Penalized Intrinsic Quadratic Spline on the Sphere

• Activity Number: 440 - SLDS CSpeed 8
• Type: Contributed
• Date/Time: Thursday, August 12, 2021 : 4:00 PM to 5:50 PM
• Sponsor: Section on Statistical Learning and Data Science
• Author(s): Jae-Kyung Shin and Kwan-Young Bak* and Ja-Yong Koo
• Companies: Department of Statistics, Korea University, Seoul 02841, Korea and Department of Statistics, Korea University, Seoul 02841, Korea and Department of Statistics, Korea University, Seoul 02841, Korea
• Keywords: coordinate descent; penalization; sparsity; spherical data; spherical spline; Riemannian geometry

Abstract

This study considers a problem of constructing a spherical path on the 2-sphere based on a given dataset. We propose an intrinsic penalized smoothing method using a quadratic spherical spline defined by concatenating spherical Bezier curves. In order to control smoothness of the quadratic spline in a data-dependent way, the norm of the velocity and acceleration differences of the adjacent Bezier curves at each knot is adopted as a penalty function. A smooth quadratic spherical spline is obtained by minimizing the sum of the squared spherical distances and penalty function. As an optimization scheme, we devise a Riemannian block coordinate-wise descent algorithm in which each control point corresponds to a block. For this purpose, we derive the Riemannian gradient of the objective function in terms of each control point. To investigate performance of the proposed method, numerical studies based on real and simulated data are provided. Our numerical experiments show that the proposed method captures local characteristics as well as global trends of spherical data.