diff --git a/tex/chapters/estimation.tex b/tex/chapters/estimation.tex index 23530ec..86ff55f 100644 --- a/tex/chapters/estimation.tex +++ b/tex/chapters/estimation.tex @@ -1,11 +1,62 @@ \subsection{State Estimation} \label{sec:estimation} +% 1/2 bis 3/4 Seite +% particles describe posterior as samples +% (MAP) estimate => max particle +% very jumpy +% MMSE estimate => weighted average +% most of the time very good +% goes out of the window with multi modalities +% estimation of the pdf could help +% computational cheap methods are based on a parametric family +% not neccesserly given in our case +% non parametric => slow +% solution boxKDE +% Problems: larger error compared to WA and bandwidth selection + + +Each particle is a realization of one possible system state, here, the position of a pedestrian within a building. +The set of all particles represents the posterior of the system. +In other words, the particle filter naturally generates a sample based representation of the posterior. +With this representation a point estimator can directly be applied to the sample data to derive a sample statistic severing as a \qq{best guess}. + +A popular point estimate, which can be directly obtained from the sample set, is the minimum mean squared error (MMSE) estimate. +In the case of particle filters the MMSE estimate equals to the weighted-average over all samples, \ie{} the sample mean +% TODO Notation prüfen +\begin{equation} + \hat{\mStateVec}_t := \frac{1}{W_t} \sum_{i=1}^{N} w^i_t \mStateVec^i_t \, \text{,} +\end{equation} +where $W_t=\sum_{i=1}^{N}w^i_t$ is the sum of all weights. +While producing an overall good result in many situations, it fails when the posterior is multimodal. +In these situations the weighted-average estimate will find the estimate somewhere between the modes. +Clearly, such a position between modes could never be the real position of the pedestrian. +The real position is more likely to be found at the position of one of the modes, but never somewhere between. + +In the case of a multimodal posterior the system should estimate the position based on the most highest mode. +Therefore, the maximum a posteriori (MAP) estimate is a suitable choice for such a situation. +A straightforward approach is to select the particle with the highest weight. +However, this is in fact not necessarily a valid MAP estimate, because only the weight of the particle is taken into account. +In order to compute the true MAP estimate the local density of the particles needs to be considered as well \cite{cappe2007overview}. + +It is obvious, that a computation of the probability density function of the posterior could solve the above, but finding such an analytical solution is clearly an intractable problem, which is the reason for applying a sample representation in the first place. +A feasible alternative is to estimate the parameters of a specific parametric model based on the sample set, assuming that the unknown distribution is approximately a parametric distribution or a mixture of parametric distributions, \eg{} Gaussian mixture distributions. +Given the estimated parameters the most probable state can be obtained from the parameterised density function. +%In the case of multi-modalities several parametric distributions can be combined into a mixture distribution. +However, parametric models fail when the assumption does not fit the underlying model. +For our application assuming a parametric distribution is too limiting as the posterior is changing in a non-predictable way over time. +%As a result, those techniques are not able to provide an accurate statement about the most probable state, rather causing misleading or false outcomes. + +On the other side a non-parametric approach directly obtains an estimate of the entire density function driven by the structure of the data. +A classic non-parametric method is the kernel density estimator (KDE), where a kernel function with given bandwidth is placed at each particle to approximate the posterior. +While the kernel estimate inherits all the properties of the kernel, usually it is not of crucial matter if a non-optimal kernel was chosen. +As a matter of fact, the quality of the kernel estimate is primarily determined by the bandwidth. % TODO \cite{scott2015} ? +For our system we choose the Gaussian kernel in favour of computational efficiency. + +The great flexibility of the KDE comes at the cost of a high computational time, which renders it unpractical for real time scenarios. +The complexity of a naive implementation of the KDE is \landau{MN}, given by $M$ evaluations and $N$ particles as input size. +Our rapid computation scheme of the KDE has a linear time complexity and is fast enough to estimate the density of the posterior in each time step \cite{Bullmann-18}. + +\commentByMarkus{To be continued} -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/egbib.bib b/tex/egbib.bib index 64a1c49..88606d6 100644 --- a/tex/egbib.bib +++ b/tex/egbib.bib @@ -2917,8 +2917,7 @@ address = {{Rothenburg, Germany}}, title={{Fast Kernel Density Estimation using Gaussian Filter Approximation}}, year={2018}, IGNOREmonth={October}, - pages={1-8}, - note={under review} + pages={1245-1252} } @book{goldberg89, diff --git a/tex/misc/functions.tex b/tex/misc/functions.tex index 0fa1eb6..1215713 100644 --- a/tex/misc/functions.tex +++ b/tex/misc/functions.tex @@ -107,7 +107,7 @@ \fcolorbox{black}{green}{% \parbox[position]{0.95\textwidth}{% \footnotesize% - {\bf Lukas:} #1% + {\bf Markus:} #1% }% }% } @@ -148,4 +148,5 @@ %\newcommand{\mEdge}{\ensuremath{e}} +\newcommand{\landau}[1]{\ensuremath{ \mathcal{O}\left( #1 \right) }} diff --git a/tex/misc/keywords.tex b/tex/misc/keywords.tex index 9c7f64f..954661b 100644 --- a/tex/misc/keywords.tex +++ b/tex/misc/keywords.tex @@ -1,5 +1,7 @@ \usepackage{xspace} -\newcommand{\eg}{e.g.\@\xspace} +\newcommand{\eg}{e.\,g.\@\xspace} +\newcommand{\ie}{i.\,e.\@\xspace} +\newcommand{\qq} [1]{``#1''} % keyword macros \newcommand{\docIBeacon}{iBeacon}