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I don't know of where this would come up in a practical sense, but I had this crazy idea this morning that maybe if given various pairs $(r, \theta)$ in polar coordinates that one could attempt to fit a curve through them, using $\theta$ as the response variable and $r$ as the explanatory variable.

For simplicity, we could start with a "linear regression" of sorts through a set of points $\{(r_i, \theta_i)\}$.

I suppose one could naively assume a model of the form $$r_i\cos(\theta_i) = \beta_0+\beta_1 r_i\sin(\theta_i) + \epsilon_i$$ (recalling that $x_i = r_i \sin(\theta_i)$ and $y_i = r_i \cos(\theta_i)$ in rectangular coordinates), but this ruins the point of making $\theta_i$ the "response" variable of $r_i$, given that functions of $\theta_i$ appear on both sides of the equation.

Has there been any literature developed on any sort of regression in polar coordinates?

Of course, since various values of $\theta$ are coterminal, the domain of $\theta$ will probably need to be truncated. What may also complicate things slightly is that $r > 0$.

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    $\begingroup$ I believe this is what directional statistics is about, but I've never spent much time reading about it, so I couldn't say much more than that. $\endgroup$ – Jake Westfall Jul 19 '18 at 14:10
  • $\begingroup$ If the original data comes as (x,y) pairs, why not just convert it to polar and then regress r = a + b * theta (or whatever relevant functional form for your application)? I seem to be getting sensible results in a simple test I did, though I'm a little worried that I'm not really taking into account the non-linearity. $\endgroup$ – John Jul 19 '18 at 15:06
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There is an enormous amount of statistics related to this under the name circular or directional statistics.

Your case is not so well described though.

What is the 'switch' that you were thinking of by introducing polar coordinates?

Do you mean to solve a situation of a non-negative variable, e.g. some count, as function of a cyclical variable, e.g. hour of the day, by picturing it as a curve and solve it with the mechanisms for curve fitting in euclidean coordinates?

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  • $\begingroup$ I would have to think about your question some more. But at least I know what to search if I wanted to explore the literature. Thanks. (+1) $\endgroup$ – Clarinetist Jul 19 '18 at 14:16
  • $\begingroup$ There is also an R package named 'circular' containing lots of tools for analysis. In comparison to your case more interesting problems arise when the dependent variable is circular. You got the problem that regular techniques like solving least squares by matrix equations, do not work (imagine the residual distance between 23:59 and 00:01). $\endgroup$ – Sextus Empiricus Jul 19 '18 at 14:19

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