A. H. Abuzaid a, A. G. Hussin b & I. B. Mohamed a a Institute of Mathematical Sciences, University of Malaya, 50603,

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1 This article was downloaded by: [University of Malaya] On: 09 June 2013, At: 19:32 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journal of Statistical Computation and Simulation Publication details, including instructions for authors and subscription information: Detection of outliers in simple circular regression models using the mean circular error statistic A. H. Abuzaid a, A. G. Hussin b & I. B. Mohamed a a Institute of Mathematical Sciences, University of Malaya, 50603, Kuala Lumpur, Malaysia b Centre for Foundation Studies in Science, University of Malaya, 50603, Kuala Lumpur, Malaysia Published online: 17 Aug To cite this article: A. H. Abuzaid, A. G. Hussin & I. B. Mohamed (2013): Detection of outliers in simple circular regression models using the mean circular error statistic, Journal of Statistical Computation and Simulation, 83:2, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Journal of Statistical Computation and Simulation Vol. 83, No. 2, February 2013, Detection of outliers in simple circular regression models using the mean circular error statistic A.H. Abuzaid a *, A.G. Hussin b and I.B. Mohamed a a Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur, Malaysia; b Centre for Foundation Studies in Science, University of Malaya, Kuala Lumpur, Malaysia (Received 10 January 2010; final version received 30 June 2011) The investigation on the identification of outliers in linear regression models can be extended to those for circular regression case. In this paper, we propose a new numerical statistic called mean circular error to identify possible outliers in circular regression models by using a row deletion approach. Through intensive simulation studies, the cut-off points of the statistic are obtained and its power of performance investigated. It is found that the performance improves as the concentration parameter of circular residuals becomes larger or the sample size becomes smaller. As an illustration, the statistic is applied to a wind direction data set. Keywords: circular distance; circular regression model; mean circular error; outlier; row deletion 1. Introduction One of the most common problems in any statistical analysis is the existence of some unexpected observations. Such observations may not be a part of the phenomenon under study and are known as outliers. The problem of outliers is considered to be as old as the subject of statistics itself. Studies have shown that outliers affect the performance of standard statistical methodology in modelling, forecasting and diagnostic processes. In some cases, the effect is disastrous [1 5] and needs to be dealt with prior to further analysis. Beckman and Cook [6] reviewed the literature on outliers and the available approaches to deal with them in different areas of statistics. Outliers in linear regression models received special interest from many researchers [7 9]. Furthermore, many statistical software packages, such as S-Plus and Minitab, provide different tests to identify outliers in linear regression models. Collett [10] proposed four different statistics to identify a single outlier in univariate circular samples. Recently, Abuzaid et al. [5] proposed a new test of discordance in circular data based on the summation of circular distances of the point of interest to all other points. Although the first discussion of circular regression goes back to Gould [11], there is no known published work found on the identification of outliers in circular regression models up to *Corresponding author. alizaid33@yahoo.com ISSN print/issn online 2013 Taylor & Francis

3 270 A.H. Abuzaid et al. Only recently, Abuzaid et al. [4] utilized the definition of circular distance to give a type of residual for a circular regression model that can be used in detecting outliers via graphical and numerical tools. It is our aim to further exploit the approach by introducing a new method of detecting outliers in circular regression models. In this paper, we propose a new statistic for the purpose of using a row deletion approach. This approach was developed by Belsley et al. [7] for linear regression models. The impact of deleting one row at a time from the data set on the estimated coefficients, fitted values, residuals and estimated parameters was investigated. Here, this approach is extended to the circular regression case. Thus, we investigate the effect of deleting one row at a time from the circular data on the circular distances between fitted and observed values. We consider the simple regression model for circular variables proposed by Hussin et al. [12]. The formulation of the circular regression model and the estimation of its parameter are discussed in the following section. Section 3 discusses the development and the properties of the proposed statistic. Section 4 presents the cut-off points and the performance of the statistic via simulation studies. Section 5 gives a numerical example. 2. Circular regression models 2.1. Regression of circular variables The study on circular regression was initiated four decades ago. Gould [11] proposed a regression model to predict a circular response variable from a set of linear covariates, where follows a von Mises distribution with mean μ and concentration parameter κ denoted by VM(μ, κ). The model is given by p μ = μ 0 + β j x j, (1) where μ 0 and the β s are unknown parameters and x j is a linear covariate, for j = 1,..., p. Assuming that θ 1, θ 2,..., θ n are independently distributed as the von Mises distributions with mean directions μ 1, μ 2,..., μ n, respectively, and unknown concentration parameter κ, Mardia [13] extended model (1) to give μ i = μ 0 + βt i, (2) for some known numbers t 1, t 2,..., t n and unknown parameters μ 0 and β. Jammalamadaka and Sarma [14] proposed a regression model for two circular random variables X and Y in terms of the conditional expectation of the vector e (iy) given x, namely j=1 E(e iy x) = ρ(x)e iμ(x) = g 1 (x) + ig 2 (x), where i = 1, μ(x) is the conditional mean direction of y given x with conditional concentration 0 ρ(x) 1. Due to the difficulty of estimating g 1 (x) and g 2 (x), they are expressed instead as Fourier series. In the case when X and Y are circular variables with mean directions α and β, respectively, Downs and Mardia [15] applied the following mapping: tan 1 2 (y β) = ω tan 1 (x α), 2 where ω is a slope parameter in the closed interval [ 1,1]. The mapping defines a one-to-one relationship with a unique solution given by { y = β + 2 tan 1 ω tan 1 }. 2 (x α)

4 Journal of Statistical Computation and Simulation 271 They classified the regression model according to the nature of the parameters α, β and ω. The maximum-likelihood estimates are derived and the properties of the model are discussed with an application to circadian biological rhythms and wind direction data. On the other hand, Kato et al. [16] expressed the regression curve as a form of Mobius circle transformation. For circular random covariate X and circular response variable Y, they proposed the regression curve x + β 1 y = β β 1 x ε, where β 0 and β 1 are complex parameters with β 0, β 1 C, β 1 is the conjugate of β 1 and ε follows a wrapped Cauchy distribution. Here, the angular error is assumed to follow the wrapped Cauchy distribution, while Downs and Mardia [15] assumed the angular error to follow the von Mises distribution. Due to the attractive properties of the wrapped Cauchy distribution, some desirable properties of the model were derived The simple regression model for circular variables While models (1) and (2) assume that the explanatory variables are linear while the response variables are circular, Hussin et al. [12] extended the models to the case when both variables are circular while confining the variables to the range [0, 2π] by introducing mod 2π. For any circular observations (x 1, y 1 ), (x 2, y 2 ),..., (x n, y n ) of circular variables X and Y with a linear relationship between them, they proposed a model given by y i = α + βx i + ε i (mod 2π), (3) where ε i is a circular random error having a von Mises distribution with circular mean 0 and concentration parameter κ. Due to its unique characterization and well-defined estimation procedure, the model is identifiable by using the iteration technique. It is obvious that the log-likelihood function is bounded and there exist maximum-likelihood estimates of parameters. Some applications of model (3) are in the analysis of the wind and wave direction data obtained by using two different techniques in order to compare an alternative instrument with the standard one. The log-likelihood function of model (3) is given by log L(α, β, κ; x 1,..., x n, y 1,..., y n ) = n log(2π) n log I 0 (κ) + κ (cos(y i α βx i )), where I 0 (κ) is the modified Bessel function of order one. The maximum-likelihood estimates of model parameters are given by ( ) S tan 1, S > 0, C > 0, C ( ) S ˆα = tan 1 + π, C < 0, ( C ) S tan 1 + 2π, S < 0, C > 0. C where S = sin(y i ˆβx i ) and C = cos(y i ˆβx i ). Due to the nonlinear nature of the first partial derivatives of the log-likelihood function, the parameter β can be estimated iteratively using the formula ˆ β 1 ˆ β 0 + xi sin(y i ˆα βˆ 0 x i ) x 2 i cos(y i ˆα βˆ 0 x i ).

5 272 A.H. Abuzaid et al. The concentration parameter is estimated by ( 1 ) ˆκ = A 1 cos(yi ˆα ˆβx i ), n where the function A( ) is the ratio of the modified Bessel function of the first kind of order one to the first kind of order zero. One of the inverses of function A(w) was approximated by Dobson [17]; it is given by A 1 (w) 9 8w + 3w2. 8(1 w) It is obvious that the log-likelihood function can be made arbitrarily large by putting ŷ i =ˆα + ˆβx i and taking κ large enough, that is, this likelihood is unbounded. Consequently, for maximumlikelihood estimates to exist, one has to either restrict the parameter set to a bounded subset or to impose a restriction between the parameters. Caries and Wyatt [18] recommended the same model with β = 1 for the case of the linear functional relationship model for circular variables where they imposed a restriction on the ratio of concentration parameters so that the model is identifiable. In our case, Hussin et al. [12] restricted the model to a neighbourhood of β = 1 which would give a single global maximum occurring in the interval of 0.5 <β< Mean circular error statistic Rao [19] defined the circular distance between two circular observations θ i and θ j as d ij = 1 cos(θ i θ j ). We see that d ij is a monotone increasing function of (θ i θ j ) and d ij [0, 2]. Recently, Abuzaid et al. [5] formulated a statistic in terms of d ij to detect possible outliers in circular samples. Mardia [13, p. 128] defined an angular deviation of observations from their fitted values for the circular model. In this paper, we use this statistic for the detection of possible outliers in model (3) by using the row deletion approach. Let the mean circular error (MCE) be the MCE statistic given by MCE = 1 1 n n cos(y i ŷ i ), (4) i=1 where n is the sample size and ŷ i the estimated value of y i under model (3). We notice that MCE [0, 2]. The MCE statistic may be considered as a type of arithmetic mean which is not robust to the existence of outliers. Thus, it can be used to detect the possible outliers in the circular regression. It is expected that the MCE statistic is more powerful for a small sample size n, because the estimated mean of smaller samples is more sensitive to outlier existence than that of larger samples. If an observation y i is an outlier, then the circular distance between y i and its associated fitted value ŷ i is expected to be relatively large. Thus, the existence of such observations in a data set will increase the summation of all circular distances as well as the value of the MCE statistic. Consequently, the removal of the ith observation from the data set will decrease the value of the statistic. We shall denote this decreased value by MCE ( i). Now, let the maximum absolute difference between the values of the statistics for full and reduced data sets be DMCE = max i { MCE MCE ( i) }.

6 Journal of Statistical Computation and Simulation 273 The ith observation is identified as an outlier if the difference of means circular error (DMCE) exceeds a pre-specified cut-off point. In the following section, simulation studies are carried out to investigate the cut-off points and performance for DMCE. 4. Simulation studies 4.1. Percentile points A simulation study is carried out to find the percentile (cut-off) point of DMCE by using Monte Carlo methods. Fifteen different sample sizes are used, namely n = 10, 20,..., 150. For each sample size n, a set of random circular errors is generated from the von Mises distribution with mean direction 0 and various values of concentration parameter κ = 0.5,1, 2, 5, 7 and 10. Samples of von Mises distribution VM(π/4, 10) with corresponding size n are generated to represent the values of X variable. The parameters of model (3) are fixed at α = 0 and β = 1. Observed values of the response variable Y are calculated based on model (3) and subsequently the fitted values Ŷ are obtained. We then compute the value of the MCE statistic for full data set. Sequentially, we exclude the ith observation from the generated sample, where i = 1,..., n. We refit the reduced data using model (3) and then calculate the values of MCE ( i). Then, we obtain the value of DMCE. The process is carried out 2000 times for each combination of sample size n and concentration parameter κ. A part of the results is tabulated in Table 1. The 10% and 5% upper percentile values are given in the first and second columns, respectively. The results are also displayed in Figure 1 for (a) κ = 5 and (b) n = 70. From Figure 1(a), it can be seen that the percentile values for DMCE are a decreasing function of the sample size n. As for Figure 1(b), the percentile values reach the peak point around κ = 2 and then decrease rapidly. We can now use the tabulated values as the cut-off points for DMCE Performance study To study the performance of DMCE, three different sample sizes are considered, specifically n = 30, 50 and 70. We generate the data as described in Section 4.1. For the observation y d,we Table 1. The simulated 10% and 5% points of the distribution of DMCE. κ n 90% 95% 90% 95% 90% 95% 90% 95% 90% 95% 90% 95%

7 274 A.H. Abuzaid et al. (a) (b) Figure α= per α= per n κ=5 The sampling behaviour of DMCE. define the contaminated value y d by y d = y d + λπ (mod 2π), α= per α= per κ where λ is the degree of contamination in the range 0 λ 1. When λ = 0, there is no contamination at position d, whereas when λ = 1, the observation yd is located at the anti-mode of the location of y d. The generated data are fitted using model (3) and the values of ŷ i, i = 1,2,,n, are obtained. Thus, the values of DMCE are calculated for each generated data set. The process is carried out 2000 times. The power of performance of DMCE is investigated by computing the percentage of correctly detecting the outlier at position d. A part of the simulation results is displayed in Figure 2. It is found that the power of performance of DMCE increases with the level of contamination λ. This is expected as the difference between the observed and fitted values at position d will increase, resulting in larger values of DMCE. Figure 2(a) shows the performance for n = 50 at different values of the concentration parameter κ. It is obvious that there are three main clusters of performance results based on the concentration parameter. The performance for a small concentration parameter, for instance, κ 0.5, is almost zero. For large concentration κ = 5, 7 and 10, the performances are close to each other and are better than the other clusters. On the other hand, the performance is lower than 0.05 when λ<0.2 and increases gradually with λ. This is because, the data are more dispersed around the unit circle for κ closer to zero. In general, DMCE performs very well for λ>0.6. Figure 2(b) illustrates the effect of the sample size n on the performance of DMCE for κ = 7. The performance is slightly lower for larger sample sizes. From Equation (4), as n increases, the term 1/n n i=1 cos(y i ŷ i ) generally gets smaller and thus giving larger values of MCE and removing one observation from the calculation will have lesser effect on the MCE ( i). That is, in this scenario, there could be more cases whereby the statistic DMCE would give smaller difference between MCE and MCE ( i) and, hence, more cases of DMCE values fail to exceed the corresponding cut-off point. Furthermore, it is also observed that for all sample sizes n with κ 2, DMCE performs poorly when λ<0.5 and n=70

8 Journal of Statistical Computation and Simulation 275 (a) Power 100 κ=0.5 k=0.5 κ=1 k=1 κ=2 k=2 κ=5 k=5 κ=7 k=7 80 κ=10 k=10 (b) Power 100 n=70 n=50 n= Figure λ 0 λ n=50 κ=7 Power of performance versus the level of contamination λ for DMCE. improves rapidly for λ > 0.8. This is not shown here. Since λ is the contamination level, these results are as expected because a larger λ would give an outlier much further from the rest of the observations. 5. Practical example As an illustration, we consider 129 measurements of wind direction (in radians) recorded over a period of 22.7 days along the Holderness coastline (the Humberside coast of the North Sea, UK) by using two different instruments: a high frequency (HF) radar system and an anchored wave buoy. Figure 3 shows the scatter plot of the wind direction data with the scale broken artificially at 0 = 2π. At the top left of the plot, two points seem to be outliers. However, they are actually Anchored Radar Figure 3. Scatter plot of the wind data.

9 276 A.H. Abuzaid et al MCE-MCE (-i) Figure Index The calculated values of MCE MCE ( i) for the wind data. consistent with the rest of the observations as they are close to other observations at the top right or bottom left due to the closed-range property of the circular variable. Furthermore, the scatter plot in Figure 3 shows a linear relationship between the measurements of the HF radar system and the anchored wave buoy. Since both variables are circular, model (3) is used to fit the data. The maximum-likelihood estimates are ˆα = 0.165, ˆβ = and ˆκ = 7.34 giving the fitted model ŷ i = x i (mod 2π). The value of the MCE statistic is The values of MCE MCE ( i) are plotted in Figure 4. Since the sample size is 129 and the estimated concentration parameter ˆκ = 7.34, the appropriate cut-off point at 0.05 level of significance is around It is obvious that the observation numbers 38 and 111 exceed the cut-off point indicated by the dashed line. These results coincide with those given by Abuzaid et al. [4]. 6. Conclusions A new numerical statistic to identify outliers in simple circular regression models is proposed. Circular distance is used to define the MCE statistic. This statistic, together with the row deletion approach, gives rise to the DMCE function. This function performs better in detecting outliers as the concentration parameter for circular error increases or the sample size decreases. Our approach is able to detect outliers similar to those given in Abuzaid et al. [4] References [1] R.V. Hogg, An introduction to robust estimation, inrobustness in Statistics: Proceeding of a Workshop, R.L. Launer and G.N. Wilkinson, eds., Academic Press, New York, 1979, pp [2] D. Pena, Influential observations in time series, J. Bus. Econom. Statist. 8(6) (1990), pp [3] A. Gelman, J.B. Carlin, H.S. Stern, and D.B. Rubin, Bayesian Data Analysis, Chapman and Hall, London, [4] A.H. Abuzaid, A.G. Hussin, and I.B. Mohamed, Identifying single outlier in linear circular regression model based on circular distance, J. Appl. Probab. Statist. 3(1) (2008), pp [5] A.H.Abuzaid,A.G. Hussin, and I.B. Mohamed, A new tests of discordancy in circular data, Comm. Statist. Simulation Comput. 38(4) (2009), pp [6] R.J. Beckman and R.D. Cook, Outliers, Technometrics 25(2) (1983), pp [7] D.D. Belsley, E. Kuh, and R.E. Welsch, Regression Diagnostics: Identifying Influential Data and Sources of Collinearity, John Wiley & Sons, New York, 1980.

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