Page 114 - Special Topic Session (STS) - Volume 1

STS410 Abdul Ghapor H. et al.
1  cos    
g ;( ,  ) e where I 0 ( ) is the modified Bessel function
2  I )(
0
of the first kind and order zero, which can be defined by
1 2   cos 
I 0     e   d for 0  x  2 , 0    2 and  0 where  is
 2
0
the mean direction and  is the concentration parameter.
In this paper, the circular data is described in a linear functional relationship
model given by Y    X (mod 2 ) for the rotation parameter  . In this
model, both X and Y variables are subject to random errors  and  ,
i
i
respectively (Caires ans Wyatt (2003)).
However, the existence of outlier in the data may lead to a different model.
This paper discusses on outlier detection method using difference mean
circular error cosine (FDMCEC) statistic. Section 2 describes the methodology
and Section 3 describes the result for the power of performance of the
method.

2. Methodology
The cosine statistics is known as the functional mean circular error (FMCEC)
1  n n 
in which the statistic is given by FMCEC 1   cos x  X ˆ   cos  Yy ˆ 

2 n  i 1 k k  i 1 i i 
ˆ
ˆ
where n is the sample size, Y is the estimated value of y and X is the
i
i
i
estimated value of x , under the parameter estimation of the unreplicated
i
LFRM, depending on the case either equal or unequal error concentration.
The approach of row deletion is applied in this method in which we find
the absolute difference of the mean circular error when an observation is

deleted one after another. Thereafter, FMCEC(-i) denotes the removal of i
th
observation. The absolute difference between the value of full data set and the
reduced data set is as given by FDMCEC ( ) i  FMCEC  FMCEC ( ) i .
The existence of outlier in x and y will give a large value of the FMCEC(-i)
statistics. An i observation is defined as an outlier if the value of FDMCEC(-i)
th
exceeds the cut-off equation. To detect oulier, we need to determine the cut-
off equation as the indicator for a particular observation to be remarked as
the outlier. Hence, a Monte Carlo simulation study is carried out with different
values of sample size and error concentration parameter.
In doing so, we set the number of simulation s = 500 (Ibrahim et al. (2013)).
Without loss of generality, the variable X is generated from the von Mises

distribution and we set the value of  with    . 0 7854 . The values of
4
the concentration parameters of the error term used in this study are

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