Abstract
AbstractThere has been increasing interest in statistical analysis of data lying in manifolds. This paper generalizes a smoothing spline fitting method to Riemannian manifold data based on the technique of unrolling, unwrapping and wrapping originally proposed by Jupp and Kent for spherical data. In particular, we develop such a fitting procedure for shapes of configurations in general m-dimensional Euclidean space, extending our previous work for two-dimensional shapes. We show that parallel transport along a geodesic on Kendall shape space is linked to the solution of a homogeneous first-order differential equation, some of whose coefficients are implicitly defined functions. This finding enables us to approximate the procedure of unrolling and unwrapping by simultaneously solving such equations numerically, and so to find numerical solutions for smoothing splines fitted to higher dimensional shape data. This fitting method is applied to the analysis of some dynamic 3D peptide data.
Highlights
Analysis of temporal shape data has become increasingly important for applications in many fields, and a common definition is that the shape of an object is what is left after removing the effects of rotation, translation and re-scaling (Kendall, 1984)
There is very limited methodology available for fitting models for m = 3 dimensional shape data which exhibit a large amount of variability
We briefly provide an introduction to some relevant aspects of differential geometry, and for further details there are many texts, including Bär (2010), Boothby (1986) and Dryden and Mardia (2016 section 3.1)
Summary
Analysis of temporal shape data has become increasingly important for applications in many fields, and a common definition is that the shape of an object is what is left after removing the effects of rotation, translation and re-scaling (Kendall, 1984). For an introduction to the statistical analysis of shape, see Dryden and Mardia (2016); Kendall et al (1999); Patrangenaru and Ellingson (2016) and Srivastava and Klassen (2016). An important focus is on the shapes of landmark data, where each object consists of k > m points in Rm. After removing translation, scale and rotation, the data lie in Kendall's shape space, denoted as Σkm (Kendall, 1984). Suppose that we are given a time series of landmark shape data which are measurements of a moving object.
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