This paper proposes a parametric hierarchical model for functional data with an elliptical shape, using a Gaussian process prior to capturing the data dependencies that reflect systematic errors while modeling the underlying curved shape through a von Mises-Fisher distribution. The model definition, Bayesian inference, and MCMC algorithm are discussed. The effectiveness of the model is demonstrated through the reconstruction of curved trajectories using both simulated and real-world examples. The discussion in this paper focuses on two-dimensional problems, but the framework can be extended to higher-dimensional spaces, making it adaptable to a wide range of applications.
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