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fixed related work

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toni
2018-02-24 13:38:49 +01:00
parent 9881da57e4
commit 0d4cd0ff31
1 changed files with 5 additions and 5 deletions
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tex/chapters/relatedwork.tex
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@@ -14,10 +14,10 @@ The selection of a \qq{good} bandwidth is still an open problem and heavily rese
An extensive overview regarding the topic of automatic bandwith 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 renders it very useful for many applications.
However, this comes at the cost of a relative slow computation speed.
The great flexibility of the KDE makes it very useful 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.
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.
@@ -25,7 +25,7 @@ Various methods have been proposed, which can be clustered based on different te
% k-nearest neighbor searching
An obvious way to speed up the computation is to reduce the number of evaluated kernel functions.
One possible optimization is based on k-nearest neighbour search performed on spatial data structures.
These algorithms reduce the number of evaluated kernels by taking the the spatial distance between clusters of data points into account \cite{gray2003nonparametric}.
These algorithms reduce the number of evaluated kernels by taking the distance between clusters of data points into account \cite{gray2003nonparametric}.
% 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.
@@ -33,7 +33,7 @@ The term fast Gauss transform was coined by Greengard \cite{greengard1991fast} w
% However, the complexity grows exponentially with dimension. \cite{Improved Fast Gauss Transform and Efficient Kernel Density Estimation}
% FastKDE, passed on ECF and nuFFT
Recent methods based on the \qq{self-consistent} KDE proposed by Bernacchia and Pigolotti \cite{bernacchia2011self} allow to obtain an estimate without any assumptions, i.e. the kernel and bandwidth are both derived during the estimation.
Recent methods based on the self-consistent KDE proposed by Bernacchia and Pigolotti \cite{bernacchia2011self} allow to obtain an estimate without any assumptions, i.e. the kernel and bandwidth are both derived during the estimation.
They define a Fourier-based filter on the empirical characteristic function of a given dataset.
The computation time was further reduced by \etal{O'Brien} using a non-uniform fast Fourier transform (FFT) algorithm to efficiently transform the data into Fourier space \cite{oBrien2016fast}.