95 lines
5.4 KiB
TeX
95 lines
5.4 KiB
TeX
\section{Extension to multi-dimensional data}
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So far only univariate sample sets were considered.
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This is due to the fact, that the equations of the KDE \eqref{eq:kde}, BKDE \eqref{eq:binKde}, Gaussian filter \eqref{eq:gausFilt}, and the box filter \eqref{eq:boxFilt} are quite easily extended to multi-dimensional input.
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Each method can be seen as several one-dimensional problems combined to a multi-dimensional result.
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%However, with an increasing number of dimensions the computation time significantly increases.
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In the following, the generalization to multi-dimensional input are briefly outlined.
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In order to estimate a multivariate density using KDE or BKDE a multivariate kernel needs to be used.
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Multivariate kernel functions can be constructed in various ways, however, a popular way is given by the product kernel.
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Such a kernel is constructed by combining several univariate kernels into a product, where each kernel is applied in each dimension with a possibly different bandwidth.
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Given a multivariate random variable $\bm{X}=(x_1,\dots ,x_d)$ in $d$ dimensions.
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The sample set $\mathcal{X}$ is a $n\times d$ matrix \cite[162]{scott2015}.
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The multivariate KDE $\hat{f}$ which defines the estimate pointwise at $\bm{u}=(u_1, \dots, u_d)^T$ is given as
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\begin{equation}
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\label{eq:mvKDE}
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\hat{f}(\bm{u}) = \frac{1}{W} \sum_{i=1}^{n} \frac{w_i}{h_1 \dots h_d} \left[ \prod_{j=1}^{d} K\left( \frac{u_j-x_{ij}}{h_j} \right) \right] \text{,}
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\end{equation}
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where the bandwidth is given as a vector $\bm{h}=(h_1, \dots, h_d)$.
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Note that \eqref{eq:mvKDE} does not include all possible multivariate kernels, such as spherically symmetric kernels, which are based on rotation of a univariate kernel.
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In general a multivariate product and spherically symmetric kernel based on the same univariate kernel will differ.
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The only exception is the Gaussian kernel which is spherical symmetric and has independent marginals. % TODO scott cite?!
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In addition, only smoothing in the direction of the axes are possible.
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If smoothing in other directions is necessary, the computation needs to be done on a prerotated sample set and the estimate needs to be rotated back to fit the original coordinate system \cite{wand1994fast}.
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For the multivariate BKDE, in addition to the kernel function the grid and the binning rules need to be extended to multivariate data.
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Their extensions are rather straightforward, as the grid is easily defined on many dimensions.
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Likewise, the ideas of common and linear binning rule scale with the dimensionality \cite{wand1994fast}.
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In general multi-dimensional filters are multi-dimensional convolution operations.
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However, by utilizing the separability property of convolution a straightforward and a more efficient implementation can be found.
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Convolution is separable if the filter kernel is separable, i.e. it can be split into successive convolutions of several kernels.
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In example, the Gaussian filter is separable, because of $e^{x^2+y^2} = e^{x^2}\cdot e^{y^2}$.
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Likewise digital filters based on such kernels are called separable filters.
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They are easily applied to multi-dimensional signals, because the input signal can be filtered in each dimension individually by an one-dimensional filter.
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% KDE:
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%So far only the univariate case was considered.
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%This is due to the fact, that univariate kernel estimators can quite easily be extended to multivariate distributions.
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%A common approach is to apply an univariate kernel with a possibly different bandwidth in each dimension.
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%These kind of multivariate kernel is called product kernel as the multivariate kernel result is the product of each individual univariate kernel.
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%
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%Given a multivariate random variable $X=(x_1,\dots ,x_d)$ in $d$ dimensions.
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%The sample $\bm{X}$ is a $n\times d$ matrix defined as \cite[162]{scott2015}
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%\begin{equation}
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% \bm{X}=
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% \begin{pmatrix}
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% X_1 \\
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% \vdots \\
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% X_n \\
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% \end{pmatrix}
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% =
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% \begin{pmatrix}
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% x_{11} & \dots & x_{1d} \\
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% \vdots & \ddots & \vdots \\
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% x_{n1} & \dots & x_{nd}
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% \end{pmatrix} \text{.}
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%\end{equation}
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%
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%The multivariate kernel density estimator $\hat{f}$ which defines the estimate pointwise at $\bm{x}=(x_1, \dots, x_d)^T$ is given as \cite[162]{scott2015}
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%\begin{equation}
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% \hat{f}(\bm{x}) = \frac{1}{nh_1 \dots h_d} \sum_{i=1}^{n} \left[ \prod_{j=1}^{d} K\left( \frac{x_j-x_{ij}}{h_j} \right) \right] \text{.}
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%\end{equation}
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%where the bandwidth is given as a vector $\bm{h}=(h_1, \dots, h_d)$.
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% Product kernel allows our method
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% Spherically symmetric kernel not supported, but Gaussian kernel == product & spehrically symmetric
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% smoothing not in the direction of the axes -> rotate data, kde, rotate back
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%Multivariate Gauss-Kernel
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%\begin{equation}
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%K(u)=\frac{1}{(2\pi)^{d/2}} \expp{-\frac{1}{2} \bm{x}^T \bm{x}}
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%\end{equation}
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% Gaus:
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%If the filter kernel is separable, the convolution is also separable i.e. multi-dimensional convolution can be computed as individual one-dimensional convolutions with a one-dimensional kernel.
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%Because of $e^{x^2+y^2} = e^{x^2}\cdot e^{y^2}$ the Gaussian filter is separable and can be easily applied to multi-dimensional signals. \todo{quelle}
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%wie benutzen wir das ganze jetzt? auf was muss ich achten?
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% Am Beispiel 2D Daten
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% Histogram erzeugen (== data binnen)
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% Hierzu wird min/max benötigt
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% Anschließend Filterung per Box Filter über das Histogram
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% - Wenn möglich parallel (SIMD, GPU)
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% - separiert in jeder dim einzeln
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% Maximum aus Filter ergebnis nehmen
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