From 401108268027845e7cc6c5d1619d2e6093113f97 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Gi=C3=B9=20Marcer?= Date: Fri, 12 Jun 2020 14:31:08 +0200 Subject: [PATCH] sections: fix things here and there --- slides/sections/1.md | 16 +++------------- slides/sections/2.md | 2 +- slides/sections/3.md | 18 ++---------------- slides/sections/4.md | 2 +- slides/sections/6.md | 14 +++++++------- slides/sections/7.md | 8 ++------ slides/sections/8.md | 4 ++-- slides/sections/9.md | 12 +++++++----- 8 files changed, 25 insertions(+), 51 deletions(-) diff --git a/slides/sections/1.md b/slides/sections/1.md index aa351a7..f2948fb 100644 --- a/slides/sections/1.md +++ b/slides/sections/1.md @@ -24,19 +24,9 @@ $H_0$: sample following Landau PDF The Landau and Moyal PDFs are really similar. Historically, the latter was utilized as an approximation of the former. -:::: {.columns} -::: {.column width=33%} - ![](images/moyal-photo.jpg){height=130pt} -::: - -::: {.column width=33%} - ![](images/mondau-photo.jpg){height=130pt} -::: - -::: {.column width=33%} - ![](images/landau-photo.jpg){height=130pt} -::: -:::: +\includegraphics<1>[height=5.5cm]{images/moyal-photo.jpg} +\includegraphics<2>[height=5.5cm]{images/mondau-photo.jpg} +\includegraphics<3>[height=5.5cm]{images/landau-photo.jpg} ## Two similar distributions diff --git a/slides/sections/2.md b/slides/sections/2.md index ac014ef..6556c24 100644 --- a/slides/sections/2.md +++ b/slides/sections/2.md @@ -40,7 +40,7 @@ m = F^{-1}\left(\frac{1}{2}\right) $$ - - Numerical integration or QDF is needed + - PDF Numerical integration up to $1/2$ or QDF is needed ::: :::: diff --git a/slides/sections/3.md b/slides/sections/3.md index 6710a46..9be2b2e 100644 --- a/slides/sections/3.md +++ b/slides/sections/3.md @@ -38,21 +38,7 @@ $$ . . . -With the change of variable $z = e^{-\frac{y}{2}}/\sqrt{2}$: -$$ - F_M(x) = - \frac{-2 \sqrt{2}}{\sqrt{2 \pi}} \int\limits_{+ \infty}^{f(x)} dz \, e^{- z^2} - \with f(x) = \frac{e^{- \frac{x}{2}}}{\sqrt{2}} -$$ - - -## Moyal CDF - -Remembering the error function -$$ - \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x dy \, e^{-y^2} -$$ -one finally gets: +after a bit of math, one finally gets: $$ F_M(x) = 1 - \text{erf} \left( \frac{e^{- \frac{x}{2}}}{\sqrt{2}} \right) $$ @@ -125,7 +111,7 @@ $$ We need to compute the maximum value: $$ - M(\mu) = \frac{1}{\sqrt{2 \pi e}} \thus M(x_{\pm}) = \frac{1}{\sqrt{8 \pi e}} + M(\mu_M\ex) = \frac{1}{\sqrt{2 \pi e}} \thus M(x_{\pm}) = \frac{1}{\sqrt{8 \pi e}} $$ . . . diff --git a/slides/sections/4.md b/slides/sections/4.md index 6f1879b..35c3120 100644 --- a/slides/sections/4.md +++ b/slides/sections/4.md @@ -75,7 +75,7 @@ How to estimate sample median, mode and FWHM? ::: incremental 1. Find the smallest interval containing half points 2. Repeat on the new interval (called modal) -3. If the interval has less than three points, take average +3. If the interval has less than two points, take average ::: \End{block} diff --git a/slides/sections/6.md b/slides/sections/6.md index 492bc6f..5548143 100644 --- a/slides/sections/6.md +++ b/slides/sections/6.md @@ -127,11 +127,11 @@ $$ \draw [thick, cyclamen] (8.5,-0.2) -- (8.5,0.2); \node [below right, cyclamen] at (8.5,-0.2) {$a_{j+4}$}; % notes - \node [below] at (1,-1) {0}; - \node [below] at (2,-1) {0}; - \node [below] at (5.2,-1) {1}; - \node [below] at (6,-1) {1}; - \node [below] at (8.5,-1) {1}; + \node [below] at (1,-1) {1}; + \node [below] at (2,-1) {1}; + \node [below] at (5.2,-1) {0}; + \node [below] at (6,-1) {0}; + \node [below] at (8.5,-1) {0}; \draw [thick, ->] (1,-0.5) -- (1,-1); \draw [thick, ->] (2,-0.5) -- (2,-1); \draw [thick, ->] (5.2,-0.5) -- (5.2,-1); @@ -162,7 +162,7 @@ $$ If $a_j$ uniformly distributed, for the CLT: $$ \sum_j \zeta_j (u) \hence - G \left( \frac{r}{2}, \frac{r}{4} \right) + G \left( \frac{r}{2}, \frac{\sqrt{r}}{2} \right) \thus \vartheta (u) \hence G \left( 0, 1 \right) $$ @@ -195,7 +195,7 @@ $$ ## Trapani test -If $\mu_k \ne + \infty \hence \left\{ a_j \right\}$ are not uniformly distributed +If $\mu_k < + \infty \hence \left\{ a_j \right\}$ are not uniformly distributed \vspace{20pt} diff --git a/slides/sections/7.md b/slides/sections/7.md index 4063d65..aa20058 100644 --- a/slides/sections/7.md +++ b/slides/sections/7.md @@ -22,18 +22,14 @@ A $M(x)$ similar to $L(x)$ can be found by imposing: - equal mode $$ - \mu_M\ex = \mu_L\ex \approx −0.22278298... + \mu_M\ex = \mu_L\ex \thus \mu \approx −0.22278298... $$ . . . - equal width $$ - w_M\ex = w_L\ex = \sigma \cdot a -$$ - -$$ - \implies \sigma_M \approx 1.1191486... + w_M\ex = w_L\ex = \sigma \cdot a \thus \sigma \approx 1.1191486... $$ diff --git a/slides/sections/8.md b/slides/sections/8.md index 5bded85..30f1bcc 100644 --- a/slides/sections/8.md +++ b/slides/sections/8.md @@ -59,7 +59,7 @@ $$ \setbeamercovered{transparent} -## Landau sample results +## L sample results \begin{center} \begin{tabular}{rcccc} @@ -79,7 +79,7 @@ $$ \end{center} -## Moyal sample results +## M sample results \begin{center} \begin{tabular}{rcccc} diff --git a/slides/sections/9.md b/slides/sections/9.md index 876c3ac..96a5a64 100644 --- a/slides/sections/9.md +++ b/slides/sections/9.md @@ -2,16 +2,18 @@ ## Summary -- All five tests properly work for a high number of points ($N = 50000$); +- All six tests properly work for a high number of points ($N = 50000$); - Properties comparison: - - The median estimation decreases in significe as $N$ decreases; - - The KDE for FWHM is the least stable test (not working for $N \leq 1000$); - - the mode estimation swtill properly works for very few points ($N = 200$); + - Median estimation decreases in significance as $N$ decreases; + - KDE for FWHM is the least stable test (not working for $N \leq 1000$); + - HSM swtill properly works for very few points ($N = 200$); - KS still properly works for very few points ($N = 200$); -- The Trapani test decreases in significe as $N$ decreases. +- Trapani test (less informative) decreases in significance as $N$ decreases. ## Any questions? {.standout} +Any questions? + ## Bibliograph