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I have a 2D field of an Eulerian scalar and I'm looking for a statistical index to quantify how clustered or grouped the field is.

Consider the figure below for example (which is not mine, since I can't disclose my data; I got it from google):

enter image description here

If I do the mean or standard deviation on all 3 pictures, the result might be the same, but the arrangement of the values clearly isn't.

A PDF also is not appropriate, since the same data (not in the above picture, though) could have the same PDF but different arrangements for the values, and therefore large and small clustering for the same PDF. An FFT also isn't ideal, since you only change one plot for another.

Ideally I'd like a "clustering number" for each picture. Where it would be close to zero in the leftmost figure, and close to one in the rightmost.

I have checked answers like this but from what I got they only apply to lagrangian data, that is, individual points in continuum space.

EDIT

To explain better, they only work if you have a list of particles and their exact position (lagrangian data). What I have is a grid in space, and each grid cell has a concentration of particles in it (eulerian data).

Cheers

PS: I don't really have much idea on which tags to put on this. Please feel free to improve my tags.

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TomCho
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  • I'm not sure I really follow this. Do you want to have an estimate of the number of clusters in different datasets? I don't see why probability densities would be bad, either: the plots appear to be contours of probability densities, which yields a clustering method. What do you mean that you have "a discretized space, with a 'concentration' in each point", instead of "individual points in continuum space"? Can you post a small example dataset (even if not your actual data) to illustrate your situation? – gung - Reinstate Monica Feb 03 '17 at 17:54
  • There is a point of view in which all three pictures have the same degree of "clustering": as you move from left to right, you are merely zooming in to see more detail of the same clustering phenomenon. This suggests that you really need a measure of *spatial correlation distance,* such as the one offered by a *variogram.* It's difficult to tell, though, because "Eulerian scalar" and "lagrangian data" do not appear to be statistical terms: could you explain what you mean by these? – whuber Feb 03 '17 at 17:57
  • @whuber A variogram seems it would have a similar effect as an FFT. Which is just change one plot by another. Although, I didn't know that term and will look into it. And yeah, I don't generally work with statistics. I added an explanation at the end. Let me know if it still needs more clarification. – TomCho Feb 03 '17 at 18:02
  • @gung Probability would be bad because a figure could have the same PDF and different number of clusters in space. Please see my explanation of the terms I used. I didn't realized those weren't used in statistics. And I'll try to make an example out of scratch, but it might take a while. Thanks! – TomCho Feb 03 '17 at 18:04
  • I think you should look up what a variogram actually is. – whuber Feb 03 '17 at 18:51
  • @whuber I am. Isn't it a function that converges to the variance? (Is my question clearer after the edit?) – TomCho Feb 03 '17 at 20:30
  • See https://en.wikipedia.org/wiki/Variogram. But since locations of particles are nothing like your illustrations, you very well might be asking about clustering: I cannot tell. – whuber Feb 03 '17 at 20:37

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