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Background

Circular correlation is given by:

$$R_{\operatorname{circular}} \triangleq \frac{\sum_{i=1}^m \sin (x_i - \bar x) \sin (y_i - \bar y)}{\sqrt{\sum_{i=1}^m \sin (x_i - \bar x)^2} \sqrt{\sum_{i=1}^m \sin (y_i - \bar y)^2}}$$

Example 1

I generated a data set with a circular correlation of 1.0000 and it looked like this:

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Example 2

I similarly generated a dataset with near-perfect circular correlation, but the initialization of parameters included a great dispersion in where to place points:

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Example 3

Dialing up the amount of dispersion and the number of points, we get something that to my eye doesn't have any grid-like structure.

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Zooming in doesn't seem to reveal any fine grid structure either:

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And zoom in further:

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And further:

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Question

Is circular correlation maximized by points laying on a square grid on the plane?

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  • $\begingroup$ Used similar methods of data generation here: stats.stackexchange.com/a/589221/69508 $\endgroup$
    – Galen
    Sep 19, 2022 at 3:51
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    $\begingroup$ Is the unit radians? In that case, probably all of these are a line on the bivariate plot after performing $\theta mod 2 \pi$. $\endgroup$ Sep 19, 2022 at 10:45
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    $\begingroup$ Hint: when $a \approx 0,$ $\sin(a) \approx a.$ Consider, then, a dataset where the $x_i$ and $y_i$ don't vary much. To a good approximation, then, you can drop all appearances of "sin" from the formula, giving the usual correlation for $(x_i,y_i).$ What kind of configuration gives rise to large correlations? $\endgroup$
    – whuber
    Sep 19, 2022 at 13:03
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    $\begingroup$ I think I see what the real problem is: your formula is incorrect. In all your examples, the definition of the mean angle is problematic, because it requires arbitrary choices of the phase for its definition. Instead, there's a version inspired by the characterization of covariance I give at stats.stackexchange.com/a/18200/919 as a mean over all possible pairs of data of a product of the sines of their differences. See formula (2.2) in Fisher, N. I., and A. J. Lee. 1983. “A Correlation Coefficient for Circular Data.” Biometrika 70 (2): 327–32. doi.org/10.2307/2335547 . $\endgroup$
    – whuber
    Sep 20, 2022 at 11:52

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