Statistics of Shape, Direction and Cylindrical Variables

In statistical shape analysis, the shape of an object is understood to be what remains after the effects of location, scale and rotation are removed. We consider the distributional problem of triangular shape and an associated direction; motivated by a data set of microscopic fossils. We begin by constructing a parallel transport system such that the data transform onto the space S2×S2. A joint shape distribution on S2×S1 is proposed based on Jupp & Mardia's bivariate distribution on S2×S1. For concentrated data, an approximation to the distribution on S2×S1 is given by a distribution on 1×S1, and we explore a distribution on this space by extending Mardia & Sutton's distribution on 2×S1. In this distribution, the expected edgel direction varies linearly in the shape coordinates. This is found to be a useful model for the microfossil data.