\section{Experiments}

\subsection{Mean Integrated Squared Error}
We now empirically evaluate the accuracy of our method, using the mean integrated squared error (MISE).
The ground truth is given as $N=1000$ 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}} + \G{\VecTwo{-3}{0} }{\bm{I}} + \G{\VecTwo{0}{-3}}{\bm{I}}
\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.

Both the BKDE and the extended box filter estimate 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 box filter estimate has noticeable jumps at $h=(0.4; 0.252; 0.675; 0.825)$.
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 $\sigma$, it lacks these discontinues.

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.
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}.


\begin{figure}
\label{fig:evalBandwidth}
\end{figure}

Other test cases of theoretical relevance are error as a function of the grid size $G$ and the sample size $N$.
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.

% kde, box filter, exbox in abhänigkeit von h (bild)
% sample size und grid size text
% fastKDE fehler vergleich macht kein sinn weil kernel und bandbreite unterschiedlich sind


\subsection{Performance}
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{oBrien2016fast}.
The KernSmooth packages provides a FFT-based BKDE implementation based on optimized C functions at its core.

\input{chapters/realworld}
