Reconstruction of foliations from directional information

Abstract

In many areas of science, especially geophysics, geography and meteorology, the data are often directions or axes rather than scalars or unrestricted vectors. Directional statistics considers data which are mainly unit vectors lying in two- or three-dimensional space (R² or R³). One way in which directional data arise is as normals to foliations. A (codimension-1) foliation of {R}^{d} is a system of non-intersecting (d-1)-dimensional surfaces filling out the whole of {R}^{d}. At each point z of {R}^{d}, any given codimension-1 foliation determines a unit vector v normal to the surface through z. The problem considered here is that of reconstructing the foliation from observations ({z}{i}, {v}{i}), i=1,...,n. One way of doing this is rather similar to fitting smooth splines to data. That is, the reconstructed foliation has to be as close to the data as possible, while the foliation itself is not too rough. A tradeoff parameter is introduced to control the balance between smoothness and closeness. The approach used in this thesis is to take the surfaces to be surfaces of constant values of a suitable real-valued function h on {R}^{d}. The problem of reconstructing a foliation is translated into the language of Schwartz distributions and a deep result in the theory of distributions is used to give the appropriate general form of the fitted function h. The model parameters are estimated by a simplified Newton method. Under appropriate distributional assumptions on v{1},...,v{n}, confidence regions for the true normals are developed and estimates of concentration are given.