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tex/chapters/introduction.tex
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@@ -6,9 +6,9 @@ A powerful method to obtain numerical results for this approach are particle fil
%einführen von partikel filter ganz allgemein
Especially in indoor localisation, particle filter can lately be considered as the standard method for solving complex non-linear problems \cite{Doucet11:ATO}.
By using a set of weighted random samples, they approximate a probability distribution describing the pedestrian's possible whereabouts and therefore the uncertainty of the system.
By using a set of weighted random samples (particles), they approximate a probability distribution describing the pedestrian's possible whereabouts and therefore the uncertainty of the system.
In its most basic form, the particle filter operates three main steps:
At first, new samples are drawn according to some importance distribution, those samples are then weighted by an incremental importance weight distribution and finally a resampling step is deployed to prevent that only a small number of samples have a signifcant weight and all the other will have negligible small weights instead \cite{orhan2012particle}.
At first, new particles are drawn according to some importance distribution, those particles are then weighted by an incremental importance weight distribution and finally a resampling step is deployed to prevent that only a small number of particles have a signifcant weight and all the other will have negligible small weights instead \cite{orhan2012particle}.
%transition und evaluation einführen
In practice the importance distribution is often represented by the state transition, modelling the dynamics of the system.
@@ -16,11 +16,11 @@ A new weight is then obtained by the state evaluation given different sensor mea
Most localisation approaches differ mainly in how the transition and evaluation steps are implemented and the available sensors are incorporated \cite{Nurminen13-PSI, Ebner-15, Hilsenbeck2014}.
However, as \cite{Li2014} already mentioned, particle filter (and nearly all of its modifications) continue to suffer from two notorious problems: sample degeneracy and impoverishment.
As one can imagine, after a few iterations with continuously reweighting samples, the weight will concentrate on a few samples only.
As one can imagine, after a few iterations with continuously reweighting particles, the weight will concentrate on a few particles only.
This is why the resampling step was presented in the first place.
Here, a new set of equally weighted samples is drawn by multiplying high weighted samples while abandoning low weighted ones.
However, this leads to an decreasing diversity of samples after a resampling step, also known as sample impoverishment.
This high concentration of samples follows a bad approximation of the underlying probability distribution and therefore worse estimation results.
Here, a new set of equally weighted particles is drawn by multiplying high weighted particles while abandoning low weighted ones.
However, this leads to an decreasing diversity of particles after a resampling step, also known as sample impoverishment.
This high concentration of particles follows a bad approximation of the underlying probability distribution and therefore worse estimation results.
The effect of impoverishment is not solely caused by resampling only.
Restrictive transition models, as they are used in indoor localisation applications, also enhance this effect significantly.
@@ -36,10 +36,10 @@ An example is illustrated in fig. \ref{fig:multimodalPath}, where a graph-based
\end{figure}
%
Due to uncertain measurements the posterior distribution of the particle filter is captured within a room.
Between time $t-1$ and $t$, the resampling step abandons all samples on the corridor and drawing new samples outside the room is not possible due to the restricted transition.
Between time $t-1$ and $t$, the resampling step abandons all particles on the corridor and drawing new particles outside the room is not possible due to the restricted transition.
At this point, standard filtering methods are not able to recover.
A simple solution would be drawing a handful new samples randomly in the building.
A simple solution would be drawing a handful new particles randomly in the building.
However, it is obvious that this leads to a higher uncertainty and possible a highly multimodal posterior distribution.
Additionally, very uncertain absolute measurements, like attenuated Wi-Fi signals, can cause unpredictable jumps to such a newly drawn position, which would otherwise be not possible.
Especially, methods using relative measurements like pedestrian dead reckoning approaches are losing their importance.
@@ -47,13 +47,11 @@ Especially, methods using relative measurements like pedestrian dead reckoning a
As mentioned before, sample degeneracy and impoverishment are a pair of contradictions that can be described as a trade-off between the need of diversity and the need of focus \cite{Li2014}.
We tackle this problem in context of indoor localisation by deploying an interacting multiple model particle filter (IMMPF) for jump Markov non-linear systems \cite{Driessen2005}.
This enables a merging between posterior probability distributions approximated by particle filters, refereed as modes within this context.
combining two particles filters, one with a very restrictive transition scheme sehr genaue ergebnisse, and one with a more flexible but schlechtere ergebnisse scheme werden gewinnrbingen gemixed within the IMMPF,
Therefore a non-trivial Markov switching process, depending upon the Kullback-Leibler divergence between the modes and a
The main benefit of this approach is that it be easily adapted to other existing localization approaches based on particle filters.
We combine two similar particle filters using a non-trivial Markov switching process, depending upon the Kullback-Leibler divergence between the modes and a Wi-Fi quality factor.
One with a very restrictive transition scheme, providing very accurate results.
The other with more flexible and simple dynamics, resulting in a higher sample diversity.
Both are then successfully combined, to satisfy the need of diversity and the need of focus.
The main benefit of this approach is that it can be easily adapted to other existing localization approaches based on particle filters.

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tex/chapters/relatedwork.tex
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@@ -1,28 +1,51 @@
\section{Related Work}
\label{sec:relatedWork}
% 3/4 Seite ca.
% 1/2 - 3/4 Seite ca.
%klassisch resampling
A common way to handle degeneracy and impoverishment is to apply suitable resampling methods.
The four most popular and well established approaches found in literature are multinomial, stratified, systematic and residual resampling.
They are also referred to as traditional methods, since a single distribution is used for resampling and the number of times a particle is re-drawn is always proportional to is weight \cite{Li2015b}.
%advanced resampling
A more advanced method, with an adaptive particle size instead of a fixed one, is KLD-resampling.
It determines the number of particles to resample so that the Kullback-Leibler divergence between the distribution before resampling and after resampling does not exceed a pre-specified error bound \cite{Sun2013}.
The key idea is to choose a small number of samples if the distribution is focused on a small part of the state space and a large number of particles if the distribution is much more spread out and requires a higher diversity of samples.
In \cite{Li2015b} an overview of different resampling approaches are given.
%allgemien auf andere methoden überleiten
As seen, resampling methods are able to reduce impoverishment to a certain degree by themselves.
However, in practice sample impoverishment is also a problem of environmental restrictions and system dynamics.
Here, classical resampling schemes fail, since they are not able to propagate new particles into the state space.
More promising and intelligent solutions are given by techniques of Particle Distribution Optimization (PDO).
These variations of techniques are acting in different ways to optimize the spatial distribution of particles and are particularly effective in alleviating sample degeneracy and impoverishment \cite{Li2014}.
For example in \cite{Xiaoqin2008} a Particle Swarm Optimization is used as importance distribution for visual tracking.
Particles are iteratively updated according to their own experience and the experience of the swarm (or neighboring particles).
This allows for a multi-layer importance sampling and incorporation of the current measurement into the importance distribution, dealing with the sample impoverishment.
Other PDO methods are presented in \cite{Li2014}.
%hinführen zu IMM
In context of this work, our aim is to present a general solution that can be easily adapted to common localisation systems.
A promising approach for an easy to deploy PDO are Interacting Multiple Models (IMM) \cite{Bar-Shalom1988}.
IMM are able to mix appropriate dynamical systems based on a Bayesian probability metric and Gaussian noise.
Therefore, a set of modes like Kalman Filters are running in parallel.
The mixing between modes is done by using a Markov Chain process, providing a probability for every mode and a transition matrix for switching between them.
The most proper mode is then chosen for the current state estimation, what allows
the right choice to the right time.
For example \cite{Zhang2013} uses IMM to switch between a line-of-sight and a non-line-of-sight filtering procedure for indoor localisation.
Thereby, they are able to provide a robust and stable position estimation in both environments.
An extension to particle filters and therefore to non-linear and non-Gaussian system was presented by \cite{Boers2003}.
The so called Interacting Multiple Model Particle Filter (IMMPF) was then further developed by \cite{Driessen2005}, adding a direct sampling approach.
This allows a merging between different particle filters by providing a possibility for each filter to additional sample particles from all available particle sets and not just from its own.
It is obvious that the possibility to draw from other particle sets is based on the mode probability and the transition matrix provided by the Markov Chain process and therefore does not violate the Markov property.
Now, the key idea of this work is to satisfy the trade-off between diversity and focus by using appropriate modes within the IMMPF.
resampling approaches presented in and ... not able to solve impovershement, especially the one caused by a restrictive transition model.
trade of effective sample size
%Therefore, two different dynamical models are utilized and a novel approach for a non-trivial Markov switching Process based on Kullback-Leibler divergence and a Wi-Fi quality factor are presented.
andere arbeiten die particle depletion verhindern wollte -> was haben die so gemacht?
basic idea of this work is to combine two different filters. on depending upon realistic movement and the other observing absolut positions to prefent particle depletion due to relative measurements.
combining different filters.
jump markov non linear system.
interacting multiple model
particle filter