added section recursive state estimation, and 3/4 of related work
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Toni
@@ -1,11 +1,8 @@
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\section{Recursive State Estimation}
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\commentByFrank{schon mal kopiert, dass es da ist.}
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\commentByFrank{die neue activity in die observation eingebaut}
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\commentByFrank{magst du hier auch gleich smoothing ansprechen? denke es würde sinn machen weils ja zum kompletten systemablauf gehört und den hatten wir hier ja immer drin. oder was meinst du?}
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We consider indoor localisation as a time-sequential, non-linear and non-Gaussian state estimation problem.
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Using a recursive Bayes filter that satisfies the Markov property, the posterior distribution at time $t$ can be written as
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As mentioned before, most smoothing methods require a preceding filtering.
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In our previous work \cite{Ebner-16}, we consider indoor localisation as a time-sequential, non-linear and non-Gaussian state estimation problem.
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Therefore, a Bayes filter that satisfies the Markov property is used to calculate the posterior, which is given by
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%
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\begin{equation}
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\arraycolsep=1.2pt
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@@ -13,40 +10,33 @@
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&p(\mStateVec_{t} \mid \mObsVec_{1:t}) \propto\\
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&\underbrace{p(\mObsVec_{t} \mid \mStateVec_{t})}_{\text{evaluation}}
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\int \underbrace{p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})}_{\text{transition}}
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\underbrace{p(\mStateVec_{t-1} \mid \mObsVec_{1:t-1})d\vec{q}_{t-1}}_{\text{recursion}} \enspace,
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\underbrace{p(\mStateVec_{t-1} \mid \mObsVec_{1:t-1})d\vec{q}_{t-1}}_{\text{recursion}} \enspace.
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\end{array}
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\label{equ:bayesInt}
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\end{equation}
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%
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where $\mObsVec_{1:t} = \mObsVec_{1}, \mObsVec_{1}, ..., \mObsVec_{t}$ is a series of observations up to time $t$.
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The hidden state $\mStateVec$ is given by
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Here, the previous observation $\mObsVec_{t-1}$ is included into the state transition \cite{Koeping14-PSA}.
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For approximating eq. \eqref{equ:bayesInt} by means of MC methods, the transition is used as proposal distribution, also known as CONDENSATION algorithm \cite{isard1998smoothing}.
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In context of indoor localisation, the hidden state $\mStateVec$ is defined as follows:
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\begin{equation}
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\mStateVec = (x, y, z, \mStateHeading, \mStatePressure),\enskip
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x, y, z, \mStateHeading, \mStatePressure \in \R \enspace,
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\end{equation}
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%
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where $x, y, z$ represent the position in 3D space, $\mStateHeading$ the user's heading and $\mStatePressure$ the
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relative atmospheric pressure prediction in hectopascal (hPa).
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\commentByFrank{hier einfach kuerzen und aufs fusion paper verweisen? auch wenn das noch ned durch ist?}
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The recursive part of the density estimation contains all information up to time $t-1$.
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Furthermore, the state transition $p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$ models the pedestrian's movement as described in section \ref{sec:trans}.
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%It should be noted, that we also include the current observation $\mObsVec_{t}$ in it.
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As proven in \cite{Koeping14-PSA}, we may include the observation $\mObsVec_{t-1}$ into the state transition.
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Containing all relevant sensor measurements to evaluate the current state, the observation vector is defined as follows:
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where $x, y, z$ represent the position in 3D space, $\mStateHeading$ the user's heading and $\mStatePressure$ the relative atmospheric pressure prediction in hectopascal (hPa). Further, the observation is given by
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%
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\begin{equation}
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\mObsVec = (\mRssiVec_\text{wifi}, \mRssiVec_\text{ib}, \mObsHeading, \mObsSteps, \mObsPressure, \mObsActivity) \enspace,
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\mObsVec = (\mRssiVec_\text{wifi}, \mRssiVec_\text{ib}, \mObsHeading, \mObsSteps, \mObsPressure, x) \enspace,
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\end{equation}
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%
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where $\mRssiVec_\text{wifi}$ and $\mRssiVec_\text{ib}$ contain the measurements of all nearby \docAP{}s (\docAPshort{})
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and \docIBeacon{}s, respectively. $\mObsHeading$ and $\mObsSteps$ describe the relative angular change and the number
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of steps detected for the pedestrian. $\mObsPressure$ is the relative barometric pressure with respect to a fixed reference.
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Finally, $\mObsActivity$ contains the activity, currently estimated for the pedestrian, which is one of: unknown, standing, walking or
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walking stairs.
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%For further information on how to incorporate such highly different sensor types,
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%one should refer to the process of probabilistic sensor fusion \cite{Khaleghi2013}.
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By assuming statistical independence of all sensors, the probability density of the state evaluation is given by
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covering all relevant sensor measurements.
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Here, $\mRssiVec_\text{wifi}$ and $\mRssiVec_\text{ib}$ contain the measurements of all nearby \docAP{}s (\docAPshort{}) and \docIBeacon{}s, respectively.
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$\mObsHeading$ and $\mObsSteps$ describe the relative angular change and the number of steps detected for the pedestrian.
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$\mObsPressure$ is the relative barometric pressure with respect to a fixed reference.
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Finally, $x$ contains the activity, currently estimated for the pedestrian, which is one of: unknown, standing, walking or walking stairs.
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The probability density of the state evaluation is given by
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%
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\begin{equation}
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%\begin{split}
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@@ -54,23 +44,16 @@
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p(\vec{o}_t \mid \vec{q}_t)_\text{baro}
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\,p(\vec{o}_t \mid \vec{q}_t)_\text{ib}
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\,p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}
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\enspace.
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\enspace
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%\end{split}
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\label{eq:evalBayes}
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\end{equation}
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%
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Here, every single component refers to a probabilistic sensor model.
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The barometer information is evaluated using $p(\vec{o}_t \mid \vec{q}_t)_\text{baro}$,
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whereby absolute position information is given by $p(\vec{o}_t \mid \vec{q}_t)_\text{ib}$ for
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\docIBeacon{}s and by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ for \docWIFI{}.
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%It is well known that finding analytic solutions for densities is very difficult and only possible in rare cases.
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%Therefore, numerical solutions like Gaussian filters or the broad class of Monte Carlo methods are deployed \cite{sarkka2013bayesian}.
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Since we assume indoor localisation to be a time-sequential, non-linear and non-Gaussian process,
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a particle filter is chosen as approximation of the posterior distribution.
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\commentByFrank{smoothing?}
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%Within this work the state transition $p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$ is used as proposal distribution,
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%also known as CONDENSATION algorithm \cite{Isard98:CCD}.
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and therefore similar to \cite{Ebner-16}.
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Here, we assume a statistical independence of all sensors and every single component refers to a probabilistic sensor model.
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The barometer information is evaluated using $p(\vec{o}_t \mid \vec{q}_t)_\text{baro}$, whereby absolute position information is given by $p(\vec{o}_t \mid \vec{q}_t)_\text{ib}$ for \docIBeacon{}s and by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ for \docWIFI{}.
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