Statistics Papers

Document Type

Technical Report

Date of this Version

2010

Publication Source

Journal of Computational and Graphical Studies

Volume

19

Issue

1

Start Page

205

Last Page

220

DOI

10.1198/jcgs.2010.09046

Abstract

The computational complexity of evaluating the kernel density estimate (or its derivatives) at m evaluation points given n sample points scales quadratically as O(nm)—making it prohibitively expensive for large datasets. While approximate methods like binning could speed up the computation, they lack a precise control over the accuracy of the approximation. There is no straightforward way of choosing the binning parameters a priori in order to achieve a desired approximation error. We propose a novel computationally efficient ε-exact approximation algorithm for the univariate Gaussian kernel-based density derivative estimation that reduces the computational complexity from O(nm) to linear O(n+m). The user can specify a desired accuracy ε. The algorithm guarantees that the actual error between the approximation and the original kernel estimate will always be less than ε. We also apply our proposed fast algorithm to speed up automatic bandwidth selection procedures. We compare our method to the best available binning methods in terms of the speed and the accuracy. Our experimental results show that the proposed method is almost twice as fast as the best binning methods and is around five orders of magnitude more accurate. The software for the proposed method is available online.

Copyright/Permission Statement

This is an Accepted Manuscript of an article published by Taylor & Francis in the Journal of Computational and Graphical Studies in 2010, available online: http://dx.doi.org/10.1198/jcgs.2010.09046

Keywords

bandwidth estimation, binning, fast Fourier transform, kernel density derivative estimation, kernel density estimation

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Date Posted: 25 October 2018

This document has been peer reviewed.