\section{Related Work}
\label{sec:relatedWork}
% 3/4 Seite ca.

%kurze einleitung zum smoothing
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 $\vec{X}^i_{t}$.
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.

Within this work we utilise two types of smoothing: fixed-lag and fixed-interval smoothing. 
In fixed-lag smoothing, one tries to estimate the current state, given measurements up to a time $t + \tau$, where $\tau$ is a predefined lag. 
This makes the fixed-lag smoother able to run online. 
On the other hand, fixed-interval smoothing requires all observations until time $T$ and therefore only runs offline, after the filtering procedure is finished \cite{chen2003bayesian}. 


%historie des smoothings und entwicklung der methoden.
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. 
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 $\mathcal{O}(N)$. 
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. 
Algorithmic details will be shown in section \ref{sec:smoothing}.

%wo werden diese eingesetzt, paar beispiele. offline, online 
In recent years, smoothing gets attention mainly in other areas as indoor localisation. 
The early work of \cite{isard1998smoothing} demonstrates the possibilities of smoothing for visual tracking.
They used a combination of the CONDENSATION particle filter with a forward-backward smoother. 
Based on this pioneering approach, many different solutions for visual and multi-target tracking have been developed \cite{Perez2004}.
For example, in \cite{Platzer:2008} a particle smoother is used to reduce multimodalities in a blood flow simulation for human vessels. Or \cite{Hu2014} uses a smoother to overcoming the problem of particle impoverishment while predicting the Remaining Useful Life (RUL) of equipment (e.g. a Lithium-ion battery).

%smoothing im bezug auf indoor
Nevertheless, their are some promising approaches for indoor localisation systems as well. 
For example \cite{Nurminen2014} deployed a fixed-interval forward-backward smoother to improve the position estimation for non-real-time applications. 
They combined Wi-Fi, step and turn detection, a simple line-of-sight model for floor plan restrictions and the barometric change within a particle filter. 
The state transition samples a new state based on the heading change, altitude change and a fixed step length. 
The experiments of \cite{Nurminen2014} clearly emphasize the benefits of smoothing techniques. The estimation error could be decreased significantly. 
However, a fixed-lag smoother was treated only in theory.

In the work of \cite{Paul2009} both fixed-interval and fixed-lag smoothing were presented. 
They implemented Wi-Fi, binary infra-red motion sensors, binary foot-switches and a potential field for floor plan restrictions. 
Those sensors were incorporated using a sigma-point Kalman filter in combination with a forward-backward smoother.
It was also proven by \cite{Paul2009}, that the fixed-lag smoother is slightly less accurate then the fixed-interval smoother. 
As one would expect from the theoretical foundation.
Unfortunately, even a sigma-point Kalman filters is after all just a linearisation and therefore not as flexible and suited for the complex problem of indoor localisation as a non-linear estimator like a particle filter. 
\commentByToni{Kann das jemand nochmal verifizieren? Das mit dem Kalman Filter. Danke.}
Additionally, the Wi-Fi RSSI model requires known calibration points and is deployed using a remarkable number of access points for very small spaces.  
In our opinion this is not practical and we would further recommend adding a PDR-based transition instead of a random one. 

In contrast, the here presented approach is able to use two different smoothing algorithm, both implemented as fixed-interval and fixed-lag versions. 
Further, our localisation system presented in \cite{Ebner-16} enables us to walk stairs and therefore going into the third dimension.
Therefore, a regularly tessellated graph is utilized to avoid walls, detecting doors and recognizing stairs. 
Within this work, this is additionally supported by a simple classification that detects the activities  unknown, standing, walking and walking stairs. 
Finally, we incorporate prior navigation knowledge by using syntactically calculated realistic human walking paths \cite{Ebner-16}. 
This method makes use of the given destination and thereby provides a more targeted movement.