diff --git a/tex/chapters/experiments.tex b/tex/chapters/experiments.tex index d8712cd..d9e52c7 100644 --- a/tex/chapters/experiments.tex +++ b/tex/chapters/experiments.tex @@ -3,7 +3,7 @@ We now empirically evaluate the accuracy of our method and compare its runtime p To conclude our findings we present a real world example from a indoor localisation system. All tests are performed on a Intel Core \mbox{i5-7600K} CPU with a frequency of $4.5 \text{GHz}$, which supports the AVX2 instruction set, hence 256-bit wide SIMD registers are available. -We compare our C++ implementation of the box filter based KDE to the KernSmooth R package and the \qq{FastKDE} implementation \cite{fastKDE}. +We compare our C++ implementation of the box filter based KDE to the KernSmooth R package and the \qq{FastKDE} implementation \cite{oBrien2016fast}. The KernSmooth packages provides a FFT-based BKDE implementation based on optimized C functions at its core. \subsection{Error} diff --git a/tex/chapters/mvg.tex b/tex/chapters/mvg.tex index 5d9ea51..6e72bb1 100644 --- a/tex/chapters/mvg.tex +++ b/tex/chapters/mvg.tex @@ -55,7 +55,7 @@ Furthermore, only one addition and subtraction is required to calculate a single Given a fast approximation scheme, it is necessary to construct a box filter analogous to a given Gaussian filter. -As given in \eqref{eq:gausfx} the solely parameter of the Gaussian kernel is the standard deviation $\sigma$. +The solely parameter of the Gaussian kernel is the standard deviation $\sigma$. In contrast, the moving average filter is parametrized by its width $w$. Therefore, in order to approximate the Gaussian filter of a given $\sigma$ a corresponding value of $w$ must be found. Given $n$ iterations of moving average filters with identical widths the ideal width $\wideal$, as suggested by Wells~\cite{wells1986efficient}, is diff --git a/tex/chapters/relatedwork.tex b/tex/chapters/relatedwork.tex index d530075..6b6e9d1 100644 --- a/tex/chapters/relatedwork.tex +++ b/tex/chapters/relatedwork.tex @@ -11,7 +11,7 @@ It was subject to extensive research and its theoretical properties are well und A comprehensive reference is given by Scott \cite{scott2015}. 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. -An extensive overview regarding the topic of automatic bandwith selection is given by \cite{heidenreich2013bandwith}. +An extensive overview regarding the topic of automatic bandwith selection is given by \cite{heidenreich2013bandwidth}. %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. @@ -33,10 +33,9 @@ The term fast Gauss transform was coined by Greengard \cite{greengard1991fast} w % However, the complexity grows exponentially with dimension. \cite{Improved Fast Gauss Transform and Efficient Kernel Density Estimation} % 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. +Recent methods based on the \qq{self-consistent} KDE proposed by Bernacchia and Pigolotti \cite{bernacchia2011self} allow to obtain an estimate without any assumptions, i.e. the kernel and bandwidth are both derived during the estimation. 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 fast Fourier transform (FFT) algorithm to efficiently transform the data into Fourier space. -Therefore, the data is not required to be on a grid. +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 \cite{oBrien2016fast}. % binning => FFT In general, it is desirable to omit a grid, as the data points do not necessary fall onto equally spaced points. diff --git a/tex/egbib.bib b/tex/egbib.bib index 24a1016..659b164 100644 --- a/tex/egbib.bib +++ b/tex/egbib.bib @@ -2899,6 +2899,15 @@ year = {2003} publisher={IEEE} } +@inproceedings{gwosdek2011theoretical, + title={Theoretical foundations of gaussian convolution by extended box filtering}, + author={Gwosdek, Pascal and Grewenig, Sven and Bruhn, Andr{\'e}s and Weickert, Joachim}, + booktitle={International Conference on Scale Space and Variational Methods in Computer Vision}, + pages={447--458}, + year={2011}, + publisher={Springer} +} + @book{turlach1993bandwidth, title={Bandwidth selection in kernel density estimation: A review}, author={Turlach, Berwin A.}, @@ -2958,6 +2967,71 @@ year = {2003} publisher={Taylor \& Francis Group} } +@article{wand1994fast, + title={Fast computation of multivariate kernel estimators}, + author={Wand, M. P.}, + journal={Journal of Computational and Graphical Statistics}, + volume={3}, + number={4}, + pages={433--445}, + year={1994}, + publisher={Taylor \& Francis Group} +} + +@article{hall1996accuracy, + title={On the accuracy of binned kernel density estimators}, + author={Hall, Peter and Wand, Matt P.}, + journal={Journal of Multivariate Analysis}, + volume={56}, + number={2}, + pages={165--184}, + year={1996}, + publisher={Elsevier} +} + +@article{holmstrom2000accuracy, + title={The accuracy and the computational complexity of a multivariate binned kernel density estimator}, + author={Holmstr{\"o}m, Lasse}, + journal={Journal of Multivariate Analysis}, + volume={72}, + number={2}, + pages={264--309}, + year={2000}, + publisher={Elsevier} +} + +@article{silverman1982algorithm, + title={Algorithm AS 176: Kernel density estimation using the fast Fourier transform}, + author={Silverman, BW}, + journal={Journal of the Royal Statistical Society. Series C (Applied Statistics)}, + volume={31}, + number={1}, + pages={93--99}, + year={1982}, + publisher={JSTOR} +} + +@article{bernacchia2011self, + title={Self-consistent method for density estimation}, + author={Bernacchia, Alberto and Pigolotti, Simone}, + journal={Journal of the Royal Statistical Society: Series B (Statistical Methodology)}, + volume={73}, + number={3}, + pages={407--422}, + year={2011}, + publisher={Wiley Online Library} +} + +@article{oBrien2016fast, + title={A fast and objective multidimensional kernel density estimation method: fastKDE}, + author={O’Brien, Travis A and Kashinath, Karthik and Cavanaugh, Nicholas R and Collins, William D and O’Brien, John P}, + journal={Computational Statistics \& Data Analysis}, + volume={101}, + pages={148--160}, + year={2016}, + publisher={Elsevier} +} + @book{dspGuide1997, Author = {Steven W. Smith}, Title = {The Scientist and Engineer's Guide to Digital Signal Processing}, @@ -2965,4 +3039,37 @@ year = {2003} Year = {1999}, Isbn = {978-0-9660176-7-0}, Publisher = {California Technical Pub} +} + +@article{rosenblatt1956remarks, + title={Remarks on some nonparametric estimates of a density function}, + author={Rosenblatt, Murray}, + journal={The Annals of Mathematical Statistics}, + volume={27}, + number={3}, + pages={832--837}, + year={1956}, + publisher={Institute of Mathematical Statistics} +} + +@article{parzen1962estimation, + title={On estimation of a probability density function and mode}, + author={Parzen, Emanuel}, + journal={The Annals of Mathematical Statistics}, + volume={33}, + number={3}, + pages={1065--1076}, + year={1962}, + publisher={Institute of Mathematical Statistics} +} + +@book{scott2015, + author = {Scott, David W.}, + year = {2015}, + title = {Multivariate Density Estimation: Theory, Practice, and Visualization}, + address = {Hoboken, NJ}, + edition = {2}, + publisher = {Wiley}, + isbn = {978-0-471-69755-8}, + series = {Wiley series in probability and statistics} } \ No newline at end of file