# Circular distributions Kolmogorov-Smirnov vs. Watson test vs. Rayleigh test

I am interested in assessing whether my angular data distribution is satisfactorily described by a von Mises distribution.

In scale data, one can potentially use a Kolmogorov-Smirnov test. However, with circular data, it is usual to use the Watson test. The Rayleigh, Watson or Rao Spacing tests are used if a uniform distribution on the circle is suspected.

Question: Why is a Kolmogorov-Smirnov test not appropriate?

Also, I notice differences when plotting Probability and Quantile plots in routine software packages, e.g., Mathematica etc. vs. dedicated software for circular data? How are PP and QQ plots distorted in the circular data setting?

## 1 Answer

The Kolmogorov-Smirnov test compares the theoretical CDF with the empirical CDF. Specifically, it looks at the maximum difference between the two.

For circular data, it is not as straightforward to define the CDF. With circular data, we have to choose a reference point $0^\circ$, arbitrarily. If we rotate the data, say by $180^\circ$, we should find the same answer as to whether the von Mises distribution is an appropriate distribution. However, the maximum difference between the empirical and theoretical CDF (which is the Kolmogorov-Smirnov test) depends on the arbitrary reference point, and will change when we rotate the data.

Differences in plotting could be because of the same reason: we might rotate the data to maximize or minimize the maximal difference between plotted CDFs.

Kuiper's and Watson's test account for this, which is why they are more relevant here. For extensions of these tests to test goodness-of-fit for von Mises data, you could refer to Circular Statistics in R by Arthur Pewsey, Markus Neuhauser and Graeme D. Ruxton.