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Linguistic checking

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Lukas Koeping
2016-05-12 10:40:24 +02:00
parent ff56649a5b
commit b936668818
7 changed files with 44 additions and 45 deletions
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22
tex/chapters/smoothing.tex
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@@ -17,7 +17,7 @@ p(\vec{q}_t \mid \vec{o}_{1:T}) \approx \sum^N_{i=1} W^i_{t \mid T} \delta_{\vec
\end{equation}
%\commentByFrank{support?}
where $p(\vec{q}_t \mid \vec{o}_{1:T})$ has the same support as the filtering distribution $p(\vec{q}_t \mid \vec{o}_{1:t})$, but the weights are different.
This means, that the FBS maintains the original particle locations and just reweights the particles to obtain a smoothed density.
This means that the FBS maintains the original particle locations and just reweights the particles to obtain a smoothed density.
$\delta_{\vec{X}^i_{t}}$ denotes the Dirac delta function.
The complete FBS can be seen in algorithm \ref{alg:forward-backwardSmoother} in pseudo-algorithmic form.
%\commentByFrank{forward step vlt etwas genauer erklaeren weil 1. mal benutzt? oder is das hinlaenglich bekannt? :P}
@@ -66,7 +66,7 @@ For smoothing applications with a high number of particles, it is often not nece
This decision can, for example, be made due to a high sample impoverishment and/or highly accurate sensors.
By choosing a good sub-set for representing the posterior distribution, it is theoretically possible to further improve the estimation.
Therefore, \cite{Godsill04:MCS} presented the backward simulation (BS). Where a number of independent sample realisations
Therefore, \cite{Godsill04:MCS} presented the backward simulation (BS), where a number of independent sample realisations
from the entire smoothing density are used to approximate the smoothing distribution.
%
\begin{algorithm}[t]
@@ -91,7 +91,7 @@ from the entire smoothing density are used to approximate the smoothing distribu
\end{algorithm}
%
This method can be seen in algorithm \ref{alg:backwardSimulation} in pseudo-algorithmic form.
Again, a particle filter is performed at first and then the smoothing procedure gets applied.
Again, a particle filter is performed at first and then the smoothing procedure is applied.
%\commentByFrank{das klingt so, als waeren particle-filter und smoothing zwei komplett verschiedene sachen.}
%\commentByToni{Sind sie doch auch irgendwo.}
%\commentByFrank{was heisst 'drawn approximately'? nach welchen gesichtspunkte?}
@@ -109,7 +109,7 @@ Unlike the transition presented in section \ref{sec:transition}, it is not possi
Here, $p(\vec{q}_{t+1} \mid \vec{q}_{t})$ needs to provide the probability of the \textit{known} future state $\vec{q}_{t+1}$ under the condition of its ancestor $\vec{q}_{t}$.
The smoothing transition model therefore calculates the probability of being in a state $\vec{q}_{t+1}$ in regard to previous states and the pedestrian's walking behaviour.
This means that a state $\vec{q}_t$ is more likely if it is a proper ancestor (realistic previous position) of a future state $\vec{q}_{t+1}$.
In the following a simple and inexpensive approach for receiving this information will be described.
In the following, a simple and inexpensive approach for receiving this information will be described.
By writing
\begin{equation}
@@ -122,9 +122,9 @@ Of course, based on the graph structure, one could calculate the shortest path b
However, this requires tremendous calculation time for negligible improvements.
Therefore this is not further discussed within this work.
The average step length $\mu_{\text{step}}$ is based on the pedestrian's walking speed and $\sigma_{\text{step}}^2$ denotes the step length's variance.
Both values are chosen depending on the activity $\mObsActivity$ recognized at time $t$.
For example $\mu_{\text{step}}$ gets smaller while a pedestrian is walking upstairs, than just walking straight.
This requires to extend the smoothing transition by the current observation $\mObsVec_t$.
Both values are chosen depending on the activity $\mObsActivity$ recognised at time $t$.
For example $\mu_{\text{step}}$ becomes smaller while a pedestrian is walking upstairs than just walking straight.
This requires an extension of the smoothing transition by the current observation $\mObsVec_t$.
Since $\mStateVec$ is hidden and the Markov property is satisfied, we are able to do so.
@@ -143,10 +143,10 @@ To further improve the results, especially in 3D environments, the vertical (non
p(\vec{q}_{t+1} \mid \vec{q}_t, \mObsVec_t)_{\text{baro}} = \mathcal{N}(\Delta z \mid \mu_z, \sigma^2_{z}) \enspace .
\label{eq:smoothingTransPressure}
\end{equation}
This assigns a low probability to false detected or misguided floor changes.
This assigns a low probability to falsely detected or misguided floor changes.
Similar to \refeq{eq:smoothingTransDistance} we set $\mu_z$ and $\sigma^2_{z}$ based on the activity recognised at time $t$.
Therefore, $\mu_z$ is the expected change in $z$-direction between two time steps.
This means, if the pedestrian is walking alongside a corridor, we set $\mu_z = 0$.
This means that if the pedestrian is walking alongside a corridor, we set $\mu_z = 0$.
In contrast, $\mu_z$ is positive while walking downstairs or otherwise negative for moving upstairs.
The size of $\mu_z$ and also $\mu_{\text{step}}$ could be a predefined value or set dynamically based on the measured vertical and linear acceleration.
@@ -162,7 +162,7 @@ Looking at \refeq{eq:smoothingTransDistance} to \refeq{eq:smoothingTransPressure
\enspace .
\end{equation}
%
It is important to notice, that all particles at each time step $t$ of the forward filtering need to be saved.
Therefore, the memory requirement increases proportional to the processing time.
It is important to notice that all particles at each time step $t$ of the forward filtering need to be saved.
Therefore, the memory requirement increases proportionally to the processing time.