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Minor changes to wording

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Markus Bullmann
2018-05-08 11:08:48 +02:00
parent 35e1f4e0b3
commit 9f358d69c9
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tex/chapters/relatedwork.tex
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@@ -6,21 +6,20 @@
% -> Fourier transfom
The Kernel density estimator is a well known non-parametric estimator, originally described independently by Rosenblatt \cite{rosenblatt1956remarks} and Parzen \cite{parzen1962estimation}.
The kernel density estimator is a well known non-parametric density estimator, originally described independently by Rosenblatt \cite{rosenblatt1956remarks} and Parzen \cite{parzen1962estimation}.
It was subject to extensive research and its theoretical properties are well understood.
A comprehensive reference is given by Scott \cite{scott2015}.
Although classified as non-parametric, the KDE depends on two free parameters, the kernel function and its bandwidth.
The selection of a \qq{good} bandwidth is still an open problem and heavily researched.
An extensive overview regarding the topic of automatic bandwith selection is given by \cite{heidenreich2013bandwidth}.
An extensive overview regarding the topic of automatic bandwidth selection is given by \cite{heidenreich2013bandwidth}.
%However, the automatic selection of the bandwidth is not subject of this work and we refer to the literature \cite{turlach1993bandwidth}.
The great flexibility of the KDE makes it very useful for many applications.
The great flexibility of the KDE makes it suitable for many applications.
However, this comes at the cost of a slow computation speed.
%
The complexity of a naive implementation of the KDE is \landau{MN}, given by $M$ evaluations of $N$ data samples as input size.
%The complexity of a naive implementation of the KDE is \landau{NM} evaluations of the kernel function, given $N$ data samples and $M$ points of the estimate.
Therefore, a lot of effort was put into reducing the computation time of the KDE.
Various methods have been proposed, which can be clustered based on different techniques.
Various methods have been proposed to reduce the computation time of the KDE.
% k-nearest neighbor searching
An obvious way to speed up the computation is to reduce the number of evaluated kernel functions.
@@ -29,7 +28,7 @@ These algorithms reduce the number of evaluated kernels by taking the distance b
% fast multipole method & Fast Gaus Transform
Another approach is to reduce the algorithmic complexity of the sum over Gaussian functions, by employing a specialized variant of the fast multipole method.
The term fast Gauss transform was coined by Greengard \cite{greengard1991fast} who suggested this approach to reduce the complexity of the KDE to \landau{N+M}.
The term fast Gauss transform was coined by Greengard \cite{greengard1991fast} who suggested this approach to reduce the complexity to \landau{N+M}.
% However, the complexity grows exponentially with dimension. \cite{Improved Fast Gauss Transform and Efficient Kernel Density Estimation}
% FastKDE, passed on ECF and nuFFT