Helix modelling through the Mardia-Holmes model framework and an extension of the Mardia-Holmes model

For noisy two-dimensional data, which are approximately uniformly distributed near the circumference of an ellipse, Mardia and Holmes (1980) developed a model to fit the ellipse. In this paper we adapt their methodology to the analysis of helix data in three dimensions. If the helix axis is known, then the Mardia-Holmes model for the circular case can be fitted after projecting the helix data onto the plane normal to the helix axis. If the axis is unknown, an iterative algorithm has been developed to estimate the axis. The methodology is illustrated using simulated protein alpha-helices. We also give a multivariate version of the Mardia-Holmes model which will be applicable for fitting an ellipsoid and in particular a cylinder.