From 3c39538e92c8bab30680a7757d6925dfa2863a62 Mon Sep 17 00:00:00 2001
From: MBulli <markusbullmann@gmail.com>
Date: Sat, 24 Feb 2018 13:09:22 +0100
Subject: [PATCH] Fixed multivariate notation

---
 tex/chapters/kde.tex          | 2 +-
 tex/chapters/multivariate.tex | 8 ++++----
 2 files changed, 5 insertions(+), 5 deletions(-)

diff --git a/tex/chapters/kde.tex b/tex/chapters/kde.tex
index b26e5a8..ae2ecf1 100644
--- a/tex/chapters/kde.tex
+++ b/tex/chapters/kde.tex
@@ -13,7 +13,7 @@
 %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 $X=\{X_1, \dots, X_n\}$, where $X$ has the density function $f$ and let $w_1, \dots w_n$ be associated weights.
+Given a univariate random sample set $X=\{X_1, \dots, X_n\}$, 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}
diff --git a/tex/chapters/multivariate.tex b/tex/chapters/multivariate.tex
index bdab24c..23447f4 100644
--- a/tex/chapters/multivariate.tex
+++ b/tex/chapters/multivariate.tex
@@ -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 $\bm{h}=(h_1, \dots, h_d)$.