text

tex/bare_conf.tex

@@ -18,6 +18,9 @@
 
 \newcommand{\dop}   [1]{\ensuremath{ \mathop{\mathrm{d}#1}                 }}
 \newcommand{\R}        {\ensuremath{ \mathbf{R}                            }}
+\newcommand{\expp}  [1]{\ensuremath{ \exp \left( #1 \right)                }}
+
+\newcommand{\qq}    [1]{``#1''}
 
 \begin{document}
 %
@@ -94,20 +97,43 @@ However, its inability to produce a continuous estimate dismisses it for many ap
 In contrast, the KDE is often the preferred tool because of its ability to produce a continuous estimate and its flexibility.
 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
 \begin{equation}
-\label{eq:kede}
+\label{eq:kde}
 \hat{f}_n = \frac{1}{nh} \sum_{i=1}^{n} K \left(  \frac{x-X_i}{h} \right)  \text{,} %= \frac{1}{n} \sum_{i=1}^{n} K_h(x-x_i)
 \end{equation}
 where $K$ is the kernel function and $h\in\R^+$ is an arbitrary smoothing parameter called bandwidth.
 While any density function can be used as the kernel function $K$ (such that $\int K(u) \dop{u} = 1$), a variety of popular choices of the kernel function $K$ exits.
-Commonly the Gaussian kernel is used.
+In practice the Gaussian kernel is commonly used:
+\begin{equation}
+K(u)=\frac{1}{\sqrt{2\pi}} \expp{- \frac{u^2}{2} }
+\end{equation}
+
+\begin{equation}
+\hat{f}_n = \frac{1}{nh\sqrt{2\pi}} \sum_{i=1}^{n} \expp{-\frac{(x-X_i)^2}{2h^2}}
+\end{equation}
 
 
-\section{Box Filter}
+\section{Moving Average Filter}
 % Basic box filter formula
 % Recursive form
 % Gauss Blur Filter
 % Repetitive Box filter to approx Gauss
 % Simple multipass, n/m approach, extended box filter
+The moving average filter is a simplistic filter which takes an input function $x$ and produces a second function $y$.
+A single output value is computed by taking the average of a number of values symmetrical around a single point in the input.
+The number of values in the average can also be seen as the width $w=2r+1$, where $r$ is the \qq{radius} of the filter.
+The computation of an output value using a moving average filter of radius $r$ is defined as
+\begin{equation}
+\label{eq:symMovAvg}
+y[i]=\frac{1}{2r+1} \sum_{j=-r}^{r}x[i+j] \text{.}
+\end{equation}
+
+It is well-known that a moving average filter can approximate a Gaussian filter by repetitive recursive computations.
+
+
+
+As is known the Gaussian filter is parametrized by its standard deviation $\sigma$.
+To approximate a Gaussian filter one needs to express a given $\sigma$ in terms of moving average filters.
+
 
 \section{Combination}