1. SSCP: Mining Statistically Significant Co-location Patterns Sajib Barua and Jörg Sander Dept. of Computing Science University of Alberta, Canada

2. SSCP: Mining Statistically Significant Co-location Patterns 2 Outline Introduction  Related work  Motivation Proposed Method Experimental evaluation  Synthetic data  Real data Conclusions

3. SSCP: Mining Statistically Significant Co-location Patterns 3 Definition Co-location patterns are subsets of Boolean spatial features whose instances are often seen to be located at close spatial proximity. Examples: {Nile crocodile, Egyptian plover} {Shopping mall, parking}

4. SSCP: Mining Statistically Significant Co-location Patterns 4 Event Centric Model Co-location is defined based on a spatial relationship R A co-location type C is a set of n different spatial features f 1, f 2, …, and f n. A2A2 B1B1 C1C1 D1D1 B2B2 C2C2 C3C3 {A 2, B 1, C 1, D 1 } form a clique under a relation R. C3C3 A2A2 B1B1 C1C1 D1D1 B2B2 C2C2 {A 2, B 1, C 1 } is an instance of co-location {A,B,C} C3C3 A2A2 B1B1 C1C1 D1D1 B2B2 C2C2 {A 2, B 1, D 1 } is an instance of co-location {A,B,D} {A 2, C 1, D 1 } is an instance of co-location {A,C,D} {B 1, C 1, D 1 } is an instance of co-location {B,C,D} {A 2, B 1, C 1, D 1 } is an instance of co-location {A, B,C,D}

5. SSCP: Mining Statistically Significant Co-location Patterns 5 Prevalence Measure Participation ratio (PR) of a feature in a co- location type C, is the fraction of its instances participating in any instance of C. Participation index (PI) is the minimum participation ratio in C. A2A2 B1B1 C1C1 A1A1 B2B2 C2C2 C3C3 PI ({A, B}) = min {1/2, 1/2} = 0.5 PI ({B, C}) = min {1, 2/3} = 0.66 PI ({A, C}) = min {1/2, 1/3} = 0.33 PI ({A, B, C}) = min {1/2, 1/2, 1/3} = 0.33 PI({A,B,C}) <= PI ({A, B}) or PI ({B, C}) or PI ({A, C}) PR and PI are anti-monotonic PI ({A, B}) = min {1/2, 1/2} = 0.5 A2A2 B1B1 C1C1 A1A1 B2B2 C2C2 C3C3

6. SSCP: Mining Statistically Significant Co-location Patterns 6 Related Work Spatial statistics  Ripley’s K function, distance based measure, co-variogram function. Spatial data mining  Koperski et al. [4] mine spatial association rules.  Morimoto [5] also look for frequently occurring patterns.  Shekhar et al. [2] introduce three models to materialize transaction.  Huang, et al. [3], Yo et al. [6,7], and Xiao et al. [8].

7. SSCP: Mining Statistically Significant Co-location Patterns 7 Limitations of the Existing Methods Spatial statistics  Defined only for pairs. Co-location mining  Only one global threshold for PI is used.  No guideline to setup PI-threshold  Do not address the spatial auto-correlation and feature abundance effects. A simple threshold can report meaningless patterns or can miss meaningful patterns.

8. SSCP: Mining Statistically Significant Co-location Patterns 8 Motivation Existing co-location mining algorithms will not report {A,B}. A has fewer instances B is abundant A & B have true spatial dependency. Assume PI-threshold = 0.4

9. SSCP: Mining Statistically Significant Co-location Patterns 9 Motivation Existing co-location mining algorithms will report {A,B}. A & B are abundant. Both randomly distributed. Do not have any true spatial dependency. Assume PI-threshold = 0.4

10. SSCP: Mining Statistically Significant Co-location Patterns 10 Motivation A & B are auto-correlated. Do not have any true spatial dependency. Existing co-location mining algorithms will report {A,B}. Assume PI-threshold = 0.4

11. SSCP: Mining Statistically Significant Co-location Patterns 11 Our Idea Our approach uses statistical test. Spatial dependency is measured using PI. #○ = 12 #∆ = 12 If features ○ and ∆ were spatially independent of each other, what is the chance of seeing the PI-value of {○, ∆} equal or higher than the observed PI- value (0.41)?

12. SSCP: Mining Statistically Significant Co-location Patterns 12 Generate Artificial Data Sets Observed data Artificial data sets generated under null model

13. SSCP: Mining Statistically Significant Co-location Patterns 13 p-value computation α = 0.05 PI obs = 0.41 p-value = 0.163 If p <= α, PI obs is statistically significant at level α.

14. SSCP: Mining Statistically Significant Co-location Patterns 14 Auto-correlated Feature A & B are auto-correlated. Do not have any true spatial dependency.

15. SSCP: Mining Statistically Significant Co-location Patterns 15 Modeling Auto-correlation Auto-correlation is modeled as a cluster process. Poisson Cluster Process [9] Autocorrelation is measured in terms of intensity and type of distribution of a parent process and offspring process around each parent.

16. SSCP: Mining Statistically Significant Co-location Patterns 16 Estimating Summary Statistics Estimate the summary statistics.  Auto-correlated feature: intensity of parent and offspring process (κ, and µ values).  Randomly distributed feature: Poisson intensity (either homogenous (a constant) or non-homogenous (a function of x and y)).

17. SSCP: Mining Statistically Significant Co-location Patterns 17 Null Model Design The artificial data sets maintain the following properties of the observed data:  same number of instances for each feature, and  similar spatial distribution for each individual feature.

18. SSCP: Mining Statistically Significant Co-location Patterns 18 p-value computation Estimate Use randomization tests, where a large number of datasets conforming to the null hypothesis is generated. How many simulations do we need?  Diggle suggested 500 simulations for α = 0.01 [10].

19. SSCP: Mining Statistically Significant Co-location Patterns 19 Improving Runtime: Data Generation In a simulation, we only generate feature instances of those clusters which are close enough to other different features (either auto-correlated or non auto-correlated) This saves time of the artificial data generation step of a simulation.

20. SSCP: Mining Statistically Significant Co-location Patterns 20 Improving Runtime: PI-value Computation In a simulation R i, for a co-location C Procedure: In each simulation, compute -values of all possible 2-size subsets For a co-location C of size k ( > 2), we lookup PI-values of its 2-size subsets of C. If a subset C' is found for which < PI obs (C), is not required to be computed. Otherwise is computed for simulation R i.  No need to compute

21. SSCP: Mining Statistically Significant Co-location Patterns 21 An Example Four features A, B, C, D {A,B,C}: If {A,B} < PI obs {A,B,C}, {A,B,C} < PI obs {A,B,C}. No need to compute {A,B,C}. {A,B,C} < PI obs {A,B,C} does not imply {A,B,C,D} < PI obs {A,B,C,D}. {A,B,C,D}: by checking 2-size subsets The worst case complexity is O(2 n )  The size of the largest co-location is much smaller.  Largest co-location size is predictable  if PI obs (C) = 0, we do not compute -value of C,  Our pruning strategies All these keep the actual cost in practice less than the worst case cost.

22. SSCP: Mining Statistically Significant Co-location Patterns 22 Experimental Results (1) Negative association: Features ○ and ∆ with 40 instances of each. This synthetic data set is generated using multi-type Strauss process to impose a negative association (inhibition) between these two features. Result PI obs = 0.55 and p-value = 0.931 > 0.05 (α), hence (○, ∆) will not be reported.

23. SSCP: Mining Statistically Significant Co-location Patterns 23 Experimental Results (2) Autocorrelation: #○ = 100, and #∆ = 120. ∆: independently and uniformly distributed over the space ○: spatially auto-correlated In our generated data, ∆ is found in most clusters of ○. The summary statistics of ○ is estimated by fitting the model of Matérn Cluster process[9] (κ= 40, µ = 5, r = 0.05). Results: PI obs {○, ∆} = 0.49, existing algorithm will report the pattern if a threshold <= 0.49 is chosen. p-value = 0.383 > 0.05 (α); {○, ∆} is not reported.

24. SSCP: Mining Statistically Significant Co-location Patterns 24 Experimental Results (3) Multiple features: #○ = 40, #∆ = 40, #+ = 118, #x = 40, and = #30. Study area = Unit square, co- location neighborhood radius = 0.1 Features ○ and ∆ are negatively associated. Feature + is spatially auto- correlated. Features +, ○, and x are positively associated. Feature is randomly distributed. Significant co-location patterns = {○, +}, {○, x}, {+, x}, {○, +, x}, {○, +, }, {○, x, }, {+, x, }, and {○, +, x, }.

25. SSCP: Mining Statistically Significant Co-location Patterns 25 Runtime Comparison (1) Features ○, ∆, +: are auto-correlated, strongly associated. Each has 400 instances. Feature x: is randomly distributed, and has 20 instances. Our algorithm finds all co-locations of features ○, ∆, and x. Instances of each auto-correlated features is increased  cluster numbers is kept same  number of instances per cluster is increased by a factor k. Runtime comparison Speedup

26. SSCP: Mining Statistically Significant Co-location Patterns 26 Runtime Comparison (2) The number of clusters for features ○, ∆, and + is increased by a factor k but the number of instances per cluster is kept same. Total instances of x is increased by the same factor k. Runtime comparison Speedup

27. SSCP: Mining Statistically Significant Co-location Patterns 27 Ants Data ○ = Cataglyphis ants (29) and ∆ = Messor ants (68). PI obs {Cataglyphis, Messor} = {24/29, 30/68} = 0.44. p-value = 0.142 > 0.05 (α); Co- location {○, ∆} is not significant. R. D. Harkness also did not find any clear association between these two species. Existing algorithm will report {○, ∆} if PI-threshold <= 0.44.

28. SSCP: Mining Statistically Significant Co-location Patterns 28 Toronto Address Repository Data

29. SSCP: Mining Statistically Significant Co-location Patterns 29 Found Co-locations

30. SSCP: Mining Statistically Significant Co-location Patterns 30 Conclusions A new definition for co-location pattern. Does not depend on a global threshold. Statistically meaningful. Runtime cost of randomization tests is reduced. Investigate other prevalence measures to check if they allow additional pruning techniques. Removing redundant patterns.

31. SSCP: Mining Statistically Significant Co-location Patterns 31 References 1. Agrawal, R., Srikant, R.: Fast Algorithms for Mining Association Rules in Large Databases. In: Proc. VLDB, pp. 487-499 (1994) 2. Shekhar, S. et al.: Discovering Spatial Co-location Patterns: A Summary of Results, In Proc. SSTD, pp. 236-256 (2001) 3. Huang, Y. et al.: Discovering Colocation Patterns from Spatial Data Sets: A General Approach. IEEE TKDE 16(12), 1472-1485 (2004) 4. Koperski, K. et al.: Discovery of Spatial Association Rules in Geographic Information Databases. In SSD, pp. 47-66 (1995) 5. Morimoto, Y.: Mining Frequent Neighboring Class Sets in Spatial Databases. In SIGKDD, pp. 353-358 (2001) 6. Yoo, J. S. et al.: A Partial Join Approach for Mining Co-location Patterns. In Proc. GIS, pp. 241-249 (2004) 7. Yoo, J. S. et al.: A joinless Apporach for Mining Spatial Co-location Patterns. IEEE TKDE 18(10), 1323-1337 (2006) 8. Xiao, X. et al.: Density Based Co-location Pattern Discovery. In Proc. GIS, pp. 250- 259 (2008). 9. Ilian et al: Statistical Analysis and Modeling of Spatial Point Patterns. 10. Diggle P.J.: Statisitcal Analysis of Spatial Point Pattern, 2003

32. SSCP: Mining Statistically Significant Co-location Patterns 32 Questions?