slides: reshape a bit section 5-6

slides/sections/5.md

@@ -3,7 +3,9 @@
 
 ## KS
 
-Quantify distance between expected and observed CDF. KS statistic:
+Quantify distance between expected and observed CDF.
+
+KS statistic:
 
 :::: {.columns}
 :::  {.column width=50% .c}
@@ -11,9 +13,8 @@ Quantify distance between expected and observed CDF. KS statistic:
     D_N = \text{sup}_x |F_N(x) - F(x)|
   $$
 
-  \vspace{20pt}
-
   - $F(x)$ is the expected CDF
+
   - $F_N(x)$ is the empirical CDF
     - sort points in ascending order
     - number of points preceding the point normalized by $N$
@@ -58,21 +59,18 @@ Quantify distance between expected and observed CDF. KS statistic:
 
 ## KS
 
-$H_0$: points sampled according to $F(x)$
+$\bold{H_0}$: points sampled from $F$
 
-. . .
+::: incremental
 
-If $H_0$ is true: $\sqrt{N}D_N \xrightarrow{N \rightarrow + \infty} K$
+- $\sqrt{N}D_N \xrightarrow{N \rightarrow + \infty} K$, independent of $F$
 
-$K$ Kolmogorov variable with CDF:
+- Kolmogorov variable $K$ with CDF:
+  $$
+    P(K \leqslant K_0) = \frac{\sqrt{2 \pi}}{K_0}
+    \sum_{j = 1}^{+ \infty} e^{-(2j - 1)^2 \pi^2 / 8 K_0^2}
+  $$
 
-$$
+- $p$-value given by: $p = 1 - P(K \leq K_0)$
-  P(K \leqslant K_0) = \frac{\sqrt{2 \pi}}{K_0}
-  \sum_{j = 1}^{+ \infty} e^{-(2j - 1)^2 \pi^2 / 8 K_0^2}
-$$
 
-. . .
+:::
-
-A $p$-value can be computed
-
-- At 95% confidence level, $H_0$ cannot be disproved if $p > 0.05$

slides/sections/6.md

@@ -20,12 +20,12 @@
 
 ## Infinite moments
 
-- Generate a sample $L$ from a Landau PDF
+- Sample $L$ from a Landau PDF
-- Generate a sample $M$ from a Moyal PDF
+- Sample $M$ from a Moyal PDF
 
 . . .
 
-\vspace{20pt}
+\vspace{2em}
 
 :::: {.columns}
 :::  {.column width=50% .c}
@@ -50,24 +50,22 @@
 
 ## Infinite moments
 
-- Previous tests: points sampled from Landau PDF?
+::: incremental
 
-. . .
+- Trapani test: check whether a moment is infinite
 
-- Trapani test: check whether a moment is finite or infinite
+- Consistency test:
 \begin{align*}
-  \text{infinite} &\thus \text{Landau} \\
+  \text{infinite} &\thus \text{may be Landau} \\
   \text{finite}   &\thus \text{not Landau}
 \end{align*}
 
-. . .
-
 - Compatibility test with $\mu_k = + \infty$
 
-. . .
-
 - If points were sampled from a Cauchy distribution...
 
+:::
+
 
 ## Trapani test
 
@@ -163,7 +161,7 @@ $$
                   \left[ \sum_{j} \zeta_j (u) - \frac{r}{2} \right]
 $$
 
-- If $a_j$ uniformly distributed, for the CLT:
+- If $a_j$ uniformly distributed, by the CLT:
 $$
   \sum_j \zeta_j (u) \hence
     G \left( \frac{r}{2}, \frac{\sqrt{r}}{2} \right)
@@ -205,18 +203,19 @@ $$
 
 ## Trapani test
 
-According to L. Trapani [@trapani15]:
+MC simulations [@trapani15] gives:
 
 - $r = o(N) \hence r = N^{0.75}$
-
 - $\underbar{u} = -1 \quad \wedge \quad \bar{u} = 1$
-
+- $\psi(u)$ uniform
-- $\psi(u) = \frac{1}{2} \, \chi_{[\underbar{u}, \bar{u}]}$
 
 . . .
 
-$\mu_k$ must be scale invariant for $k > 1$:
+\Begin{alertblock}{Important}
-$$
+
-  \mu_k^* = \frac{\mu_k}{ \left( \mu_{\phi} \right)^{k/\phi} }
+  For $k > 1$, $\mu_k$ must be made scale-invariant:
-  \with \phi \in (0, k)
+  $$
-$$
+    \mu_k^* = \frac{\mu_k}{ \left( \mu_{\phi} \right)^{k/\phi} }
+    \with \phi \in (0, k)
+  $$
+\End{alertblock}