Box & boxKDE algos + notation fixes

tex/chapters/multivariate.tex

@@ -1,5 +1,4 @@
 \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.
@@ -11,22 +10,7 @@ Multivariate kernel functions can be constructed in various ways, however, a pop
 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 sample $\bm{X}$ is a $n\times d$ matrix defined as \cite[162]{scott2015}.
 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}
@@ -41,16 +25,17 @@ 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?}
-
+Their extensions are rather straightforward, as the grid is easily defined on many dimensions.
+Likewise, the ideas of common and linear binning rule scale with the dimensionality \cite{wand1994fast}.
 
 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.
+In example, the Gaussian filter is separable, because of $e^{x^2+y^2} = e^{x^2}\cdot e^{y^2}$.
 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.
+They are easily applied to multi-dimensional signals, because the input signal can be filtered in each dimension individually 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: