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I have a scalar quantity that is distributed on a sphere. I would like to quantify the asymmetry in this scalar field. is there any standard method to do this?

Let's say that the function on the sphere is the number density of some point-like events. In an experiment we will have a finite number of points. An obvious method to estimate the asymmetry would be:

-- Consider a plane dividing the sphere in two semi-spheres, this plane is defined by two angles. For each choice of the angles, we compute the difference in the number of points in the two semi-spheres, and scan the two angles until the difference in the number of points is maximal. This will define the direction of asymmetry.

I do not know if this estimator is optimal, and I would like to know what other estimators have been proposed in the literature.

EDIT: as noted in the comments, one could just use the definition of Fourier decomposition on the sphere as estimator. My question is mainly about the optimality of this estimator. For this reason I am interested also on other estimators that may be carrying more information.

simona
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  • Could you add more detail? Would looking at the moments of the spherical angles suffice? – Alex R. Aug 10 '19 at 06:57
  • "Asymmetry" appears to mean *deviation from uniformity.* There are many possible methods, so we need more guidance concerning the purpose of this analysis and the intended interpretation. Could you provide that information in an edit to the question? – whuber Aug 10 '19 at 15:08
  • Maybe you could fit some distribution, see https://stats.stackexchange.com/search?q=von+Mises-Fisher – kjetil b halvorsen Aug 10 '19 at 22:37
  • There's a [paper](https://arxiv.org/abs/1512.02865) of mine that seems to be a good fit. – corey979 Aug 12 '19 at 20:26
  • The objective of this question could be made clearer if you would edit it to explain whether you are interested in (a) quantifying the apparent deviation between a particular point set and symmetry or (b) testing the hypothesis that the point set arose as the realization of a uniform Poisson process on a sphere or (c) exploring the specific ways in which this point set appears to deviate from a uniform distribution. The answers will vary quite a bit depending on this objective. – whuber Aug 12 '19 at 20:56
  • @whuber I think I do not understand very well the difference between a) an c). I am not interested in b) – simona Aug 12 '19 at 21:16
  • The difference may be subtle, but the concepts of symmetry and uniformity differ. One would address the latter in a natural way with [spherical Ripley K and L functions,](https://stats.stackexchange.com/a/7984/919) for instance, which have little to do with symmetry; and one would address the former by comparing the pattern to rotated versions thereof. Incidentally, your "direction of asymmetry," although in general indefinite, seems conceptually akin to a dipole moment--and therein lies a fruitful line of inquiry (via Fourier analysis on the sphere and expansions in spherical harmonics). – whuber Aug 12 '19 at 21:18
  • @whuber you are right on distinction between symmetry and uniformity. this interaction is very useful, thank you for helping me in clarifying my question. I would like to ask if I apply the definition of dipole moment (and then quadrupole etc), is it an optimal estimator of the dipole? – simona Aug 12 '19 at 21:35
  • It would depend on how you assess optimality. This is a nontrivial question whose answer would depend on how you plan to use the estimator and on what probability model you are adopting for this point process. There is a natural estimator that ought to work well in many applications though: just use the dipole moment of the points. – whuber Aug 12 '19 at 21:45

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It appears to me you need a form of information entropy on a sphere. Consider a finite interval. The uniform distribution has the highest entropy on it, see Section 3.1 in the above link. In this regard uniform distribution is symmetric in relation to transnational symmetry.

In your case you have a sphere instead of a finite interval. The uniform distribution on a sphere is symmetric to rotation, of course. The uniform distribution should have the highest entropy too if you formulate it properly.

So, you can formulate the information entropy on a sphere. The tools are given in the above link in "Introduction to Continuous Entropy" by Charles Marsh, 2013

Aksakal
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