Nonparametric Kernel Density Estimation and Its Computational Aspects by Artur Gramacki

Author:Artur Gramacki
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

(5.8)

and now the number of kernel evaluations is O(nM). For a large n, this value can grow too big for practical use. So, the next natural step could be to make use of binning, that is for every sample point to be replaced by a pair of two values: the grid point and the grid count , as it was explained in Sect. 5.2.1. Thus we obtain

(5.9)

In that case, (5.7) can be again rewritten as below. Note also that now, instead of was used as (5.10) is, in a way, an approximation of (5.7).

(5.10)

Now, the number of kernel evaluations is . If, however, the grid points are equally spaced (which is in practice almost always true, not equally spaced gridding is simply impractical), then the number of kernel evaluations is . This is because the kernel is a symmetric function and thus . Therefore, since these two values are the same, it is enough to calculate this value only once. But the number of multiplications is still . To reduce this value to , the FFT-based technique can be used. This technique is presented in the subsequent parts of this chapter.

To use the FFT for a fast computation of (5.10), this equation must be rewritten again as

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