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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}. 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}. %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. The great flexibility of the KDE makes it very useful for many applications.
However, this comes at the cost of a relative slow computation speed. 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. %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. 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, 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 % k-nearest neighbor searching
An obvious way to speed up the computation is to reduce the number of evaluated kernel functions. 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. 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 % 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. 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} % However, the complexity grows exponentially with dimension. \cite{Improved Fast Gauss Transform and Efficient Kernel Density Estimation}
% FastKDE, passed on ECF and nuFFT % 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. 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}. 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}.