Added missing cites

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}

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

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.
+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}.
-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.

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},
@@ -2966,3 +3040,36 @@ year = {2003}
     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}
+}