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Keywords: curve fitting; De Casteljau algorithm; generalized Bézier curve; Riemannian coordinate descent; sparsity; spherical data
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. We present a simulation study to illustrate the order-identification performance of the method. In addition, we apply the method to Triassic and Jurassic polar wander data for North America and elephant seal data that record the daily position of a migrating elephant seal. The obtained paths coincide with the findings of previous research and illustrate that our method provides appropriate orders of Bézier curves in a data-adaptive way. The results also show that the penalized Bézier curve adapts well to local data trends without compromising overall smoothness.