From a6c467bb0934832fc35ef0cd528000ebda90be5d Mon Sep 17 00:00:00 2001 From: toni Date: Wed, 23 May 2018 14:22:53 +0200 Subject: [PATCH] sample impoverishment weiter gemacht + bulli chapter in eigene datei --- tex/chapters/estimation.tex | 10 ++++++++++ tex/chapters/eval.tex | 2 ++ tex/chapters/misc.tex | 22 +++++++++++++--------- tex/misc/functions.tex | 2 ++ 4 files changed, 27 insertions(+), 9 deletions(-) create mode 100644 tex/chapters/estimation.tex diff --git a/tex/chapters/estimation.tex b/tex/chapters/estimation.tex new file mode 100644 index 0000000..4b9c950 --- /dev/null +++ b/tex/chapters/estimation.tex @@ -0,0 +1,10 @@ +\subsection{State Estimation} + +1/2 bis 3/4 Seite + +\begin{itemize} + \item weighted average + \item max particle + \item bulli methode (gleich citen :)) +\end{itemize} + diff --git a/tex/chapters/eval.tex b/tex/chapters/eval.tex index 5128fa8..9435fc6 100644 --- a/tex/chapters/eval.tex +++ b/tex/chapters/eval.tex @@ -1,5 +1,6 @@ \section{Evaluation} + The probability density of the state evaluation in \eqref{equ:bayesInt} is given by % \begin{equation} @@ -17,6 +18,7 @@ The barometer readings are used to determine the current activity $\mObsActivity Absolute positioning information is given by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ for \docWIFI{}. \subsection{\docWIFI{}} +\label{sec:wifi} As stated in section \ref{sec:relatedWork}, we use the smartphone's \docWIFI{} component to provide an absolute location estimation based on a comparison between recent RSSI measurements of nearby AP's and signal strength predictions. The probability given those measurements $\mRssiVec_\text{wifi}$ and a prediction, corresponding to a well-known location $\mPosVec = (x,y,z)^T$ provided by $\vec{q}_t$, can thus be written as diff --git a/tex/chapters/misc.tex b/tex/chapters/misc.tex index 6004a04..081c070 100644 --- a/tex/chapters/misc.tex +++ b/tex/chapters/misc.tex @@ -14,6 +14,8 @@ p(\mStateVec_{t} \mid \mObsVec_{1:t}) \approx \sum^N_{i=1} w^i_t \delta_{\vec{X} As one can imagine, after a few iterations with continuously reweighting particles, the weight will concentrate on a few particles only. To handle this phenomenon of weight degeneracy, a resampling procedure is performed after every filter step \cite{robotics}. +\input{chapters/estimation} + \subsection{Sample Impoverishment} As we have extensively discussed in \cite{Fetzer-17}, besides sample degeneracy, particle filters (and nearly all of its modifications) continue to suffer from another notorious problem: sample impoverishment. It refers to a situation, in which the filter is unable to sample enough particles into proper regions of the building, caused by a high concentration of misplaced particles. @@ -50,9 +52,19 @@ This allows the primary filter to recover, while retaining prior knowledge. However, we believe that such a combination of two independent filters is not necessary for most scenarios and thus the resulting overhead can be avoided. %neue methode: +For the simplified version we distribute \SI{10000}{} samples uniformly within the complete building to approximate $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ as presented in section \ref{sec:wifi}. +From the resulting probability grid $\probGrid_{t, \text{wifi}}$ we are able to identify the areas where the \docWIFI{} model assumes the pedestrian is most likely located. +Of course, this often results in a multimodal representation of the probability density and thus multiple possible whereabouts. +However, compared to the used particle filter, this representation enables us to monitor the complete building without any environmental restrictions and can thus be deployed as an indicator to detect sample impoverishment. +If $\probGrid_{t, \text{wifi}}$ and the current posterior $p(\mStateVec_{t} \mid \mObsVec_{1:t})$ show a significant difference, we can assume that either the posterior got stuck and suffers from impoverishment or the \docWIFI{} quality is low due to factors like attenuation or bad coverage. +A good measure of how one probability distribution differs from a second is the well-established Kullback-Leibler divergence $D_\text{KL}$ \cite{Fetzer-17}. +To calculate $D_\text{KL}$, we need to sample densities from both probability density functions likewise. +For the posterior we use the results provided by our rapid kernel density estimation performed in the state estimation procedure, while $\probGrid_{t, \text{wifi}}$ is already in the desired form. + + +We compare the particles provided by the posterior and the samples of $\probGrid_{t, \text{wifi}}$ -we sample 10k samples uniformaly within the whole building.. \begin{itemize} \item zufällig einen partikel streuen @@ -60,15 +72,7 @@ we sample 10k samples uniformaly within the whole building.. \item KLD zwischen wifi und aktuellen particeln des filters. \end{itemize} -\subsection{State Estimation} -1/2 bis 3/4 Seite - -\begin{itemize} - \item weighted average - \item max particle - \item bulli methode (gleich citen :)) -\end{itemize} diff --git a/tex/misc/functions.tex b/tex/misc/functions.tex index 84dfb32..e61fec3 100644 --- a/tex/misc/functions.tex +++ b/tex/misc/functions.tex @@ -27,6 +27,8 @@ \newcommand{\mParticle}{X} \newcommand{\mParticleVec}{\vec{X}} +\newcommand{\probGrid}{\vec{Q}} + \newcommand{\mProb}{p} % char for probability \newcommand{\mMovingAvgWithSize}[1]{\ensuremath{\text{avg}_{#1}}}