218 lines
7.2 KiB
Markdown
218 lines
7.2 KiB
Markdown
# Results
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## Compatibility test
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Comparing sample properties:
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$$
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p = 1 - \text{erf} \left( \frac{t}{\sqrt{2}} \right)\ \with
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t = \frac{|x\ex - x\ob|}{\sqrt{\sigma\ex^2 + \sigma\ob^2}}
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$$
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- $x\ex$ and $x\ob$ are the expected and observed values
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- $\sigma\ex$ and $\sigma\ob$ are their absolute errors
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. . .
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At 95% confidence level, the values are compatible if:
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$$
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p > 0.05
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$$
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## Compatibility test
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}[>=Stealth]
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%notes
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\draw [very thick, gray] (0,0) -- (0,3);
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\draw [very thick, gray] (-1.45,1.5) -- (1.45,1.5);
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\draw [very thick, gray] (-1.35,1.3) -- (-1.55,1.7);
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\draw [very thick, gray] ( 1.35,1.3) -- ( 1.55,1.7);
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\node at (0,-0.4) {$x\ex$};
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\node [right] at (1.7,1.7) {$2 \, \sqrt{\sigma\ex^2 + \sigma\ob^2}$};
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% axes
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\draw [very thick, <->] (-5,4) -- (-5,0) -- (5,0);
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\node [right] at (5,0) {$x$};
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% Gaussian
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\draw [domain=-5:5, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {3*exp(-(\x*\x/3))});
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\pause
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% area
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\fill [domain=2:5, smooth, variable=\x, cyclamen!20!white, very thick]
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(2,0) -- plot ({\x}, {3*exp(-(\x*\x/3))}) -- (5,0) -- cycle;
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\fill [domain=-5:-2, smooth, variable=\x, cyclamen!20!white, very thick]
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(-5,0) -- plot ({\x}, {3*exp(-(\x*\x/3))}) -- (-2,0) -- cycle;
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% axes
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\draw [very thick, <->] (-5,4) -- (-5,0) -- (5,0);
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% Gaussian
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\draw [domain=-5:5, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {3*exp(-(\x*\x/3))});
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%notes
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\draw [thick, cyclamen] (-2,0) -- (-2,0.8);
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\draw [thick, cyclamen] ( 2,0) -- ( 2,0.8);
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\node at (2,-0.4) {$x\ob$};
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\end{tikzpicture}
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\end{center}
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\setbeamercovered{transparent}
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## L sample results
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\vspace{20pt}
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\begin{center}
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\begin{tabular}{rcccc}
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\toprule
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& & median & mode & fwhm \\
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\midrule
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\multirow{2}{*}{50000} & $t$ & 0.027 & 1.012 & 1.338 \\
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& $p$ & \gre{0.979} & \gre{0.311} & \gre{0.181} \\
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\midrule
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\multirow{2}{*}{5000} & $t$ & 0.912 & 0.200 & 3.033 \\
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& $p$ & \gre{0.362} & \gre{0.841} & \red{0.002} \\
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\midrule
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\multirow{2}{*}{1000} & $t$ & 1.189 & 0.232 & 0.208 \\
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& $p$ & \gre{0.235} & \gre{0.817} & \gre{0.835} \\
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\toprule
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\end{tabular}
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\end{center}
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}[>=Stealth, overlay]
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% Rectangles
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\draw <2> [gray, very thick] (-1.4,0.3) rectangle (0.4,5.5);
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\draw <3> [gray, very thick] (0.4,0.3) rectangle (2,5.5);
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\draw <4> [gray, very thick] (2,0.3) rectangle (3.8,5.5);
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% Estimations ellipses
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\draw <5-> [cyclamen, thick] (2.8,3.57) ellipse (0.8cm and 0.3cm);
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\draw <5-> [cyclamen, thick] (2.8,2.18) ellipse (0.8cm and 0.3cm);
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\draw <5-> [cyclamen, thick] (2.8,0.8) ellipse (0.8cm and 0.3cm);
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% Estimations notes
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\draw <6> [fill=white] (-5.5,-0.1) rectangle (0.3,2.3);
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\node <6> [above right, align=right] at (-5.5,-0.1) {
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$w\ex = 4.02... \hspace{22pt} $\\
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$50000 \hence w\ob = 4.11 \pm 0.07$\\
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$5000 \hence w\ob = 4.40 \pm 0.13$\\
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$1000 \hence w\ob = 3.96 \pm 0.27$};
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% FHWM notes
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\draw <7-> [fill=white] (-5.5,2.9) rectangle (-1.4,5.1);
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\node <7-> [above right, align=center] at (-5.5,3) {
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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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$\varepsilon = 0.88 \, S_N
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\left( \frac{d + 2}{4}N \right)^{-1/(d + 4)}$};
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\draw <8> [cyclamen, thick] (-4.45,3.45) ellipse (0.4cm and 0.3cm);
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\end{tikzpicture}
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\end{center}
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\setbeamercovered{transparent}
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## M sample results
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\vspace{20pt}
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\begin{center}
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\begin{tabular}{rcccc}
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\toprule
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& & median & mode & fwhm \\
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\midrule
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\multirow{2}{*}{50000} & $t$ & 58.01 & 0.732 & 1.329 \\
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& $p$ & \gre{0.000} & \gre{0.464} & \gre{0.184} \\
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\midrule
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\multirow{2}{*}{5000} & $t$ & 16.80 & 0.014 & 0.215 \\
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& $p$ & \gre{0.000} & \gre{0.989} & \gre{0.830} \\
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\midrule
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\multirow{2}{*}{1000} & $t$ & 11.61 & 0.028 & 1.001 \\
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& $p$ & \gre{0.000} & \gre{0.978} & \gre{0.317} \\
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\toprule
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\end{tabular}
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\end{center}
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}[>=Stealth, overlay]
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% Rectangles
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\draw <2> [gray, very thick] (-1.4,0.3) rectangle (0.4,5.5);
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\draw <3> [gray, very thick] (0.4,0.3) rectangle (2,5.5);
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\draw <4,5,6> [gray, very thick] (2,0.3) rectangle (3.8,5.5);
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\node <5> at (-2,2.5) {\includegraphics{images/gamo-pdf.pdf}};
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\node <6> at (-2,2.5) {\includegraphics{images/gala-pdf.pdf}};
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\draw <5,6> [thick] (-5.15,0) rectangle (1.22,5.05);
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\end{tikzpicture}
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\end{center}
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\setbeamercovered{transparent}
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## KS samples results
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\vspace{10pt}
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\begin{center}
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\begin{tabular}{rccc}
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\toprule
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& & L sample & M sample \\
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\midrule
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\multirow{2}{*}{50000} & $D$ & 0.004 & 0.153 \\
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& $p$ & \gre{0.379} & \gre{0.000} \\
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\midrule
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\multirow{2}{*}{5000} & $D$ & 0.010 & 0.148 \\
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& $p$ & \gre{0.714} & \gre{0.000} \\
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\midrule
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\multirow{2}{*}{1000} & $D$ & 0.026 & 0.162 \\
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& $p$ & \gre{0.478} & \gre{0.000} \\
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\midrule
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\multirow{2}{*}{200} & $D$ & 0.054 & 0.205 \\
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& $p$ & \gre{0.579} & \gre{0.000} \\
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\toprule
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\end{tabular}
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\end{center}
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}[>=Stealth, overlay]
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% Rectangles
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\draw <2> [gray, very thick] (-1,0.3) rectangle (1,6.9);
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\draw <3> [gray, very thick] (1,0.3) rectangle (3.3,6.9);
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% 200 points ellipse
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\draw <4> [gray, very thick] (-3.5,0.3) rectangle (3.5,1.9);
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\end{tikzpicture}
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\end{center}
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\setbeamercovered{transparent}
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## Trapani samples results
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\vspace{20pt}
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\begin{center}
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\begin{tabular}{rccccc}
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\toprule
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& & \multicolumn{2}{c}{L sample} & \multicolumn{2}{c}{M sample} \\
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& & $\mu_1$ & $\mu_2$ & $\mu_1$ & $\mu_2$ \\
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\midrule
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\multirow{2}{*}{50000} & $\Theta$ & 0.255 & 0.432 & 106.1 & 162.5 \\
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& $p$ & \gre{0.614} & \gre{0.511} & \gre{0.000} & \gre{0.000} \\
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\midrule
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\multirow{2}{*}{5000} & $\Theta$ & 2.178 & 2.556 & 23.36 & 36.48 \\
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& $p$ & \gre{0.140} & \gre{0.110} & \gre{0.000} & \gre{0.000} \\
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\midrule
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\multirow{2}{*}{1000} & $\Theta$ & 0.694 & 0.060 & 5.985 & 7.302 \\
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& $p$ & \gre{0.405} & \gre{0.549} & \gre{0.014} & \gre{0.007} \\
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\toprule
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\end{tabular}
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\end{center}
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}[>=Stealth, overlay]
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% Rectangles
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\draw <2,3> [gray, very thick] (-2,0.3) rectangle (1,6.1);
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\draw <4> [gray, very thick] (1,0.3) rectangle (4.2,6.1);
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% Estimations ellipses
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\draw <3> [cyclamen, thick] (-1.28,2.18) ellipse (0.7cm and 0.3cm);
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\draw <3> [cyclamen, thick] (0.29,2.18) ellipse (0.7cm and 0.3cm);
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\end{tikzpicture}
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\end{center}
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\setbeamercovered{transparent}
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