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MBulli
2018-03-12 22:21:39 +01:00
parent c224967b19
commit 316b1d2911
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38
tex/chapters/experiments.tex
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@@ -3,7 +3,7 @@
\subsection{Mean Integrated Squared Error}
We now empirically evaluate the accuracy of our boxKDE method, using the mean integrated squared error (MISE).
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}
@@ -17,20 +17,20 @@ Therefore, the particular choice of the ground truth is only of minor importance
\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}
\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.
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.
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 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 $h=(0.4; 0.252; 0.675; 0.825)$.
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.
Consequently, it reduces the overall error of the approximation, but only marginal in this scenario.
As the extend box filter is able to approximate an exact $\sigma$, these discontinues don't appear.
Consequently, it reduces the overall error of the approximation, but only marginally in this scenario.
The global average MISE over all values of $h$ is $0.0049$ for the regular box filter and $0.0047$ in case of the extended version.
Likewise, the maximum MISE is $0.0093$ and $0.0091$, respectively.
The choice between the extended and regular box filter algorithm depends on how large the acceptable error should be, thus on the particular application.
@@ -42,7 +42,7 @@ However, both cases do not give a deeper insight of the error behavior of our me
\begin{figure}[t]
%\includegraphics[width=\textwidth,height=6cm]{gfx/tmpPerformance.png}
\input{gfx/perf.tex}
\caption{Logarithmic plot of the runtime performance with increasing grid size $G$ and bivariate data. The weighted-average estimate (blue) performs fastest followed by the boxKDE (orange) approximation. Both the BKDE (red) and the fastKDE (green) are magnitudes slower, especially for $G<10^3$.}\label{fig:performance}
\caption{Logarithmic plot of the runtime performance with increasing grid size $G$ and bivariate data. The weighted-average estimate (blue) performs fastest followed by the BoxKDE (orange) approximation. Both the BKDE (red) and the FastKDE (green) are magnitudes slower, especially for $G<10^3$.}\label{fig:performance}
\end{figure}
% kde, box filter, exbox in abhänigkeit von h (bild)
@@ -53,18 +53,18 @@ However, both cases do not give a deeper insight of the error behavior of our me
\subsection{Performance}
In the following, we underpin the promising theoretical linear time complexity of our method with empirical time measurements compared to other methods.
All tests are performed on a Intel Core \mbox{i5-7600K} CPU with a frequency of \SI{4.2}{\giga\hertz}, and \SI{16}{\giga\byte} main memory.
We compare our C++ implementation of the boxKDE approximation as shown in algorithm~\ref{alg:boxKDE} to the \texttt{ks} R package and the fastKDE Python implementation \cite{oBrien2016fast}.
We compare our C++ implementation of the BoxKDE approximation as shown in algorithm~\ref{alg:boxKDE} to the \texttt{ks} R package and the FastKDE Python implementation \cite{oBrien2016fast}.
The \texttt{ks} package provides a FFT-based BKDE implementation based on optimized C functions at its core.
With state estimation problems in mind, we additionally provide a C++ implementation of a weighted-average estimator.
As both methods are not using a grid, an equivalent input sample set was used for the weighted-average and the fastKDE.
As both methods are not using a grid, an equivalent input sample set was used for the weighted-average and the FastKDE.
The results for performance comparison are presented in fig.~\ref{fig:performance}.
The results of the performance comparison are presented in fig.~\ref{fig:performance}.
% O(N) gut erkennbar für box KDE und weighted average
The linear complexity of the boxKDE and the weighted average is clearly visible.
% Gerade bei kleinen G bis 10^3 ist die box KDE schneller als R und fastKDE, aber das WA deutlich schneller als alle anderen
Especially for small $G$ up to $10^3$ the boxKDE is much faster compared to BKDE and fastKDE.
The linear complexity of the BoxKDE and the weighted average is clearly visible.
% Gerade bei kleinen G bis 10^3 ist die box KDE schneller als R und FastKDE, aber das WA deutlich schneller als alle anderen
Especially for small $G$ up to $10^3$ the BoxKDE is much faster compared to BKDE and FastKDE.
% Bei zunehmend größeren G wird der Abstand zwischen box KDE und WA größer.
Nevertheless, the simple weighted-average approach performs the fastest and with increasing $G$ the distance to the boxKDE grows constantly.
Nevertheless, the simple weighted-average approach performs the fastest and with increasing $G$ the distance to the BoxKDE grows constantly.
However, it is obvious that this comes with major disadvantages, like being prone to multimodalities, as discussed in section \ref{sec:intro}.
% (Das kann auch daran liegen, weil das Binning mit größeren G langsamer wird, was ich mir aber nicht erklären kann! Vlt Cache Effekte)
@@ -82,13 +82,13 @@ The termination of BKDE graph at $G=4406^2$ is caused by an out of memory error
% Sowohl der box filter als auch der extended box filter haben ein sehr ähnliches Laufzeit Verhalten und somit einen sehr ähnlichen Kurvenverlauf.
% Während die durschnittliche Laufzeit über alle Werte von G beim box filter bei 0.4092s liegt, benötigte der extended box filter im Durschnitt 0.4169s.
Both discussed Gaussian filter approximations, namely box filter and extended box filter, yield a similar runtime behavior and therefore a similar curve progression.
While the average runtime over all values of $G$ for the standard box filter is \SI{0.4092}{\second}, the extended one provides an average of \SI{0.4169}{\second}.
To keep the arrangement of fig. \ref{fig:performance} clear, we only illustrated the results of the boxKDE with the regular box filter.
While the average runtime over all values of $G$ for the standard box filter is \SI{0.4092}{\second}, the extended one has an average of \SI{0.4169}{\second}.
To keep the arrangement of fig. \ref{fig:performance} clear, we only illustrated the results of the BoxKDE with the regular box filter.
The weighted-average has the great advantage of being independent of the dimensionality of the input and can be implemented effortlessly.
In contrast, the computation of the boxKDE approach increases exponentially with increasing number of dimensions.
However, due to the linear time complexity and the very simple computation scheme, the overall computation time is still sufficient fast for many applications and much smaller compared to other methods.
The boxKDE approach presents a reasonable alternative to the weighted-average and is easily integrated into existing systems.
In contrast, the computation of the BoxKDE approach increases exponentially with increasing number of dimensions.
However, due to the linear time complexity and the very simple computation scheme, the overall computation time is still sufficiently fast for many applications and much smaller compared to other methods.
The BoxKDE approach presents a reasonable alternative to the weighted-average and is easily integrated into existing systems.
In addition, modern CPUs do benefit from the recursive computation scheme of the box filter, as the data exhibits a high degree of spatial locality in memory and the accesses are reliable predictable.
Furthermore, the computation is easily parallelized, as there is no data dependency between the one-dimensional filter passes in algorithm~\ref{alg:boxKDE}.