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Markus Bullmann
2018-02-20 10:43:34 +01:00
parent 6ffa7d1c15
commit e49c7a1cbf
5 changed files with 114 additions and 118 deletions
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2
tex/bare_conf.tex
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@@ -245,6 +245,8 @@
\input{chapters/mvg}
\input{chapters/multivariate}
\input{chapters/usage}
\input{chapters/experiments}

5
tex/chapters/kde.tex
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@@ -1,4 +1,4 @@
\section{Binned Kernel Density Estimation}
\section{Kernel Density Estimation}
% KDE by rosenblatt and parzen
% general KDE
% Gauss Kernel
@@ -121,9 +121,6 @@ In terms of DSP this is analogous to filter the binned data with a Gaussian filt
This finding allows to speedup the computation of the density estimate by using a fast approximation scheme based on iterated box filters.
\commentByToni{hier vielleicht nochmal explizit erwähnen, also mit Namen, das der Gauss jetzt die BKDE approximiert und das diese erkenntniss toll und wichtig ist, weil wir so ein komplexes problem total einfach und schnell dargestellt haben. \commentByMarkus{Reicht das so?}}
%Given $n$ independently observed realizations of the observation set $X=(x_1,\dots,x_n)$, the kernel density estimate $\hat{f}_n$ of the density function $f$ of the underlying distribution is given with

109
tex/chapters/multivariate.tex Normal file
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@@ -0,0 +1,109 @@
\section{Extension to multi-dimensional data}
\todo{WIP}
So far only univariate sample sets were considered.
This is due to the fact, that the equations of the KDE \eqref{eq:kde}, BKDE \eqref{eq:binKde}, Gaussian filter \eqref{eq:gausFilt}, and the box filter \eqref{eq:boxFilt} are quite easily extended to multi-dimensional input.
Each method can be seen as several one-dimensional problems combined to a multi-dimensional result.
%However, with an increasing number of dimensions the computation time significantly increases.
In the following, the generalization to multi-dimensional input are briefly outlined.
In order to estimate a multivariate density using KDE or BKDE a multivariate kernel needs to be used.
Multivariate kernel functions can be constructed in various ways, however, a popular way is given by the product kernel.
Such a kernel is constructed by combining several univariate kernels into a product, where each kernel is applied in each dimension with a possibly different bandwidth.
Given a multivariate random variable $X=(x_1,\dots ,x_d)$ in $d$ dimensions.
The sample $\bm{X}$ is a $n\times d$ matrix defined as \cite[162]{scott2015}
\begin{equation}
\bm{X}=
\begin{pmatrix}
X_1 \\
\vdots \\
X_n \\
\end{pmatrix}
=
\begin{pmatrix}
x_{11} & \dots & x_{1d} \\
\vdots & \ddots & \vdots \\
x_{n1} & \dots & x_{nd}
\end{pmatrix} \text{.}
\end{equation}
The multivariate KDE $\hat{f}$ which defines the estimate pointwise at $\bm{x}=(x_1, \dots, x_d)^T$ is given as \cite[162]{scott2015}
\begin{equation}
\label{eq:mvKDE}
\hat{f}(\bm{x}) = \frac{1}{W} \sum_{i=1}^{n} \frac{w_i}{h_1 \dots h_d} \left[ \prod_{j=1}^{d} K\left( \frac{x_j-x_{ij}}{h_j} \right) \right] \text{.}
\end{equation}
where the bandwidth is given as a vector $\bm{h}=(h_1, \dots, h_d)$.
Note that \eqref{eq:mvKDE} does not include all possible multivariate kernels, such as spherically symmetric kernels, which are based on rotation of a univariate kernel.
In general a multivariate product and spherically symmetric kernel based on the same univariate kernel will differ.
The only exception is the Gaussian kernel which is spherical symmetric and has independent marginals. % TODO scott cite?!
In addition, only smoothing in the direction of the axes are possible.
If smoothing in other directions is necessary, the computation needs to be done on a prerotated sample set and the estimate needs to be rotated back to fit the original coordinate system \cite{wand1994fast}.
For the multivariate BKDE, in addition to the kernel function the grid and the binning rules need to be extended to multivariate data.
\todo{Reicht hier text oder müssen Formeln her?}
In general multi-dimensional filters are multi-dimensional convolution operations.
However, by utilizing the separability property of convolution a straightforward and a more efficient implementation can be found.
Convolution is separable if the filter kernel is separable, i.e. it can be split into successive convolutions of several kernels.
Likewise digital filters based on such kernels are called separable filters.
They are easily applied to multi-dimensional signals, because the input signal can be filtered in each dimension separately by an one-dimensional filter.
The Gaussian filter is separable, because of $e^{x^2+y^2} = e^{x^2}\cdot e^{y^2}$.
% KDE:
%So far only the univariate case was considered.
%This is due to the fact, that univariate kernel estimators can quite easily be extended to multivariate distributions.
%A common approach is to apply an univariate kernel with a possibly different bandwidth in each dimension.
%These kind of multivariate kernel is called product kernel as the multivariate kernel result is the product of each individual univariate kernel.
%
%Given a multivariate random variable $X=(x_1,\dots ,x_d)$ in $d$ dimensions.
%The sample $\bm{X}$ is a $n\times d$ matrix defined as \cite[162]{scott2015}
%\begin{equation}
% \bm{X}=
% \begin{pmatrix}
% X_1 \\
% \vdots \\
% X_n \\
% \end{pmatrix}
% =
% \begin{pmatrix}
% x_{11} & \dots & x_{1d} \\
% \vdots & \ddots & \vdots \\
% x_{n1} & \dots & x_{nd}
% \end{pmatrix} \text{.}
%\end{equation}
%
%The multivariate kernel density estimator $\hat{f}$ which defines the estimate pointwise at $\bm{x}=(x_1, \dots, x_d)^T$ is given as \cite[162]{scott2015}
%\begin{equation}
% \hat{f}(\bm{x}) = \frac{1}{nh_1 \dots h_d} \sum_{i=1}^{n} \left[ \prod_{j=1}^{d} K\left( \frac{x_j-x_{ij}}{h_j} \right) \right] \text{.}
%\end{equation}
%where the bandwidth is given as a vector $\bm{h}=(h_1, \dots, h_d)$.
% Product kernel allows our method
% Spherically symmetric kernel not supported, but Gaussian kernel == product & spehrically symmetric
% smoothing not in the direction of the axes -> rotate data, kde, rotate back
%Multivariate Gauss-Kernel
%\begin{equation}
%K(u)=\frac{1}{(2\pi)^{d/2}} \expp{-\frac{1}{2} \bm{x}^T \bm{x}}
%\end{equation}
% Gaus:
%If the filter kernel is separable, the convolution is also separable i.e. multi-dimensional convolution can be computed as individual one-dimensional convolutions with a one-dimensional kernel.
%Because of $e^{x^2+y^2} = e^{x^2}\cdot e^{y^2}$ the Gaussian filter is separable and can be easily applied to multi-dimensional signals. \todo{quelle}
%wie benutzen wir das ganze jetzt? auf was muss ich achten?
% Am Beispiel 2D Daten
% Histogram erzeugen (== data binnen)
% Hierzu wird min/max benötigt
% Anschließend Filterung per Box Filter über das Histogram
% - Wenn möglich parallel (SIMD, GPU)
% - separiert in jeder dim einzeln
% Maximum aus Filter ergebnis nehmen

2
tex/chapters/mvg.tex
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@@ -1,4 +1,4 @@
\section{Moving Average Filter}
\section{Gaussian Filter Approximation}
% Basic box filter formula
% Recursive form
% Gauss Blur Filter

114
tex/chapters/usage.tex
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@@ -1,117 +1,4 @@
\section{Usage}
\subsection{Extension to multi-dimensional data}
\todo{Absatz zum Thema 2D - Extension to multi-dimensional data}
So far only univariate sample sets were considered.
This is due to the fact, that the equations of the KDE \eqref{eq:kde}, BKDE \eqref{eq:binKde}, Gaussian filter \eqref{eq:gausFilt}, and the box filter \eqref{eq:boxFilt} are quite easily extended to multi-dimensional input.
Each method can be seen as several one-dimensional problems combined to a multi-dimensional result.
In the following, the generalization to multi-dimensional input are briefly outlined.
In order to estimate a multivariate density using KDE or BKDE, a multivariate kernel needs to be used.
Multivariate kernel functions can be constructed in various ways, however, a popular way is given by the product kernel.
Such a kernel is constructed by combining several univariate kernels into a product, where each kernel is applied in each dimension with a possibly different bandwidth.
Given a multivariate random variable $X=(x_1,\dots ,x_d)$ in $d$ dimensions.
The sample $\bm{X}$ is a $n\times d$ matrix defined as \cite[162]{scott2015}
\begin{equation}
\bm{X}=
\begin{pmatrix}
X_1 \\
\vdots \\
X_n \\
\end{pmatrix}
=
\begin{pmatrix}
x_{11} & \dots & x_{1d} \\
\vdots & \ddots & \vdots \\
x_{n1} & \dots & x_{nd}
\end{pmatrix} \text{.}
\end{equation}
The multivariate KDE $\hat{f}$ which defines the estimate pointwise at $\bm{x}=(x_1, \dots, x_d)^T$ is given as \cite[162]{scott2015}
\begin{equation}
\label{eq:mvKDE}
\hat{f}(\bm{x}) = \frac{1}{W} \sum_{i=1}^{n} \frac{w_i}{h_1 \dots h_d} \left[ \prod_{j=1}^{d} K\left( \frac{x_j-x_{ij}}{h_j} \right) \right] \text{.}
\end{equation}
where the bandwidth is given as a vector $\bm{h}=(h_1, \dots, h_d)$.
Note that \eqref{eq:mvKDE} does not include all possible multivariate kernels, such as spherically symmetric kernels, which are based on rotation of a univariate kernel.
In general a multivariate product and spherically symmetric kernel based on the same univariate kernel will differ.
The only exception is the Gaussian kernel which is spherical symmetric and has independent marginals. % TODO scott cite?!
In addition, only smoothing in the direction of the axes are possible.
If smoothing in other directions is necessary, the computation needs to be done on a prerotated sample set and the estimate needs to be rotated back to fit the original coordinate system \cite{wand1994fast}.
For the multivariate BKDE, in addition to the kernel function the grid and the binning rules need to be extended to multivariate data.
\todo{Reicht hier text oder müssen Formeln her?}
In general multi-dimensional filters are multi-dimensional convolution operations.
However, by utilizing the separability property of convolution a straightforward and a more efficient implementation can be found.
Convolution is separable if the filter kernel is separable, i.e. it can be split into successive convolutions of several kernels.
Likewise digital filters based on such kernels are called separable filters.
They are easily applied to multi-dimensional signals, because the input signal can be filtered in each dimension separately by an one-dimensional filter.
The Gaussian filter is separable, because of $e^{x^2+y^2} = e^{x^2}\cdot e^{y^2}$.
% KDE:
%So far only the univariate case was considered.
%This is due to the fact, that univariate kernel estimators can quite easily be extended to multivariate distributions.
%A common approach is to apply an univariate kernel with a possibly different bandwidth in each dimension.
%These kind of multivariate kernel is called product kernel as the multivariate kernel result is the product of each individual univariate kernel.
%
%Given a multivariate random variable $X=(x_1,\dots ,x_d)$ in $d$ dimensions.
%The sample $\bm{X}$ is a $n\times d$ matrix defined as \cite[162]{scott2015}
%\begin{equation}
% \bm{X}=
% \begin{pmatrix}
% X_1 \\
% \vdots \\
% X_n \\
% \end{pmatrix}
% =
% \begin{pmatrix}
% x_{11} & \dots & x_{1d} \\
% \vdots & \ddots & \vdots \\
% x_{n1} & \dots & x_{nd}
% \end{pmatrix} \text{.}
%\end{equation}
%
%The multivariate kernel density estimator $\hat{f}$ which defines the estimate pointwise at $\bm{x}=(x_1, \dots, x_d)^T$ is given as \cite[162]{scott2015}
%\begin{equation}
% \hat{f}(\bm{x}) = \frac{1}{nh_1 \dots h_d} \sum_{i=1}^{n} \left[ \prod_{j=1}^{d} K\left( \frac{x_j-x_{ij}}{h_j} \right) \right] \text{.}
%\end{equation}
%where the bandwidth is given as a vector $\bm{h}=(h_1, \dots, h_d)$.
% Product kernel allows our method
% Spherically symmetric kernel not supported, but Gaussian kernel == product & spehrically symmetric
% smoothing not in the direction of the axes -> rotate data, kde, rotate back
%Multivariate Gauss-Kernel
%\begin{equation}
%K(u)=\frac{1}{(2\pi)^{d/2}} \expp{-\frac{1}{2} \bm{x}^T \bm{x}}
%\end{equation}
% Gaus:
%If the filter kernel is separable, the convolution is also separable i.e. multi-dimensional convolution can be computed as individual one-dimensional convolutions with a one-dimensional kernel.
%Because of $e^{x^2+y^2} = e^{x^2}\cdot e^{y^2}$ the Gaussian filter is separable and can be easily applied to multi-dimensional signals. \todo{quelle}
%wie benutzen wir das ganze jetzt? auf was muss ich achten?
% Am Beispiel 2D Daten
% Histogram erzeugen (== data binnen)
% Hierzu wird min/max benötigt
% Anschließend Filterung per Box Filter über das Histogram
% - Wenn möglich parallel (SIMD, GPU)
% - separiert in jeder dim einzeln
% Maximum aus Filter ergebnis nehmen
\subsection{Our method}
The objective of our method is to allow a reliable recover of the most probable state from a time-sequential Monte Carlo sensor fusion system.
Assuming a sample based representation, our method allows to estimate the density of the unknown distribution of the state space in a narrow time frame.
Such systems are often used to obtain an estimation of the most probable state in near real time.
@@ -153,3 +40,4 @@ Finally, the most likely state can be obtained from the filtered data, i.e. from
Würde es Sinn machen das obere irgendwie Algorithmisch darzustellen? Also mit Pseudocode? Weil irgendwie/wo müssen wir ja "DAS IST UNSER APPROACH" stehen haben}.