-Rectangle(MBR), Predi- cate Approximation Ordering Plane Sweeping, Space Filling Curves possible info loss Hierarchical Structures Spatial Index, Tree Matching Parameter Estimation Parameter estimation with spatial autocorre- lation Table 10: Algorithmic Strategies in Spatial Data Mining Challenges • Does spatial domain provide computational efficiency? – Low dimensionality: 2-3 – Spatial autocorrelation – Spatial indexing methods • Generalize to solve spatial problems – Linear regression vs SAR ∗ Continuity matrix W is assumed known for SAR, however, estimation of anisotropic W is non-trivial – Spatial outlier detection: spatial join – Co-location: bunch of joins & % What’s Special About Spatial Data Mining? 36
  • 38. ' $ Example of Computational Process Teleconnection • Find locations with climate correlation over θ – e.g., El Nino affects global climate Figure 13: Global Influence of El Nino during the Northern Hemisphere Winter(D: Dry; W:Warm; R:Rainfall) Challenge: high dim(e.g., 600) feature space Computational Efficiency Idea • Observation: Spatial autocorrelation • Spatial indexing to organize locations – Top-down tree traversal is a strong filter – Spatial join query: filter-and-refine ∗ 50 year long monthly data on 67k land locations and 100k ocean locations ∗ save 40% to 98% computational cost at θ = 0.3 to 0.9 & % What’s Special About Spatial Data Mining? 37
  • 39. ' $ Summary What’s Special About Spatial Data Mining? • Input Data • Statistical Foundation • Output Patterns • Computational Process Classical DM Spatial DM Input All explicit, simple types often Implicit relationships, complex types Stat Foundation Independence of samples spatial autocorrelation Output Interest Measures: set-based Location-awareness Computational Process Combinatorial optimization Computational efficiency opportunity Spatial autocorrelation, plane-sweeping Numerical alg. New complexity: SAR, co-location mining Estimation of anisotropic W is nontrivial Table 11: Summary of Spatial Data Mining A Hard Problem: • Estimate W besides ρ and β for y = ρW y + Xβ + y ρ W y X β ε = + + nx1 nxn nx1 nxmmx1 nx1 & % What’s Special About Spatial Data Mining? 38
  • 40. ' $ Research Needs Research Issues: • Classical DM techniques vs. SDM techniques • Statistical interpretation models for spatial patterns – e.g., co-location and Ripley’s K-function • Spatial interest measures: e.g., spatial accuracy • Modeling semantically rich spatial properties • Visualization • Improving computational efficiency • Preprocessing & % What’s Special About Spatial Data Mining? 39
  • 41. ' $ Conclusions Applications of Spatial Data Mining • Businesses, e.g. logistics, marketing, ... • Government - almost all branches e.g. defense, public safety, ... Rationale for spatial data mining • Simpson’s paradox and 2nd law of Geography • Space as a surrogate variable – Ex. co-location(water, cholera) led to Germ theory • Unique properties of spatial data, e.g. auto-correlation Approaches to mine spatial data • A. Traditional DM methods + spatial feature selection + Easy to start with − But results are weak due to spatial-autocorrelation etc. • B. Novel spatial DM methods + Better models unique properties of spatial data + Often improves results + Sometime reduces computation costs & % What’s Special About Spatial Data Mining? 40
  • 42. ' $ References Email: shekhar@cs.umn.edu More – http://www.cs.umn.edu/research/shashi-group References • [Cressie, 1991], N. Cressie, Statistics for Spatial Data, John Wiley and Sons, 1991 • [Miller, Han, 2001], H. Miller and J. Han(eds), Geographic Data Mining and Knowledge Discovery, Taylor and Francis, 2001 • [Kazar at al., 2003], B. Kazar, S. Shekhar, and D. Lilja, Parallel Formulation of Spatial Auto-Regression, Army High-Performance Computing Research Center (AHPCRC) Technical Report no. 2003-125, August 2003 • [Koperski, Han, 1995], K. Kopperski and J. Han, Discovery of Spatial Asso- ciation Rules in Geographic Information Database, SSTD, 1995 • [Koperski et al, 1996], K. Kopperski, J. Adhikary, and J. Han, Spatial Data Mining: Progress and Challenges, DMKD, 1996 • [Roddick, 2001], J. Roddick, K. Hornsby and M. Spiliopoulou, Yet Another Bibliography of Temporal, Spatial Spatio-temporal Data Mining Research, KDD Workshop, 2001 • [Shekhar et al, 2003], S. Shekhar, C. T. Lu, and P. Zhang, A Unified Ap- proach to Detecting Spatial Outliers, GeoInformatica, 7(2), Kluwer Academic Publishers, 2003 • [Shekhar, Chawla, 2003], S. Shekhar and S. Chawla, Spatial Databases: A Tour, Prentice Hall, 2003 & % What’s Special About Spatial Data Mining? 41
  • 43. ' $ References References • [Shekhar et al, 2002], S. Shekhar, P. Schrater, R. Vatsavai, W. Wu, and S. Chawla, Spatial Contextual Classification and Prediction Models for Mining Geospatial Data, IEEE Transactions on Multimedia (special issue on Mul- timedia Databases), 2002 • [Shekhar et al, 2001], S. Shekhar and Y. Huang, Discovering Spatial Co- location Patterns: A Summary of Results ,SSTD, 2001 • [Tan et al, 2001], P. Tan and M. Steinbach and V. Kumar and C. Potter and S. Klooster and A. Torregrosa, Finding Spatio-Temporal Patterns in Earth Science Data, KDD Workshop on Temporal Data Mining, 2001 • [Tobler, 1970], W. Tobler, A Computer Movie Simulating Urban Growth of Detroit Region, Economic Geography, 46:236-240, 1970 • [Zhang at al, 2003], P. Zhang, Y. Huang, S. Shekhar, and V. Kumar, Exploiting Spatial Autocorrelation to Efficiently Process Correlation-Based Sim- ilarity Queries, SSTD, 2003 & % What’s Special About Spatial Data Mining? 42
  • 44. ' $ Spatial Databases: A Tour & % What’s Special About Spatial Data Mining? 43

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