fixed comments

tex/chapters/relatedwork.tex

@@ -6,7 +6,7 @@
 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 $\vec{X}^i_{t}$.
-In context of particle filtering $\{\vec{X}^i_{1:t}, W^i_{1:t} \}_{i=1}^N$ is a weighted set of samples, also called particles.
+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.
 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.
@@ -22,7 +22,7 @@ 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.
 %\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 $\{(\vec{X}_{1:t}^i , W^i_t)\}^N_{i=1}$, producing an accurate approximation of the filtering posterior $p(\vec{q}_{t} \mid  \vec{o}_{1:t})$ with a computational complexity of only $\mathcal{O}(N)$. 
+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}.
 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.