Bak Kwan-Young, Shin Jae-Kyung, Koo Ja-Yong
School of Mathematics, Statistics and Data Science, Sungshin Women's University, Seoul, Republic of Korea.
Data Science Center, Sungshin Women's University, Seoul, Republic of Korea.
J Appl Stat. 2022 Mar 28;50(9):1942-1961. doi: 10.1080/02664763.2022.2054962. eCollection 2023.
This study examines an intrinsic penalized smoothing method on the 2-sphere. We propose a method based on the spherical Bézier curves obtained using a generalized de Casteljau algorithm to provide a degree-based regularity constraint to the spherical smoothing problem. A smooth Bézier curve is found by minimizing the least squares criterion under the regularization constraint. The de Casteljau algorithm constructs higher-order Bézier curves in a recursive manner using linear Bézier curves. We introduce a local penalization scheme based on a penalty function that regularizes the velocity differences in consecutive linear Bézier curves. The imposed penalty induces sparsity on the control points so that the proposed method determines the number of control points, or equivalently the order of the Bézier curve, in a data-adaptive way. An efficient Riemannian block coordinate descent algorithm is devised to implement the proposed method. Numerical studies based on real and simulated data are provided to illustrate the performance and properties of the proposed method. The results show that the penalized Bézier curve adapts well to local data trends without compromising overall smoothness.
本研究考察了二维球面上的一种内在惩罚平滑方法。我们提出了一种基于使用广义德卡斯特里奥算法获得的球面贝塞尔曲线的方法，为球面平滑问题提供基于度数的正则性约束。通过在正则化约束下最小化最小二乘准则来找到一条平滑的贝塞尔曲线。德卡斯特里奥算法使用线性贝塞尔曲线以递归方式构造高阶贝塞尔曲线。我们引入了一种基于惩罚函数的局部惩罚方案，该方案对连续线性贝塞尔曲线中的速度差异进行正则化。施加的惩罚在控制点上诱导稀疏性，使得所提出的方法以数据自适应的方式确定控制点的数量，或者等效地确定贝塞尔曲线的阶数。设计了一种高效的黎曼块坐标下降算法来实现所提出的方法。提供了基于真实数据和模拟数据的数值研究，以说明所提出方法的性能和特性。结果表明，惩罚贝塞尔曲线能够很好地适应局部数据趋势，而不会损害整体平滑性。
