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added mc sampling and fixed some stuff in smoothing

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Toni
2016-04-26 20:46:32 +02:00
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tex/chapters/relatedwork.tex
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@@ -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.