toni/IPIN2016
2
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

added mc sampling and fixed some stuff in smoothing

Browse Source
This commit is contained in:
Toni
2016-04-26 20:46:32 +02:00
parent 33f8acbcab
commit df1be331a6
2 changed files with 6 additions and 16 deletions
Download Patch File Download Diff File Expand all files Collapse all files

7
tex/chapters/relatedwork.tex
View File

@@ -3,8 +3,9 @@
% 3/4 Seite ca.
%kurze einleitung zum smoothing
Filtering algorithm, like aforementioned particle filter, use all observations $\mObsVec_{1:t}$ until the current time $t$ for computing an estimation of the state $\mStateVec_t$.
In a Bayesian setting, this can be formalized as the computation of the posterior distribution $p(\mStateVec_t \mid \mObsVec_{1:t})$.
Sequential MC filter, like aforementioned particle filter, use all observations $\mObsVec_{1:t}$ until the current time $t$ for computing an estimation of the state $\mStateVec_t$.
In a Bayesian setting, this can be formalized as the computation of the posterior distribution $p(\mStateVec_t \mid \mObsVec_{1:t})$ using a sample of $N$ independent random variables, $\vec{X}^i_{t} \sim (\mStateVec_t \mid \mObsVec_{1:t})$ for $i = 1,...,N$ for approximation.
Due to importance sampling, a weight $W^i_t$ is assigned to each sample $\vec{X}^i_{t}$.
By considering a situation given all observations $\vec{o}_{1:T}$ until a time step $T$, where $t \ll T$, standard filtering methods are not able to make use of this additional data for computing $p(\mStateVec_t \mid \mObsVec_{1:T})$.
This problem can be solved with a smoothing algorithm.
@@ -18,7 +19,7 @@ On the other hand, fixed-interval smoothing requires all observations until time
The origin of MC smoothing can be traced back to Genshiro Kitagawa.
In his work \cite{kitagawa1996monte} he presented the simplest form of smoothing as an extension to the particle filter.
This algorithm is often called the filter-smoother since it runs online and a smoothing is provided while filtering.
This approach uses the particle filter steps to update weighted paths $\{(\vec{q}_{1:t}^i , w^i_t)\}^N_{i=1}$, producing an accurate approximation of the filtering posterior $p(\vec{q}_{t} \mid \vec{o}_{1:t})$ with a computational complexity of only $\mathcal{O}(N)$.
This approach uses the particle filter steps to update weighted paths $\{(\vec{X}_{1:t}^i , W^i_t)\}^N_{i=1}$, producing an accurate approximation of the filtering posterior $p(\vec{q}_{t} \mid \vec{o}_{1:t})$ with a computational complexity of only $\mathcal{O}(N)$.
However, it gives a poor representation of previous states due a monotonic decrease of distinct particles caused by resampling of each weighted path \cite{Doucet11:ATO}.
Based on this, more advanced methods like the forward-backward smoother \cite{doucet2000} and backward simulation \cite{Godsill04:MCS} were developed.
Both methods are running backwards in time to reweight a set of particles recursively by using future observations.

15
tex/chapters/smoothing.tex
View File

@@ -1,16 +1,13 @@
\section{Smoothing}
\label{sec:smoothing}
The main purpose of this work is to provide MC smoothing methods in context of indoor localisation.
As mentioned before, those algorithm are able to compute probability distributions in the form of $p(\mStateVec_t \mid \mObsVec_{1:T})$ and are therefore able to make use of future observations between $t$ and $T$.
The main purpose of this work is to provide smoothing methods in context of indoor localisation.
As mentioned before, those algorithm are able to compute probability distributions in the form of $p(\mStateVec_t \mid \mObsVec_{1:T})$ and are therefore able to make use of future observations between $t$ and $T$.
%Especially fixed-lag smoothing is very promising in context of pedestrian localisation.
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}.
@@ -31,8 +28,6 @@ The weights are obtained through the backward recursion in line 9.
\caption{Forward-Backward Smoother}
\label{alg:forward-backwardSmoother}
\begin{algorithmic}[1] % The number tells where the line numbering should start
\Statex{\textbf{Input:} Prior $\mu(\vec{X}^i_1)$}
\Statex{~}
\For{$t = 1$ \textbf{to} $T$} \Comment{Filtering}
\State{Obtain the weighted trajectories $ \{ W^i_t, \vec{X}^i_t\}^N_{i=1}$}
\EndFor
@@ -65,8 +60,6 @@ Therefore, \cite{Godsill04:MCS} presented the backward simulation (BS). Where a
\caption{Backward Simulation Smoothing}
\label{alg:backwardSimulation}
\begin{algorithmic}[1] % The number tells where the line numbering should start
\Statex{\textbf{Input:} Prior $\mu(\vec{X}^i_1)$}
\Statex{~}
\For{$t = 1$ \textbf{to} $T$} \Comment{Filtering}
\State{Obtain the weighted trajectories $ \{ W^i_t, \vec{X}^i_t\}^N_{i=1}$}
\EndFor
@@ -88,7 +81,3 @@ This method can be seen in algorithm \ref{alg:backwardSimulation} in pseudo-algo
\subsection{Transition for Smoothing}
%komplexität eingehen
The reason for not behandeln liegt ...
However, \cite{} and \cite{} have proven this wrong and reduced the complexity of different smoothing methods.