1. Discovering Interesting Regions in Spatial Data Sets Christoph F. Eick for the Data Mining Class 1.Motivation: Examples of Region Discovery 2.Region Discovery Framework 3.A Fitness For Hotspot Discovery 4.Other Fitness Functions 5.A Family of Clustering Algorithms for Region Discovery 6.Summary

2. Discovering Interesting Regions in Spatial Data Sets Christoph F. Eick for Data Mining Class 1.Motivation: Examples of Region Discovery 2.Region Discovery Framework 3.A Fitness For Hotspot Discovery 4.Other Fitness Functions 5.A Family of Clustering Algorithms for Region Discovery 6.Summary

3. Ch. Eick: Introduction Region Discovery 1. Motivation: Examples of Region Discovery RD-Algorithm Application 1: Supervised Clustering [EVJW07] Application 2: Regional Association Rule Mining and Scoping [DEWY06, DEYWN07] Application 3: Find Interesting Regions with respect to a Continuous Variables [CRET08] Application 4: Regional Co-location Mining Involving Continuous Variables [EPWSN08] Application 5: Find “representative” regions (Sampling) Application 6: Regional Regression [CE09] Application 7: Multi-Objective Clustering [JEV09] Application 8: Change Analysis in Spatial Datasets [RE09] Wells in Texas: Green: safe well with respect to arsenic Red: unsafe well  =1.01  =1.04 References: http://www2.cs.uh.edu/~ceick/pub.htmlhttp://www2.cs.uh.edu/~ceick/pub.html

4. Ch. Eick: Introduction Region Discovery 2. Region Discovery Framework We assume we have spatial or spatio-temporal datasets that have the following structure: (x,y,[z],[t]; ) 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

5. Ch. Eick: Introduction Region Discovery 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={c 1,…,c k } as follows: q(X)=  c  X reward(c)=  c  X interestingness(c)  size(c)  with  >1 Objective: Find c 1,…,c k  O such that: 1.c i  c j =  if i  j 2.X={c 1,…,c k } maximizes q(X) 3.All cluster c i  X are contiguous (each pair of objects belonging to c i has to be delaunay-connected with respect to c i and to d) 4.c 1 ,…,  c k  O 5.c 1,…,c k are usually ranked based on the reward each cluster receives, and low reward clusters are frequently not reported

6. Ch. Eick: Introduction Region Discovery Challenges for Region Discovery 1.Recall and precision with respect to the discovered regions should be high 2.Definition of measures of interestingness and of corresponding parameterized reward-based fitness functions that capture “what domain experts find interesting in spatial datasets” 3.Detection of regions at different levels of granularities (from very local to almost global patterns) 4.Detection of regions of arbitrary shapes 5.Necessity to cope with very large datasets 6.Regions should be properly ranked by relevance (reward); in many application only the top-k regions are of interest 7.Design and implementation of clustering algorithms that are suitable to address challenges 1, 3, 4, 5 and 6.

7. Ch. Eick: Introduction Region Discovery 3. Fitness Function for Supervised Clustering Cluster 1Cluster 2Cluster 3Cluster 4Cluster 5 |c| 50200 350200 P(c, Unsafe) 20/50 = 40%40/200 = 20%10/200 = 5%30/350 = 8.6%100/200=50% Reward Class of Interest: Unsafe_Well Prior Probability: 20% γ1 = 0.5, γ2 = 1.5; R+ = 1, R-= 1; β = 1.1,  =1. 10%30%

8. Ch. Eick: Introduction Region Discovery 4. Fitness Functions for Other Region Discovery Tasks 4.1 Creating Contour Maps for Water Temperature (Temp) 1.Examples in the data set WT have the form: (x,y,temp); var(c,temp) denotes the variance of variable temp in region c 2.interestingness(c)= IF var(c,temp)>  THEN 0 ELSE (  -var(c,temp))  with  +  {0} being a form parameter (with default 1) and  being a theshold parameter (  0). 3.Many other possible fitness functions could be used. Fig. 1: Sea Surface Temperature on July 7 2002 Var=2.2 Reward: 48.5 Rank: 3 A single region and its summary Mean=11.2

9. Ch. Eick: Introduction Region Discovery 4.2 Finding Regions with High Water Temperature Differences 1.Examples in the data set WT have the form: (x,y,Temp); Var(WT, temp) denotes the variance of the dataset for attribute temp. 2.Fitness function: Let c be a cluster to be evaluated interestingness(c)= IF var(c,temp)<(var(WT,temp)+  ) THEN 0 ELSE ((var(c,temp)/(var(WT,temp)+  ) -1)  with  being a form parameter (with default 1) and  0 threshold parameter (with default 0)

10. Ch. Eick: Introduction Region Discovery 4.3 Programming Project Fitness Functions Purity r1 r2 (6, 2, 2) (0, 0, 5) 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*10  + 0.25*5  (2,2,1) r3 We assume we have 3 classes; in r1 we have 6 objects of class1, 3 objects of class 2, and 2 objects of class1 p c (r)= (number of instance of class c in region r)/(number of instances in r)

11. Ch. Eick: Introduction Region Discovery Programming Project 2008 Fitness Function Variance We assume  =1 and th=1.5 i(r1)= 0 i(r2)=(2-1.5)=0.5 i(r3)=(11-1.5)=9.5 i(r4)=0 O Var(O)=100 r1 var(r1)=80 r2 Var(r2)=200 r3 Var(r3)=1100 r4 Var(r4)=20

12. Ch. Eick: Introduction Region Discovery Co-location Interesting Measure for 2-Continuous Variables The values of attributes A1 and A2 are converted into z- scores Interestingness of an object: Remark: i(A,o) can be negative Interestingness of a region: Remark: Patterns {A 1 , A 2  } and {A 1 , A 2  } are treated as same. Same is true for {A 1 , A 2  } and {A 1 , A 2  } Remark: will be called Binary Co-location Interestingness Function in the following.

13. Ch. Eick: Introduction Region Discovery Example: Using the Binary Co-location Fitness Function We assume  =1, z th =0.1 and A={B1,B2} i(r1)= |1-1-0.6|/3 -0.1=0.1 i(r2)=|4+0.5+0|/3-0.1=1.4 i(r3)=… i(r4)=0 because |-1+1-0.03|/3=0.01<0.1 r1 (1,1) (-1, 1) (1, 0.6) r2 (-1, -4) (-.0.5, -1) (-0.5,0) r3 R4 (1,-1) (1, 1) (0.3, -0.1) Meaning: z-value of B1 is -1, and z-value of B2 is -4 Binary Co-location: i(o,{B1,B2})=z B1 (o)*z B2 (o) Remark: Let A be an attribute and a value of that attribute z-score(a)= (a-mean(a))/standard-deviation(a))

14. Ch. Eick: Introduction Region Discovery Finding Regional Co-location Patterns in Spatial Datasets Objective: Find co-location regions using various clustering algorithms and novel fitness functions. Applications: 1. Finding regions on planet Mars where shallow and deep ice are co-located, using point and raster datasets. In figure 1, regions in red have very high co-location and regions in blue have anti co- location. 2. Finding co-location patterns involving chemical concentrations with values on the wings of their statistical distribution in Texas ’ ground water supply. Figure 2 indicates discovered regions and their associated chemical patterns. Figure 1: Co-location regions involving deep and shallow ice on Mars Figure 2: Chemical co-location patterns in Texas Water Supply

15. Ch. Eick: Introduction Region Discovery Programming Project Function MSE r1 r2 (2,2) (4,4) (-1,-1) (-7,-7) (-4,-4) MSE(r1)=(2**2+2**2)/2=4 MSE(r2)=(6**2+6**2+0**0)/3=24 X={r1,r2} MSE(X)= (8+72)/5=16 Assume Manhattan is used: (12,12) outlier

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

17. Ch. Eick: Introduction Region Discovery Categorical Binary Co-location Task: Find regions in which the density of 2 or more classes is elevated. In general, multipliers C 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 C1  C2 Example: Binary Co-Location Reward Framework; C (r)=p(C,r)/prior(C)  C1,C2 = 1/((prior(C1)+prior(C2)) “maximum multiplier”  C1,C2 (r) = IF C1 (r)<1 or C2 (r )<1 THEN 0 ELSE sqrt(( C1 (r)–1)*( C2 (r)–1))/(  C1,C2 –1) interestingness(r)= max C1, C2;C1  C2 (  C1, C2 (c))

18. Ch. Eick: Introduction Region Discovery 2006: The Ultimate Vision of the Presented Research Spatial Databases Data Set Domain Expert Measure of Interestingness Acquisition Tool Fitness Function Family of Clustering Algorithms Visualization Tools Ranked Set of Interesting Regions and their Properties Region Discovery Display Database Integration Tool Architecture Region Discovery Engine

19. Ch. Eick: Introduction Region Discovery How to Apply the Suggested Methodology 1.With the assistance of domain experts determine structure of dataset to be used. 2.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) 3.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. 4.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.

20. Ch. Eick: Introduction Region Discovery 5. A Family of Clustering Algorithms for Region Discovery 1.Supervised Partitioning Around Medoids (SPAM). 2.Representative-based Clustering Using Randomized Hill Climbing (CLEVER) 3.Supervised Clustering using Evolutionary Computing (SCEC) 4.Single Representative Insertion/Deletion Hill Climbing with Restart (SRIDHCR) 5.Supervised Clustering using Multi-Resolution Grids (SCMRG) 6.Agglomerative Clustering (MOSAIC) 7.Supervised Clustering using Density Estimation Techniques (SCDE) 8.Clustering using Density Contouring (DCONTOUR) Remark: For a more details about SCEC, SPAM, SRIDHCR see [EZZ04, ZEZ06]; the PKDD06 paper briefly discusses SCMRG

21. Ch. Eick: Introduction Region DiscoveryCLEVER  Separate Slideshow

22. Ch. Eick: Introduction Region Discovery 22 Steps of Grid-based Clustering Algorithms Basic Grid-based Algorithm 1.Define a set of grid-cells 2.Assign objects to the appropriate grid cell and compute the density of each cell. 3.Eliminate cells, whose density is below a certain threshold . 4.Form clusters from contiguous (adjacent) groups of dense cells (usually minimizing a given objective function) Simple version of a grid-based algorithm: Merge cells greedily as long as merging improves q(X).

23. Ch. Eick: Introduction Region Discovery 23 Advantages of Grid-based Clustering Algorithms fast: –No distance computations –Clustering is performed on summaries and not individual objects; complexity is usually O(#populated-grid-cells) and not O(#objects) –Easy to determine which clusters are neighboring Shapes are limited to union of grid-cells

24. Ch. Eick: Introduction Region Discovery 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)

25. Ch. Eick: Introduction Region Discovery Code SCMRG

26. Ch. Eick: Introduction Region Discovery Parameters SCMRG  Separate Transparency!

27. Ch. Eick: Introduction Region Discovery 6. Summary 1.A framework for region discovery that relies on additive, reward-based fitness functions and views region discovery as a clustering problem has been introduced. 2.The framework find interesting places and their associated patterns. 3.The framework extracts regional knowledge from spatial datasets 4.The ultimate vision of this research is the development of region discovery engines that assist earth scientists in finding interesting regions in spatial datasets.

28. Ch. Eick: Introduction Region Discovery 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