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Discovering Interesting Regions in Spatial Data Sets

Discovering Interesting Regions in Spatial Data Sets. Christoph F. Eick for Data Mining Class Motivation: Examples of Region Discovery Region Discovery Framework A Fitness For Hotspot Discovery Other Fitness Functions A Family of Clustering Algorithms for Region Discovery Summary.

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Discovering Interesting Regions in Spatial Data Sets

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  1. Discovering Interesting Regions inSpatial Data Sets Christoph F. Eick for Data Mining Class Motivation: Examples of Region Discovery Region Discovery Framework A Fitness For Hotspot Discovery Other Fitness Functions A Family of Clustering Algorithms for Region Discovery Summary

  2. Discovering Interesting Regions inSpatial Data Sets Christoph F. Eick for Data Mining Class Motivation: Examples of Region Discovery Region Discovery Framework A Fitness For Hotspot Discovery Other Fitness Functions A Family of Clustering Algorithms for Region Discovery Summary

  3. Next 2-3 Classes Region Discovery Framework DBSCAN Hierarchical Clustering Clustering Algorithms for Region Discovery: Clever,… Critical Issues with Respect to Clustering Programming Project-specific Discussion Similarity Assessment

  4. 1. Motivation: Examples of Region Discovery Application 1: Hot-spot Discovery [this presentation, [EVJW07] Application 2: Regional Association Rule Miningand Scoping [DEWY06, DEYWN07] Application 3: Find Interesting Regions with respect to a Continuous Variable Application 4: Regional Co-location Mining [EPWSN07] Application 5: Find “representative” regions (Sampling) b=1.01 RD-Algorithm b=1.04 Wells in Texas: Green: safe well with respect to arsenic Red: unsafe well

  5. 2. Region Discovery Framework • We assume we have spatial or spatio-temporal datasets that have the following structure: (x,y,[z],[t];<non-spatial attributes>) e.g. (longitude, lattitude, class_variable) or (longitude, lattitude, continous_variable) • Clustering occurs in the (x,y,[z],[t])-space; regions are found in this space. • The non-spatial attributes are used by the fitness function but neither in distance computations nor by the clustering algorithm itself. • For the remainder of the talk, we view region discovery as a clustering task and assume that regions and clusters are the same

  6. Region Discovery Framework Continued The algorithms we currently investigate solve the following problem: Given: A dataset O with a schema R A distance function d defined on instances of R A fitness function q(X) that evaluates clustering X={c1,…,ck} as follows: q(X)= cXreward(c)=cXinterestingness(c)*size(c) with b>1 Objective: Find c1,…,ck O such that: • cicj= if ij • X={c1,…,ck} maximizes q(X) • All cluster ciX are contiguous (each pair of objects belonging to ci has to be delaunay-connected with respect to ci and to d) • c1,…,ck  O • c1,…,ck are usually ranked based on the reward each cluster receives, and low reward clusters are frequently not reported

  7. Challenges for Region Discovery • Recall and precision with respect to the discovered regions should be high • Definition of measures of interestingness and of corresponding parameterized reward-based fitness functions that capture “what domain experts find interesting in spatial datasets” • Detection of regions at different levels of granularities (from very local to almost global patterns) • Detection of regions of arbitrary shapes • Necessity to cope with very large datasets • Regions should be properly ranked by relevance (reward); in many application only the top-k regions are of interest • Design and implementation of clustering algorithms that are suitable to address challenges 1, 3, 4, 5 and 6.

  8. 3. Fitness Function for Hot Spot Discovery Class of Interest: Unsafe_Well Prior Probability: 20% γ1 = 0.5, γ2 = 1.5; R+ = 1, R-= 1; β = 1.1, =1. 10% 30%

  9. 4. Fitness Functions for Other Region Discovery Tasks 4.1 Creating Contour Maps for Water Temperature (Temp) Fig. 1: Sea Surface Temperature on July 7 2002 Var=2.2 Reward: 48,5 Rank: 3 Mean=11.2 A single region and its summary Examples in the data set WT have the form: (x,y,temp); var(c,temp) denotes the variance of variable temp in region c interestingness(c)= IF var(c,temp)>var(WT,temp) THEN 0 ELSE min(1, log20(var(WT,temp)/var(c,temp))) with  being a parameter (with default 1) Basically, regions receive rewards if their variance is lower than the variance of the variable temperature for the whole data set, and regions whose variance is at least 20 times less receive the maximum reward of 1.

  10. 4.2 Finding Regions with High Water Temperature Differences Examples in the data set WT have the form: (x,y,Temp) Fitness function: Let c be a cluster to be evaluated interestingness(c)= IF var(c,temp)<var(WT,temp) THEN 0 ELSE min(1, log20(var(c,temp)/var(WT,temp))) ) with  being a parameter (with default 1)

  11. 4.3 Programming Project Fitness Functions Purity r1 r2 (6, 2, 2) r3 (0, 0, 5) (2,2,1) We assume we have 3 classes; in r1 we have 6 objects of class1, 3 objects of class 2, and 2 objects of class1 We assume th=0.5 and =2 i(r1)= (0.6-0.5)**2=0.01 i(r2)=(1-0.5)**2=0.25 i(r3)=0 q(X)=q({r1,r2,r3})= 0.01*10b + 0.25*5b

  12. Programming Project Fitness Functions Variance r3 Var(r3)=1100 r1 var(r1)=80 r2 Var(r2)=200 O Var(O)=100 r4 Var(r4)=20 We assume =1 and b=10 i(r1)= 0 i(r2)=log10(2)=0.3010 i(r3)=1 i(r4)=0

  13. Programming Project Function MSE r1 r2 (2,2) (4,4) (-1,-1) (-7,-7) (-4,-4) MSE(r1)=(1**2+1**2+1**2+1**2+1**2)/2=2 MSE(r2)=(3**2+3**2+3**2+3**2+1**2+0+0)/3=12

  14. 4.4 Regional Co-location Mining R1 R2 Regional Co-location R3 R4 Task: Find Co-location patterns for the following data-set. Global Co-location: and are co-located in the whole dataset

  15. A Reward Function for Binary Co-location Task: Find regions in which the density of 2 or more classes is elevated. In general, multipliers lC are computed for every region r, indicating how much the density of instances of class C is elevated in region r compared to C’s density in the whole space, and the interestness of a region with respect to two classes C1 and C2 is assessed proportional to the product lC1*lC2 Example: Binary Co-Location Reward Framework; lC(r)=p(C,r)/prior(C) C1,C2 = 1/((prior(C1)+prior(C2)) “maximum multiplier” kC1,C2(r) = IF lC1(r)<1 or lC2(r )<1 THEN 0 ELSE sqrt((lC1(r)–1)*(lC2(r)–1))/(C1,C2 –1) interestingness(r)= maxC1,C2;C1C2 (kC1,C2(c))

  16. The Ultimate Vision of the Presented Research DomainExpert Spatial Databases Measure ofInterestingness Acquisition Tool Database Integration Tool Fitness Function Data Set Family of Clustering Algorithms Region DiscoveryDisplay Ranked Set of Interesting Regions and their Properties Visualization Tools Architecture Region Discovery Engine

  17. How to Apply the Suggested Methodology • With the assistance of domain experts determine structure of dataset to be used. • Acquire measure of interestingness for the problem of hand (this was purity, variance, MSE, probability elevation of two or more classes in the examples discussed before) • Convert measure of interestingness into a reward-based fitness function. The designed fitness function should assign a reward of 0 to “boring” regions. It is also a good idea to normalize rewards by limiting the maximum reward to 1. • After the region discovery algorithm has been run, rank and visualize the top k regions with respect to rewards obtained (interestingness(c)*size(c)), and their properties which are usually task specific.

  18. 5. A Family of Clustering Algorithms for Region Discovery • Supervised Partitioning Around Medoids (SPAM). • Representative-based Clustering Using Randomized Hill Climbing (CLEVER) • Supervised Clustering using Evolutionary Computing (SCEC) • Agglomerative Hierarchical Supervised Clustering (SCAH) • Hierarchical Grid-based Supervised Clustering (SCHG) • Supervised Clustering using Multi-Resolution Grids (SCMRG) • Representative-based Clustering with Gabriel Graph Based Post-processing (MOSAIC) • Supervised Clustering using Density Estimation Techniques (SCDE) Remark: For a more details about SCEC, SPAM, SRIDHCR see [EZZ04, ZEZ06]; the PKDD06 paper briefly discusses SCAH, SCHG, SCMRG

  19. SCAH (Agglomerative Hierarchical) Inputs: A dataset O={o1,...,on} A distance Matrix D = {d(oi,oj) | oi,oj O }, Output: Clustering X={c1,…,ck} Algorithm: 1) Initialize: Create single object clusters: ci = {oi}, 1≤ i ≤ n; Compute merge candidates based on “nearest clusters” 2) DO FOREVER a) Find the pair (ci, cj) of merge candidates that improves q(X) the most b) If no such pair exist terminate, returning X={c1,…,ck} c) Delete the two clusters ci and cjfrom X and add the cluster ci  cj to X d) Update inter-cluster distances incrementally e) Update merge candidates based on inter-cluster distances

  20. SCHG (Hierarchical Grid-based) Remark: Same as SCAH, but uses grid cells as initial clusters Inputs: A dataset O={o1,...,on} A grid structure G Output: Clustering X={c1,…,ck} Algorithm: 1) Initialize: Create clusters making each single non-empty grid cell a cluster Compute merge candidates (all pairs of neighboring grid cells) 2) DO FOREVER a) Find the pair (ci, cj) of merge candidates that improves q(X) the most b) If no such pair exist terminate, returning X={c1,…,ck} c) Delete the two clusters ci and cjfrom X and add the cluster c’=ci  cj to X d) Update merge candidates: cX (MC(c’,c)  MC(c, ci)  MC(c, cj ))

  21. Ideas SCMRG (Divisive, Multi-Resolution Grids) Cell Processing Strategy 1. If a cell receives a reward that is larger than the sum of its rewards its ancestors: return that cell. 2. If a cell and its ancestor do not receive any reward: prune 3. Otherwise, process the children of the cell (drill down)

  22. Code SCMRG

  23. Problems with SCAH Too restrictive definition of merge candidates: XXXOOO OOOXXX No look ahead: Non-contiguous clusters:

  24. 6. Summary • A framework for region discovery that relies on additive, reward-based fitness functions and views region discovery as a clustering problem has been introduced. • Evidence concerning the usefulness of the framework for hot spot discovery problems has been presented. • As a by-product some known and not so well known flaws of hierarchical clustering algorithms have been identified. • The ultimate vision of this research is the development of region discovery engines that assist earth scientists in finding interesting regions in spatial datasets.

  25. Why should people use Region Discovery Engines (RDE)? RDE: finds sub-regions with special characteristics in large spatial datasets and presents findings in an understandable form. This is important for: • Focused summarization • Find interesting subsets in spatial datasets for further studies • Identify regions with unexpected patterns; because they are unexpected they deviate from global patterns; therefore, their regional characteristics are frequently important for domain experts • Without powerful region discovery algorithms, finding regional patters tends to be haphazard, and only leads to discoveries if ad-hoc region boundaries have enough resemblance with the true decision boundary • Exploratory data analysis for a mostly unknown dataset • Co-location statistics frequently blurred when arbitrary region definitions are used, hiding the true relationship of two co-occurring phenomena that become invisible by taking averages over regions in which a strong relationship is watered down, by including objects that do not contribute to the relationship (example: High crime-rates along the major rivers in Texas) • Data set reduction; focused sampling

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