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slides: final touches to section 4

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Michele Guerini Rocco 2020-06-12 16:18:16 +02:00
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1 changed files with 9 additions and 19 deletions
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slides/sections/4.md
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@ -81,8 +81,8 @@ How to estimate sample median, mode and FWHM?
. . . . . .
\centering
\setbeamercovered{} \setbeamercovered{}
\begin{center}
\begin{tikzpicture}[remember picture, >=Stealth] \begin{tikzpicture}[remember picture, >=Stealth]
% line % line
\draw [line width=3, ->, cyclamen] (-5,0) -- (5,0); \draw [line width=3, ->, cyclamen] (-5,0) -- (5,0);
@ -107,43 +107,34 @@ How to estimate sample median, mode and FWHM?
\node at (2.45,-0.7) (f1) {}; \node at (2.45,-0.7) (f1) {};
\node at (2.45,0.7) (f2) {}; \node at (2.45,0.7) (f2) {};
\end{tikzpicture} \end{tikzpicture}
\end{center}
. . . . . .
\begin{center}
\begin{tikzpicture}[remember picture, overlay] \begin{tikzpicture}[remember picture, overlay]
% region % region
\draw [orange, fill=orange, opacity=0.5] (1a) rectangle (1b); \draw [gray, fill=gray, opacity=0.5] (1a) rectangle (1b);
\end{tikzpicture} \end{tikzpicture}
\end{center}
. . . . . .
\begin{center}
\begin{tikzpicture}[remember picture, overlay] \begin{tikzpicture}[remember picture, overlay]
% region % region
\draw [orange, fill=orange, opacity=0.5] (2a) rectangle (1b); \draw [gray, fill=gray, opacity=0.6] (2a) rectangle (1b);
\end{tikzpicture} \end{tikzpicture}
\end{center}
. . . . . .
\begin{center}
\begin{tikzpicture}[remember picture, overlay] \begin{tikzpicture}[remember picture, overlay]
% region % region
\draw [orange, fill=orange, opacity=0.5] (3a) rectangle (1b); \draw [gray, fill=gray, opacity=0.7] (3a) rectangle (1b);
\end{tikzpicture} \end{tikzpicture}
\end{center}
. . . . . .
\begin{center}
\begin{tikzpicture}[remember picture, overlay] \begin{tikzpicture}[remember picture, overlay]
% region % region
\draw [ultra thick] (f1) -- (f2); \draw [ultra thick] (f1) -- (f2);
\end{tikzpicture} \end{tikzpicture}
\end{center}
## Sample FWHM ## Sample FWHM
@ -166,7 +157,7 @@ $$
G \left( \frac{x-x_i}{\varepsilon} \right) G \left( \frac{x-x_i}{\varepsilon} \right)
$$ $$
The parameter $\varepsilon$ controls the strength of the smoothing - The parameter $\varepsilon$ controls the strength of the smoothing
::: :::
::: {.column width=50%} ::: {.column width=50%}
@ -210,22 +201,21 @@ $$
## Sample FWHM ## Sample FWHM
Silverman's rule of thumb [@silver86]: **Silverman's rule of thumb** [@silver86]:
$$ $$
\varepsilon = 0.88 \, S_N \varepsilon = 0.88 \, S_N
\left( \frac{d + 2}{4}N \right)^{-1/(d + 4)} \left( \frac{d + 2}{4}N \right)^{-1/(d + 4)}
$$ $$
where:
with:
- $S_N$ is the sample standard deviation - $S_N$ is the sample standard deviation
- $d$ is number of dimensions ($d = 1$) - $d$ is number of dimensions ($d = 1$)
. . . . . .
Numerical minimization (Brent) for $\quad f_{\varepsilon_{\text{max}}}$ Minimization (Brent) for $\quad f_{\varepsilon_{\text{max}}}$
Numerical root finding (Brent) for $\quad f_{\varepsilon}(x_{\pm}) = Root finding (Brent-Dekker) for $\quad f_{\varepsilon}(x_{\pm}) =
\frac{f_{\varepsilon_{\text{max}}}}{2}$ \frac{f_{\varepsilon_{\text{max}}}}{2}$