I think you need to use something like Kuiper's test. A one-sample example using R is:

http://rgm3.lab.nig.ac.jp/RGM/r_function?p=CircStats&f=kuiper

Hope this helps,

Tim

I'm using Watson's U-square test. I think the circular stat book from N.I. Fisher explains it, or at least provides references. I'll upload my code here later today.

Old-old code of mine, but should still work. I tried putting all pieces of code in one m file for you; let me know if something is missing. Best, Balazs

Woow Balázs! I don't know what to say, thanks a lot!

However I have tested the code and there is something that is not working. The test passes also for an uniform distribution, producing a good, low, U^2 stat. value, as low as testing a von mises distribution! Do I guess there is something that is not working well.

As suggested by Timothy, I have also tested the kuiper test , and I have used the kuiper test available within the CircStat for Matlab. It seems to work consistently although the estimated p-value in the test is not enough really accurate.

Here the details.

I generated a von mises distribution with the mu and kappa estimated from my angles, say x, i.e.:

[mu kappa] = circ_vmpar(x)

and then

vonmis = circ_randvm(mu,kappa,length(x))

Then I used the kuiper test to see whether the two distributions x and vonmis differ significantly (the difference can be in any property, such as mean, location and dispersion):

[H,pValue] = circ_kuipertest(x, vonmis)

It works, I tested different distributions. However, how I was saying the pvalue is not really accurate, too rough. They are taken from tabulated values up to 500 samples for each distribution (I'm in the order of thousands samples).

If anyone would have any other tips would be great!

Anyway, many thanks again!

Ciao!!!

sergio

I have written an R function to calculate any quantile of the Watson's U^2 statistic for circular data (page 104 of Mardia and Jupp 2000). Here you go:

# To find the upper quantile of Watson's U^2 statistic for circular data

# Mardia and Jupp (2000) - page 104.

find.c

Well, sure it should pass for uniform, as it is a very flat von Mises. (More accurately, for every epsilon, you will find a von Mises closer to uniform than epsilon.) It should not pass for a mixture of two von Mises distributions, for instance.

It first estimates ML parameters for the von Mises from the data, and tests whether that fits the data. If you want to test whether a particular von Mises is fitting the data instead of the ML estimate, it's not difficult to modify the code to do that.

So I keep up the statement that it should work :-) However, I have not checked it in the past five years, so no warranties :-)

Maybe you are looking for tests for circular uniformity. There are quite many of those, like Rayleigh's test, Rao's spacing test, Kuiper-test, Watson's test for uniformity, something called Aine's test, if I remember well, etc.

You will try to review set avaliable

http://cran.r-project.org/web/packages/movMF/movMF.pdf

Dear all, thank you very much.

I didn't test all solutions that you suggested, as already with both the Kuiper-test and with the Watson's U-square test it is possible to perform easily a goodness-of-fit test for the Von Mises-Fisher distribution.

Many thanks to you all.

To conclude, I have a last question. I was generating my angles on the circle following a normal distribution (using the chi2gof from matlab).

Now, using the tests, I see that my angles passes also always a goodness-of-fit test for the Von Mises-Fisher distribution.

That is, I constructed it following a normal, and they also follow a von mises-fisher.

Is this because the Von Mises-fisher distribution is roughly analogous on the circle to a normal distribution?

Can I say that if the angles are distributed normally then they are also distributed as a von mises-fisher on the circle?

Sorry for the bothering, maybe these questions may be easy and intuitive for you, but I'm new to circular statistics.

Ciao,

Sergio

They are quite similar as long as you keep the SD low. I think when a von Mises is more flat, it starts looking less like a Gaussian (that's when it matters you're on a circle - otherwise you can cut the circle at a near-zero probability point and you're back on the line). In fact, if you wind a flat Gaussian on the circle, what you get is not a von Mises but a 'wrapped Gaussian' distribution. They do not look tremendously different from a distance, but it's probably possible to distinguish them; will likely need more tricky statistics though; I never actually needed it, so I don't know more.

Balazs

Dear Sergio,

together with two colleagues, I have recently proposed a simple QQ-plot valid for directional distributions, hence also for circular ones. Here is the link to the paper: http://homepages.ulb.ac.be/~chrisley/AboutMe_files/LSV13.pdf. If you want to use it, just let me know, we can give you the code (my coauthors should be fine with that). This is a visual inspection, of course we also have a goodness-of-fit test that goes along with it.

Christophe