From e02e4a9557b5025e35440bd6fb0aeccffbef9f7d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Gi=C3=B9=20Marcer?= Date: Fri, 12 Jun 2020 22:18:11 +0200 Subject: [PATCH] sec-4: change order of images MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Aleso add a ε=0.5 plot in the KDE slide. --- slides/sections/4.md | 219 +++++++++++++++++++++---------------------- 1 file changed, 109 insertions(+), 110 deletions(-) diff --git a/slides/sections/4.md b/slides/sections/4.md index 119e714..198db64 100644 --- a/slides/sections/4.md +++ b/slides/sections/4.md @@ -20,48 +20,46 @@ How to estimate sample median, mode and FWHM? \Begin{block}{Algorithm} ::: incremental - 1. Sort sample in ascending order - 2. + 1. Sample points + 2. Sort sample in ascending order + 3. Take middle element if odd Take average of two middle elements if even ::: - \End{block} -. . . - \setbeamercovered{} \begin{center} \begin{tikzpicture}[remember picture, >=Stealth] + % place holder + \draw [ultra thick, white] (-0.35,0.7) -- (-0.35,-0.7); % line - \draw [line width=3, ->, cyclamen] (-5,0) -- (5,0); - \node [right] at (5,0) {$x$}; + \draw <1-> [line width=3, ->, cyclamen] (-5,0) -- (5,0); + \node <1-> [right] at (5,0) {$x$}; % points - \draw [yellow!50!black, fill=yellow] (-4.6,-0.1) rectangle (-4.8,0.1); - \draw [yellow!50!black, fill=yellow] (-4,-0.1) rectangle (-4.2,0.1); - \draw [yellow!50!black, fill=yellow] (-3.3,-0.1) rectangle (-3.5,0.1); - \draw [yellow!50!black, fill=yellow] (-2.3,-0.1) rectangle (-2.5,0.1); - \draw [yellow!50!black, fill=yellow] (-0.6,-0.1) rectangle (-0.8,0.1); - \draw [yellow!50!black, fill=yellow] (-0.1,-0.1) rectangle (0.1,0.1); - \draw [yellow!50!black, fill=yellow] (1.1,-0.1) rectangle (1.3,0.1); - \draw [yellow!50!black, fill=yellow] (2,-0.1) rectangle (2.2,0.1); - \draw [yellow!50!black, fill=yellow] (2.7,-0.1) rectangle (2.9,0.1); - \draw [yellow!50!black, fill=yellow] (4,-0.1) rectangle (4.2,0.1); - \pause + \draw <1-> [yellow!50!black, fill=yellow] (-4.6,-0.1) rectangle (-4.8,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (-4,-0.1) rectangle (-4.2,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (-3.3,-0.1) rectangle (-3.5,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (-2.3,-0.1) rectangle (-2.5,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (-0.6,-0.1) rectangle (-0.8,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (-0.1,-0.1) rectangle (0.1,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (1.1,-0.1) rectangle (1.3,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (2,-0.1) rectangle (2.2,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (2.7,-0.1) rectangle (2.9,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (4,-0.1) rectangle (4.2,0.1); % nodes - \node [below] at (-4.7,-0.1) {1}; - \node [below] at (-4.1,-0.1) {2}; - \node [below] at (-3.4,-0.1) {3}; - \node [below] at (-2.4,-0.1) {4}; - \node [below] at (-0.7,-0.1) {5}; - \node [below] at ( 0 ,-0.1) {6}; - \node [below] at ( 1.2,-0.1) {7}; - \node [below] at ( 2.1,-0.1) {8}; - \node [below] at ( 2.8,-0.1) {9}; - \node [below] at ( 4.1,-0.1) {10}; - \pause - \draw [ultra thick] (-0.35,0.7) -- (-0.35,-0.7); + \node <2-> [below] at (-4.7,-0.1) {1}; + \node <2-> [below] at (-4.1,-0.1) {2}; + \node <2-> [below] at (-3.4,-0.1) {3}; + \node <2-> [below] at (-2.4,-0.1) {4}; + \node <2-> [below] at (-0.7,-0.1) {5}; + \node <2-> [below] at ( 0 ,-0.1) {6}; + \node <2-> [below] at ( 1.2,-0.1) {7}; + \node <2-> [below] at ( 2.1,-0.1) {8}; + \node <2-> [below] at ( 2.8,-0.1) {9}; + \node <2-> [below] at ( 4.1,-0.1) {10}; + \draw <3-> [ultra thick] (-0.35,0.7) -- (-0.35,-0.7); \end{tikzpicture} \end{center} \setbeamercovered{transparent} @@ -73,67 +71,42 @@ How to estimate sample median, mode and FWHM? \Begin{block}{Algorithm} ::: incremental - 1. Find the smallest interval containing half points - 2. Repeat on the new interval (called modal) - 3. If the interval has less than two points, take average + 1. Sample points + 2. Find the smallest interval containing half points + 3. Repeat on the new interval (iterative) + 4. If less than four points, take average of the closest two ::: \End{block} -. . . - \centering \setbeamercovered{} \begin{tikzpicture}[remember picture, >=Stealth] % line - \draw [line width=3, ->, cyclamen] (-5,0) -- (5,0); - \node [right] at (5,0) {$x$}; + \draw <1-> [line width=3, ->, cyclamen] (-5,0) -- (5,0); + \node <1-> [right] at (5,0) {$x$}; % points - \draw [yellow!50!black, fill=yellow] (-4.6,-0.1) rectangle (-4.8,0.1); - \draw [yellow!50!black, fill=yellow] (-4,-0.1) rectangle (-4.2,0.1); - \draw [yellow!50!black, fill=yellow] (-3.3,-0.1) rectangle (-3.5,0.1); - \draw [yellow!50!black, fill=yellow] (-2.3,-0.1) rectangle (-2.5,0.1); - \draw [yellow!50!black, fill=yellow] (-0.6,-0.1) rectangle (-0.8,0.1); - \draw [yellow!50!black, fill=yellow] (-0.1,-0.1) rectangle (0.1,0.1); - \draw [yellow!50!black, fill=yellow] (1.1,-0.1) rectangle (1.3,0.1); - \draw [yellow!50!black, fill=yellow] (2,-0.1) rectangle (2.2,0.1); - \draw [yellow!50!black, fill=yellow] (2.7,-0.1) rectangle (2.9,0.1); - \draw [yellow!50!black, fill=yellow] (4,-0.1) rectangle (4.2,0.1); - % future nodes - \node at (-1,-0.3) (1a) {}; - \node at (3.1,0.3) (1b) {}; - \node at (0.9,-0.3) (2a) {}; - \node at (1.8,-0.3) (3a) {}; - % result nodes - \node at (2.45,-0.7) (f1) {}; - \node at (2.45,0.7) (f2) {}; -\end{tikzpicture} - -. . . - -\begin{tikzpicture}[remember picture, overlay] - % region - \draw [gray, fill=gray, opacity=0.5] (1a) rectangle (1b); -\end{tikzpicture} - -. . . - -\begin{tikzpicture}[remember picture, overlay] - % region - \draw [gray, fill=gray, opacity=0.6] (2a) rectangle (1b); -\end{tikzpicture} - -. . . - -\begin{tikzpicture}[remember picture, overlay] - % region - \draw [gray, fill=gray, opacity=0.7] (3a) rectangle (1b); -\end{tikzpicture} - -. . . - -\begin{tikzpicture}[remember picture, overlay] - % region - \draw [ultra thick] (f1) -- (f2); + \draw <1-> [yellow!50!black, fill=yellow] (-4.6,-0.1) rectangle (-4.8,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (-4,-0.1) rectangle (-4.2,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (-3.3,-0.1) rectangle (-3.5,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (-2.3,-0.1) rectangle (-2.5,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (-0.6,-0.1) rectangle (-0.8,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (-0.1,-0.1) rectangle (0.1,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (1.1,-0.1) rectangle (1.3,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (2,-0.1) rectangle (2.2,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (2.7,-0.1) rectangle (2.9,0.1); + \draw <1-> [yellow!50!black, fill=yellow] (4,-0.1) rectangle (4.2,0.1); + % nodes + \node <1-> at (-1,-0.3) (1a) {}; + \node <1-> at (3.1,0.3) (1b) {}; + \node <1-> at (0.9,-0.3) (2a) {}; + \node <1-> at (1.8,-0.3) (3a) {}; + \node <1-> at (2.45,-0.7) (f1) {}; + \node <1-> at (2.45,0.7) (f2) {}; + % algorithm + \draw <2-> [gray, fill=gray, opacity=0.5] (1a) rectangle (1b); + \draw <3-> [gray, fill=gray, opacity=0.6] (2a) rectangle (1b); + \draw <4-> [cyclamen, thick] (3a) rectangle (1b); + \draw <5-> [ultra thick] (f1) -- (f2); \end{tikzpicture} @@ -157,41 +130,67 @@ $$ G \left( \frac{x-x_i}{\varepsilon} \right) $$ - - The parameter $\varepsilon$ controls the strength of the smoothing + - The parameter $\varepsilon$ controls the + sharpness of the empirical PDF ::: ::: {.column width=50%} \setbeamercovered{} \begin{center} \begin{tikzpicture} + % placeholder + \draw [white] (-2.7,-0.2) rectangle (3,3.3); + % bandwidth 1 + \node <4,5> [left] at (2.9,3) {$\epsilon = 1$}; % points - \draw [yellow!50!black, fill=yellow] (-2,-0.1) rectangle (-1.8,0.1); - \draw [yellow!50!black, fill=yellow] (-0.1,-0.1) rectangle (0.1,0.1); - \draw [yellow!50!black, fill=yellow] (1.3,-0.1) rectangle (1.5,0.1); - \draw [yellow!50!black, fill=yellow] (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 + \draw <3-> [yellow!50!black, fill=yellow] (-1.2,-0.2) rectangle (-1,0); + \draw <3-> [yellow!50!black, fill=yellow] (-0.1,-0.2) rectangle (0.1,0); + \draw <3-> [yellow!50!black, fill=yellow] (0.7,-0.2) rectangle (0.9,0); + \draw <3-> [yellow!50!black, fill=yellow] (1.3,-0.2) rectangle (1.5,0); + % lines 1 + \draw <4,5> [cyclamen, dashed] (-1.1,0.1) -- (-1.1,1); + \draw <4,5> [cyclamen, dashed] (0,0.1) -- (0,1); + \draw <4,5> [cyclamen, dashed] (1.4,0.1) -- (1.4,1); + \draw <4,5> [cyclamen, dashed] (0.8,0.1) -- (0.8,1); + % Gaussians 1 + \draw <4,5> [domain=-2.6:0.4, smooth, variable=\x, cyclamen, very thick] + plot ({\x}, {exp(-(\x + 1.1)*(\x + 1.1)) + 0.1}); + \draw <4,5> [domain=-1.5:1.5, smooth, variable=\x, cyclamen, very thick] + plot ({\x}, {exp(-\x*\x + 0.1}); + \draw <4,5> [domain=-0.7:2.3, smooth, variable=\x, cyclamen, very thick] + plot ({\x}, {exp(-(\x - 0.8)*(\x - 0.8)) + 0.1}); + \draw <4,5> [domain=-0.1:2.9, smooth, variable=\x, cyclamen, very thick] + plot ({\x}, {exp(-(\x - 1.4)*(\x - 1.4)) + 0.1}); + % sum 1 + \draw <5> [fill=white, white, opacity=0.5] (-2.7,0.1) rectangle (3,2.7); + \draw <5> [domain=-2.6:2.9, smooth, variable=\x, yellow, very thick] + plot ({\x}, {exp(-(\x + 1.1)*(\x + 1.1)) + + exp(-\x*\x) + + exp(-(\x - 1.4)*(\x - 1.4)) + + exp(-(\x - 0.8)*(\x - 0.8)) + 0.1}); + % bandwidth 2 + \node <6> [left] at (2.9,3) {$\epsilon = 0.5$}; + % lines 2 + \draw <6> [cyclamen, dashed] (-1.1,0.1) -- (-1.1,2); + \draw <6> [cyclamen, dashed] (0,0.1) -- (0,2); + \draw <6> [cyclamen, dashed] (1.4,0.1) -- (1.4,2); + \draw <6> [cyclamen, dashed] (0.8,0.1) -- (0.8,2); + % Gaussians 2 + \draw <6> [domain=-2.6:0.4, smooth, variable=\x, cyclamen, very thick] + plot ({\x}, {exp(-(\x + 1.1)*(\x + 1.1)/0.25)/0.5 + 0.1}); + \draw <6> [domain=-1.5:1.5, smooth, variable=\x, cyclamen, very thick] + plot ({\x}, {exp(-\x*\x/0.25)/0.5 + 0.1}); + \draw <6> [domain=-0.7:2.3, smooth, variable=\x, cyclamen, very thick] + plot ({\x}, {exp(-(\x - 0.8)*(\x - 0.8)/0.25)/0.5 + 0.1}); + \draw <6> [domain=-0.1:2.9, smooth, variable=\x, cyclamen, very thick] + plot ({\x}, {exp(-(\x - 1.4)*(\x - 1.4)/0.25)/0.5 + 0.1}); % 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, yellow, 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}); + \draw <6> [fill=white, white, opacity=0.5] (-2.7,0.1) rectangle (3,2.7); + \draw <6> [domain=-2.6:2.9, smooth, variable=\x, yellow, very thick] + plot ({\x}, {exp(-(\x + 1.1)*(\x + 1.1)/0.25)/0.5 + + exp(-\x*\x/0.25)/0.5 + + exp(-(\x - 1.4)*(\x - 1.4)/0.25)/0.5 + + exp(-(\x - 0.8)*(\x - 0.8)/0.25)/0.5 + 0.1}); \end{tikzpicture} \end{center} \setbeamercovered{transparent}