Different Distributional Errors-in-Circular- Variables Models

The modeling of functional relationship between circular variables is gaining an increasing interest. The existing models assume the errors have same probability distributions, but the case of different distributional errors is not yet investigated. This paper considers the modeling of functional relationship for circular variables with different distributional errors. Two functional relationship models are proposed by assuming a combination of von Mises and wrapped Cauchy errors, with a distinction between known and unknown ratio of error concentrations. Parameters of the proposed models are estimated using the maximum likelihood method based on numerical iterative procedures. The properties of parameters' estimators are investigated via an extensive simulation study. The results show a direct relationship between the performance of parameters' estimates and the sample size, as well as the concentration parameters. For illustration, the proposed models are applied on wind directions data in two main cities in the Gaza Strip, Palestine.


Introduction
Adcock (1877, 1878) explored Errors-in-variables model (EIVM) as a practical statistical approach for modelling different problem (Gillard, 2010). EIVM is known as the measurement error or random regression model.
The main difference between ordinary regression and EIVM is that the response and explanatory variables in EIVM have no distinction; both are measured with errors, unlike in the regression model, where only the response variable is measured with errors. EIVM has two types, namely, functional and structural relationship (see Kendall 1951Kendall , 1952

Parameters estimation:
The maximum likelihood estimates of model's parameters are obtained as follows:

a) Estimation of  :
To estimate the intercept parameter,  we derive the log likelihood in Equation (2.4) with respect to 1  and 2  , and equating them by 0 as follows: The first partial derivative of the log likelihood function in (2.4) with respect to i X is given by (2.6) After equating Equation (2.6) by 0, the solution is obtained iteratively for a given initial guesses of i X . Suppose 0 i X is an initial estimate for i X such as, For linear relationship between X and Y , the slope parameter  should be close to 1, and for small 1i Thus, the MLE of i X is obtained iteratively as follows: Possible initial guesses for iteration are 0

c) Estimation of
 : The first partial derivative of the log likelihood function in (2.4) with respect to  is given by: After equating Equation (2.8) by 0, then its solution is obtained iteratively for a given initial guess of 0  , The following component is reformulated as follow: Hence, Equation (2.8) can be rewritten as One possible initial guess for iteration is 0 1   ; this is sensible since both X and Y are assumed to be linearly dependent.

 :
The first partial derivative of the log likelihood function (2.4) with respect to  is given by:

Circular functional relationship model with different distributional errors and unknown ratio of concentrations
In this section we assume unknown relationship between errors' concentration parameters  and  . Thus, we follow the same procedures used in Section 2. However, we replace  with  in Equation (2.2). Therefore, the loglikelihood function of the probability for this model is given by The estimation of the parameters is briefly presented as follow:  The MLE estimates of the models' parameters can be numerically obtained by using any statistical software such as R. The iterative re-weighting algorithm for the maximum likelihood estimation is working as follows: Step 1: Initialize [ 1 k   , [ 1] 2 k   , [ 1] k X  , [ 1] 0 k   and [ 1] k   .
Step 4: Obtain the values of  and  by the formulas given in Sections 2 and 3.
The following section discusses the settings and results of the simulation study on the characteristics of parameters' estimates, including bias, mean and mean square errors.

Simulation study
This section presents the settings and results of the simulation study to assess the accuracy and biasedness of the parameters' estimates in the two proposed models.

Settings:
The simulation results are obtained on the basis of 1,000 generated samples, where the values of the random variable X follows the von Mises distribution with mean 0.5 and concentration parameter 2, i.e. The following settings are considered to generate the data sets of the considered models 1) A random sample X of size n is generated from von Mises with mean 0. 5  , where s=1,000.
The simulation study is conducted using R statistical software package.

Results :
Simulation results show a direct relationship between the goodness estimates of all considered parameters and the sample size n, as well as the concentration parameter  for all considered cases.
The performances of the two proposed models are compared at   (i.e. 1   ). For the two considered models a homogeneity is observed in the performance of the slope parameter estimates, especially for 30 n  .

Application
As an illustration, a total of 96 measurements of wind directions that have been collected from Gaza and  Given the reasonably linear relationship between wind directions, the proposed circular functional relationship models are suggested to fit the data. The parameters estimates are obtained by applying the iterative procedures and shown in Table 1.
a) Spoke plot b) Scatter plot  The estimate of intercept parameter  in Model 1 is 5.724, which is equivalent to -0.555 and closed to 0. The estimate of  in Model 2 is 3.558, which is far from the 0 direction. Furthermore, the estimate of the slope parameter  in Model 1 is 1.117 which is closed to the true value one, but the estimate of  in Model 2 is 3.6512 which is far from 1. Moreover, it is an unjustified value in the circular context.  The plot of the circular errors in the Y variable versus the observations and its histogram in the two considered models are given in Figures 4 and 5, respectively. Both figures show that the circular means of errors in both models are close to zero. Moreover, the histogram of errors in Model 1 is symmetric with suspected outliers in its right tail. The errors of the other Model are asymmetric.  The obtained residuals were tested to follow wrapped Cauchy distribution and von Mises distribution via Kolmogorov-Simirnov test and Watson test, respectively, at 0.05 level of significance. Table 3 provides the summary. Pak  The results in Table 3 reveal that the obtained residuals from the two models follow their proposed models. Therefore, we conclude that Model 1 is the best fit between the two considered models.

Conclusions
This paper has addressed the problem of functional modeling of two circular variables with different distributions of errors, namely, von Mises and wrapped Cauchy distributions. The maximum likelihood estimations of parameters are obtained in both cases when the ratios of error concentrations are known and unknwn. The results of the simulation study show that the estimators perform well for large sample sizes and for large concentration parameters.