Fixed FE 1

tex/chapters/multivariate.tex

@@ -10,15 +10,15 @@ 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 $\bm{X}=(x_1,\dots ,x_d)$ in $d$ dimensions.
-The sample set $\mathcal{X}$ is a $n\times d$ matrix \cite[162]{scott2015}.
+The sample set $\mathcal{X}$ is a $n\times d$ matrix \cite{scott2015}.
 The multivariate KDE $\hat{f}$ which defines the estimate pointwise at $\bm{u}=(u_1, \dots, u_d)^T$ is given as
 \begin{equation}
 \label{eq:mvKDE}
-    \hat{f}(\bm{u}) = \frac{1}{W} \sum_{i=1}^{n} \frac{w_i}{h_1 \dots h_d} \left[  \prod_{j=1}^{d} K\left( \frac{u_j-x_{ij}}{h_j} \right)  \right]  \text{,}
+    \hat{f}(\bm{u}) = \frac{1}{W} \sum_{i=1}^{n} \frac{w_i}{h_1 \dots h_d} \left[  \prod_{j=1}^{d} K\left( \frac{u_j-x_{i,j}}{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.
+Note that \eqref{eq:mvKDE} does not include all possible multivariate kernels, such as spherically symmetric kernels, which are based on rotation of an 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 spherically symmetric and has independent marginals. % TODO scott cite?!
 In addition, only smoothing in the direction of the axes is possible.
@@ -30,7 +30,7 @@ Likewise, the ideas of common and linear binning rule scale with dimensionality
 
 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.
+Convolution is separable if the filter kernel is separable, \ie{} 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 individually by an one-dimensional filter \cite{dspGuide1997}.
@@ -45,7 +45,7 @@ They are easily applied to multi-dimensional signals, because the input signal c
 %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}
+%The sample $\bm{X}$ is a $n\times d$ matrix defined as \cite{scott2015}
 %\begin{equation}
 %    \bm{X}=
 %    \begin{pmatrix}
@@ -61,7 +61,7 @@ They are easily applied to multi-dimensional signals, because the input signal c
 %    \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}
+%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{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}
@@ -77,7 +77,7 @@ They are easily applied to multi-dimensional signals, because the input signal c
 %\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.
+%If the filter kernel is separable, the convolution is also separable \ie{} 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}