# Very fast optimal bandwidth selection for univariate kernel density estimation

 Title Very fast optimal bandwidth selection for univariate kernel density estimation Publication Type Journal Articles Year of Publication 2006 Authors Raykar V C, Duraiswami R Journal Technical Reports from UMIACS UMIACS-TR-2005-73 Date Published 2006/01/03/T15:1 Keywords Approximation algorithms, computational statistics, fast gauss transform, kernel density estimation, projection pursuit Abstract Most automatic bandwidth selection procedures for kernel density estimates require estimation of quantities involvingthe density derivatives. Estimation of modes and inflexion points of densities also require derivative estimates. The computational complexity of evaluating the density derivative at M evaluation points given N sample points from the density is O(MN). In this paper we propose a computationally efficient $\epsilon$-exact approximation algorithm for univariate, Gaussian kernel based, density derivative estimation that reduces the computational complexity from O(MN) to linear order (O(N+M)). The constant depends on the desired arbitrary accuracy, $\epsilon$. We apply the density derivative evaluation procedure to estimate the optimal bandwidth for kernel density estimation, a process that is often intractable for large data sets. For example for N = M = 409,600 points while the direct evaluation of the density derivative takes around 12.76 hours the fast evaluation requires only 65 seconds with an error of around $10^{-12)$. Algorithm details, error bounds, procedure to choose the parameters and numerical experiments are presented. We demonstrate the speedup achieved on the bandwidth selection using the solve-the-equation plug-in method'' [18]. We also demonstrate that the proposed procedure can be extremely useful for speeding up exploratory projection pursuit techniques. URL http://drum.lib.umd.edu/handle/1903/3028