A SCALE-SENSITIVE TEST OF ATTRACTION AND REPULSION BETWEEN SPATIAL POINT PATTERNS Tony E. Smith University of Pennsylvania Diggle-Cox Test Lotwick-Hartwick Test A Combined Test Selected Applications

0 100 S Point Patterns: generated by independent processes Diggle-Cox Test: random reference point distance from to nearest neighbor in distance profile for rank correlation between and under Attraction Repulsion

0 100 Repulsion-Attraction Model random Diggle-Cox Test: significant repulsion Lotwick-Silverman Test: Global Attraction not detectable Make Mosaic of Random Shifts of Marginal Processes are preserved. shifts:

K-Functions: Radii: Reference Pairs: P-Value Plots: Radii REPULSION P-Values ATTRACTION

Myrtles Example: = Healthy Trees = Diseased Trees P-Value Plot: Radii REPULSION P-Values ATTRACTION Why isn’t the observed clumping picked up ?

Unintended Structure: New clumps formed Shifts look too clumpy Original clumps are no longer significant A Combined Procedure: Generate random reference points Select set of reference radii Construct reference cell counts for patterns Correlate cell count profiles for patterns

Conditional Null Hypothesis: Fix the set of joint locations Hypothesize that pattern is equally likely to occupy any subset of locations. All random relabelings are equally likely Permutation Testing Procedure: Simulate random relabelings Form cell-count profile pairs (observed: ) Compute rank correlations Sort correlations and compute P-values

Applications of the Procedure: Radii REPULSION P-Values ATTRACTION Attraction-Repulsion Model Radii REPULSION P-Values ATTRACTION PATTERN DATA LOTWICK-SILVERMAN NEW P-VALUE PLOT

Diseased versus Healthy Myrtles Radii REPULSION P-Values ATTRACTION PATTERN DATALOTWICK-SILVERMAN NEW P-VALUE PLOT Radii REPULSION P-Values ATTRACTION

Heterogeneous Pattern Spaces: MYRTLES DATA Conditioning excludes empty regions Appropriate for non-viable regions rocky regions swampy regions What about viable regions ?

Supermarkets versus Convenience Stores: = CONVENIENCE STORES = SUPERMARKETS Philadelphia Pop Density ° ° Normalize Pop Density to probability measure Construction of Reference Measures: Sample reference points from this measure

Comparison of Reference Points: POP DENSITY POP REF POINTS RAND REF POINTSPHIL COUNTY

Comparison of Results: Radii REPULSION P-Values ATTRACTION Radii REPULSION P-Values ATTRACTION P-Value Plot for Random Reference Points P-Value Plot for Population Reference Points