sections: fix and add a lot of things
This commit is contained in:
parent
2ea9b8d36f
commit
4ea618d8ee
@ -4,7 +4,7 @@
|
|||||||
## Landau PDF
|
## Landau PDF
|
||||||
|
|
||||||
:::: {.columns}
|
:::: {.columns}
|
||||||
::: {.column width=50% .c}
|
::: {.column width=50% align=center}
|
||||||
$$
|
$$
|
||||||
L(x) = \frac{1}{\pi} \int \limits_{0}^{+ \infty}
|
L(x) = \frac{1}{\pi} \int \limits_{0}^{+ \infty}
|
||||||
dt \, e^{-t \ln(t) -xt} \sin (\pi t)
|
dt \, e^{-t \ln(t) -xt} \sin (\pi t)
|
||||||
|
@ -83,6 +83,17 @@ Defined by $F(m) = \frac{1}{2}$ or $m = Q \left( \frac{1}{2} \right)$:
|
|||||||
\text{erf}^{-1} \left( \frac{1}{2} \right) \right]
|
\text{erf}^{-1} \left( \frac{1}{2} \right) \right]
|
||||||
\end{align*}
|
\end{align*}
|
||||||
|
|
||||||
|
\setbeamercovered{}
|
||||||
|
|
||||||
|
\begin{center}
|
||||||
|
\begin{tikzpicture}[overlay]
|
||||||
|
\pause
|
||||||
|
\node [opacity=0.5, xscale=0.55, yscale=0.4 ] at (1.85,1.1) {\includegraphics{images/high.png}};
|
||||||
|
\end{tikzpicture}
|
||||||
|
\end{center}
|
||||||
|
|
||||||
|
\setbeamercovered{transparent}
|
||||||
|
|
||||||
## Moyal mode
|
## Moyal mode
|
||||||
|
|
||||||
Peak of the PDF:
|
Peak of the PDF:
|
||||||
@ -99,6 +110,16 @@ $$
|
|||||||
\partial_x M_{\mu \sigma}(x) = 0 &\thus \mu_M\ex = \mu \\
|
\partial_x M_{\mu \sigma}(x) = 0 &\thus \mu_M\ex = \mu \\
|
||||||
\end{align*}
|
\end{align*}
|
||||||
|
|
||||||
|
\setbeamercovered{}
|
||||||
|
|
||||||
|
\begin{center}
|
||||||
|
\begin{tikzpicture}[overlay]
|
||||||
|
\pause
|
||||||
|
\node [opacity=0.5, xscale=0.18, yscale=0.25 ] at (2.4,1.8) {\includegraphics{images/high.png}};
|
||||||
|
\end{tikzpicture}
|
||||||
|
\end{center}
|
||||||
|
|
||||||
|
\setbeamercovered{transparent}
|
||||||
|
|
||||||
## Moyal FWHM
|
## Moyal FWHM
|
||||||
|
|
||||||
@ -129,3 +150,14 @@ $$
|
|||||||
M_{\mu \sigma}(x)
|
M_{\mu \sigma}(x)
|
||||||
&\thus w_M^{\text{exp}} = \sigma \cdot a \\
|
&\thus w_M^{\text{exp}} = \sigma \cdot a \\
|
||||||
\end{align*}
|
\end{align*}
|
||||||
|
|
||||||
|
\setbeamercovered{}
|
||||||
|
|
||||||
|
\begin{center}
|
||||||
|
\begin{tikzpicture}[overlay]
|
||||||
|
\pause
|
||||||
|
\node [opacity=0.5, xscale=0.2, yscale=0.25 ] at (1.9,1.9) {\includegraphics{images/high.png}};
|
||||||
|
\end{tikzpicture}
|
||||||
|
\end{center}
|
||||||
|
|
||||||
|
\setbeamercovered{transparent}
|
||||||
|
@ -133,6 +133,8 @@ $$
|
|||||||
|
|
||||||
Kernel Density Estimation
|
Kernel Density Estimation
|
||||||
|
|
||||||
|
:::: {.columns}
|
||||||
|
::: {.column width=50% .c}
|
||||||
- empirical PDF construction:
|
- empirical PDF construction:
|
||||||
|
|
||||||
$$
|
$$
|
||||||
@ -141,6 +143,46 @@ $$
|
|||||||
$$
|
$$
|
||||||
|
|
||||||
The parameter $\varepsilon$ controls the strength of the smoothing
|
The parameter $\varepsilon$ controls the strength of the smoothing
|
||||||
|
:::
|
||||||
|
|
||||||
|
::: {.column width=50%}
|
||||||
|
\setbeamercovered{}
|
||||||
|
\begin{center}
|
||||||
|
\begin{tikzpicture}
|
||||||
|
% points
|
||||||
|
\draw [blue, fill=blue] (-2,-0.1) rectangle (-1.8,0.1);
|
||||||
|
\draw [blue, fill=blue] (-0.1,-0.1) rectangle (0.1,0.1);
|
||||||
|
\draw [blue, fill=blue] (1.3,-0.1) rectangle (1.5,0.1);
|
||||||
|
\draw [blue, fill=blue] (0.7,-0.1) rectangle (0.9,0.1);
|
||||||
|
\pause
|
||||||
|
% lines
|
||||||
|
\draw [cyclamen, dashed] (-1.9,0.1) -- (-1.9,1);
|
||||||
|
\draw [cyclamen, dashed] (0,0.1) -- (0,1);
|
||||||
|
\draw [cyclamen, dashed] (1.4,0.1) -- (1.4,1);
|
||||||
|
\draw [cyclamen, dashed] (0.8,0.1) -- (0.8,1);
|
||||||
|
% Gaussians
|
||||||
|
\draw[domain=-3.4:-0.4, smooth, variable=\x, cyclamen, very thick]
|
||||||
|
plot ({\x}, {exp(-(\x + 1.9)*(\x + 1.9)) + 0.1});
|
||||||
|
\draw[domain=-1.5:1.5, smooth, variable=\x, cyclamen, very thick]
|
||||||
|
plot ({\x}, {exp(-\x*\x + 0.1});
|
||||||
|
\draw[domain=-0.1:2.9, smooth, variable=\x, cyclamen, very thick]
|
||||||
|
plot ({\x}, {exp(-(\x - 1.4)*(\x - 1.4)) + 0.1});
|
||||||
|
\draw[domain=-0.7:2.3, smooth, variable=\x, cyclamen, very thick]
|
||||||
|
plot ({\x}, {exp(-(\x - 0.8)*(\x - 0.8)) + 0.1});
|
||||||
|
\pause
|
||||||
|
% sum
|
||||||
|
\draw [fill=white, white, opacity=0.5] (-3.5,0.1) rectangle (3,1.3);
|
||||||
|
\draw[domain=-3.4:3.4, smooth, variable=\x, blue, very thick]
|
||||||
|
plot ({\x}, {exp(-(\x + 1.9)*(\x + 1.9)) +
|
||||||
|
exp(-\x*\x) +
|
||||||
|
exp(-(\x - 1.4)*(\x - 1.4)) +
|
||||||
|
exp(-(\x - 0.8)*(\x - 0.8)) + 0.1});
|
||||||
|
|
||||||
|
\end{tikzpicture}
|
||||||
|
\end{center}
|
||||||
|
\setbeamercovered{transparent}
|
||||||
|
:::
|
||||||
|
::::
|
||||||
|
|
||||||
|
|
||||||
## Sample FWHM
|
## Sample FWHM
|
||||||
@ -148,9 +190,6 @@ The parameter $\varepsilon$ controls the strength of the smoothing
|
|||||||
Silverman's rule of thumb:
|
Silverman's rule of thumb:
|
||||||
|
|
||||||
$$
|
$$
|
||||||
f_\varepsilon(x) = \frac{1}{N\varepsilon} \sum_{i = 1}^N
|
|
||||||
G \left( \frac{x-x_i}{\varepsilon} \right)
|
|
||||||
\with
|
|
||||||
\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)}
|
||||||
$$
|
$$
|
||||||
|
Loading…
Reference in New Issue
Block a user