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