diff --git a/slides/sections/2.md b/slides/sections/2.md index 6c49058..55d75a6 100644 --- a/slides/sections/2.md +++ b/slides/sections/2.md @@ -4,7 +4,7 @@ ## Landau PDF :::: {.columns} -::: {.column width=50% .c} +::: {.column width=50% align=center} $$ L(x) = \frac{1}{\pi} \int \limits_{0}^{+ \infty} dt \, e^{-t \ln(t) -xt} \sin (\pi t) diff --git a/slides/sections/3.md b/slides/sections/3.md index f6b779c..6710a46 100644 --- a/slides/sections/3.md +++ b/slides/sections/3.md @@ -83,6 +83,17 @@ Defined by $F(m) = \frac{1}{2}$ or $m = Q \left( \frac{1}{2} \right)$: \text{erf}^{-1} \left( \frac{1}{2} \right) \right] \end{align*} +\setbeamercovered{} + +\begin{center} + \begin{tikzpicture}[overlay] + \pause + \node [opacity=0.5, xscale=0.55, yscale=0.4 ] at (1.85,1.1) {\includegraphics{images/high.png}}; + \end{tikzpicture} +\end{center} + +\setbeamercovered{transparent} + ## Moyal mode Peak of the PDF: @@ -99,6 +110,16 @@ $$ \partial_x M_{\mu \sigma}(x) = 0 &\thus \mu_M\ex = \mu \\ \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 @@ -129,3 +150,14 @@ $$ M_{\mu \sigma}(x) &\thus w_M^{\text{exp}} = \sigma \cdot a \\ \end{align*} + +\setbeamercovered{} + +\begin{center} + \begin{tikzpicture}[overlay] + \pause + \node [opacity=0.5, xscale=0.2, yscale=0.25 ] at (1.9,1.9) {\includegraphics{images/high.png}}; + \end{tikzpicture} +\end{center} + +\setbeamercovered{transparent} diff --git a/slides/sections/4.md b/slides/sections/4.md index bd9b247..9dce7ec 100644 --- a/slides/sections/4.md +++ b/slides/sections/4.md @@ -133,14 +133,56 @@ $$ Kernel Density Estimation -- empirical PDF construction: +:::: {.columns} +::: {.column width=50% .c} + - empirical PDF construction: -$$ - f_\varepsilon(x) = \frac{1}{N\varepsilon} \sum_{i = 1}^N - G \left( \frac{x-x_i}{\varepsilon} \right) -$$ + $$ + f_\varepsilon(x) = \frac{1}{N\varepsilon} \sum_{i = 1}^N + G \left( \frac{x-x_i}{\varepsilon} \right) + $$ -The parameter $\varepsilon$ controls the strength of the smoothing + The parameter $\varepsilon$ controls the strength of the smoothing +::: + +::: {.column width=50%} + \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 @@ -148,9 +190,6 @@ The parameter $\varepsilon$ controls the strength of the smoothing Silverman's rule of thumb: $$ - f_\varepsilon(x) = \frac{1}{N\varepsilon} \sum_{i = 1}^N - G \left( \frac{x-x_i}{\varepsilon} \right) - \with \varepsilon = 0.88 \, S_N \left( \frac{d + 2}{4}N \right)^{-1/(d + 4)} $$