I have a set of points $(x_i,y_i)$ where each x & y value is circular & can take on a value from -pi to pi. (The topology of the data is a torus, but I am not sure how relevant that is to the question). An example dataset is shown below:
My goal is to see if there is a significant relationship between x & y. My issue is that the data is sparse, any relationship would be nonlinear (my hypothesis is that it would show some kind of peak), and the data is non evenly sampled over x.
My first approach has been to bin the x data into segments, calculate the mean y value in each segment, and then run a test of uniformity on whether the average y-values are uniform or not, as can be seen below:
However, my issue is that due to uneven sampling, some of those average y-values have a very low variance, while others have a high variance, thus leading to noise. The uniformity tests I am using (KS & Kuiper) will be deceived by that & may give artificially small P-values.
Any ideas on how to do this better?