# Is there any such thing as “polar regression”?

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 $r > 0$.

• 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. – Jake Westfall Jul 19 '18 at 14:10
• 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. – John Jul 19 '18 at 15:06