Simultaneous functional relationship model and outlier detection using clustering technique for circular variables

This study focuses on simultaneous linear functional relationship model and outlier detection for circular variables in a simple linear functional relationship model. A new simultaneous model is extended from a simple linear functional relationship model for circular data proposed by Caires and Wyatt (2003) by assuming equal error variances. The maximum likelihood estimator of the parameters in the simultaneous model are obtained and the covariance between the parameters is derived using the Fisher Information Matrix. Results from the simulation study indicate that the estimated parameters have small bias. The second part of the study, an estimation of the concentration parameter for simultaneous linear functional relationship model for circular variables when the variances of the error term are assumed not to be equal. The modified Bessel function was expanded by using the asymptotic power series which in turn becomes a cubic equation of the concentration parameter. Simulation study was done the result shows that the estimated concentration parameter has smaller bias for large concentration parameter and large sample size based on performance measure of estimated bias, estimated standard error and a few other measures. The final part of the study considers the problem in detecting multiple outliers in circular variables for functional relationship model. A clustering-based procedure is developed for the predicted and residual values obtained for the Caires and Wyatt model. Single linkage of hierarchical clustering method is used to obtain a tree diagram in detecting outliers. Based on simulation study, it can be concluded that the probability of success is good with increasing n and level of contamination. The level of masking and swamping decreases as n and  gets bigger. In all of the proposed methods, we applied to a real
data set to illustrate the applicability. The significant contribution of the study is the development of a simultaneous functional relationship model for circular variables that can be applied for both equal and unequal variance of error term. Another contribution of this study is a method of identifying multiple outliers in circular variables for functional relationship model based on the dendrogram plot.