Efficient nonparametric density estimation on the sphere with applications in fluid mechanics

The application of nonparametric probability density function estimation for the purpose of data analysis is well established. More recently, such methods have been applied to fluid flow calculations since the density of the fluid plays a crucial role in determining the ow. Furthermore, when the calculations involve directional or axial data, the domain of interest falls on the surface of the sphere. Accurate and fast estimation of probability density functions is crucial for these calculations since the density estimation is performed at each iteration during the computation. In particular the values fn(X-1), f(n)(X-2),..., f(n)(X-n) of the density estimate at the sampled points X-i are needed to evolve the system. Usual nonparametric estimators make use of kernel functions to construct f(n). We propose a special sequence of weight functions for nonparametric density estimation that is especially suitable for such applications. The resulting method has a computational advantage over kernel methods in certain situations and also parallelizes easily. Conditions for convergence turn out to be similar to those required for kernel-based methods. We also discuss experiments on different distributions and compare the computational efficiency of our method with kernel based estimators.