Circular functional relationships

This paper develops techniques for fitting a circular functional relationship. The model is formulated assuming the centre of the circle is known and taken to be the origin. This incurs some loss of generality, but considerably simplifies the analysis. Methods of estimating the parameters of the relationship, assuming that the co-ordinates have uncorrelated errors with equal variance, are developed and compared by way of simulations. Consistent estimators are obtained from unbiased estimating equations and by maximising the marginal likelihood. Approximate estimators are obtained by approximating the estimating equations and by maximising a modified profile likelihood function. The ideas and methodology are then applied to the relationship with the more realistic assumption of an unknown centre, with an example highlighting the differences between estimation techniques.     

A comparison of regularization methods applied to the linear discriminant function with high-dimensional microarray data

Classification of gene expression microarray data is important in the diagnosis of diseases such as cancer, but often the analysis of microarray data presents difficult challenges because the gene expression dimension is typically much larger than the sample size. Consequently, classification methods for microarray data often rely on regularization techniques to stabilize the classifier for improved classification performance. In particular, numerous regularization techniques, such as covariance-matrix regularization, are available, which, in practice, lead to a difficult choice of regularization methods. In this paper, we compare the classification performance of five covariance-matrix regularization methods applied to the linear discriminant function using two simulated high-dimensional data sets and five well-known, high-dimensional microarray data sets. In our simulation study, we found the minimum distance empirical Bayes method reported in Srivastava and Kubokawa [Comparison of discrimination methods for high dimensional data, J. Japan Statist. Soc. 37(1) (2007), pp. 123–134], and the new linear discriminant analysis reported in Thomaz, Kitani, and Gillies [A Maximum Uncertainty LDA-based approach for Limited Sample Size problems – with application to Face Recognition, J. Braz. Comput. Soc. 12(1) (2006), pp. 1–12], to perform consistently well and often outperform three other prominent regularization methods. Finally, we conclude with some recommendations for practitioners.