analistica/slides/sections/8.md

# Results


## Compatibility test

Comparing sample properties:

$$
  p = 1 - \text{erf} \left( \frac{t}{\sqrt{2}} \right)\ \with
  t = \frac{|x\ex - x\ob|}{\sqrt{\sigma\ex^2 + \sigma\ob^2}}
$$

- $x\ex$ and $x\ob$ are the expected and observed values
- $\sigma\ex$ and $\sigma\ob$ are their absolute errors

. . .

At 95% confidence level, the values are compatible if:
$$
  p > 0.05
$$

## Compatibility test

\setbeamercovered{}
\begin{center}
\begin{tikzpicture}[>=Stealth]
  %notes
  \draw [very thick, gray] (0,0) -- (0,3);
  \draw [very thick, gray] (-1.45,1.5) -- (1.45,1.5);
  \draw [very thick, gray] (-1.35,1.3) -- (-1.55,1.7);
  \draw [very thick, gray] ( 1.35,1.3) -- ( 1.55,1.7);
  \node at (0,-0.4) {$x\ex$};
  \node [right] at (1.7,1.7) {$2 \, \sqrt{\sigma\ex^2 + \sigma\ob^2}$};
  % axes
  \draw [very thick, <->] (-5,4) -- (-5,0) -- (5,0);
  \node [right] at (5,0) {$x$};
  % Gaussian
  \draw [domain=-5:5, smooth, variable=\x, cyclamen, very thick]
      plot ({\x}, {3*exp(-(\x*\x/3))});
  \pause
  % area
  \fill [domain=2:5, smooth, variable=\x, cyclamen!20!white, very thick]
        (2,0) -- plot ({\x}, {3*exp(-(\x*\x/3))}) -- (5,0) -- cycle;
  \fill [domain=-5:-2, smooth, variable=\x, cyclamen!20!white, very thick]
        (-5,0) -- plot ({\x}, {3*exp(-(\x*\x/3))}) -- (-2,0) -- cycle;
  % axes
  \draw [very thick, <->] (-5,4) -- (-5,0) -- (5,0);
  % Gaussian
  \draw [domain=-5:5, smooth, variable=\x, cyclamen, very thick]
      plot ({\x}, {3*exp(-(\x*\x/3))});
  %notes
  \draw [thick, cyclamen] (-2,0) -- (-2,0.8);
  \draw [thick, cyclamen] ( 2,0) -- ( 2,0.8);
  \node at (2,-0.4) {$x\ob$};
\end{tikzpicture}
\end{center}
\setbeamercovered{transparent}


## Other tests

**KS test**:

  - $K_0 = \sqrt{N}D_N$
  - $p = 1 - P(K \leq K_0)$

. . .

**Trapani test**:

  - $\Theta \to \chi^2$
  - $p = 1 - P(\chi^2 \leq \Theta)$

. . .

In either case: 95\% confidence level
$$
  p > 0.05
$$

## L sample results

\vspace{20pt}
\begin{center}
\begin{tabular}{rcccc}
  \toprule
                         &     & median      & mode        & fwhm        \\
  \midrule
  \multirow{2}{*}{50000} & $t$ & 0.027       & 1.012       & 1.338       \\
                         & $p$ & \gre{0.979} & \gre{0.311} & \gre{0.181} \\
  \midrule
  \multirow{2}{*}{5000}  & $t$ & 0.912       & 0.200       & 3.033       \\
                         & $p$ & \gre{0.362} & \gre{0.841} & \red{0.002} \\
  \midrule
  \multirow{2}{*}{1000}  & $t$ & 1.189       & 0.232       & 0.208       \\
                         & $p$ & \gre{0.235} & \gre{0.817} & \gre{0.835} \\
  \toprule
\end{tabular}
\end{center}

\setbeamercovered{}
\begin{center}
\begin{tikzpicture}[>=Stealth, overlay]
  % Rectangles
  \draw <2> [gray, very thick] (-1.4,0.3) rectangle (0.4,5.5);
  \draw <3> [gray, very thick] (0.4,0.3)  rectangle (2,5.5);
  \draw <4> [gray, very thick] (2,0.3)    rectangle (3.8,5.5);
  % Estimations ellipses
  \draw <5-> [cyclamen, thick] (2.8,2.18) ellipse (0.8cm and 0.3cm);
  \draw <5-> [cyclamen, thick] (2.8,0.8)  ellipse (0.8cm and 0.3cm);
  % Estimations notes
  \draw <6> [fill=white] (-5.5,-0.1) rectangle (0.3,2.3);
  \node <6> [above right, align=right] at (-5.5,-0.1) {
    $w\ex = 4.02... \hspace{22pt} $\\
    $50000 \hence w\ob = 4.11 \pm 0.07$\\
    $5000 \hence w\ob = 4.40 \pm 0.13$\\
    $1000 \hence w\ob = 3.96 \pm 0.27$};
  % FHWM notes
  \draw <7-> [fill=white] (-5.5,2.9) rectangle (-1.4,5.1);
  \node <7-> [above right, align=center] at (-5.5,3) {
  $f_\varepsilon(x) = \frac{1}{N\varepsilon} \sum_{i = 1}^N
   G \left( \frac{x-x_i}{\varepsilon} \right)$\\\\
  $\varepsilon = 0.88 \, S_N
   \left( \frac{d + 2}{4}N \right)^{-1/(d + 4)}$};
  \draw <8> [cyclamen, thick] (-4.45,3.45) ellipse (0.4cm and 0.3cm);
\end{tikzpicture}
\end{center}
\setbeamercovered{transparent}


## M sample results

\vspace{20pt}
\begin{center}
\begin{tabular}{rcccc}
  \toprule
                         &     & median      & mode        & fwhm        \\
  \midrule
  \multirow{2}{*}{50000} & $t$ & 58.01       & 0.732       & 1.329       \\
                         & $p$ & \gre{0.000} & \gre{0.464} & \gre{0.184} \\
  \midrule
  \multirow{2}{*}{5000}  & $t$ & 16.80       & 0.014       & 0.215       \\
                         & $p$ & \gre{0.000} & \gre{0.989} & \gre{0.830} \\
  \midrule
  \multirow{2}{*}{1000}  & $t$ & 11.61       & 0.028       & 1.001       \\
                         & $p$ & \gre{0.000} & \gre{0.978} & \gre{0.317} \\
  \toprule
\end{tabular}
\end{center}

\setbeamercovered{}
\begin{center}
\begin{tikzpicture}[>=Stealth, overlay]
  % Rectangles
  \draw <2,3>   [gray, very thick] (-1.4,0.3) rectangle (0.4,5.5);
  \draw <4>     [gray, very thick] (0.4,0.3)  rectangle (2,5.5);
  \draw <5,6,7> [gray, very thick] (2,0.3)    rectangle (3.8,5.5);
  \node <6>     at (-2,2.5) {\includegraphics{images/gamo-pdf.pdf}};
  \node <7>     at (-2,2.5) {\includegraphics{images/gala-pdf.pdf}};
  \draw <6,7>   [thick] (-5.15,0) rectangle (1.22,5.05);
  % Estimations notes
  \draw <3> [fill=white] (1,2.6) rectangle (3.4,3.9);
  \node <3> [above right, align=right] at (1,2.6) {
    $m_M = 0.658...$\\
    $m_L = 1.355...$};
\end{tikzpicture}
\end{center}
\setbeamercovered{transparent}


## KS samples results

\vspace{10pt}

\begin{center}
\begin{tabular}{rccc}
  \toprule
                         &     & L sample     & M sample     \\
  \midrule
  \multirow{2}{*}{50000} & $D$ & 0.004        & 0.153        \\
                         & $p$ & \gre{0.379}  & \gre{0.000}  \\
  \midrule
  \multirow{2}{*}{5000}  & $D$ & 0.010        & 0.148        \\
                         & $p$ & \gre{0.714}  & \gre{0.000}  \\
  \midrule
  \multirow{2}{*}{1000}  & $D$ & 0.026        & 0.162        \\
                         & $p$ & \gre{0.478}  & \gre{0.000}  \\
  \midrule
  \multirow{2}{*}{200}   & $D$ & 0.054        & 0.205        \\
                         & $p$ & \gre{0.579}  & \gre{0.000}  \\
  \toprule
\end{tabular}
\end{center}

\setbeamercovered{}
\begin{center}
\begin{tikzpicture}[>=Stealth, overlay]
  % Rectangles
  \draw <2> [gray, very thick] (-1,0.3) rectangle (1,6.9);
  \draw <3> [gray, very thick] (1,0.3)  rectangle (3.3,6.9);
  % 200 points ellipse
  \draw <4> [gray, very thick] (-3.5,0.3)  rectangle (3.5,1.9);
\end{tikzpicture}
\end{center}
\setbeamercovered{transparent}


## Trapani samples results

\vspace{20pt}
\begin{center}
\begin{tabular}{rccccc}
  \toprule
                         & & \multicolumn{2}{c}{L sample} & \multicolumn{2}{c}{M sample}    \\
                         &          & $\mu_1$     & $\mu_2$     & $\mu_1$     & $\mu_2$     \\
  \midrule
  \multirow{2}{*}{50000} & $\Theta$ & 0.255       & 0.432       & 106.1       & 162.5       \\
                         & $p$      & \gre{0.614} & \gre{0.511} & \gre{0.000} & \gre{0.000} \\
  \midrule
  \multirow{2}{*}{5000}  & $\Theta$ & 2.178       & 2.556       & 23.36       & 36.48       \\
                         & $p$      & \gre{0.140} & \gre{0.110} & \gre{0.000} & \gre{0.000} \\
  \midrule
  \multirow{2}{*}{1000}  & $\Theta$ & 0.694       & 0.060       & 5.985       & 7.302       \\
                         & $p$      & \gre{0.405} & \gre{0.549} & \gre{0.014} & \gre{0.007} \\
  \toprule
\end{tabular}
\end{center}

\setbeamercovered{}
\begin{center}
\begin{tikzpicture}[>=Stealth, overlay]
  % Rectangles
  \draw <2> [gray, very thick] (-2,0.3) rectangle (1,6.1);
  \draw <3> [gray, very thick] (1,0.3)  rectangle (4.2,6.1);
\end{tikzpicture}
\end{center}
\setbeamercovered{transparent}