64 lines
5.0 KiB
TeX
64 lines
5.0 KiB
TeX
\subsection{State Estimation}
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\label{sec:estimation}
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% 1/2 bis 3/4 Seite
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% particles describe posterior as samples
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% (MAP) estimate => max particle
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% very jumpy
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% MMSE estimate => weighted average
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% most of the time very good
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% goes out of the window with multi modalities
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% estimation of the pdf could help
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% computational cheap methods are based on a parametric family
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% not neccesserly given in our case
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% non parametric => slow
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% solution boxKDE
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% Problems: larger error compared to WA and bandwidth selection
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Each particle is a realization of one possible system state, here the position of a pedestrian within a building.
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The set of all particles represents the posterior of the system.
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In other words, the particle filter naturally generates a sample based representation of the posterior.
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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}.
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A popular point estimate, which can be directly obtained from the sample set, is the minimum mean squared error (MMSE) estimate.
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In the case of particle filters the MMSE estimate equals to the weighted-average over all samples, \ie{} the sample mean
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\begin{equation}
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\hat{\mStateVec}_t := \frac{1}{W_t} \sum_{i=1}^{N} w^i_t \vec{X}^i_{t} \, \text{,}
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\end{equation}
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%\commentByMarkus{Passt die Notation so?}
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%\commentByFrank{sieht fuer mich auf den ersten blick nach korrektem weighted average aller partikel aus. was stoert dich?}
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where $W_t=\sum_{i=1}^{N}w^i_t$ is the sum of all weights.
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While producing an overall good result in many situations, it fails when the posterior is multimodal.
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In these situations the weighted-average estimate will find the estimate somewhere between the modes.
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\del{Clearly}\add{It is expected that} such a position between modes is extremely unlikely the position of the pedestrian.
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The real position is more likely to be found at the position of one of the modes, but virtually never somewhere between.
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In the case of a multimodal posterior the system should estimate the position based on the highest mode.
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Therefore, the maximum a posteriori (MAP) estimate is a suitable choice for such a situation.
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A straightforward approach is to select the particle with the highest weight.
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However, this is in fact not necessarily a valid MAP estimate, because only the weight of the particle is taken into account.
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In order to compute the true MAP estimate the local density of the particles needs to be considered as well \cite{cappe2007overview}.
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\del{It is obvious,} A computation of the probability density function of the posterior could solve the above, but finding such an analytical solution is \del{clearly} an intractable problem, which is the reason for applying a sample representation in the first place.
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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.
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Given the estimated parameters the most probable state can be obtained from the parameterised density function.
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%In the case of multi-modalities several parametric distributions can be combined into a mixture distribution.
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However, parametric models fail when the assumption does not fit the underlying model.
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For our application assuming a parametric distribution is too limiting as the posterior is changing in a non-predictable way over time.
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%As a result, those techniques are not able to provide an accurate statement about the most probable state, rather causing misleading or false outcomes.
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On the other side a non-parametric approach directly obtains an estimate of the entire density function driven by the structure of the data.
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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.
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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.
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As a matter of fact, the quality of the kernel estimate is primarily determined by the bandwidth. % TODO \cite{scott2015} ?
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For our system we choose the Gaussian kernel in favour of computational efficiency.
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The great flexibility of the KDE comes at the cost of a high computational time, which renders it unpractical for real time scenarios.
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The complexity of a naive implementation of the KDE is \landau{MN}, given by $M$ evaluations and $N$ particles as input size.
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A fast approximation of the KDE can be applied if the data is stored in equidistant bins as suggested by \cite{silverman1982algorithm}.
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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}.
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Our \del{rapid computation} \add{approximation} scheme of the KDE is fast enough to estimate the density of the posterior in each time step.
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This allows us to recover the most prober state from occurring multimodal posterior.
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