\section{Introduction}

%It is well known, that signals given by Global Navigation Satellite Systems (GNSS) like the Global Positioning System (GPS) do not move easily through solid objects. Therefore, they work best in outdoor environments, when the device has a clear line of sight to the sky. 
%Obviously, GNSS are of no practical use in the context of indoor localisation.  

Determining a position indoors is a challenging task. 
Besides the complex architecture of many buildings, a high accuracy needs to be achieved, especially for buildings with many small separated areas like shopping malls or office blocks.
In recent years, many different systems were presented to meet those requirements. 
Especially Wi-Fi positioning and pedestrian dead reckoning (PDR) are very popular solutions. 
Approaches based on PDR try to estimate the current position given the previous position and thus require an initial state. 
However, this allows for cumulative errors and leads to an erroneous position estimation within a very short period. 
By incorporating the absolute position information of Wi-Fi this drift can be corrected.
Additional improvements can be achieved by using environmental information about walls and obstacles provided by a floor map.

In most cases, probabilistic methods are used to incorporate those highly different sensor types. 
Here, a probability distribution describes the pedestrian's possible whereabouts and therefore the uncertainty of the system. 
Drawing from a probability distribution and finding an analytical solution for densities is in most cases a difficult task, especially in case of time sequential, non-linear and non-Gaussian models. 
Due to the high complexity of the human movement, we consider indoor localisation as such.
A broad class to obtain numerical results instead are the Monte Carlo methods. 
Here, a set of weighted random samples is used to solve any problem having a probabilistic interpretation. 
By applying the time sequential hidden Markov process of Bayes filtering, one of the most important Monte Carlo techniques results: particle filtering. 
A particle filter updates the state estimation recursively in time with every new incoming measurement using the state transition and state evaluation step. 

Based on this general methodology, many different approaches for estimating a position in indoor environments have been developed. 
All these approaches differ mainly in how the dynamics are modelled in the transition step and how a specific sensor measurement can be used for evaluation. 
For example, recent approaches are using a graph-based structure to consider environmental restrictions (walking through walls) and the characteristics of human movement (walking speed) within the transition model \cite{Ebner-15}. 
The evaluation model is mostly separated into any number of sensor models, each representing the probability for a noisy measurement in regard to the current position. 
For example, a barometer can be used to determine the probability of being on a certain floor \cite{Binghao13-UBI}. 
%Another example that demonstrates the big differences between single approaches is the large number of sensor models using Wi-Fi signal strengths. There are fingerprinting methods, which require an extensive offline calibration phase, signal strength prediction models like the log-distance model or wall-attenuation-factor model and many others \cite{Ville09, Fang09, Ebner:Thesis:2013}.

Despite the many advances made in the last years, nearly all systems suffer from more or less the same problems. 
Of course, every sensor model brings its very own weaknesses. 
Like mentioned before, PDR suffers from an accumulating bias, the signal of Wi-Fi gets attenuated by walls and the barometric pressure is highly affected by weather patterns and humidity \cite{Binghao13-UBI}. 
That is the reason for the use of statistical methods in the first place. Nevertheless, there are even more profound problems regarding the whole position estimation procedure. 

Current transition models, which aim to approximate the movement, are still very restrictive and unable to handle unforeseen events. 
Faulty sensor measurements, like a falsely detected turn, can cause the estimation to lose track.
For example by taking a turn too soon and walking into a room instead of another big hallway \cite{Ebner-15}. 
Due to this, the filter needs some time to recover, which again takes a while because of the restrictive model (e.g. no walking through walls and only realistic walking speed). 
This temporal delay worsens the estimate immensely. 
A solution to recover from such filter divergences faster, is using methods for re-initializing the filtering procedure \cite{Nurminen14-MMF}. 
However, even this can not completely prevent delays. 
Another reason for possible time delays are slow sensor updates. 
For example, most mobile devices restrict the Wi-Fi module to update only every few seconds, to save on battery. 

\begin{figure}[t]
	\centering
	\def\svgwidth{0.9\columnwidth}
  \input{gfx/multimodalpath.eps_tex}
	\caption[An example of the occurrence of a multimodal distribution.]{An example of the occurrence of a multimodal distribution. At time $t-1$ the floor gets separated by a wall and the mode of the distribution (colored circle), representing the current position, splits apart. The most likely position (green line) is estimated somewhere in-between. After a right turn, the distribution slowly starts to recover its unimodality.}
	\label{fig:multimodalPath}
\end{figure}

Further critical problems arise from multimodal distributions. 
Those are caused by multiple possible position estimates. 
Fig. \ref{fig:multimodalPath} illustrates an example where a floor gets separated by a wall.
Due to inaccurate measurements and a PDR approach for evaluating the movement, the distribution splits apart. 
Therefore, the most likely position is somewhere in-between. 
Only after the pedestrian turns right, the distribution is again unimodal, since moving through walls is prohibited. 
As one can imagine, this can lead to serious problems in big indoor environments. 
Such a situation can be improved by incorporating future measurements (e.g. the right turn) or predictive information (e.g. the most likely path) to the filtering procedure \cite{Ebner-16}. 
However, standard filtering methods are not able to use any future information and the possibilities to make a distant forecast are also limited \cite{robotics, Doucet11:ATO, chen2003bayesian}.

One very promising way to deal with these problems is smoothing.


Within this work, we try to address those problems