Two-dimensional positive spline smoothing and its application to probability density estimation

Spline is a piecewise polynomial and has been widely used for interpolation and smoothing of observed data. In this paper, with the use of the sufficient condition, derived by Heß and Schmidt, for the nonnegativity of bivariate splines on square grid, we propose two-dimensional positive spline interpolation/smoothing on square grid for estimation of positive continuous functions. Moreover, we newly derive a sufficient condition for the nonnegativity on triangular grid and propose positive spline interpolation/smoothing on triangular grid. Then we estimate a two-dimensional probability density function (PDF) from its histogram by using the idea of the positive spline smoothing. Numerical experiments show the effectiveness of the newly derived sufficient condition and the proposed PDF estimator.