File Exchange

image thumbnail

Circular Statistics Toolbox (Directional Statistics)

version (46.2 KB) by Philipp Berens
Compute descriptive and inferential statistics for circular or directional data.


Updated 08 Jun 2012

View Version History

View License

Editor's Note: This file was selected as MATLAB Central Pick of the Week

CircStat for Matlab

Toolbox for circular statistics with Matlab.

Authors: Philipp Berens

Marc Velasco, Tal Krasovsky

P. Berens, CircStat: A Matlab Toolbox for Circular Statistics, Journal of Statistical Software, Volume 31, Issue 10, 2009

Please cite this paper when the provided code is used (not the technical report!). See licensing terms for details.

circ_r Resultant vector length
circ_mean Mean direction of a sample of circular data
circ_axial Mean direction for axial data
circ_median Median direction of a sample of circular data
circ_std Dispersion around the mean direction (std, mardia)
circ_var Circular variance
circ_skewness Circular skewness
circ_kurtosis Circular kurtosis
circ_moment Circular p-th moment
circ_dist Distances around a circle
circ_dist2 Pairwise distances around a circle
circ_confmean Confidence intervals for mean direction
circ_stats Summary statistics

circ_rtest Rayleigh's test for nonuniformity
circ_otest Hodges-Ajne test (omnibus test) for nonuniformity
circ_raotest Rao's spacing test for nonuniformity
circ_vtest V-Test for nonuniformity with known mean direction
circ_medtest Test for median angle
circ_mtest One-sample test for specified mean direction
circ_wwtest Multi-sample test for equal means, one-factor ANOVA
circ_hktest Two-factor ANOVA
circ_ktest Test for equal concentration parameter
circ_symtest Test for symmetry around median angle
circ_kuipertest Test whether two distributions are identical (like KS test)

circ_corrcc Circular-circular correlation coefficient
circ_corrcl Circular-linear correlation coefficient

circ_kappa Compute concentration parameter of a vm distribution

circ_plot Visualization for circular data
circ_clust Simple clustering for circular data
circ_samplecdf Evaluate CDF of a sample of angles

rad2ang Convert radian to angular values
ang2rad Convert angular to radian values

All functions take arguments in radians (expect for ang2rad). For a detailed description of arguments and outputs consult the help text in the files.

Since 2010, most functions for descriptive statistics can be used in Matlab style matrix computations. As a last argument, add the dimension along which you want to average. This changes the behavior slightly from previous relaeses, in that input is not reshaped anymore into vector format. Per default, all computations are performed columnwise (along dimension 1).

- E. Batschelet, Circular Statistics in Biology, Academic Press, 1981
- N.I. Fisher, Statistical analysis of circular data, Cambridge University Press, 1996
- S.R. Jammalamadaka et al., Topics in circular statistics, World Scientific, 2001
- J.H. Zar, Biostatistical Analysis, Prentice Hall, 1999

If you have suggestions, bugs or feature requests or want to contribute code, please email me.

Cite As

Philipp Berens (2021). Circular Statistics Toolbox (Directional Statistics) (, MATLAB Central File Exchange. Retrieved .

Comments and Ratings (130)

Simon Weber

Max Wichmann

I have a question concerning your avarage functions circ_median, circ_mean: For which intervall the functions are working? I want to average my elevation and azimuth angles (spherical coordinates), however, elevetion is defined from [0,Pi] ,whereas azimuth is defined from [-pi, pi]. I saw your functions using the mod function. Do I have to change all the mod functions from 2Pi to Pi when I use elevation angles?

Thank you


Max Wichmann

Thanks for sharing this toolbox! It is really helpful. However, sometimes I got following error message due to array size and cache problems:

Error using repmat
Requested 44551x44551 (14.8GB) array exceeds maximum array size preference. Creation of arrays greater than this
limit may take a long time and cause MATLAB to become unresponsive.

Error in circ_dist2 (line 35)
r = angle(repmat(exp(1i*x),1,length(y)) ...

Error in circ_median (line 69)
dd = circ_dist2(beta,beta);

I was wondering if it is possible to avoid this problem by changing the code syntax thus there is not such a big use of cache?
Thanks again for sharing and for any kind of help.

Jae Moon

Hi, great toolbox.
But I keep getting the following warning:

Warning: Test not applicable. Average resultant vector length < 0.45.

Could you explain why this might be happening and how I can fix it?

Jean hude MOUDINGO

Adam Danz

This toolbox fills a void in matlab statistical tools and comes with great documentation. Great work.



Mahlega Hassanpour

Thanks for sharing. Which function calculates the minor and major eigenvalues?

Alvaro Rivera-Rei

Omer Dauber



These functions have been extremely helpful, given that I am very new to circular statistics.

One question - I'm trying to calculate the circular mean and 95% CI of a vector of phase values that go from -pi to pi. Circ_mean calculates mu fine, but my upper/lower limits return NaNs with the message

Warning: Requirements for confidence levels not met.
> In circ_confmean (line 70)
In circ_mean (line 53)
In Phase_Lock_Recall (line 150)

Would you be able to help with this? Do values need to begin at 0?

Thank you!


Suddha Sourav

This is one of the most well-written code I have ever come across. Deeply impressed by the numbering of equations in the code. Thank you very much for the contribution!

Josephine Benjamin

Hi i'm new in cthis line of topic but can you help me generate a mixture of von-Mises distribution with p=[0.3 0.7], mu=[1.5pi 3.2pi] and kappa=[3.6 2.8] i think circ_vmrnd function only inputs mean directions and concetration parameters. Thanks a lot

Keerthi Nagothu

Thank you, I am now able to easily calculate circular median! However I see the below warning:
Warning: Ties detected.
> In circ_median (line 58)

Can you help me understand what it means?

Yu Du

These are fantastic codes! The helped me a lot! Thanks!

I have a question about the cluster function circ_clust. I think there is something wrong with it. For example, I tried to use [0 0 0 pi pi pi] to get two clusters. But I did not get cid as [1 1 1 2 2 2]. Instead, I got cid = [1 2 1 1 1 1], which is very weird to me. Could you check the code and add more explanation about it? Thank you in advance.

Vincent BARALE

Thanks for the work, realy usefull and simple to use !!

Daniel de Malmazet

Thanks for the work! I've noticed a problem with the circ_median function. At line 54, if there are more than 2 elements that are equal to m, you might end up computing an inaccurate mean. It seems better replacing it with: " idx = find(dm==m,n); "
Daniel de Malmazet

Giovanni Esposito

hello ,
I have the directions expressed in degrees North but it is possible to use the circ_corr function to calculate the correlation or do you have to go in degrees east?
Thanks in advance

anisse khald

can you help me please about formatSubplot function? I can't find her. thank you very much

andrea bertana

Sorry, the previous rating was a mistake that I didn't notice until a colleague just pointed out for me!

andrea bertana


Hey Philipp,

In the circ_vmpdf function, line 45, it currently says this -

C = 1/(2*pi*bessi0(kappa));

I assume it should be besseli(0,kappa) ?

Yavor Kamer

@Adrian Bondy you da real MVP


Very intriguing ... but circ_r and circ_stats (and presumably other functions) return "NaN" if there are any NaN in the input data! This should at the very least be clearly mentioned in the documentation.


Absolutely fantastic, thank you so much for putting all of this in one place.

Alessia Cacace


I am getting an error with the circ_wwtest that says: "warning: test not applicable. Average resultant vector length less than 0.45 Check assumption line 109"

I am using data points coming from two different rose plots of angular data.
Can anyone help with this?



Philipp Berens

@andrea bertana: What did you not like?

@caseyraj: You convert the angle to radians and make a long vector. You need to additional vectors: factor1 and factor 2. In your case, factor1 indicates target and takes values 1, 2 and 3. factor 2 would be condition and takes values 1 and 2.

andrea bertana



I am currently trying to analyze some data consisting of angular data. Therefore I am trying to use the following function from Circular Statistics toolbox:

[p,F] = CircularANOVA(angles,[factor1 factor2],method)

The sample data is provided below as well. Any help on how to use this function would be greatly appreciated


Target 1 Target 2 Target 3

Condition 1 30.0 40.0 50.0
Condition 1 40.0 50.0 30.0
Condition 2 30.0 34.0 25.0
Condition 2 20.0 34.0 30.0

Philipp Berens

Dear all,

please not that feature requests, bug fixes or performance improvements should be made on github at

Thank you!

Tim Kietzmann

Perfect toolbox, so helpful. What I am missing is an implementation of the covariance (similar to the "cov" function in matlab). Could this be added by any chance?

Renaud JG

I have a question about how to use the function circ_hktest. I want to use it to analyse the difference in the mean of phase relation value between movement of the right vs left leg during locomotion at different time points following a spinal cord injury across different groups. The data comes from several subjects, so when I put all the data in a single column, the degrees of freedom are consistent with the number of groups and timepoints I have in the experimental design put the error match the total number of phas relation value I got, which equals the number of steps taken by all my subjects, which don't make sense. Also, the F is really high because I think the test mix intra and intersubject variability. Does I understand correctly? Thanks for helping me using your great toolbox!

Tristan Chaplin

Thanks for the great toolbox.

I have question about the applications to neuroscience. I had low firing cell (3Hz above spontaneous) that looked quite direction selective (CV = 0.3) but it just failed the Rayleigh test (p = 0.07). I notice the magnitude of weights (spike rates) used in the Rayleigh test affect the result - if I multiply the rates by 100 the result is highly significant, so it would seem it might not work well for low firing cells?

Then, I tried using the firing rate of each trial, rather than the mean firing rate for each direction, which your examples seem to use. The result is now high significant.

Do you have any recommendation on handling low firing rates and whether it is more appropriate to use trial rates or mean rates?

Alberto Dallolio

Great Job!
I'm trying to compute the standard deviation for angles..
Is there any way to do this with this toolbox?
Thanks a lot!


Michael Melnychuk

Hello, thanks for all your work putting this together. I have a question.

In the script 'circ_wwtest' there are a set of assumptions that are checked. Sample size and average vector length are the parameters. I am wondering what the rationale for this is. Should the F-test result (say on a very large sample for example) not outweigh arbitrary, hard-coded assumptions?

If you have a reference or a mathematical reason for these assumptions please let me know. :)

Thank you.

Philipp Berens

I have added the CircStat toolbox to github and would like you to add all feature requests and bug reports as issues there.

If you have fixes for bugs you can also create pull requests and I will incorporate them.

steve rogers

How do I get this toolbox installed in matlab?

Adrian Bondy

circ_vmpdf has a serious problem with numerical stability for large kappa (>700 for double precision returns NaN).

This is because the code tries to explicitly evaluate the bessel function in the formula for the von Mises, which is intractable for large kappa. This is unnecessary, since what is ultimately being computed is a ratio between the Bessel function and a large number in the numerator. By calling the bessel function with an additional argument:


Matlab computes the ratio of the Bessel function and exp(kappa) which is numerically stable.

Then you can simply replace the last two lines with:
p = exp( kappa*(cos(alpha-thetahat)-1)) / (2*pi*besseli(0,kappa,1));

This produces identical results within machine precision and is numerically stable for large kappa.


Thanks for the great toolbox.
I have been trying to use the function circ_clust. However, I keep getting the following error message:
In an assignment A(I) = B, the number of elements in B and I must be the same.

Error in circ_clust (line 53)
mu(j) = circ_mean(alpha(cid==j)');

Can you please advise how to overcome this problem?

Chethan Pandarinath

C Potgieter

Great package.

I found a small inconsistency in the circ_vmrnd program. On line 50, it should read:

alpha = 2*pi*rand(n,1)-pi;

Currently the "-pi" isn't in the code, which means for large values of kappa, the distribution takes values on the interval (-pi,pi), while for kappa close to 0 it takes values between (0,2pi).


This is a great toolbox. However, I have an issue with circ_corrcl, or perhaps I'm misunderstanding its proper use. When I feed it perfectly correlated data, I do not get rho=1:

ans = .7785

I noticed this while trying to see if I could get a sign for the rho value (which is always positive by the definition in the function). Unfortunately, I don't have a copy of the Zar text available.

Owen Brimijoin


Ch. Lat.

you meant "circ_rtest Rayleigh's test for nonuniformity " this is not what you look for ?

Ch. Lat.

@ Wasim Malik. I believe the toolbox fieldtrip has it, but their implementation is a bit more bothersome.

Wasim Malik

On a quick look, the Moore-Rayleigh test for uniformity of vector data (B.R. Moore, Biometrika, 1980) does not seem to be available in this toolbox. Philipp, do you have any plans to implement it? Alternatively, does anyone know if a Matlab implementation of that test is available elsewhere? Thanks.

Wasim Malik


Hi everybody!

I have a question about circ_plot.m; When I execute this code the angles appear from 0 to 360 degrees.

I only want represent values from 0 to 180. How I can do it? Thanks in advance!

Richard Edmonds

Actually, ignore the inverse_cdf function I have provided. It should generate a vlaue for kappa and it needs adjusting for values of thetahat other than zero.

Richard Edmonds

Great submission. It would be nice to have cdf and inversion cdf for the vmpdf functions. Here's what I wrote for my needs
function p = circ_vmcdf(alpha, thetahat, kappa)
%integrates the pdf from an angle of -pi to an angle alpha
F = @(x)circ_vmpdf(x, thetahat, kappa);
p = quad(F,-pi(),alpha);

function theta = circ_vminv(p, thetahat, kappa)
%computes the inverse of the abovecirc_vmcdf.
fun =@(alpha)(circ_vmcdf(alpha, thetahat, kappa)-p);
theta = fzero(fun,[-pi pi]);


Thanks for the great contribution.
Can you please let me know if it is ok to get negative mean or median?
Shall I add 360 to the final angle to make it positive?


Thanks for the toolkit.

Does anybody have a clue how to do multiple-regression with circular data?


Hi everyone,
Quick question regarding circ_hktest - I quite often get NaNs as an output in the 'Interaction' row. Any idea what am I doing wrong?
Thanks in advance for help.

Philipp Berens

Mear - could you be more specific regarding your doubts about the wwtest?

The negative values for circ_mean are a result of the way circ_mean is implemented. If you prefer them to be between 0 and 2pi, just edit the function to provide the data in that format.

I'll update the von Mises function in a future release.


Just a quick question concerning the circ_mean function: I usually get the results in negative values, despite all the input angles being in positive degrees (conv. to rads). It's hardly a big deal to translate this to [0,360] degrees, but it is a bit annoying and seems unneccesary. Is this how it should be? I'm also getting some results for the wwtest which seem very wrong to me (but make sense in light of the negative mean values), and it's making me question the accuracy of this toolbox.


Marc, thanks, you are right.
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 use the kuiper test to see whether the two distribution x and vonmis differ significantly (the difference can be in any property, such as mean, location and dispersion):
[H,pValue] = circ_kuipertest(x, vonmis)

However I was wondering if it is possible to have more accurate p-value estimates in the Kuiper test, as already asked by another user before.


Dear Mark,
thanks for your tip, however I'm not really convinced.
Both the circ_ktest and the circ_kuipertest are not described in the pdf:

Anyway, circ_ktest is a parametric two-sample test to determine whether two concentration parameters are different.
The circ_kuipertest is a two-sample test which allow to test whether two input samples differ significantly. The difference can be in any property, such as mean location and dispersion. It is a circular analogue of the Kolmogorov-Smirnov test.

I do not understand how these tests could help me with a goodness-of-fit test for the Von Mises-Fisher distribution, but probably is my limit.

Could anyone being of any help?




sergio - did you see the pdf with descriptions? (

You probably want either the ktest of the kuipertest.


Hi guys, I'm new to circular statistics and I've downloaded this package.

Given some vectors, I'd like to test if they are distributed following a Von Mises-Fisher distribution.
Do you know what instructions of the package I should use?
Can you help?

Aviram Gelblum

After some testing I figured the previous bug has to do with recurrence of unique values in the data. I took care of it by using
alpha=alpha+0.00001*(1:numel(alpha)), but this is obviously a workaround which isn't satisfactory for a self-respecting algorithm.

At any rate, I forgot to mention how great this toolbox is. It has been of great help, and saved me a lot of time and work.

Aviram Gelblum

Hi, I'm getting wrong clustering using circ_clust.
for example, if I give as an input circ_clust([1 1 1 1 3.5 4 5.5 0.5],2)

I get
ans =

sometimes the clustering does work, but I don't know why it does/doesn't...

I'm using Matlab 2012b...

Dylan Muir

Great toolbox. I was wondering if it is possible to have more accurate p-value estimates in the Kuiper test?

Francesco Montorsi

great toolbox, thanks.
By the way, I agree with Allan's comment (see below) that regarding the Von Mises distribution, it may be useful to have an implementation with higher numerical stability. In particular, I added this trivial function, which returns the log-pdf of the Von Mises distrib:

function [p alpha] = circ_vm_logpdf(alpha, thetahat, kappa)

% if no angles are supplied, 100 evenly spaced points around the circle are
% chosen
if nargin < 1 || isempty(alpha)
alpha = linspace(0, 2*pi, 101)';
alpha = alpha(1:end-1);
if nargin < 3
kappa = 1;
if nargin < 2
thetahat = 0;

alpha = alpha(:);

% evaluate pdf
C = -log( 2*pi*besseli(0,kappa) );
p = C + kappa*cos(alpha-thetahat);

Thanks to the greater numerical stability log-pdfs are often used in place of pdfs, so this little function may be of help to others...


Hi there, great toolbox. I propose a change to avoid numerical instability in circ_vmpdf.m.
Current code to evaluate the pdf:

C = 1/(2*pi*besseli(0,kappa));
p = C * exp(kappa*cos(alpha-thetahat));

Proposed replacement code:
C = log(1)-log(2*pi*besseli(0,kappa,1))+(kappa*cos(alpha-thetahat))-kappa;
p = exp(C);



Old code result: NaN
New code result: 12.6141

Philipp Berens

Ryan, the average is in the dot product w'*exp(...) which in the simplest case is a vector of ones - so this is the sum operation. exp(i*angle) decomposes the angle into its sine and cosine components. Finally, angle is atan2. Compare the results of your and my code - they should be identical with my code likely running a bit fast due to matrix style computations.



I haven't run through this toolbox yet, so I apologize if I am missing something with this question (I just glanced through the source code because I am interested in directional stats).

When you calculate the mean, the formula you use is:

% compute weighted sum of cos and sin of angles
r = w'*exp(1i*alpha);

% obtain mean by
mu = angle(r);

Now, correct me if I'm wrong, but this doesn't seem to calculate the average at all? It seems to me that here we are inputting a data array into the angle command, which will output the phase angle of each element of that array, not a singular mean.

Wouldn't a better way of calculating the average be to use atan2? Something like:

for i = 1:w
S(i) = sin(alpha(i));
C(i) = cos(alpha(i));

X = sum(S)*(1/w);
Y = sum(C)*(1/w);

mu = atan2(X,Y);

Marnix Maas

Thanks for the great toolbox! I have a question: I have a set of directional stochastic variables that are mutually correlated. I have used circ_corrcc to construct a correlation matrix for these variables, but I’m also interested in their covariance matrix. There does not appear to be a function for this in the current toolbox.

Not having any previous experience with circular statistics, I’m wondering if it makes sense to construct a covariance matrix by de-normalizing the correlation matrix, multiplying each element by the two corresponding circular standard deviations? Perhaps a covariance matrix could be a useful addition to the toolbox.


Philipp Berens

Hi Francesco, if you have orientations, multiply all orientations by 2 to obtain directions. If you want to obtain the mean resultant vector, devide its orientation by 2 again.


excuse my previous post! I just realize what p-axial truly meant.

For further reference this will solve the previously cited problem

%% uniform distribution test
% in the interval [0, 180)
y180 = circ_axial(circ_ang2rad(0 + 179*rand(4000,1)),2);
p180 = circ_otest(y180)
% in the interval [0 360)
y360 = deg2rad(0 + 359*rand(4000,1));
p360 = circ_otest(y360)


excuse my previous post! I just realize what p-axial truly meant.

For further reference this will solve the previously cited problem

%% uniform distribution test
% in the interval [0, 180)
y180 = circ_axial(circ_ang2rad(0 + 179*rand(4000,1)),2);
p180 = circ_otest(y180)
% in the interval [0 360)
y360 = deg2rad(0 + 359*rand(4000,1));
p360 = circ_otest(y360)


I am testing the toolbox out with not much of a prior knowledge on the subject. It seems a really good piece of software and it's helping me out grasping some of the theory.

I have a question: if I am dealing with orientations [0, 180) degrees more than directions [0 360), is there a proper way to transform may data prior to using the function in the toolbox?

For example, if I am trying to test for circular uniformity with a population that is uniformly distributed in [0 180) - which I'd like to have a p>0.05 - I obtained a very small value, which is consistent with the test looking over the full interval.

Suggestions? Thanks

---Example code ----
y180 = circ_ang2rad(0 + 179*rand(4000,1));
p180 = circ_otest(y180)
% in the interval [0 360)
y360 = deg2rad(0 + 359*rand(4000,1));
p360 = circ_otest(y360)

Jer Walley

Great toolbox! Exactly what I needed. However, my data has many NaN's - do you have a way to work around data with gaps?

Matt Davis

This is a great toolbox - very helpful. A few bug reports:

1. formatSubPlot calls "parseVarArgs" that's not standard matlab, or part of this toolbox. Could you add a pointer to where to download this.

2. In Example 2 the descriptive stats cell needs updating to respect the matrix style computations. So, line 67 should read:

stats(i,1) = circ_mean(ori,spk,2);

and similar for all the other lines of code.

Thanks for supporting this toolbox.


I confirm Dillon's report on circ_wwtest bug.


Great toolbox but I think there is an error in the logic used at circ_wwtest.m -> checkAssumption() lines 107-115.

The code currently reads:
if n > 10 && rw<.45
warning('Test not applicable. Average resultant vector length < 0.45.') %#ok<WNTAG>
elseif n > 6 && rw<.5
warning('Test not applicable. Average number of samples per population < 11 and average resultant vector length < 0.5.') %#ok<WNTAG>
elseif n >=5 && rw<.55
warning('Test not applicable. Average number of samples per population < 7 and average resultant vector length < 0.55.') %#ok<WNTAG>
elseif n < 5
warning('Test not applicable. Average number of samples per population < 5.') %#ok<WNTAG>

Notice that the if/else statements do not match the warning text. Particularly when n>5 the user will always be warned when the resultant vector, rw<0.55 which is not captured by the warning. The corrected if/else statements are as follows:
if n >= 11 && rw<.45
warning('Test not applicable. Average resultant vector length < 0.45.') %#ok<WNTAG>
elseif n<11 && n >= 7 && rw<.5
warning('Test not applicable. Average number of samples per population < 11 and average resultant vector length < 0.5.') %#ok<WNTAG>
elseif n<7 && n >=5 && rw<.55
warning('Test not applicable. Average number of samples per population < 7 and average resultant vector length < 0.55.') %#ok<WNTAG>
elseif n < 5
warning('Test not applicable. Average number of samples per population < 5.') %#ok<WNTAG>

I've assumed that the warning statements are correct but if the if/else statements are correct it would be more compact to warn the user under only 2 conditions: n<5 and rw<0.55.

Thanks again for the very useful toolbox.

Philipp Berens

Thanks for the recent feedback and bugreports. I was away for a while and will start taking care of them soon.

Omar Mian

Suggestion for addition:
Parametric and nonparametric paired sample tests, Zar (2010) Biostatistical Analysis, sections 27.13 and 27.14

Pablo Martinez

Sorry wrong line number. The error is at line 169!
in the function circ_hktest.m
pI = 1 - chi2pdf(chiI, df_i);
It should be
pI = 1 - chi2cdf(chiI, df_i);

Great toolbox.

Pablo Martinez

Great toolbox!

Pablo Martinez

I found an error in the function circ_hktest.m at line 160
pI = 1 - chi2pdf(chiI, df_i);
It should be
pI = 1 - chi2cdf(chiI, df_i);


Thank you very much for such a useful toolbox. Now, I have a question related to circ_ktest (two-smple test to compare concentration). The F statistic is defined only in case of rbar>.7, Mardia (pag 133, 1999) compute F in the case where resultant vector length is <0.45 :
n1 = length(alpha1);
n2 = length(alpha2);

R2avg = circ_r(alpha2);

R1 = n1*circ_r(alpha1);
R2 = n2*circ_r(alpha2);

%make sure that rbar > .7
rbar = (R1+R2)/(n1+n2);

if rbar > .7

f = ((n2-1)*(n1-R1))/((n1-1)*(n2-R2));

elseif rbar< .45 %taken from Mardia 1999 p.133 (Baschelet report: Mardia 1972 pag 161)

g11= asin(2*sqrt(3/8)*(R1avg));
g12= asin(2*sqrt(3/8)*(R2avg));

f= (2/sqrt(3))*((g11-g12)/(1/(n1-4)+ 1/(n2-4)).^(1/2));

But here Sample 1 and Sample 2 define the sign of F... and so S1 and S2 will be defined depending on Ravg value being S1>S2 for computation of F. Is this right?
Thank you!


Great tool.
I do have to say that circ_mtest is a bit weird.
The input is [pval, z] but output is set as [h,mu,ul,ll]

Balazs Barkoczi

Thanks for this excellent toolbox!
I have only some problems with the example files, that I downloaded from
??? Undefined function or method 'parseVarArgs', therefore the figure 2 isn't complete, and it hasn't axis labels.

??? Error using ==> mtimes
Inner matrix dimensions must agree.

Error in ==> example2 at 42
zm = r*exp(i*phi);
Perhaps a dot is absent, but after this modification zm = r.*exp(i*phi); the same error occurs:
??? Undefined function or method 'parseVarArgs'
Can somebody help me to fix this problems?
Thank you very much!


@Christopher: Thank you so much for your kindness and help. I really appreciate it.


@Fuh: indeed it should and when I step carefully through the function, sometimes the result comes out correct and sometimes it doesn't, somewhat dependent on the numbers in alpha. To fix the problem go to lines 45 and 46 of circ_median (ver 2011f). You see two inequalities, dd>=0 and dd<0. The two inequalities should be identical for consistency and the correct result. Edit line 46 to read:
m2 = sum(dd<=0,1);
Now the function seems to behave as expected.


I am new in circular statistics, so don't laugh at me... But I do have a question about the circ_median() function.

Say I have a data set that contains six angles [0.1 0.2 0.3 0.4 0.5 0.6]. when I feed these data into circ_median(), the function returns a median = 0.4

I thought that, when a data set contains an even number of observations, the median would be calculated as the average of the middle two numbers (i.e., (0.3+0.4)/2 = 0.35).

My code is listed below.

alpha = [0.1 0.2 0.3 0.4 0.5 0.6]';
med = circ_median(alpha)

Can anyone help me with this?


Philipp Berens

Thanks for the comments.

@Christopher: The ~ has been introduced as a placeholder in the latest MATLAB versions for output arguments that are not needed. I will go back to some dummy variable with the next upload.

@Heida: I don't see an easy way of doing what you suggest with the functions implemented.

@Omzaz: The multi-sample tests assume independent samples. I don't know about repeated-measures ANOVA etc. for circular data. If you find anything let me know.

The option to ignore NaNs... I think this is a tricky thing, because you always make a specific choice how NaNs are treated and each user might have different preferences. I will think about it though.

Omar Mian

Can any of the multi-sample tests in this toolbox be used with repeated measures data or do they all assume independent samples?

Omar Mian

Very useful toolbox. Option to ignore NaNs in the calculations would make it even better.

Vlad Atanasiu

Good toolbox! I added a function for kernel smoothing density estimate for circular data here: .

Thanks people, this toolbox is really helpful and easy to use.

I have one stats question -- forgive me that it is not a direct question about this toolbox but perhaps someone could help nonetheless.

I have repeated measures of circular data for multiple participants, lets say 15 participants where each participant contributes four angles. I have reason to believe that the angular distributions are going to be multipolar and not von Mises distributed. This would in essence require some kind of non-parametric repeated measures test which I am not sure has been developed for circular data. Is there a way to test for circular uniformity in this data set by using the procedures in the circstat toolbox, perhaps using some kind of p-value correction?


Um, what happened to my comment? What I wrote was that an typo error appears to have been introduced in circ_kuipertest.m in the advance to version 2011f. In 2010e, line 48 of the file reads:
[phis2 cdf2 phiplot2 cdfplot2] = circ_samplecdf(alpha2, res);

and in version 2011f, that line reads:
[~, cdf2 phiplot2 cdfplot2] = circ_samplecdf(alpha2, res);

and matlab complains of incorrect statement or expression.


Sorry, the error is in circ_kuipertest.m, not circ_kuiper.m as previously written!

Quang Nguyen Dinh

I'm sorry for this silly question. When I use circ_vmpdf to calculate for the case kappa is big and data has close value with the mean, it return the value bigger than 1. For example:

ans =

Did I mis-understand anything here?
I saw in the graphs of two functions based on kappa:

f1 = 1/(2*pi*besseli(0,kappa))
f2 = exp(kappa)

With high value of kappa, the increasing rate of second one is much higher than the first one so that it's not strange if the above case happened.

Bart Geurten

Needed a circular statistic means and got what I needed... thanks!

Philipp Berens

Hi Christopher, thanks for your feedback. I will update circ_plot with the next upload.

Omar Mian


Another issue with circ_plot.m:
A typo in line 121, should read
s = varargin{3};
(instead of vargin{1})


I am seeing an anomaly in circ_plot.m that does not make sense to me, can someone please explain or concur that it is a bug.

According to the help hist can plot either count or normalized bins.
In line 110 of circ_plot rose is called to calculate the bins
110] [t,r] = rose(alpha,x);
and in line 112 the normalized bins are plotted:
112] polar(t,r/sum(r),formats)
Now the vectors t and r returned by rose are such that they can be used to plot the bins directly and have the layout [0 n1 n1 0 0 n2 n2 0 0 ...] i.o.w. each element of bin count appears twice, and sum(r) is equal to 2.*length(alpha). So to truly normalize the r, we should divide by half of sum(r) and each bin should be twice as tall. Check by comparison to hist, which returns the same kind of information, but for the first and last bins.
I propose the following replacements for lines 112 and 113
112] polar(t,2.*r./sum(r),formats)
113] mr = max(2.*r./sum(r));


I found circ_median very slow. There might be a more efficient algorithm to get the median, but at least note that around line 42 m1 and m2 are each determined by calculating the same circ_dist2(beta,beta). circ_dist2 takes a LONG time. I would use an intermediate variable, and calculate circ_dist2 only once (to almost halve the time for the function to run, down from 2 seconds to 1 second for 2000 data points).


Nice work


Excellent toolbox!!! Thanks Philipp!

Philipp Berens

I fixed the bug in circ_clust.


This toolbox is great!! I'm getting an error with circ_clust though. Is there a bug? I haven't been able to solve it myself. This is the error message i get.

??? In an assignment A(I) = B, the number of elements in B and
I must be the same.

Error in ==> circ_clust at 53
mu(j) = circ_mean(alpha(cid==j));

Philipp Berens

There has been a slight (and unfortunately undocumented) change in semantics from 2009 to the later versions.

vmpdf computes the density, i.e. it evaluates the probability density function of the von mises distribution at the designated points. The earlier version computed the approximate probability in a small bin with width (alpha(2)-alpha(1) ), as is needed if you want to plot histograms. As you will see, you can easily recover the old behavior by

p = circ_vmpdf(points(1:end-1).', mu, kappa);
p = p * diff(points(1:2));

to obtain approximate probabilities.


Ops sorry I must have meesed up with the posts....
I'll repost my doubt properly:

Try to run

points = -pi:((4*pi)/(2*Nbin)):3*pi;
mu = -2.838;
kappa = 0.5125;

p = circ_vmpdf(points(1:end-1).', mu, kappa);

It seems to me that results from version circStat2009 (which I think were correct) differ from those of CircStat2009d and CircStat2010b. Maybe I'm doing something wrong. Please let me know.
Thanks a lot,


also with the 2010b.

Philipp Berens

I fixed the bugs reported in December in the first upload of 2010.

I also added the functionality asked for by chairmanK. The functions for descriptive statistics now handle N dimensional arrays and the computations can be performed 'Matlab-style'. As a backup, the new release comes with a folder 'old', which contains the functions that are thus replaced as backup. If you experience problems or issues with the new functions or would like to see additional functions converted let me know. Unused arguments in between can be left empty.

circ_vmpdf seems to work fine with me and produces data with the correct moments. Please be more specific.


I think, in circ_skewness, the eponent for the denominator of the last equation should be (3/2), rather than (2/3), according to formula (2.29) of Fisher 1993.


something's wrong in the new circ_vmpdf....can't replicate results from previous release


Functions do not gracefully handle N-dimensional arrays for powerful MATLAB-style computations; instead, inputs are coerced to be column vectors. There are also numerous errors. One example, in circ_moment.m:
cbar = sum(cos(p*alpha'*w))/n;
(p*alpha'*w) is a SCALAR dot product, so clearly this is not a weighted sum of cosines as it ought to be.
There are many other bugs like this. Please fix!

Philipp Berens

Issue is fixed in the upload of 11/5/09.

Omar Mian

Encountering same issue as Shiquan

Shiquan Wang

bug report:
function stats = circ_stats(alpha, w, d)
line50:stats.std_mardia = circ_std(alpha,w,d,'mardia');
the function circ_std(alpha, w, d) doesn't accept parameter 'mardia'.

Shiquan Wang

This is great work! Thanks.

Philipp Berens

Dear Florin,

thanks for the error report.

With regards to 1: Fixed all bugs. I tested the output on the example in Harrison & Kanji.

With regards to 2: This is unfortunate. The current (and more recent) toolbox version returns both, angular deviation and circular standard deviation as first and second return argument.

Appreciate your feedback,


Flurin Honegger

Good work! So far it helped me a lot. But there are some
errors arround!

1.) function circ_hktest

Line 55 found -> qm = zeros(p,1); qr = qm; qn = pm;
corrected? -> qm = zeros(p,1); qr = qm; qn = qm;

Line 94 found -> eff_2 = sum(qr.^2 ./ sum(cn,2)) - tr.^2/n;
corrected? -> eff_2 = sum(qr.^2 ./ sum(cn,1)') - tr.^2/n;

Line 107 found -> beta = 1/(1-1/(5*kk)-1/(10*(kk^2)));
comment -> beta overloads the beta function (help beta)
An other name like betaF should be used

Line 144 found -> F1 = beta * ms_1 / ms_r;
comment -> if inter is set to 0/false beta is not defined!

2.) function circ_std

The documentation in the paper

In CircStat , theangular deviation is computed as
and the circular standard deviation as

does not fit with neither - paper and Matlab Central - implementation.

Philipp Berens

Both definitions are around... I will optionally add computing both with the next update.

Richard Heitz

Sorry, but I think circ_var returns s = (1-r) when it should be s = 2*(1-r).

for this reason, circ_std and circ_var will not agree

This is a great submission, filling an obvious gap in the statistical world out there. Easy to use, well done, and the author provides great feedback.


Tal Krasovsky

Excellent toolbox, helped me a lot. Greatly appreciated!

Richard Heitz

this is a great toolbox

Lorenzo Guerrasio

I think there is an error in the circ_dist function

I think this

r = angle(repmat(exp(1i*x(:)'),length(y),1) ...
./ repmat(exp(1i*y(:)),1,length(x)));

Let me know if it's correct

Lorenzo Guerrasio

I wish I saw it before :)
Very nicely done.

Adrian B

I take that back -- guilty of confusing 'r' and 'R'

Adrian Bartlett

I believe I have found an error in one of your functions:
Line 51: z = R^2 / n;
should be
Line 51: z = R^2 * n;

Lyle Muller

Would include skewness and kurtosis!

Sooho Park

Exactly what I needed

sanjay sane


MATLAB Release Compatibility
Created with R2012a
Compatible with any release
Platform Compatibility
Windows macOS Linux

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!