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toni
2016-05-08 20:22:02 +02:00
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10
tex/chapters/relatedwork.tex
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% 3/4 Seite ca.
%kurze einleitung zum smoothing
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.
Sequential MC filters, like the 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, $\vec{X}^i_{t} \sim p(\mStateVec_t \mid \mObsVec_{1:t})$ for $i = 1,...,N$ for approximation.
Due to importance sampling, a weight $W^i_t$ is assigned to each sample $\vec{X}^i_{t}$.
In context of particle filtering $\{W^i_{1:t}, \vec{X}^i_{1:t} \}_{i=1}^N$ is a weighted set of samples, also called particles.
In the context of particle filtering $\{W^i_{1:t}, \vec{X}^i_{1:t} \}_{i=1}^N$ is a weighted set of samples, also called particles.
Therefore a particle is a representation of one possible system state $\mStateVec$.
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.
@@ -23,7 +23,7 @@ In his work \cite{kitagawa1996monte} he presented the simplest form of smoothing
This algorithm is often called the filter-smoother since it runs online and a smoothing is provided while filtering.
%\commentByFrank{das mit dem weighted paths irritiert mich etwas. war das original work auch fuer etwas, wo pfade im spiel waren? weils halt gar so gut passt. ned dass da begrifflichkeiten durcheinander kommen. beim lesen fehlt mir das beim 1. anlauf was damit gemeint ist}
This approach uses the particle filter steps to update weighted paths $\{(W^i_t, \vec{X}_{1:t}^i)\}^N_{i=1}$, 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}.
However, it gives a poor representation of previous states due to 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.
Algorithmic details will be shown in section \ref{sec:smoothing}.
@@ -45,7 +45,7 @@ The experiments of \cite{Nurminen2014} clearly emphasize the benefits of smoothi
However, a fixed-lag smoother was discussed only in theory.
In the work of \cite{Paul2009} both fixed-interval and fixed-lag smoothing were presented.
They implemented Wi-Fi, binary infra-red motion sensors, binary foot-switches and a potential field for floor plan restrictions.
They implemented Wi-Fi, binary infrared motion sensors, binary foot-switches and a potential field for floor plan restrictions.
Those sensors were incorporated using a sigma-point Kalman filter in combination with a forward-backward smoother.
It was also proven by \cite{Paul2009}, that the fixed-lag smoother is slightly less accurate than the fixed-interval smoother, as one would expect from the theoretical foundation.
Unfortunately, even a sigma-point Kalman filters is after all just a linearisation and therefore not as flexible and suited for the complex problem of indoor localisation as a non-linear estimator like a particle filter.