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I know that there are variants of the arithmetic mean and the median that are applicable to three dimensional data (centroid and medoid), but have not been able to find such a thing for the modal average.

Now I know that modal averaging could just as well be applied to 3d data (it would just yield the coordinate that occurs most frequently), but the data i'm averaging probably won't have any values that occur more than once.

So what i'm looking for would not require the datapoints to be exactly alike, but alike to a certain degree (say X% similarity), as a tradeoff (if a numerical majority cannot be found). Is there such a thing? Or would this yield exactly the same result as the medoid?

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I suspect the most practical approach would be multidimensional kernel density estimation, and then choose the point with highest kernel density. This will depend on your assumptions, particularly over bandwidth.

The np package in R may help.

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And do you think that using kernel density estimation would yield very different results from the medoid? I am trying to think of datasets for which results from both would differ, any ideas? – user25251 Feb 29 '12 at 13:19
It all depends on the distribution. As with the difference between the mode and the median in one dimension, if the distribution is highly skewed then I would expect a noticeable difference. – Henry Feb 29 '12 at 14:06
Excellent, i'll get to it then, thanks for the answer! – user25251 Feb 29 '12 at 17:18

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