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Added weights to KDE and BKDE

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MBulli
2018-02-18 21:11:42 +01:00
parent 62fceaa815
commit a46fe242e8
2 changed files with 18 additions and 18 deletions
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32
tex/chapters/kde.tex
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@@ -13,12 +13,14 @@
%In contrast,
The KDE is often the preferred tool to estimate a density function from discrete data samples because of its ability to produce a continuous estimate and its flexibility.
%
Given a univariate random sample $X=\{X_1, X_2, \dots, X_n\}$, the kernel estimator $\hat{f}$ which defines the estimate at the point $x$ is given as
Given a univariate random sample $X=\{X_1, \dots, X_n\}$, where $X$ has the density function $f$ and let $w_1, \dots w_n$ be associated weights.
The kernel estimator $\hat{f}$ which estimates $f$ at the point $x$ is given as
\begin{equation}
\label{eq:kde}
\hat{f}(x) = \frac{1}{nh} \sum_{i=1}^{n} K \left(\frac{x-X_i}{h}\right)
\hat{f}(x) = \frac{1}{W} \sum_{i=1}^{n} \frac{w_i}{h} K \left(\frac{x-X_i}{h}\right)
\end{equation}
where $K$ is a kernel function such that $\int K(u) \dop{u} = 1$ and $h\in\R^+$ is an arbitrary smoothing parameter called bandwidth \cite[138]{scott2015}.
where $W=\sum_{i=1}^{n}w_i$ and $h\in\R^+$ is an arbitrary smoothing parameter called bandwidth.
$K$ is a kernel function such that $\int K(u) \dop{u} = 1$ \cite[138]{scott2015}.
In general any kernel can be used, however the general advice is to chose a symmetric and low-order polynomial kernel.
Thus, several popular kernel functions are used in practice, like the Uniform, Gaussian, Epanechnikov, or Silverman kernel \cite[152.]{scott2015}.
@@ -54,40 +56,38 @@ In general, reducing the size of the sample negatively affects the accuracy of t
Still, the sample size is a suitable parameter to speedup the computation.
Since each single sample is combined with its adjacent samples into bins, the BKDE approximates the KDE.
Each bin represents the \qq{weight} of the sample set at a given point of a equidistant grid with spacing $\delta$.
A binning rule distributes a sample $x$ among the grid points $g_j=j\delta$ indexed by $j\in\Z$.
Each bin represents the \qq{count} of the sample set at a given point of a equidistant grid with spacing $\delta$.
A binning rule distributes a sample $x$ among the grid points $g_j=j\delta$, indexed by $j\in\Z$.
% and can be represented as a set of functions $\{ w_j(x,\delta), j\in\Z \}$.
Computation requires a finite grid on the interval $[a,b]$ containing the data, thus the number of grid points is $G=(b-a)/\delta+1$.
Given a binning rule $w_j$ the BKDE $\tilde{f}$ of a density $f$ computed pointwise at the grid point $g_x$ is given as
Given a binning rule $b_j$ the BKDE $\tilde{f}$ of a density $f$ computed pointwise at the grid point $g_x$ is given as
\begin{equation}
\label{eq:binKde}
\tilde{f}(g_x) = \frac{1}{nh} \sum_{j=1}^{G} C_j K \left(\frac{g_x-g_j}{h}\right)
\tilde{f}(g_x) = \frac{1}{W} \sum_{j=1}^{G} \frac{C_j}{h} K \left(\frac{g_x-g_j}{h}\right)
\end{equation}
where $G$ is the number of grid points and
\begin{equation}
\label{eq:gridCnts}
C_j=\sum_{i=1}^{n} w_j(x_i,\delta)
C_j=\sum_{i=1}^{n} b_j(x_i,\delta)
\end{equation}
is the count at grid point $g_j$ \cite{hall1996accuracy}.
is the count at grid point $g_j$, such that $\sum_{j=1}^{G} C_j = W$ \cite{hall1996accuracy}.
\commentByMarkus{Wording: Count vs. Weight}
In theory, any function which assigns weights to grid points is a valid binning rule.
In theory, any function which determines the count at grid points is a valid binning rule.
However, for many applications it is recommend to use the simple binning rule
\begin{align}
\label{eq:simpleBinning}
w_j(x,\delta) &=
b_j(x,\delta) &=
\begin{cases}
1 & \text{if } x \in ((j-\frac{1}{2})\delta, (j-\frac{1}{2})\delta ] \\
w_j & \text{if } x \in ((j-\frac{1}{2})\delta, (j-\frac{1}{2})\delta ] \\
0 & \text{else}
\end{cases}
\end{align}
or the common linear binning rule which divides the sample into two fractional weights shared by the nearest grid points
\begin{align}
w_j(x,\delta) &=
b_j(x,\delta) &=
\begin{cases}
1-|\delta^{-1}x-j| & \text{if } |\delta^{-1}x-j|\le1 \\
w_j(1-|\delta^{-1}x-j|) & \text{if } |\delta^{-1}x-j|\le1 \\
0 & \text{else.}
\end{cases}
\end{align}