The difference between spatial median and a group centroid in multivariate space

Question

Can someone clarify the difference for me between the spatial median and a group centroid, in the context of a group of points in multivariate space.

I have 9 locations each with 30 samples each. Each sample is a quadrat with species abundance data for multiple species.

I want to determine the group centroid for each location, but now I am confused about the difference between the group centroid (what i understand to be like a multivariate mean) and the spatial median of the points in multivariate space.

Related: https://stats.stackexchange.com/questions/120194/what-is-the-multivariate-analog-of-the-median — EngrStudent, Jun 02 '17 at 13:27
Related: https://stats.stackexchange.com/questions/89676/k-medians-formula-to-compute-the-median/271441#271441 — kjetil b halvorsen, Jun 02 '17 at 14:52

score 1 · Answer 1 · answered Jun 02 '17 at 14:26

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In my memories of my courses, centroid is the point minimizing the L2 norm, and spatial median is the point minimizing the L1 norm.

cent = arg min_x sum_i ||x - x_i||_2

med = arg min_x sum_i ||x - x_i||_1

answered Jun 02 '17 at 14:26

MOB

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The difference between spatial median and a group centroid in multivariate space

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