shortend smoothing

tex/chapters/smoothing.tex

@@ -9,12 +9,15 @@ As mentioned before, those algorithm are able to compute probability distributio
 In the following we discuss the algorithmic details of the forward-backward smoother and the backward simulation. 
 Further, two novel approaches for incorporating them into the localisation system are shown.
 
+\todo{Einfuehren von $X$ und etwas konkreter schreiben.}
+
 \subsection{Forward-backward Smoother}
 
 The forward-backward smoother (FBS) of \cite{Doucet00:OSM} is a well established alternative to the simple filter-smoother. The foundation of this algorithm was again laid by Kitagawa in \cite{kitagawa1987non}. 
 An approximation is given by
 \begin{equation}
 p(\vec{q}_t \mid \vec{o}_{1:T}) \approx \sum^N_{i=1} W^i_{t \mid T} \delta_{\vec{X}^i_{t}}(\vec{q}_{t}) \enspace, 
+\label{eq:approxFBS}
 \end{equation}
 where $p(\vec{q}_t \mid \vec{o}_{1:T})$ has the same support as the filtering distribution $p(\vec{q}_t \mid \vec{o}_{1:t})$, but the weights are different. 
 This means, that the FBS maintains the original particle locations and just reweights the particles to obtain a smoothed density. 
@@ -56,30 +59,8 @@ By reweighting the filter particles, the FBS improves the simple filter-smoother
 \subsection{Backward Simulation}
 For smoothing applications with a high number of particles, it is often not necessary to use all particles for smoothing. This decision can for example be made due to a high sample impoverishment and/or highly certain sensors. By choosing a good sub-set for representing the posterior distribution, it is theoretically possible to further improve the estimation. 
 
-Therefore, \cite{Godsill04:MCS} presented the \textit{backward simulation}. Where a number of independent sample realizations from the entire smoothing density are used to approximate the smoothing distribution. To derive the backward simulation algorithm we need to factorize the joint smoothing distribution as follows:
-\begin{equation}
-p(\vec{q}_{1:T} \mid \vec{o}_{1:T}) = p(\vec{q}_{T} \mid \vec{o}_{1:T}) \prod^{T-1}_{t=1} p(\vec{q}_{t} \mid \vec{q}_{t+1:T}, \vec{o}_{1:T})
-\end{equation}
-By using the Markov property of the model, we can write
-\begin{equation}
-\begin{split}
-p(\vec{q}_{t} \mid \vec{q}_{t+1:T}, \vec{o}_{1:T}) &= p(\vec{q}_{t} \mid \vec{q}_{t+1}, \vec{o}_{1:t})\\
-&= \frac{p(\vec{q}_{t} \mid \vec{o}_{1:t}) p(\vec{q}_{t+1} \mid \vec{q}_{t})}{p(\vec{q}_{t+1} \mid \vec{o}_{1:t})} \\
-& \propto p(\vec{q}_{t} \mid \vec{o}_{1:t}) p(\vec{q}_{t+1} \mid \vec{q}_{t})
-\enspace ,
-\end{split}
-\label{eq:backwardSimulation-02}
-\end{equation}
-where $p(\vec{q}_t \mid \vec{o}_{1:t})$ is obtained using any particle filter in the forward step. Using \ref{eq:backwardSimulation-02} the particles can be approximated as follows:
-\begin{equation}
-p(\vec{q}_{t} \mid \vec{q}_{t+1:T}, \vec{o}_{1:T}) \approx \sum^N_{i=1} W^i_{t \mid t+1} \delta_{\vec{X}^i_{t}}(\vec{q}_{t}) \enspace,
-\end{equation}
-where the weights are determined with
-\begin{equation}
-W^i_{t \mid t+1} = \frac{ W^i_t ~ p(\vec{q}_{t+1} \mid \vec{X}^i_{t})}{\sum^N_{j=1} W^j_t p(\vec{q}_{t+1} \mid \vec{X}^j_{t})}\enspace .
-\end{equation}
-This can now be used to generate states successively in the reverse-time direction, conditioning upon future states. 
-
+Therefore, \cite{Godsill04:MCS} presented the backward simulation (BS). Where a number of independent sample realizations from the entire smoothing density are used to approximate the smoothing distribution. 
+%
 \begin{algorithm}[t]
     \caption{Backward Simulation Smoothing}
     \label{alg:backwardSimulation}
@@ -102,8 +83,8 @@ This can now be used to generate states successively in the reverse-time directi
 			\EndFor
     \end{algorithmic}
 \end{algorithm}
-
-The backward simulation method can be seen in algorithm \ref{alg:backwardSimulation} in pseudo-algorithmic form. Again, a particle filter is performed at first and then the smoothing procedure gets applied. Here, $\tilde{\vec{q}}_t$ is a random sample drawn approximately from $p(\vec{q}_{t} \mid \tilde{\vec{q}}_{t+1}, \vec{o}_{1:T})$. Therefore $\tilde{\vec{q}}_{1:T} = (\tilde{\vec{q}}_{1}, \tilde{\vec{q}}_{2}, ...,\tilde{\vec{q}}_{T})$ is one particular sample realization from $p(\vec{q}_{1:T} \mid \vec{o}_{1:T})$. Further independent realizations are obtained by repeating the algorithm until the desired number $N_{\text{sample}}$ is reached. The computational complexity for one particular realization is $\mathcal{O}(N)$. However, the computations are then repeated for each realization drawn \cite{Godsill04:MCS}.
+%
+This method can be seen in algorithm \ref{alg:backwardSimulation} in pseudo-algorithmic form. Again, a particle filter is performed at first and then the smoothing procedure gets applied. Here, $\tilde{\vec{q}}_t$ is a random sample drawn approximately from $p(\vec{q}_{t} \mid \tilde{\vec{q}}_{t+1}, \vec{o}_{1:T})$. Therefore $\tilde{\vec{q}}_{1:T} = (\tilde{\vec{q}}_{1}, \tilde{\vec{q}}_{2}, ...,\tilde{\vec{q}}_{T})$ is one particular sample realization from $p(\vec{q}_{1:T} \mid \vec{o}_{1:T})$. Further independent realizations are obtained by repeating the algorithm until the desired number $N_{\text{sample}}$ is reached. The computational complexity for one particular realization is $\mathcal{O}(N)$. However, the computations are then repeated for each realization drawn \cite{Godsill04:MCS}.
 
 \subsection{Transition for Smoothing}