# Sample statistics


## Sample statistics

How to estimate sample median, mode and FWHM?

. . .

- \only<3>\strike{Binning data $\hence$ depends wildly on bin-width}

. . .

- Alternative solutions
  - Robust estimators
  - Kernel density estimation


## Sample median

:::: {.columns}
:::  {.column width=50% .c}
  $$
    F(m) = \frac{1}{2}
  $$

  \vspace{20pt}

  . . .

  - Sort points in ascending order

  . . .

  - Middle element if odd

    Average of the two central elements if even
:::

:::  {.column width=50%}
  ![](images/median.pdf)
:::
::::


## Sample mode

Most probable value

. . .

Half Sample Mode

- Iteratively identify the smallest interval containing half points
- Once the sample is reduced to less than three points, take average

. . .

\setbeamercovered{}

\begin{center}
\begin{tikzpicture}[remember picture]
  % line
  \draw [line width=3, ->, cyclamen] (-5,0) -- (5,0);
  \node [right] at (5,0) {$x$};
  % points
  \draw [blue, fill=blue] (-4.6,-0.1) rectangle (-4.8,0.1);
  \draw [blue, fill=blue] (-4,-0.1)   rectangle (-4.2,0.1);
  \draw [blue, fill=blue] (-3.3,-0.1) rectangle (-3.5,0.1);
  \draw [blue, fill=blue] (-2.3,-0.1) rectangle (-2.5,0.1);
  \draw [blue, fill=blue] (-0.6,-0.1) rectangle (-0.8,0.1);
  \draw [blue, fill=blue] (-0.1,-0.1) rectangle (0.1,0.1);
  \draw [blue, fill=blue] (1.1,-0.1)  rectangle (1.3,0.1);
  \draw [blue, fill=blue] (2  ,-0.1)  rectangle (2.2,0.1);
  \draw [blue, fill=blue] (2.7,-0.1)  rectangle (2.9,0.1);
  \draw [blue, fill=blue] (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}
\end{center}

. . .

\begin{center}
\begin{tikzpicture}[remember picture, overlay]
  % region
  \draw [orange, fill=orange, opacity=0.5] (1a) rectangle (1b);
\end{tikzpicture}
\end{center}

. . .

\begin{center}
\begin{tikzpicture}[remember picture, overlay]
  % region
  \draw [orange, fill=orange, opacity=0.5] (2a) rectangle (1b);
\end{tikzpicture}
\end{center}

. . .

\begin{center}
\begin{tikzpicture}[remember picture, overlay]
  % region
  \draw [orange, fill=orange, opacity=0.5] (3a) rectangle (1b);
\end{tikzpicture}
\end{center}

. . .

\begin{center}
\begin{tikzpicture}[remember picture, overlay]
  % region
  \draw [cyclamen, ultra thick] (f1) -- (f2);
\end{tikzpicture}
\end{center}


## Sample FWHM

$$
  \text{FWHM} = x_+ - x_- \with L(x_{\pm}) = \frac{L_{\text{max}}}{2}
$$

\setbeamercovered{transparent}
. . .

Kernel Density Estimation

:::: {.columns}
:::  {.column width=50% .c}
  - empirical PDF construction:

  $$
    f_\varepsilon(x) = \frac{1}{N\varepsilon} \sum_{i = 1}^N
    G \left( \frac{x-x_i}{\varepsilon} \right)
  $$

  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

Silverman's rule of thumb:

$$
  \varepsilon = 0.88 \, S_N
  \left( \frac{d + 2}{4}N \right)^{-1/(d + 4)}
$$

with:

- $S_N$ is the sample standard deviation
- $d$ is number of dimensions ($d = 1$)

. . .

Numerical minimization (Brent) for $\quad f_{\varepsilon_{\text{max}}}$  
Numerical root finding (Brent) for $\quad f_{\varepsilon}(x_{\pm}) =
\frac{f_{\varepsilon_{\text{max}}}}{2}$


## Sample FWHM

![](images/kde.pdf)
