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MBulli
2018-02-19 21:12:13 +01:00
parent a46fe242e8
commit 601a93552d
4 changed files with 112 additions and 6 deletions
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2
tex/chapters/experiments.tex
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@@ -3,7 +3,7 @@ We now empirically evaluate the accuracy of our method and compare its runtime p
To conclude our findings we present a real world example from a indoor localisation system.
All tests are performed on a Intel Core \mbox{i5-7600K} CPU with a frequency of $4.5 \text{GHz}$, which supports the AVX2 instruction set, hence 256-bit wide SIMD registers are available.
We compare our C++ implementation of the box filter based KDE to the KernSmooth R package and the \qq{FastKDE} implementation \cite{fastKDE}.
We compare our C++ implementation of the box filter based KDE to the KernSmooth R package and the \qq{FastKDE} implementation \cite{oBrien2016fast}.
The KernSmooth packages provides a FFT-based BKDE implementation based on optimized C functions at its core.
\subsection{Error}

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tex/chapters/mvg.tex
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@@ -55,7 +55,7 @@ Furthermore, only one addition and subtraction is required to calculate a single
Given a fast approximation scheme, it is necessary to construct a box filter analogous to a given Gaussian filter.
As given in \eqref{eq:gausfx} the solely parameter of the Gaussian kernel is the standard deviation $\sigma$.
The solely parameter of the Gaussian kernel is the standard deviation $\sigma$.
In contrast, the moving average filter is parametrized by its width $w$.
Therefore, in order to approximate the Gaussian filter of a given $\sigma$ a corresponding value of $w$ must be found.
Given $n$ iterations of moving average filters with identical widths the ideal width $\wideal$, as suggested by Wells~\cite{wells1986efficient}, is

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tex/chapters/relatedwork.tex
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@@ -11,7 +11,7 @@ It was subject to extensive research and its theoretical properties are well und
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{heidenreich2013bandwith}.
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.
@@ -33,10 +33,9 @@ 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 allow to obtain an estimate without any assumptions.
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.
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.
Therefore, the data is not required to be on a grid.
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}.
% binning => FFT
In general, it is desirable to omit a grid, as the data points do not necessary fall onto equally spaced points.