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Fixed multivariate notation

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MBulli
2018-02-24 13:09:22 +01:00
parent 96b1840f1b
commit 3c39538e92
2 changed files with 5 additions and 5 deletions
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8
tex/chapters/multivariate.tex
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@@ -9,12 +9,12 @@ In order to estimate a multivariate density using KDE or BKDE a multivariate ker
Multivariate kernel functions can be constructed in various ways, however, a popular way is given by the product kernel.
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.
Given a multivariate random variable $X=(x_1,\dots ,x_d)$ in $d$ dimensions.
The sample $\bm{X}$ is a $n\times d$ matrix defined as \cite[162]{scott2015}.
The multivariate KDE $\hat{f}$ which defines the estimate pointwise at $\bm{x}=(x_1, \dots, x_d)^T$ is given as \cite[162]{scott2015}
Given a multivariate random variable $\bm{X}=(x_1,\dots ,x_d)$ in $d$ dimensions.
The sample set $\mathcal{X}$ is a $n\times d$ matrix \cite[162]{scott2015}.
The multivariate KDE $\hat{f}$ which defines the estimate pointwise at $\bm{u}=(u_1, \dots, u_d)^T$ is given as
\begin{equation}
\label{eq:mvKDE}
\hat{f}(\bm{x}) = \frac{1}{W} \sum_{i=1}^{n} \frac{w_i}{h_1 \dots h_d} \left[ \prod_{j=1}^{d} K\left( \frac{x_j-x_{ij}}{h_j} \right) \right] \text{.}
\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{,}
\end{equation}
where the bandwidth is given as a vector $\bm{h}=(h_1, \dots, h_d)$.