first draft particle filter

tex/chapters/4_ftmloc.tex

@@ -109,6 +109,7 @@ Hence, the distance measurements for a given AP $i$ are given with the vector $\
 The probability to observe the given distances $\vec{d}_{1:i} =(\vec{d}_1, \dots, \vec{d}_i)$ to each AP at the position $\mPosVec$ is given by the joint probability density function
 \begin{equation}
   p(\vec{d}_{1:i} | \mPosVec ) = \prod_i p(\vec{d}_i | \mPosVec ) = \prod_i\prod_{m=1}^{M_i} p(d_{i,m} | \mPosVec )
+	\label{eq:distanceProbability}
 \end{equation}
 where $M_i$ is the number of successful measurements to AP $i$.
 For each AP its measurements form a joint distribution by itself.
@@ -120,7 +121,7 @@ An alternative approach could be to compute the mean measurement over all $M_i$
 
 %Particle Filter Introduction
 As seen above, one can use a probability distribution like a Gaussian to describe the pedestrian’s most proper position and therefore the uncertainty of the system. 
-However, drawing from a probability distribution and finding an analytical solution for densities is in more complex scenarios a difficult task. 
+However, drawing from a probability distribution and finding an analytic solution for densities is in more complex scenarios a difficult task. 
 Especially when considering time sequential, non-linear and non-Gaussian models. 
 Due to the strong variation in human movement and the complexity of different sensor modalities, positioning indoors is often seen as such.
 
@@ -161,7 +162,7 @@ In case of this work, we are only interested in the distance measure $d$ provide
 Thus $\mObsVec$ can be easily given by  
  
 \begin{equation}
-    \mObsVec = (\vec{d}_1, \dots ,\vec{d}_i) 
+    \mObsVec = (\vec{d}_{1:i}) 
 \end{equation}
 
 containing a set of distance measurements $\vec{d}$ of all access-points $i$ currently visible to the phone. 
@@ -175,7 +176,7 @@ These steps are performed based on a predefined discrete update interval, e.g. e
 As described above, we deliberately do without a full stack IPS in order to clearly demonstrate the advantages and
 disadvantages of FTM compared to RSSI. 
 We have decided to utilize a very simple transition model, where the movement of particles from time step $t-1$ to $t$ is provided by drawing from a set of Gaussian distributions. 
-New potential whereabouts $p(\mStateVec_{t} \mid \mStateVec_{t-1})$ are then
+New potential whereabouts $p(\mStateVec_{t} \mid \mStateVec_{t-1})$:
 
 \begin{equation}
 		\begin{aligned}
@@ -184,11 +185,37 @@ y_t &=&	y_{t-1}\phantom{.}&	& &+& \delta \phantom{....}& & &\cdot& \sin(\mStateH
 		\end{aligned}
 		\label{eq:navMeshTrans}
 \end{equation}
-
+%
 Note that the uniform distribution in \eqref{eq:navMeshTrans} is limited in the interval $[0; 2\pi)$ to avoid oversampling at the pole.
 Further, the parameters for the Gaussian depend on the chosen update interval, as they describe the to-be-walked distance of the pedestrian.
 Put simply, \eqref{eq:navMeshTrans} causes the particles to spread out in a (uniquely distributed) circle within a certain (Gaussian distributed) distance.
 
 %Evaluation
+As particle filter approximate the posterior using importance sampling, thus every particle gets weighted using the probability density of the evaluation in \eqref{equ:bayesInt}.
+Here, a multitude of different sensor modalities can be incorporated by calculating the product of their respective probabilistic sensor models, which are often assumed to be statistical independent \cite{Fetzer-18}.  
+Within this work we are only interested in a single sensor model, which describes the Wi-Fi range measurements in a probabilistic manner.
+The necessary formulation for this is already stated in \eqref{eq:distanceProbability} by simply assigning 
+
+\begin{equation}
+p(\mObsVec_{t} \mid \mStateVec_{t}) = p(\vec{d}_{1:i} \mid \mPosVec) 
+\end{equation}
+%
+every particle, described by its position $\mPosVec$, can be weighted accordingly.
+Note that we also need to assume a statistical independence between the respective AP’s.
+
+Finally, a weighted set of particles $\{W^i_{t}, \vec{X}^i_{t} \}_{i=1}^N$ results after every time interval.
+As indoor positioning is often seen as a time sequential problem, we want to provide the best or likeliest position of the pedestrian for the current time step $t$. 
+The fastest and most intuitive method is simply selecting the particle with the highest weight. 
+However, realistic scenarios are often represented by multimodal densities and therefore it is common that some particles are sharing the highest weight \cite{Bullmann-18}. 
+The best way to receive the pedestrian's position is to recover the probability density function from the sample set itself, by using a non-parametric estimator like a kernel density estimation (KDE).
+As we have shown in \cite{Bullmann-18}, this can be done in an computational efficiency manner.
+Despite reducing the overall variance, such a method does not significantly reduce the error. 
+Thus, we decided to keep things simple by finding the likeliest position by calculating the weighted average state $\mStateVec_{t}^{\text{wa}}$ using
+
+\begin{equation}
+\mStateVec_{t}^{\text{wa}} = \frac{\sum_{i=1}^{N} \vec{X}^i_{t} \cdot W^i_{t}}{\sum_{i=1}^{N} W^i_{t}}
+\end{equation}
+%
+Of course, this does not avoid that the calculated state is somewhere in between the local maxima, if the current approximated posterior is multimodal.