Improved kde & mvg

tex/chapters/usage.tex

@@ -1,4 +1,47 @@
 \section{Usage}
+
+\subsection{Extension to multi-dimensional data}
+\todo{Absatz zum Thema 2D - Extension to multi-dimensional data}
+
+% 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 $\bm{h}=(h_1, \dots, h_d)$.
+
+%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
@@ -9,7 +52,8 @@
 % - separiert in jeder dim einzeln
 % Maximum aus Filter ergebnis nehmen
 
-\todo{Absatz zum Thema 2D}
+
+
 
 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.