Directional Data Analysis and An Application Gazel

Directional data analysis method is used in the observed values that are obtained angularly. In this technique, every observation obtained as an angular data type is located on the unit circle axis and a mean vector in relation to the data is formed. The statistical computations are obtained through that mean vector. In this study, some of the basic directional statistical methods are aimed to be analyzed. In accordance with this purpose, the angles between teats are measured six times in the mornings before milking the goats during the lactation period. Some directional statistics were obtained from this analysis by these teat angles measurements were analyzed. As a consequence, the length of mean vector increased while the circular variance was decreasing. The lowest variance and the maximum length of the mean vector are obtained in the 6 th test milking. In addition, according to the Watson-Williams F test, which is used to determine whether there is a difference between the mean angles, the differences between the mean angles of test milking 5 and 6 are found to be statistically important (p<0.05) and the mean angles decrease through the end of the lactation period.


Introduction
Directional data analysis is the subfield of the statistics (Berens and Velasco 2009).The observations obtained from the directional data cannot be analyzed with standard statistical methods.It is because that kind of data inherently are circularly structured and they are located on the 360 degree axis (Açar 2010).The measuring instruments used to get such data are clock, compass and protractor.Every observation obtained by this way is located on unit circle (Gaile and Burt 1980).The observations of different data types (like day or month) can be transformed into angles.For instance, time arrivals of the patients going to the accidents unit of a hospital in 24 hours, directions of the winds on a daily basis, directional movements of the animals, directions of the bees after a stimulus in a hive or migration directions of the birds (Mardia and Jupp 2000;NCSS 2012).Traditional statistical methods may not characterize enough the statistical specifications of the data obtained from the circularly scaled variables (Mahan 1991).For example, the arithmetic mean of 10˚, 30˚ and 350˚ is 130˚ and they show south east as the geographical direction.However, when all the directions are located on circle, they show a direction towards the north.Therefore, the calculated classic arithmetic mean may lead a wrong direction.In this case, using special methods such as directional data analysis methods instead of the classic methods is more appropriate to determine the angular measures (Gaile and Burt 1980;Mahan 1991;Jones 2006;Berens and Velasco 2009).In this study, some of the basic directional data analysis methods are aimed to be analyzed.

Animal Sampling
Thirty Akkeçi crossbred goats used as material in the research began to be milked after kids weaned.Udder measurements and milk controls were performed for 6 times with intervals of two weeks until the end of the lactation period.The angle between right and left teats among the measurements were used in this study.The measurements were taken in the morning and immediately before milking the animals.No further measurement was taken prior to evening milking.Teat angle were measured according to Kor et al., 2004.

Directional basic statistics
To calculate the mean angle, the cartesian coordinates ( ) , S C of the data points must be computed; These values allow for the computation of the length of the mean vector ( When the data points equally distributed on the circle, length of mean vector becomes ''0''.However, when the data points are on the same location, length of mean vector is ''1''.The mean angle ( ) ) (Mardia and Jupp 2000).Circular variance is used as a measure of dispersion around the mean vector.The simplest of these is the sample circular variance defined as, (Mahan 1991;Mardia and Jupp 2000;NCSS 2012).Watson-Williams F-test is used to determine statistical different between two or more mean angles .Watson-Williams test is a modification of the linear F test (Mahan 1991;Mardia and Jupp 2000;Hassan et al. 2009).

Results and Discussion
In Table 1, circular descriptive statistics of angles between right and left teats obtained for six test milking were given.The mean angle of all time of test milking is 32.412˚.If observation values are calculated using classical arithmetic mean, the means of all time of test milking shall be obtained as 32.564.This classic arithmetic mean is a value close to mean angle (see Table 1).However, this situation is misleading.Because the circular mean calculated for the circular data set is not the measurement of the location of data such as classic arithmetic mean but rather it is a function of the zero direction and choice of the turning direction.Therefore, descriptive statistics obtained with classic statistical methods, such as the arithmetic mean are not the appropriate methods for the circular data (Gaile and Burt 1980;Mahan 1991;Jones 2006;Peker and Bacanlı 2009;NCSS 2012).Moreover, directional data analysis unlike the linear-based analysis methods does not get affected by the extreme values (outlier) in the data (Berens and Velasco 2009).The length of the mean vector ( R ) plays a determining role in the directional data analysis.Especially, the length of the R vector affects the estimate of variance.For example, when the length of the mean vector decreases from 1 to 0, variance of the distribution increases.In other words, when the length of the mean vector increases from 0 to 1, variance of the distribution decreases so it is thought to be a measurement of the variance dispersion (Mahan 1991).Table 1 shows that when the length of the mean vector is 0.995, the variance is 0.005 in test milking 1 when 998 .0  R , the variance is 0.002 in test milking 6 which means that when the length of the mean vector increases, the circular variance decreases.Nevertheless, in Figure 1.Circular plot of teat angle Figure 2 (Test milking 1) is more distributed while in Figure 7 (Test milking 6) there is a uniform dispersion around the mean vector.Circular plot (also rose diagram) of distribution teat angles measured for all time of test milking is depicted in Figure 1.
On the circle, 0 degrees is on the east, 90 degrees is at the north, 180 degrees is on the west and 270 degrees is at the south.In the Figure 1, the values of angles between teats were between 0 ˚ and 45 ˚.However since the 3 values (53 ˚, 46 ˚ and 49 ˚) obtained in test milking 1, 2 and 3 are higher than 45 ˚, they were positioned after this value.In Figure 2-7, rose plots of measurement values at 6 test milking points were given.The rose plots between Figures 2-7 provide an idea about the angular direction of data from the each test milking.The parallel line in these figures shows a group range and gives the number of observation within this group range.Furthermore, the long straight line apparent in each figure shows the direction of the mean regarding that test milking and gives an idea about the dispersion of observations around the mean direction (Hassan et al. 2009;NCSS 2012).When the length of the mean vector increases, the variation between the data decreases and it shows a uniform distribution.Therefore, the variation in Figure 7 (Test milking 6) is lesser compared to the other all time of test milking and the data shows more uniform distribution.In order to detect whether there is a difference among all time of test milking, Watson-Williams F test was applied and the results are given in Table 2.For our data set, the test statistics (Test milking 5 and 6) give the value of 6.635 and the probability value is 0.015.Hence, there are significant differences between the two test milking.The mean angle become smaller depends on decreasing the milk content of udder at the end of the lactation period.

Conclusion
Directional data analysis is not common method in animal science research.However, it is aimed to give some information about some of the basic statistics of the directional data analysis and their graphical representations.The measures obtained as angular differ from the classic methods because of the directions that the observed values are located on the unit circle in the directional data analysis and the processes are based on the length of the mean vector.Therefore, for the data sets that have circular scales, directional data analysis methods can be applied easily to the every field of life sciences.

Figure 2 .
Figure 2. Rose plot of teat angle Figure 3. Rose plot of teat angle when test milking=1 when test milking=2

Table 1 .
The results of circular descriptive statistics