From e5cced162a78f7e53fcfd12227b1fb747c6335ea Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Gi=C3=B9=20Marcer?= <giuliamarcer@yahoo.it>
Date: Mon, 8 Jun 2020 23:44:51 +0200
Subject: [PATCH] slides: do the KS section

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+# Kolmogorov - Smirnov test
+
+
+## KS
+
+Quantify distance between expected and observed CDF
+
+. . .
+
+KS statistic:
+
+$$
+  D_N = \text{sup}_x |F_N(x) - F(x)|
+$$
+
+- $F(x)$ is the expected CDF
+- $F_N(x)$ is the empirical CDF of $N$ sampled points
+  - sort points in ascending order
+  - number of points preceding the point normalized by $N$
+
+
+## KS
+
+$H_0$: points sampled according to $F(x)$
+
+. . .
+
+If $H_0$ is true:
+
+- $\sqrt{N}D_N \xrightarrow{N \rightarrow + \infty} K$
+
+Kolmogorov distribution with CDF:
+
+$$
+  P(K \leqslant K_0) = 1 - p = \frac{\sqrt{2 \pi}}{K_0}
+  \sum_{j = 1}^{+ \infty} e^{-(2j - 1)^2 \pi^2 / 8 K_0^2}
+$$
+
+. . .
+
+a $p$-value can be computed
+
+- At 95% confidence level, $H_0$ cannot be disproved if $p > 0.05$
+
+
+# Samples results
+
+
+## Samples results
+
+$N = 50000$ sampled points
+
+. . .
+
+Landau sample:
+
+:::: {.columns}
+:::  {.column width=50%}
+  - $D = 0.004$
+  - $p = 0.379$
+:::
+
+:::  {.column width=50%}
+  $$
+  \thus \text{Compatible!}
+  $$
+:::
+::::
+
+\vspace{10pt}
+
+. . .
+
+Moyal sample:
+
+:::: {.columns}
+:::  {.column width=50%}
+  - $D = 0.153$
+  - $p = 0.000$
+:::
+
+:::  {.column width=50%}
+  $$
+  \thus \text{Not compatible!}
+  $$
+:::
+::::