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Minor changes to wording

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Markus Bullmann
2018-05-08 11:08:48 +02:00
parent 35e1f4e0b3
commit 9f358d69c9
5 changed files with 17 additions and 19 deletions
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tex/chapters/introduction.tex
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@@ -14,7 +14,7 @@ While such methods are computational fast and suitable most of the time, it is n
Especially time-sequential, non-linear and non-Gaussian state spaces, depending upon a high number of different sensor types, frequently suffer from a multimodal representation of the posterior distribution.
As a result, those techniques are not able to provide an accurate statement about the most probable state, rather causing misleading or false outcomes.
For example, in a localization scenario where a bimodal distribution represents the current posterior, a reliable position estimation is more likely to be at one of the modes, instead of somewhere in-between, like provided by a simple weighted-average estimation.
Additionally, in most practical scenarios the sample size and therefore the resolution is limited, causing the variance of the sample based estimate to be high \cite{Verma2003}.
Additionally, in most practical scenarios the sample size, and hence the resolution is limited, causing the variance of the sample based estimate to be high \cite{Verma2003}.
It is obvious, that a computation of the full posterior could solve the above, but finding such an analytical solution is an intractable problem, which is the reason for applying a sample representation in the first place.
Another promising way is to recover the probability density function from the sample set itself, by using a non-parametric estimator like a kernel density estimation (KDE).
@@ -25,10 +25,10 @@ Nevertheless, the availability of a fast processing density estimate might impro
%Therefore, this paper presents a novel approximation approach for rapid computation of the KDE.
%In this paper, a well known approximation of the Gaussian filter is used to speed up the computation of the KDE.
In this paper, a novel approximation approach for rapid computation of the KDE is presented.
The basic idea is to interpret the estimation problem as a filtering operation.
The basic idea is to interpret the density estimation problem as a filtering operation.
We show that computing the KDE with a Gaussian kernel on binned data is equal to applying a Gaussian filter on the binned data.
This allows us to use a well known approximation scheme for Gaussian filters: the box filter.
By the central limit theorem, multiple recursion of a box filter yields an approximative Gaussian filter \cite{kovesi2010fast}.
This allows us to use a well known approximation scheme based on multiple recursions of a box filter, which yields an approximative Gaussian filter given by the central limit theorem \cite{kovesi2010fast}.
This process converges quite fast to a reasonable close approximation of the ideal Gaussian.
In addition, a box filter can be computed extremely fast by a computer, due to its intrinsic simplicity.