Hierarchical MK Splines: Algorithm and Applications to Data Fitting

In the era of big data, it is very important to study large-scale data fitting methods. In order to ensure the calculation speed and accuracy, we propose a new kind of hierarchical many-knot splines (hereinafter called "hierarchical MK splines," generally abbreviated as HMK splines) in this paper. The HMK splines method produces a sequence of MK spline functions. These MK spline functions are constructed into one ideal interpolation function by the MK spline refinement. In the case of regular sampling data, HMK splines can achieve the purpose of accurate approximation for the given data points without solving systems of equations. Further, in order to deal with the issues of scattered data fitting, the use of least-squares method will lead to the necessary of solving a linear system of equations. Since the ill-conditioned systems of equations often lead to unacceptable deviation of calculation results, one tries to avoid it as much as possible. The HMK splines algorithm can meet this requirement; it can avoid the intolerable deviation caused by solving systems of equations. Experimental results show that large-scale scattered data fitting can be easily achieved by the HMK splines algorithm and the reconstruction of nonuniform samples has a high accuracy.