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MBulli
2018-03-12 23:24:45 +01:00
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tex/chapters/experiments.tex
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\section{Experiments}
\subsection{Mean Integrated Squared Error}
We now empirically evaluate the feasibility of our BoxKDE method by analyzing its approximation error.
In order to evaluate the error the KDE and various approximations of it are computed and compared using the mean integrated squared error (MISE).
A synthetic sample set $\bm{X}$ with $N=1000$ obtained from a bivariate mixture normal density $f$ provides the basis of the comparison.
For each method an estimate is computed and the MISE of it relative to $f$ is calculated.
The specific structure of the underlying distribution clearly affects the error in the estimate, but only the closeness of the approximation to the KDE is of interest.
Hence, $f$ is of minor importance here and was chosen rather arbitrary to highlight the behavior of the BoxKDE.
We now empirically evaluate the accuracy of our BoxKDE method, using the mean integrated squared error (MISE).
The ground truth is given with $N=1000$ synthetic samples drawn from a bivariate mixture normal density $f$
\begin{equation}
\begin{split}
\bm{X} \sim & ~\G{\VecTwo{0}{0}}{0.5\bm{I}} + \G{\VecTwo{3}{0}}{\bm{I}} + \G{\VecTwo{0}{3}}{\bm{I}} \\
@@ -12,21 +15,20 @@ The ground truth is given with $N=1000$ synthetic samples drawn from a bivariate
\end{split}
\end{equation}
where the majority of the probability mass lies in the range $[-6; 6]^2$.
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.
Therefore, the particular choice of the ground truth is only of minor importance here.
\begin{figure}[t]
\input{gfx/error.tex}
\caption{MISE relative to the ground truth as a function of $h$. While the error curves of the BKDE (red) and the BoxKDE based on the extended box filter (orange dotted line) resemble the overall course of the error of the exact KDE (green), the regular BoxKDE (orange) exhibits noticeable jumps to rounding.} \label{fig:errorBandwidth}
\end{figure}
Evaluated at $50^2$ points the exact KDE is compared to the BKDE, BoxKDE, and extended box filter approximation, which are evaluated at a smaller grid with $30^2$ points.
The MISE between $f$ and the estimates as a function of $h$ are evaluated, and the resulting plot is given in fig.~\ref{fig:errorBandwidth}.
A minimum error is obtained with $h=0.35$, for larger oversmoothing occurs and the modes gradually fuse together.
Four estimates are computed with varying bandwidth using the exact KDE, BKDE, BoxKDE, and ExBoxKDE, which uses the extended box filter.
%Evaluated at $50^2$ points the exact KDE is compared to the BKDE, BoxKDE, and extended box filter approximation, which are evaluated at a smaller grid with $30^2$ points.
The graphs of the MISE between $f$ and the estimates as a function of $h\in[0.15; 1.0]$ are given in fig.~\ref{fig:errorBandwidth}.
A minimum error is obtained with $h=0.35$, for larger values oversmoothing occurs and the modes gradually fuse together.
Both the BKDE and the extended box filter estimate resemble the error curve of the KDE quite well and stable.
Both the BKDE and the ExBoxKDE resemble the error curve of the KDE quite well and stable.
They are rather close to each other, with a tendency to diverge for larger $h$.
In contrast, the error curve of the BoxKDE has noticeable jumps at $h=(0.4; 0.252; 0.675; 0.825)$.
In contrast, the error curve of the BoxKDE has noticeable jumps at $h=\{0.25, 0.40, 0.67, 0.82\}$.
These jumps are caused by the rounding of the integer-valued box width given by \eqref{eq:boxidealwidth}.
As the extend box filter is able to approximate an exact $\sigma$, these discontinues don't appear.