Robust estimators for circular regression models

2.1 Model formulation

For any two circular random variables $\mathbf{U}$ and $\mathbf{V}$ , Sarma and Jammalamadaka (1993) proposed a regression model to predict v for a given u, by considering the conditional expectation of the vector $e^{\mathit{iv}}$ given u such that

Then, v can be predicted such that

Due to the fact that $g_1\left(u\right)$ and $g_2\left(u\right)$ are periodic functions, thus they are approximated for a suitable degree m (Kufner and Kadlec, 1971), which have the following two observational regression-like models $$V_{1j}=\cos v_j=g_1\left(u\right)\simeq {\displaystyle \sum _{k=0}^m}\left(A_k\cos {\mathit{ku}}_j+B_k\sin {\mathit{ku}}_j\right)+{\epsilon }_{1j}$$

2.2 Least squares estimation

Let $(u_1,v_1),\dots ,(u_n,v_n)$ be a random circular sample of size n. Therefore, the observational Eqs. (4) can be summarized as $${\mathbf{V}}^{\left(1\right)}={\left(V_{11},\dots ,V_{1n}\right)}^{\prime },{\mathbf{V}}^{\left(2\right)}={\left(V_{21},\dots ,V_{2n}\right)}^{\prime },$$

The observational Eqs. (4) can be written in matrix form

The least squares estimates turn out to be

These equations can be combined into the following single matrices

The following section explains the effect of outliers on the JS circular regression model.

4.1 Robust CM-estimation of JS circular regression parameters

In Eq. (9), if ${\left({\mathbf{V}}_i-\mathbf{U}{\mathit{\lambda }}^{\left(\mathrm{p}\right)}\right)}^2$ , where $p=1,2$ , has been replaced by $F\left({\mathbf{V}}_i-\mathbf{U}{\mathit{\lambda }}^{\left(\mathrm{p}\right)}\right)$ , where, F is symmetric, non decreasing function on [0, $\infty$ ), and almost continuously differentiable anywhere, where $F\left(0\right)=0$ . Furthermore, F is a function, which is less sensitive to outliers than squares, then it yields an estimating equation, which result is the same idea of M-estimation of a linear regression model as described by Huber and Lovric (2011).

We define CM-estimates on the JS circular regression as follows:

To solve these equations, let the influence curve be $\psi =F^{\prime }$ , if this exists then we will have:

Whereas, if $F(.)={(\mathbf{V}-\mathbf{U}\lambda )}^2$ , then the solution becomes the LS estimate. Generally, Biweight or Huber functions have been widely used as $F(.)$ . However, for Biweight function we define the weight matrix $W=\mathit{diag}\left(w\right)$ with $w=\frac{\psi \left(\epsilon \right)}{\epsilon }$ then (13) can be reformulated as

Then (13) can be written as:

These equations can be combined into the following single matrices:

A popular possibility of $F(.)$ is to use the Huber’s function as introduced in Huber and Lovric (2011). For a positive real $M=1.345$ , Huber introduced the following objective function

So the $\psi$ is given as

Then the weight function W is given by:

4.2 Influence Function of Circular Regression Estimators

Let T be an estimator of $\psi$ - type, then the influence function (IF) describes the effect of an infinitesimal contamination at $\mathbf{U}$ on the estimator T, and it is defined as:

It worth to remark that, IF of circular LS estimator is an unbounded function in $\mathbf{U}$ and $\mathbf{V}$ . On the other hand, the influence function of robust CM estimator is given by:

4.3 Bounded-influence circular estimator

The CM estimators are sensitive to circular leverage observations so, we propose a bounded-influence circular estimator named robust CLTS estimator. By ordering the squared residuals ${\epsilon }_1^2$ and ${\epsilon }_2^2$ ascendingly: $${\epsilon }_{1\left(1\right)}^2,{\epsilon }_{1\left(2\right)}^2,\dots .,{\epsilon }_{1\left(n\right)}^2\mathrm{and}{\epsilon }_{2\left(1\right)}^2,{\epsilon }_{2\left(2\right)}^2,\dots .,{\epsilon }_{2\left(n\right)}^2\text{.}$$

Then the CLTS circular estimator choose the circular coefficients ${\widehat{\lambda }}^{\left(1\right)}$ and ${\widehat{\lambda }}^{\left(2\right)}$ which minimize the sum of the smallest $\frac{n}{2}$ of the squared residuals, $$\mathit{CLTS}\left({\widehat{\lambda }}^{\left(1\right)}\right)={\displaystyle \sum _{i=1}^{\frac{n}{2}}}{\left(e^{\left(1\right)}\right)}_{\left(i\right)}\mathrm{and}\mathit{CLTS}\left({\widehat{\lambda }}^{\left(2\right)}\right)={\displaystyle \sum _{i=1}^{\frac{n}{2}}}{\left(e^{\left(2\right)}\right)}_{\left(i\right)},$$ which is equivalent to find the circular estimates corresponding to the half circular sample having the smallest sum of squares of residuals. As such, breakdown point is 50%. Replacing $\frac{n}{2}$ by $[n+(2m+1)+1/2]$ , we get a robust CLMS estimator.

The following section investigates the performance of the proposed robust estimators via simulation.

5.1 Settings

A simulation study was carried out to investigate the performance of the proposed robust estimators for the JS circular regression model, namely $\mathit{CM},\mathit{CLTS}$ , and CLMS. Furthermore, to compare these estimators with classical estimator LS. For simplicity, we consider the case when $m=1$ . Hence, we have the following set of parameters to be estimated: $$\mathit{\lambda }=({\mathit{\lambda }}^{\left(1\right)},{\mathit{\lambda }}^{\left(2\right)})=({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5,{\lambda }_6)$$

We consider the set of uncorrelated random errors $({\epsilon }_1,{\epsilon }_2)$ from the bivariate Normal distribution with mean vector 0 and variances $\left({\sigma }_1^2,{\sigma }_2^2\right)$ to be (0.03,0.03). The independent circular variable $\mathbf{V}$ is generated from von Mises distribution with mean $\pi$ and concentration parameter equals 2 i.e. $\mathit{vM}(\pi ,2)$ .

For simplicity, we set the true values of $A_0$ and $C_0$ of the JS model to be 0, while $A_1,B_1$ , $C_0$ and $D_1$ are obtained by using the standard additive trigonometric polynomial equations $\cos (a+u)$ and $\sin (a+u)$ when $a=2$ . For example, $\cos (2+u)=-0.0416\cos u-0.9093\sin u$ and $\sin (2+u)=0.9093\cos u-0.04161\sin u$ . Then by comparison with Eq. (4), the true values of $A_1,B_1,C_1$ and $D_1$ are −0.0 4161, −0.09093, 0.09093 and −0.04161 respectively. Similarly, we can also get different sets of true values by choosing different values of a.

We then introduce vertical and leverage outliers into the data such that the percentages of contamination used are c%= 5%, 10%, 20%, 30%, 40% and 50% from the different sample sizes, namely n = 20, 50 and 100.

To investigate the robustness of the estimators against vertical and leverage circular outliers, the following scenarios were considered:

1. No contamination

2. Vertical (outliers in the $\mathbf{V}$ only)

3. Leverage points (Outliers in some $\mathbf{U}$ only).

For vertical outliers scenario, the observation at position d, say $v_d$ , is contaminated as follows; $v_d^{☆}=v_d+\gamma \pi \mathit{mod}\left(2\pi \right)$ , where $v_d^{☆}$ is the value after contamination and $\gamma$ is the degree of contamination in the range $0⩽\gamma ⩽1$ . The generated data of $\mathbf{U}$ and $\mathbf{V}$ are then fitted by the JS circular regression model to give the estimates of ${\widehat{A}}_0,{\widehat{A}}_1,{\widehat{B}}_0,{\widehat{B}}_1,{\widehat{C}}_1$ , and ${\widehat{D}}_1$ .

For leverage point scenario, different percentages of observation at position d, say $u_d\sim vM\left(2\pi ,6\right)$ instead of the original generated data from $\mathit{vM}(\pi ,2)$ . The performance of the proposed estimators were then determined by assessing summary of three statistics based on $s=1000$ Monte Carlo trials.

The first statistic is the median of standard error (SE) of the six parameters and it is obtained by $$\mathit{SE}\left({\widehat{\lambda }}_j\right)=\sqrt{\frac{{\displaystyle \sum _{j=1}^s}{({\widehat{\lambda }}_{i,j}-{\overline{\lambda }}_i)}^2}{s}},i=1,2,\dots ,6$$ where $\overline{\lambda }$ is the mean of the estimates which is obtained by, $\frac{{\sum }_{j=1}^s{\widehat{\lambda }}_{i,j}}{s}$ . The second statistic is the median of mean errors of the estimators given by, $\frac{{\epsilon }_{i,j}}{s}$ . Finally, the median of mean of the cosines of the circular residuals $A\left(\kappa \right)$ .

The simulations were performed by the statistical software R. To run the simulation, the function $\mathit{rlm},\mathit{ltsreg}$ and lmsreg from library MASS were used for M-estimation, LTS and LMS, respectively.

5.2 Results and discussion

Table 1 shows that the median (MSE) of the LS is relatively smaller than other estimators when the data are uncontaminated, So the LS gave the best estimator.

Table 2 shows the results for contaminated data with vertical outliers, where the MSE for CM was the smallest, and estimated the associated $A\left(\kappa \right)$ were larger than others estimators. Thus, we concluded that the robust CM is better than LS. The CLTS and CLMS performance are almost the same.

According to Table 3, CLTS and CLMS perform better than all the other estimators. They estimated models parameters with smallest MSE, but suffer from small values of $A\left(\kappa \right)$ when the leverage percentages are increased in the data set. The CM does poorly as worse as LS, and has higher median (MSE) than other robust estimators.