2020-06-10 16:23:33 +02:00
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# Sample statistics
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2020-06-06 19:40:48 +02:00
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2020-06-10 16:23:33 +02:00
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## Sample statistics
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2020-06-07 00:02:20 +02:00
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2020-06-10 16:23:33 +02:00
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How to estimate sample median, mode and FWHM?
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2020-06-07 14:32:03 +02:00
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. . .
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2020-06-10 16:23:33 +02:00
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- \only<3>\strike{Binning data $\hence$ depends wildly on bin-width}
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2020-06-07 14:32:03 +02:00
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. . .
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- Alternative solutions
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2020-06-10 16:23:33 +02:00
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- Robust estimators
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- Kernel density estimation
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2020-06-07 14:32:03 +02:00
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## Sample median
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2020-06-06 19:40:48 +02:00
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2020-06-11 19:36:14 +02:00
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:::: {.columns align=bottom}
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::: {.column width=50%}
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2020-06-10 16:23:33 +02:00
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$$
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F(m) = \frac{1}{2}
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$$
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2020-06-06 19:40:48 +02:00
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2020-06-10 16:23:33 +02:00
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\vspace{20pt}
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2020-06-07 14:32:03 +02:00
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2020-06-10 16:23:33 +02:00
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. . .
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2020-06-07 14:32:03 +02:00
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2020-06-10 16:23:33 +02:00
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- Sort points in ascending order
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. . .
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- Middle element if odd
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2020-06-07 14:32:03 +02:00
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2020-06-10 16:23:33 +02:00
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Average of the two central elements if even
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:::
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::: {.column width=50%}
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![](images/median.pdf)
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:::
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::::
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2020-06-07 14:32:03 +02:00
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2020-06-12 00:09:22 +02:00
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}[remember picture, >=Stealth]
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% line
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\draw [line width=3, ->, cyclamen] (-5,0) -- (5,0);
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\node [right] at (5,0) {$x$};
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% points
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\draw [yellow!50!black, fill=yellow] (-4.6,-0.1) rectangle (-4.8,0.1);
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\draw [yellow!50!black, fill=yellow] (-4,-0.1) rectangle (-4.2,0.1);
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\draw [yellow!50!black, fill=yellow] (-3.3,-0.1) rectangle (-3.5,0.1);
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\draw [yellow!50!black, fill=yellow] (-2.3,-0.1) rectangle (-2.5,0.1);
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\draw [yellow!50!black, fill=yellow] (-0.6,-0.1) rectangle (-0.8,0.1);
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\draw [yellow!50!black, fill=yellow] (-0.1,-0.1) rectangle (0.1,0.1);
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\draw [yellow!50!black, fill=yellow] (1.1,-0.1) rectangle (1.3,0.1);
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\draw [yellow!50!black, fill=yellow] (2,-0.1) rectangle (2.2,0.1);
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\draw [yellow!50!black, fill=yellow] (2.7,-0.1) rectangle (2.9,0.1);
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\draw [yellow!50!black, fill=yellow] (4,-0.1) rectangle (4.2,0.1);
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\pause
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% nodes
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\node [below] at (-4.7,-0.1) {1};
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\node [below] at (-4.1,-0.1) {2};
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\node [below] at (-3.4,-0.1) {3};
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\node [below] at (-2.4,-0.1) {4};
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\node [below] at (-0.7,-0.1) {5};
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\node [below] at ( 0 ,-0.1) {6};
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\node [below] at ( 1.2,-0.1) {7};
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\node [below] at ( 2.1,-0.1) {8};
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\node [below] at ( 2.8,-0.1) {9};
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\node [below] at ( 4.1,-0.1) {10};
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\pause
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\draw [ultra thick] (-0.35,0.7) -- (-0.35,-0.7);
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\end{tikzpicture}
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\end{center}
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\setbeamercovered{transparent}
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2020-06-07 14:32:03 +02:00
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## Sample mode
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2020-06-11 19:38:08 +02:00
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Half Sample Mode[@robertson74]
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2020-06-07 14:32:03 +02:00
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2020-06-11 19:38:08 +02:00
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- Find the smallest interval containing half points
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- Repeat
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- If the sample has less than three points, take average
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2020-06-07 14:32:03 +02:00
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2020-06-10 16:23:33 +02:00
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. . .
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}[remember picture]
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% line
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\draw [line width=3, ->, cyclamen] (-5,0) -- (5,0);
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\node [right] at (5,0) {$x$};
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% points
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2020-06-11 00:21:44 +02:00
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\draw [blue!50!black, fill=blue] (-4.6,-0.1) rectangle (-4.8,0.1);
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\draw [blue!50!black, fill=blue] (-4,-0.1) rectangle (-4.2,0.1);
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\draw [blue!50!black, fill=blue] (-3.3,-0.1) rectangle (-3.5,0.1);
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\draw [blue!50!black, fill=blue] (-2.3,-0.1) rectangle (-2.5,0.1);
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\draw [blue!50!black, fill=blue] (-0.6,-0.1) rectangle (-0.8,0.1);
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\draw [blue!50!black, fill=blue] (-0.1,-0.1) rectangle (0.1,0.1);
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\draw [blue!50!black, fill=blue] (1.1,-0.1) rectangle (1.3,0.1);
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2020-06-12 00:09:00 +02:00
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\draw [blue!50!black, fill=blue] (2,-0.1) rectangle (2.2,0.1);
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2020-06-11 00:21:44 +02:00
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\draw [blue!50!black, fill=blue] (2.7,-0.1) rectangle (2.9,0.1);
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\draw [blue!50!black, fill=blue] (4,-0.1) rectangle (4.2,0.1);
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2020-06-10 16:23:33 +02:00
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% future nodes
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\node at (-1,-0.3) (1a) {};
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\node at (3.1,0.3) (1b) {};
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\node at (0.9,-0.3) (2a) {};
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\node at (1.8,-0.3) (3a) {};
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% result nodes
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\node at (2.45,-0.7) (f1) {};
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\node at (2.45,0.7) (f2) {};
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\end{tikzpicture}
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\end{center}
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. . .
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\begin{center}
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\begin{tikzpicture}[remember picture, overlay]
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% region
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\draw [orange, fill=orange, opacity=0.5] (1a) rectangle (1b);
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\end{tikzpicture}
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\end{center}
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. . .
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\begin{center}
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\begin{tikzpicture}[remember picture, overlay]
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% region
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\draw [orange, fill=orange, opacity=0.5] (2a) rectangle (1b);
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\end{tikzpicture}
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\end{center}
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. . .
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\begin{center}
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\begin{tikzpicture}[remember picture, overlay]
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% region
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\draw [orange, fill=orange, opacity=0.5] (3a) rectangle (1b);
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\end{tikzpicture}
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\end{center}
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. . .
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\begin{center}
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\begin{tikzpicture}[remember picture, overlay]
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% region
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\draw [cyclamen, ultra thick] (f1) -- (f2);
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\end{tikzpicture}
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\end{center}
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2020-06-07 14:32:03 +02:00
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## Sample FWHM
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2020-06-06 19:40:48 +02:00
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$$
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2020-06-07 14:32:03 +02:00
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\text{FWHM} = x_+ - x_- \with L(x_{\pm}) = \frac{L_{\text{max}}}{2}
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2020-06-06 19:40:48 +02:00
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$$
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2020-06-10 16:23:33 +02:00
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\setbeamercovered{transparent}
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2020-06-06 19:40:48 +02:00
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. . .
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2020-06-11 00:21:44 +02:00
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**Kernel Density Estimation**
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2020-06-07 14:32:03 +02:00
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2020-06-10 18:48:17 +02:00
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:::: {.columns}
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::: {.column width=50% .c}
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- empirical PDF construction:
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2020-06-07 14:32:03 +02:00
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2020-06-10 18:48:17 +02:00
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$$
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f_\varepsilon(x) = \frac{1}{N\varepsilon} \sum_{i = 1}^N
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G \left( \frac{x-x_i}{\varepsilon} \right)
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$$
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2020-06-07 00:02:20 +02:00
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2020-06-10 18:48:17 +02:00
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The parameter $\varepsilon$ controls the strength of the smoothing
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:::
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::: {.column width=50%}
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}
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% points
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2020-06-11 00:21:44 +02:00
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\draw [blue!50!black, fill=blue] (-2,-0.1) rectangle (-1.8,0.1);
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\draw [blue!50!black, fill=blue] (-0.1,-0.1) rectangle (0.1,0.1);
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\draw [blue!50!black, fill=blue] (1.3,-0.1) rectangle (1.5,0.1);
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\draw [blue!50!black, fill=blue] (0.7,-0.1) rectangle (0.9,0.1);
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2020-06-10 18:48:17 +02:00
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\pause
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% lines
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\draw [cyclamen, dashed] (-1.9,0.1) -- (-1.9,1);
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\draw [cyclamen, dashed] (0,0.1) -- (0,1);
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\draw [cyclamen, dashed] (1.4,0.1) -- (1.4,1);
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\draw [cyclamen, dashed] (0.8,0.1) -- (0.8,1);
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% Gaussians
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\draw[domain=-3.4:-0.4, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {exp(-(\x + 1.9)*(\x + 1.9)) + 0.1});
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\draw[domain=-1.5:1.5, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {exp(-\x*\x + 0.1});
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\draw[domain=-0.1:2.9, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {exp(-(\x - 1.4)*(\x - 1.4)) + 0.1});
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\draw[domain=-0.7:2.3, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {exp(-(\x - 0.8)*(\x - 0.8)) + 0.1});
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\pause
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% sum
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\draw [fill=white, white, opacity=0.5] (-3.5,0.1) rectangle (3,1.3);
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\draw[domain=-3.4:3.4, smooth, variable=\x, blue, very thick]
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plot ({\x}, {exp(-(\x + 1.9)*(\x + 1.9)) +
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exp(-\x*\x) +
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exp(-(\x - 1.4)*(\x - 1.4)) +
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exp(-(\x - 0.8)*(\x - 0.8)) + 0.1});
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\end{tikzpicture}
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\end{center}
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\setbeamercovered{transparent}
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:::
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::::
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2020-06-07 14:32:03 +02:00
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2020-06-07 00:02:20 +02:00
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2020-06-07 14:32:03 +02:00
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## Sample FWHM
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2020-06-11 18:30:30 +02:00
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Silverman's rule of thumb [@silver86]:
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2020-06-07 14:32:03 +02:00
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$$
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2020-06-08 18:02:21 +02:00
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\varepsilon = 0.88 \, S_N
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2020-06-07 14:32:03 +02:00
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\left( \frac{d + 2}{4}N \right)^{-1/(d + 4)}
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$$
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with:
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2020-06-08 23:45:13 +02:00
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- $S_N$ is the sample standard deviation
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- $d$ is number of dimensions ($d = 1$)
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2020-06-07 14:32:03 +02:00
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. . .
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2020-06-07 00:02:20 +02:00
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2020-06-10 16:23:33 +02:00
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Numerical minimization (Brent) for $\quad f_{\varepsilon_{\text{max}}}$
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Numerical root finding (Brent) for $\quad f_{\varepsilon}(x_{\pm}) =
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\frac{f_{\varepsilon_{\text{max}}}}{2}$
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2020-06-09 16:52:28 +02:00
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## Sample FWHM
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![](images/kde.pdf)
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