46 lines
2.8 KiB
TeX
46 lines
2.8 KiB
TeX
\section{Experiments}
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\subsection{Mean Integrated Squared Error}
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We now empirically evaluate the accuracy of our method, using the mean integrated squared error (MISE).
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The ground truth is given as $N=1000$ synthetic samples drawn from a bivariate mixture normal density $f$
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\begin{equation}
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\begin{split}
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\bm{X} \sim & ~\G{\VecTwo{0}{0}}{0.5\bm{I}} + \G{\VecTwo{3}{0}}{\bm{I}} + \G{\VecTwo{0}{3}}{\bm{I}} \\
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&+ \G{\VecTwo{-3}{0} }{\bm{I}} + \G{\VecTwo{0}{-3}}{\bm{I}}
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\end{split}
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\end{equation}
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where the majority of the probability mass lies in the range $[-6; 6]^2$.
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Clearly, the structure of the ground truth affects the error in the estimate, but as our method approximates the KDE only the closeness to the KDE is of interest.
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Therefore, the particular choice of the ground truth is only of minor importance here.
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Both the BKDE and the extended box filter estimate resemble the error curve of the KDE quite well and stable.
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They are rather close to each other, with a tendency to diverge for larger $h$.
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In contrast, the error curve of the box filter estimate has noticeable jumps at $h=(0.4; 0.252; 0.675; 0.825)$.
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These jumps are caused by the rounding of the integer-valued box width given by \eqref{eq:boxidealwidth}.
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As the extend box filter is able to approximate an exact $\sigma$, it lacks these discontinues.
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The exact KDE, evaluated at $50^2$ points, is compared to the BKDE, box filter, and extended box filter approximation, which are evaluated at a smaller grid with $30^2$ points.
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The MISE between $f$ and the estimates as a function of $h$ are evaluated, and the resulting plot is given in figure~\ref{fig:evalBandwidth}.
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\begin{figure} [t]
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\label{fig:evalBandwidth}
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\includegraphics[width=\columnwidth]{gfx/Eval1Bandwidth_abs.png}
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\caption{Bla.} \label{fig:eval1GroundTruth}
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\end{figure}
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Other test cases of theoretical relevance are error as a function of the grid size $G$ and the sample size $N$.
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However, both cases do not give a deeper insight of the error behaviour of our method, as it closely mimics the error curve of the KDE and only confirm the theoretical expectations.
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% kde, box filter, exbox in abhänigkeit von h (bild)
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% sample size und grid size text
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% fastKDE fehler vergleich macht kein sinn weil kernel und bandbreite unterschiedlich sind
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\subsection{Performance}
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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.
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We compare our C++ implementation of the box filter based KDE to the KernSmooth R package and the \qq{FastKDE} implementation \cite{oBrien2016fast}.
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The KernSmooth packages provides a FFT-based BKDE implementation based on optimized C functions at its core.
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\input{chapters/realworld}
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