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Question 1: I am trying to determine which (if any) of my different park types are spatially clustered across my study area (city boundary).

There are 162 total parks, 9 nominal categories (which represent statistically determined clusters or park 'types' based on 33 social, built, and ecological variables).

Also, would I run this analysis on the polygons (park boundaries) or points (based on centroids), or some other way?

Question 2 (related but different): I am also wondering if I can run the entire analysis (including spatial) in Arc? The cluster groups I have now reflect the results of an SPSS process including 1. a factor analysis on the 33 variables to reduce redundancy then 2. a two-step cluster analysis (Hierarchical to determine clusters then k-means to form groups).