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State estimation v2

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Markus Bullmann
2018-07-31 10:07:49 +02:00
parent 86ffdf3901
commit a29aac498c
1 changed files with 8 additions and 7 deletions
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15
tex/chapters/estimation.tex
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@@ -18,19 +18,19 @@
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}.
With this representation a point estimator can directly be applied to the sample data to derive a sample statistic serving 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}
\commentByMarkus{Passt die Notation so?}
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.
Clearly, such a position between modes is extremely unlikely the real position of the pedestrian.
The real position is more likely to be found at the position of one of the modes, but virtually 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.
@@ -54,9 +54,10 @@ For our system we choose the Gaussian kernel in favour of computational efficien
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}
A fast approximation of the KDE can be applied if the data is stored in a equidistant bins.
Computation of the KDE with a Gaussian kernel on the binned data becomes analogous to applying a Gaussian filter, which can be approximated by iterated box filter in \landau{N} \cite{Bullmann-18}.
Our rapid computation scheme of the KDE is fast enough to estimate the density of the posterior in each time step.
This allows us to recover the most prober state from occurring multimodal posterior.