Curve fitting by Spherical Least Squares on two-dimensional sphere

Abstract

To measure the similarity between two high dimensional vector data, correlation coefficient is often used instead of Euclidean distance. For this purpose, the high dimensional vectors are mapped into hyperspherical points by normalization, and the distance between two hyperspherical data is measured as the length along geodesic on the hypersphere. Then estimations from high dimensional vector data should be resolved as minimizing appropriate energy function of the length along geodesic when high dimensional vector data are regarded as hyperspherical data. In this paper, for the first step of hyper surface fitting to hyperspehrical data, the method of curve fitting to two-dimensional spherical data by Spherical Least Squares is proposed. It is also shown that the proposed method is closely related to the curve fitting by Euclidenization of the metric.