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Given a finite set $A$ of points of a metric space $(X, d)$, I would like to find its mean. A Frechet mean seems appropriate here: $\arg \min_{x \in X} \sum_{a \in A} d(x, a)^2$. I also would like to find the corresponding standard deviation. What would this be?

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  • $\begingroup$ By I also would like to find the corresponding standard deviation do you mean, that you are looking for a natural way of defining a std.dev. for the Frechet mean? $\endgroup$
    – SBF
    Commented Jul 11, 2013 at 11:36
  • $\begingroup$ Yes, I am interested in a natural generalization of standard deviation, in analogy to Frechet mean generalizing ordinary mean. A reference to the literature would be ideal. $\endgroup$
    – Franck van Breugel
    Commented Jul 11, 2013 at 15:11
  • $\begingroup$ I see. Since mean (in whatever framework) is natural to define as a minimizer of a measure of spread of data/points/... it justifies the fact that the minimal such spread is called variance, as Kjetil wrote already. So the std.dev would indeed be natural to defined as a square root of the minimals sum of distances, however maybe you could say why are you interested in std.dev related to the Frechet mean? Perhaps, concentration theorem for random elements taking values in metric spaces? $\endgroup$
    – SBF
    Commented Jul 11, 2013 at 16:50

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A Frechet mean is (mostly) appropriate, but you should be aware that it is not always unique! An example of non-unicity is the circle, with data points the north and south pole, and (following Winnie Pooh), two minimzers: the east and the west pole. A condition guaranteeing always unicity is manifolds with non-positive curvature, a proof of this fact can be got from Burago, Burago & Ivanov: A Course in metric geometry, chapter 9. There are also inicity conditions based on small variance, in some sense.

As to your question: The book Nonparametric Inference on Manifolds call the minimizer of the Frechet function, the Variation. That is of course unique even if the mean itself is not. Probably one could use its square root as some sort of standard deviation, but the usability of such a definition would have to be investigated on a case basis.

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