analistica/slides/sections/5.md

Kolmogorov-Smirnov test

KS

Quantify distance between expected and observed CDFs.

KS statistic:

:::: {.columns} ::: {.column width=50% .c}

$$
  D_N = \text{sup}_x |F_N(x) - F(x)|

• F(x) is the expected CDF

• F_N(x) is the empirical CDF

  • sort points in ascending order
  • number of points preceding the point normalized by N

. . .

:::

::: {.column width=50%} \setbeamercovered{} \begin{center} \begin{tikzpicture}[>=Stealth] % empiric \draw [cyclamen, thick, fill=cyclamen!20!white] (-2.5,0) -- (-2.5,0.5) -- (-1.5,0.5) -- (-1.5,1) -- (-0.9,1) -- (-0.9,1.5) -- (-0.1,1.5) -- (-0.1,2) -- (1,2) -- (1,2.5) -- (1.2,2.5) -- (1.2,3) -- (1.3,3) -- (1.3,3.5) -- (1.6,3.5) -- (1.6,4) -- (2.3,4) -- (2.3,4.5) -- (2.5,4.5) -- (2.5,0) -- cycle; % points \draw [yellow!50!black, fill=yellow] (-2.6,-0.1) rectangle (-2.4,0.1); %-2.5 \draw [yellow!50!black, fill=yellow] (-1.6,-0.1) rectangle (-1.4,0.1); %-1.5 \draw [yellow!50!black, fill=yellow] (-1,-0.1) rectangle (-0.8,0.1); %-0.9 \draw [yellow!50!black, fill=yellow] (-0.2,-0.1) rectangle (0,0.1); %-0.1 \draw [yellow!50!black, fill=yellow] (0.9,-0.1) rectangle (1.1,0.1); % 1 \draw [yellow!50!black, fill=yellow] (1.1,-0.1) rectangle (1.3,0.1); % 1.2 \draw [yellow!50!black, fill=yellow] (1.2,-0.1) rectangle (1.4,0.1); % 1.3 \draw [yellow!50!black, fill=yellow] (1.5,-0.1) rectangle (1.7,0.1); % 1.6 \draw [yellow!50!black, fill=yellow] (2.2,-0.1) rectangle (2.4,0.1); % 2.3 % expected \pause \draw[domain=-2.5:2.5, yscale=5, smooth, variable=\x, yellow, very thick] plot ({\x}, {((atan(\x)*pi/180) + pi/2)/pi}); \pause \draw [very thick, cyclamen, <->] (1,2.5) -- (1,3.75); \end{tikzpicture} \end{center} \setbeamercovered{transparent} ::: ::::

KS

\bold{H_0}: points sampled from reference distribution

::: incremental

• \sqrt{N}D_N \xrightarrow{N \rightarrow + \infty} K, independent of F

• Kolmogorov variable K with CDF:

  
    P(K \leqslant K_0) = \frac{\sqrt{2 \pi}}{K_0}
    \sum_{j = 1}^{+ \infty} e^{-(2j - 1)^2 \pi^2 / 8 K_0^2}
  

• $p$-value given by: p = 1 - P(K \leq K_0)

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