Circular Structural Model

The listed below files contains all the code needed for implementing the ML estimation of circles from noisy data, as described in the attached paper

List of files

1. GetMoments - Computes the sample average of $X$ and $Y$, and $\widehat{\Sigma}$ the empirical covariance matrix of the pair $(X,Y)$.
2. UniformityTest - Function that apply the uniformity test and returns the $\chi^2(2)$ statistic and the corresponding p-values of the test.
3. CircleMLE_Uniform - Maximum likelihood estimation under uniformity of the observations across the circle.
4. FindMu - Estimating the 2 possible values of $\mu$ - the mean direction of the Von-Mises distribution using the spectal decomposotion of $\widehat\Sigma$.
5. bisection1 - Computing $\varkappa_L$, the lower bound of $\hat\varkappa$ using the bisection method.
6. get_mu_bounds1 - Lower and upper bounds for the 2 possible values of $\mu$.
7. VM_negLogLike3 - Function that computes the negative log-likelihood function of the sample $(x_i,y_i)$, $i=1,\dots,n$.
8. CircleMLE - Wrapper of functions 1-7 that given a sample $(x_i,y_i)$, $i=1,\dots,n$ return an estimates of the circle parameters along with the nuisance paramteres.

Instructions

Scripts 1-8 need to be downloaded to your working directory. Then, apply the function CircleMLE (i.e. script 8) which takes 2 arguments, $x$ values (vector) and $y$ (vector of the same length).

CircleMLE returns a vector of length 6, with the following order - kappa, mu, rho, a, b, sigma. This function is a wrapper which uses function 1-7 in order to estimate the vector of parameters.

There is no need to use functions 1-7 seperately.

Examples

• sol=CircleMLE(x,y) - Returns a vector of length 6, with the estimated parameters, ordered by kappa, mu, rho, a, b, sigma.
• [sol, P_val] = CircleMLE(x,y) - sol is the vector of estimated parameters and P_val is the P-value of the uniformity test.
• [sol, P_val, stat] = CircleMLE(x,y) - sol is the vector of estimated parameters, and P_val and stat are the uniformity test P-value and $\chi^2$ statistic respectively.

For any issue - feel free to contact me at adonner@campus.haifa.ac.il