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Received: 14 Feb. 2016 Accepted: 29 August 2018 Extended Abstract Paper pages (85-94) Introduction Statistical analysis of the data on the Earth's surface was a favorite subject among many researchers. Such data can be related to animal's migration from a region to another position. Then, statistical modeling of their paths helps biological researchers to predict their movements and estimate the areas that are most likely to constitute the presence of the animals. From a geometrical view, spherical data are points that take their values on the surface of a unit sphere. There are many methods to fit a curve, especially regression curves to spherical data. For example, Gould [1] used the corresponding angles of spherical data coordinates to introduce regression models. He considered Fisher distribution as a candidate density for the error in his analysis. A non-parametric version of his model was proposed by Thompson and Clark [2]. Usually, the data that are close to the North or South pole have different behavior. Hence, their proposed model was failing to work there, and so they tried to keep the data somewhat away from the pole via adopting their model. They advocated overcoming this problem by using the tangent plane and suggested the use of splines there [3]. On the other hand, Fisher et al. [4] proposed two families of the spherical spline for spherical data. They introduced two families of curves using differential geometry suitable for fitting the splines. One of the methods to predict statistics is to utilize non-parametric regression models. Another strategy is to consider some forms of smooth models. Both of these procedures, along with other approaches in non-Euclidean statistics context are somewhat an initiative method in analyzing the spherical data. It worths mentioning that the benefits of using the spline path by employing the rotation parameters were of interest in directional statistics in [5], albeit for circular data. One of the interesting techniques to construct the non-parametric regression model was to minimize the Euclidean risk function, first proposed in [6]. We also follow the same procedure in this paper. In particular, the primary objective of this paper is to introduce a nonparametric regression model based on minimizing the mean square errors risk function for spherical data. To apply this idea, we used the suggested method in [6] for data on the circle. We initiate our model by considering two separate models for two common angles on the sphere. Then, we impose a correlation among these angles using an appropriate risk function. The proposed models will be evaluated using simulated and real-life data.

Regression Modeling for Spherical Data via Non-parametric and Least Square Methods