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first draft introduction finished

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first draft related work finished
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toni
2018-02-14 18:16:29 +01:00
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tex/chapters/abstract.tex
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This is the abstract stract stract
linear complexity
This will be shown in an a theoritical bases and also realistic ... blah

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tex/chapters/introduction.tex
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Sensor fusion approaches are often based upon probabilistic descriptions like particle filters, using samples to represent the distribution of a dynamical system.
To update the system recursively in time, probabilistic sensor models process the noisy measurements and a state transition function provides the system's dynamics.
Therefore a sample or particle is a representation of one possible system state, e.g. the position of a pedestrian within a building.
In most real world scenarios one is then interested in finding the most probable state within the state space, to provide the "best estimate" of the underlying problem.
In the discrete manner of a sample representation this is often done by providing a single value, also known as sample statistic, to serve as a "best guess".
In most real world scenarios one is then interested in finding the most probable state within the state space, to provide the \qq{best estimate} of the underlying problem.
In the discrete manner of a sample representation this is often done by providing a single value, also known as sample statistic, to serve as a \qq{best guess}.
This value is then calculated by means of simple parametric point estimators, e.g. the weighted-average over all samples, the sample with the highest weight or by assuming other parametric statistics like normal distributions \cite{}.
%da muss es doch noch andere methoden geben... verflixt und zugenäht... aber grundsätzlich ist ein weighted average doch ein point estimator? (https://www.statlect.com/fundamentals-of-statistics/point-estimation)
%Für related work brauchen wir hier definitiv quellen. einige berechnen ja auch https://en.wikipedia.org/wiki/Sample_mean_and_covariance oder nehmen eine gewisse verteilung für die sample menge and und berechnen dort die parameter
@@ -12,34 +12,37 @@ This value is then calculated by means of simple parametric point estimators, e.
While such methods are computational fast and suitable most of the time, it is not uncommon that they fail to recover the state in more complex scenarios.
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.
\commentByMarkus{Vlt. noch drauf eingehen, dass avg. eben in die Mitte geht?}
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}.
It is obvious, that a computation of the full posterior could solve the above, but finding such an analytical solution is an intractable problem, what 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).
With this, it is easy to find the "real" most probable state and thus to avoid the aforementioned drawbacks.
With this, it is easy to find the \qq{real} most probable state and thus to avoid the aforementioned drawbacks.
However, non-parametric estimators tend to consume a large amount of computational time, which renders them unpractical for real time scenarios.
Nevertheless, the availability of a fast processing density estimate might improve the accuracy of today's sensor fusion systems without sacrificing their real time capability.
\commentByToni{Der nachfolgende Satz ist ziemlich wichtig. Find ich aktuell noch nicht gut. Allgemein sollte ihr jetzt noch ca eine viertel Seite ein wenig die Methode grob beschrieben werden.
The basic idea ...
We formalize this ...
Our experiments support our ..
}
%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.
We show that computing the KDE with a Gaussian kernel on pre-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 using the box filter.
Multiple recursion of a box filter yields an approximative Gaussian filter \cite{kovesi2010fast}.
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 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.
While the idea to use several box filter passes to approximate a Gaussian has been around for a long, the application to obtain a fast KDE is new.
Especially in time critical and time sequential sensor fusion scenarios, the here presented approach outperforms other state of the art solutions, due to a fully linear complexity \landau{N} and a negligible overhead, even for small sample sets.
In addition, it requires only a few elementary operations and is highly parallelizable.
%linear complexity and easy parall
%ist immer gleich schnell.
%andere rießen daten, wir weniger daten.
%low complexity, only requires a few elementar operations
%produces nearly no overhead.
% time sequential, fixed computation time, pre binned data!!
% KDE wellknown nonparametic estimation method

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tex/chapters/relatedwork.tex
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Kernel density estimation is well known non-parametric estimator, originally described independently by Rosenblatt \cite{rosenblatt1956remarks} and Parzen \cite{parzen1962estimation}.
It was subject to extensive research and its theoretical properties are well understood.
A comprehensive reference is given by Scott \cite{scott2015}.
Although classified as non-parametric, the KDE has a two free parameters, the kernel function and its bandwidth.
Although classified as non-parametric, the KDE depends on two free parameters, the kernel function and its bandwidth.
The selection of a \qq{good} bandwidth is still an open problem and heavily researched.
However, the automatic selection of the bandwidth is not subject of this work and we refer to the literature \cite{turlach1993bandwidth}.
An extensive overview regarding the topic of automatic bandwith selection is given by \cite{heidenreich2013bandwith}.
%However, the automatic selection of the bandwidth is not subject of this work and we refer to the literature \cite{turlach1993bandwidth}.
The great flexibility of the KDE renders it very useful for many applications.
However, its flexibility comes at the cost of a relative slow computation speed.
The complexity of a naive implementation of the KDE is \landau{NM} evaluations of the kernel function, given $N$ data samples and $M$ points of the estimate.
However, this comes at the cost of a relative slow computation speed.
%
The complexity of a naive implementation of the KDE is \landau{MN}, given by $M$ evaluations of $N$ data samples.
%The complexity of a naive implementation of the KDE is \landau{NM} evaluations of the kernel function, given $N$ data samples and $M$ points of the estimate.
Therefore, a lot of effort was put into reducing the computation time of the KDE.
Various methods have been proposed, which can be clustered based on different techniques.
@@ -32,16 +35,26 @@ The term fast Gauss transform was coined by Greengard \cite{greengard1991fast} w
% FastKDE, passed on ECF and nuFFT
Recent methods based on the \qq{self-consistent} KDE proposed by Bernacchia and Pigolotti allow to obtain an estimate without any assumptions.
They define a Fourier-based filter on the empirical characteristic function of a given dataset.
The computation time was further reduced by \etal{O'Brien} using a non-uniform FFT algorithm to efficiently transform the data into Fourier space.
The computation time was further reduced by \etal{O'Brien} using a non-uniform fast Fourier transform (FFT) algorithm to efficiently transform the data into Fourier space.
Therefore, the data is not required to be on a grid.
% binning => FFT
In general, it is desirable to omit a grid, as the data points do not necessary fall onto equally spaced points.
However, reducing the sample size by distributing the data on a equidistant grid can significantly reduce the computation time, if an approximative KDE is acceptable.
Silverman \cite{silverman1982algorithm} originally suggested to combine adjacent data points into data bins and apply a FFT to quickly compute the estimate.
This approximation scheme was later called binned KDE an was extensively studied \cite{fan1994fast} \cite{wand1994fast} \cite{hall1996accuracy} \cite{holmstrom2000accuracy}.
Silverman \cite{silverman1982algorithm} originally suggested to combine adjacent data points into data bins, which results in a discrete convolution structure of the KDE.
Allowing to efficiently compute the estimate using a FFT algorithm.
This approximation scheme was later called binned KDE (BKDE) and was extensively studied \cite{fan1994fast} \cite{wand1994fast} \cite{hall1996accuracy} \cite{holmstrom2000accuracy}.
The idea to approximate a Gaussian filter using several box filters was first formulated by Wells \cite{wells1986efficient}.
Kovesi \cite{kovesi2010fast} suggested to use two box filter with different widths to increase accuracy maintaining the same complexity.
Kovesi \cite{kovesi2010fast} suggested to use two box filters with different widths to increase accuracy maintaining the same complexity.
To eliminate the approximation error completely \etal{Gwosdek} \cite{gwosdek2011theoretical} proposed a new approach called extended box filter.
This work highlights the discrete convolution structure of the BKDE and elaborates its connection to digital signal processing, especially the Gaussian filter.
Accordingly, this results in an equivalence relation between BKDE and Gaussian filter.
It follows, that the above mentioned box filter approach is also an approximation of the BKDE, resulting in an efficient computation scheme presented within this paper.
This approach has a lower complexity as comparable FFT-based algorithms and adds only a negligible small error, while improving the performance significantly.