Circular data refer to a set of observations measured by angles and distributed within radians and it can be presented on the circumference of a unit circle. Circular data need special statistical methods to be described and modeled rather than the conventional linear techniques. Circular data can be found whenever periodic phenomena occur; it is the source of interest to scientists in many fields, including: biology, meteorology, physics, psychology, image analysis, medicine, astronomy, social sciences and earth sciences, see [1] . The existence of outliers is considered as one of the most common problems in statistical analysis. This can be extended to circular data due to the expected influence of outliers on the parameters estimates. Outliers in the context of circular data would be defined as a set of observations which is inconsistent with the rest of the sample. It is expected to lie far from the mean direction of the circular sample. Despite this, there are only a few numerical and graphical tests of discordancy in circular samples. The problem of outliers in different types of circular data including univariate samples, regression, functional relationship models and circular time series are addressed by several authors (see [2] -[7] ).

The rest of this paper is organized as follows: Section 2 describes the properties of the wrapped Cauchy distribution. Section 3 presents four discordance tests to detect possible outliers in circular univariate data. In Section 4, the cut-off points for tests are obtained based on samples generated from the wrapped Cauchy distribution. The power of performances is investigated via simulation studies in Section 5. Lastly, we apply the statistics on two real data sets for illustration in Section 6.

A circular random variable can be obtained from any random variable on the real line X with probability density function, and distribution function by defining

.

That’s mean wrapping the original distribution on the real line around the circle to get the wrapped distribution. The Cauchy distribution on the real line with the density

where and are the mean and standard deviation, respectively. Once we wrapped the around the circle, then we get to the wrapped Cauchy distribution with probability density function denoted by and given by:

where is the mean direction and is the concentration parameter that is called the mean resultant length. Then, the distribution function of the wrapped Cauchy is given by:

Reference [8] introduced the wrapped Cauchy distribution, and [9] illustrated that the wrapped Cauchy distribution can be obtained by mapping Cauchy distribution on to the circle by the transformation. Reference [10] quantified the dispersion measure for the wrapped Cauchy distribution by a concentration

parameter, and is given in the form, and he explained that as approaches 0, the distribution converges to the circular uniform distribution with probability density function;

and as approaches one, the distribution tends to the point distribution concentrated in the direction.

The distribution is unimodal and symmetric about, Reference [11] illustrated that the distribution enjoys the additive property and the central limit theorem, on other words, the convolution of the wrapped Cauchy distributions and is the wrapped Cauchy distribution. One of the main features of the wrapped Cauchy distribution that has a heavy tail even for large concentrations, which make the detection of outlier a hard task.

Suppose that we are given angles that are observations in a random circular sample of size from a circular population. We consider four discordance tests based on M, C, D, and A statistics to identify outliers in a univariate circular sample from the WC distribution. Under the null hypothesis that is not an outlier.

In this section, we obtain the cut-off points for the four test statistics based on simulation studies. The percentage points of the null distribution of free outliers in the generated random circular samples from the wrapped Cauchy distribution, with mean direction zero and concentration parameter,. We consider 12 values of the concentration parameter in the range of 0.1 to 0.999 and 20 different sample sizes from 5 to 150. For each generated random sample the values of the four considered statistics M, C, D and A are calculated based on the formulas in Section 3.

For each combination of the sample size n and concentration parameter, the process is repeated 3000 times to ensure the convergence of the desired percentiles (cut-off points). The obtained statistics are sorted in ascending manner and then 10%, 5% and 1% upper percentiles of free outliers samples are obtained. Tables 1-4

present part of the cut-off points at 5% percentiles. The comprehensive cut-off points are available upon request from the authors. From the obtained cut-off points we notice that:

Firstly, as one would expect, there are an inverse relationship between the cut-off points and the level of percentiles. Secondly, for M statistic the increase of the concentration parameter increases the cut-off of points, while the increase the sample size n decreases the cut-off points. Thirdly, the cut-off points of D statistic are fluctuating slightly for, and correlated indirectly with either sample size n or concentration parameter for. Fourthly, for C statistic the cut-off points are a decreasing function of the concentration parameter and there are an inverse relationship between the cut-off points and the sample size. Lastly, the cut-off points of A statistic keep increasing as the concentration parameter increase up to, and then the cut-off points are rapidly approach zero for. Furthermore, the increase of sample size reflects on the concentration parameters as follows: 1) for small concentration parameter the cut-off points decreases gradually; 2) for the cut-off points almost constant; 3) for high concentration parameter the cut-off points increases gradually.

The power of performance of discordancy tests can be evaluated via several measures. References [16] [17] stated that a good test of discordancy should have: 1) a high power function; where is the Type-II error; 2) a high probability of identifying a contaminating value as an outlier when it is in fact an extreme value, where an extreme value is defined as a point with the maximum circular deviation, denoted by; and 3) a low probability of wrongly identifying a good observation as discordant, where.

To study the performances of the four discordancy tests, we use 3000 samples based on different sizes n and concentration parameter. The samples are generated in such a way that of the observations come from and the remaining one observation comes from, where is the degree of contamination and. The M, C, D, and A statistics in each random sample are then calculated based on corresponding equations as given in Section 3. Furthermore, the values of power performances are obtained.

Figure 1 illustrates the behavior of power of performances of the tests for different cases. The main results can be summarized as follows:

Firstly, the performance for all statistics increases when we increase the contamination value (Figure 1(a) and Figure 1(d)) and tests outperform for (Figure 1(c) and Figure 1(d)). C and A statistics perform better than other statistics for large contamination levels, while M statistic is better for small contamination level. Secondly, there is an increasing function between the power of performances and the concentration parameter (see Figure 1(a) and Figure 1(c)). Thirdly: For any sample size, all considered discordancy tests at moderate or less concentration parameter, the values of P1 are very low (less than 0.1) regardless the contamination level (Figure 1(a) and Figure 1(b)). The weak performances for small concentration parameter is attributed to heavily tails of the wrapped Cauchy distribution, similar trends are observed for P3 and P5. Lastly, the difference between P1 and P3 generally are very closes to 0 for all cases.

For illustration purposes, two real data sets following the wrapped Cauchy distribution are considered to be analyzed, and to apply the proposed tests of discordancy to illustrate their performance in real data as given in the following subsections.

In this paper four tests of discordancy M, C, D and A were extended for the wrapped Cauchy distribution; the cut-off points and the power of performances were investigated via extensive simulation study. It was noticed that for any sample size, all considered discordancy tests at moderate or less concentration parameter (), the power of performances is very low (less than 0.1) regardless the contamination level λ due to the heavy tailed characteristics of the wrapped Cauchy distribution. Thus, it is recommended to propose various circular regression and functional relationship models with wrapped Cauchy error which is expected to be more robust to the existence of outliers. Moreover, the tests were applied on ants’ data set and wind direction data set.