diff --git a/slides/sections/0.md b/slides/sections/0.md index e94a124..610af6d 100644 --- a/slides/sections/0.md +++ b/slides/sections/0.md @@ -8,16 +8,38 @@ institute: - Università di Milano-Bicocca theme: metropolis +themeoptions: + - titleformat=allcaps aspectratio: 169 -fontsize: 14pt +fontsize: 12pt +mainfont: Fira Sans +mainfontoptions: + - BoldFont=Fira Sans + mathfont: FiraMath-Regular -sansfont: Fira Sans header-includes: | ```{=latex} + %% Colors + \definecolor{mDarkTeal} {HTML}{020202} + \definecolor{mLightBrown}{HTML}{C49D4A} + \definecolor{mDarkRed} {HTML}{92182B} + + \definecolor{green} {HTML}{60AC39} + \definecolor{red} {HTML}{D73737} + \definecolor{blue} {HTML}{6684E1} + \definecolor{yellow}{HTML}{CFB017} + + \setbeamercolor{frametitle}{bg=mDarkRed} + + % center images + \LetLtxMacro{\oldIncludegraphics}{\includegraphics} + \renewcommand{\includegraphics}[2][]{ + \centering + \oldIncludegraphics[#1]{#2} + } - % Misc % "thus" in formulas \DeclareMathOperator{\thus}{% \hspace{30pt} \Longrightarrow \hspace{30pt} @@ -27,6 +49,5 @@ header-includes: | \DeclareMathOperator{\with}{% \hspace{30pt} \text{with} \hspace{30pt} } - ``` ... diff --git a/slides/sections/1.md b/slides/sections/1.md index 38d172a..d77bb9a 100644 --- a/slides/sections/1.md +++ b/slides/sections/1.md @@ -1,63 +1,3 @@ ---- -title: Randomness tests of a non-uniform distribution -date: \today -author: - - Giulia Marcer - - Michele Guerini Rocco -institute: - - Università di Milano-Bicocca - -theme: metropolis -themeoptions: - - titleformat=allcaps -aspectratio: 169 - -fontsize: 12pt -mainfont: Fira Sans -mainfontoptions: - - BoldFont=Fira Sans - -mathfont: FiraMath-Regular - -header-includes: | - ```{=latex} - %% Colors - \definecolor{mDarkTeal} {HTML}{020202} - \definecolor{mLightBrown}{HTML}{C49D4A} - \definecolor{mDarkRed} {HTML}{92182B} - - \definecolor{green} {HTML}{60AC39} - \definecolor{red} {HTML}{D73737} - \definecolor{blue} {HTML}{6684E1} - \definecolor{yellow}{HTML}{CFB017} - - \setbeamercolor{frametitle}{bg=mDarkRed} - - % center images - \LetLtxMacro{\oldIncludegraphics}{\includegraphics} - \renewcommand{\includegraphics}[2][]{ - \centering - \oldIncludegraphics[#1]{#2} - } - - %% customer macros - \DeclareMathOperator{\with}{% - \hspace{30pt} \text{with} \hspace{30pt} - } - - % "thus" in formulas - \DeclareMathOperator{\thus}{% - \hspace{30pt} \Longrightarrow \hspace{30pt} - } - - % "et" in formulas - \DeclareMathOperator{\et}{% - \hspace{30pt} \wedge \hspace{30pt} - } - ``` -... - - # Goal diff --git a/slides/sections/2.md b/slides/sections/2.md index f9eb128..566dd48 100644 --- a/slides/sections/2.md +++ b/slides/sections/2.md @@ -1,135 +1,32 @@ -# Moyal distribution +# Landau PDF -## Moyal PDF +## Pathological probability distribution +Because of its fat tail: -Standard form: -$$ - M(z) = \frac{1}{\sqrt{2 \pi}} \exp - \left[ - \frac{1}{2} \left( z + e^{-z} \right) \right] -$$ +\begin{align*} + E[x] &\longrightarrow + \infty \\ + V[x] &\longrightarrow + \infty +\end{align*} -. . . +No closed form for parameters. -More generally: +## Landau median - - location parameter $\mu$ - - scale parameter $\sigma$ +The median of a PDF is defined as: $$ - z = \frac{x - \mu}{\sigma} - \thus - M(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp - \left[ - \frac{1}{2} \left( - \frac{x - \mu}{\sigma} - + e^{-\frac{x - \mu}{\sigma}} \right) \right] + Q_L(x) = \frac{1}{2} $$ +- CDF computed by numerical integration, +- QDF computed by numerical root-finding (Brent) -## Moyal CDF - -The CDF $F_M(x)$ can be derived by direct integration: -$$ - F_M(x) = \int\limits_{- \infty}^x dy \, M(y) - = \frac{1}{\sqrt{2 \pi}} \int\limits_{- \infty}^x dy \, e^{- \frac{y}{2}} - e^{- \frac{1}{2} e^{-y}} -$$ - -. . . - -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: -$$ - F_M(x) = 1 - \text{erf} \left( \frac{e^{- \frac{x}{2}}}{\sqrt{2}} \right) -$$ - - -## Moyal QDF - -The quantile (CDF\textsuperscript{-1}) is found solving: -$$ - y = 1 - \text{erf} \left( \frac{e^{- \frac{x}{2}}}{\sqrt{2}} \right) -$$ hence: -$$ - Q_M(x) = -2 \ln \left[ \sqrt{2} \, \text{erf}^{-1} (1 - F_M(x)) \right] -$$ - - -## Moyal median - -Defined by $\text{CDF}(m) = 1/2$, or $m=\text{QDF}(1/2)$. - -\begin{align*} - M(z) - &\thus m_M = -2 \ln \left[ \sqrt{2} \, - \text{erf}^{-1} \left( \frac{1}{2} \right) \right] \\ - M_{\mu \sigma}(x) - &\thus m_M = \mu -2 \sigma \ln \left[ \sqrt{2} \, - \text{erf}^{-1} \left( \frac{1}{2} \right) \right] -\end{align*} - -## Moyal mode - -Peak of the PDF: -$$ - \partial_x M(x) = \partial_x \left( \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} - \left( x + e^{-x} \right)} \right) - = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} - \left( x + e^{-x} \right)} \left( -\frac{1}{2} \right) - \left( 1 - e^{-x} \right) -$$ - -\begin{align*} - \partial_x M(z) = 0 &\thus \mu_M = 0 \\ - \partial_x M_{\mu \sigma}(x) = 0 &\thus \mu_M = \mu \\ -\end{align*} - - -## Moyal FWHM - -We need to compute the maximum value: -$$ - M(\mu) = \frac{1}{\sqrt{2 \pi e}} \thus M(x_{\pm}) = \frac{1}{\sqrt{8 \pi e}} -$$ - -. . . - -which leads to: -$$ - x_{\pm} + e^{-x_{\pm}} = 1 + 2 \ln(2) \thus - \begin{cases} - x_+ = 1 + 2 \ln(2) + W_0 \left( - \frac{1}{4 e} \right) \\ - x_- = 1 + 2 \ln(2) + W_{-1} \left( - \frac{1}{4 e} \right) - \end{cases} -$$ - -## Moyal FWHM $$ - x_+ - x_- = W_0 \left( - \frac{1}{4 e} \right) - - W_{-1} \left( - \frac{1}{4 e} \right) - = 3.590806098... - = a + m_L = 1.3557804... $$ -\begin{align*} - M(z) - &\thus \text{FWHM}_M = a \\ - M_{\mu \sigma}(x) - &\thus \text{FWHM}_M = \sigma \cdot a \\ -\end{align*} +o diff --git a/slides/sections/3.md b/slides/sections/3.md index ace417b..f9eb128 100644 --- a/slides/sections/3.md +++ b/slides/sections/3.md @@ -1,21 +1,135 @@ -# Data sample +# Moyal distribution -## Data sample -The $M(x)$ most similar to $L(x)$ is found by imposing: +## Moyal PDF -- equal mode + +Standard form: $$ - \mu_M = M_L \approx −0.22278298... -$$ - -- equal width -$$ - \text{FWHM}_M = \text{FWHM}_L = \sigma \cdot a + M(z) = \frac{1}{\sqrt{2 \pi}} \exp + \left[ - \frac{1}{2} \left( z + e^{-z} \right) \right] $$ . . . +More generally: + + - location parameter $\mu$ + - scale parameter $\sigma$ + $$ - \implies \sigma_M \approx 1.1191486 + z = \frac{x - \mu}{\sigma} + \thus + M(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp + \left[ - \frac{1}{2} \left( + \frac{x - \mu}{\sigma} + + e^{-\frac{x - \mu}{\sigma}} \right) \right] $$ + + +## Moyal CDF + +The CDF $F_M(x)$ can be derived by direct integration: +$$ + F_M(x) = \int\limits_{- \infty}^x dy \, M(y) + = \frac{1}{\sqrt{2 \pi}} \int\limits_{- \infty}^x dy \, e^{- \frac{y}{2}} + e^{- \frac{1}{2} e^{-y}} +$$ + +. . . + +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: +$$ + F_M(x) = 1 - \text{erf} \left( \frac{e^{- \frac{x}{2}}}{\sqrt{2}} \right) +$$ + + +## Moyal QDF + +The quantile (CDF\textsuperscript{-1}) is found solving: +$$ + y = 1 - \text{erf} \left( \frac{e^{- \frac{x}{2}}}{\sqrt{2}} \right) +$$ +hence: +$$ + Q_M(x) = -2 \ln \left[ \sqrt{2} \, \text{erf}^{-1} (1 - F_M(x)) \right] +$$ + + +## Moyal median + +Defined by $\text{CDF}(m) = 1/2$, or $m=\text{QDF}(1/2)$. + +\begin{align*} + M(z) + &\thus m_M = -2 \ln \left[ \sqrt{2} \, + \text{erf}^{-1} \left( \frac{1}{2} \right) \right] \\ + M_{\mu \sigma}(x) + &\thus m_M = \mu -2 \sigma \ln \left[ \sqrt{2} \, + \text{erf}^{-1} \left( \frac{1}{2} \right) \right] +\end{align*} + +## Moyal mode + +Peak of the PDF: +$$ + \partial_x M(x) = \partial_x \left( \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} + \left( x + e^{-x} \right)} \right) + = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} + \left( x + e^{-x} \right)} \left( -\frac{1}{2} \right) + \left( 1 - e^{-x} \right) +$$ + +\begin{align*} + \partial_x M(z) = 0 &\thus \mu_M = 0 \\ + \partial_x M_{\mu \sigma}(x) = 0 &\thus \mu_M = \mu \\ +\end{align*} + + +## Moyal FWHM + +We need to compute the maximum value: +$$ + M(\mu) = \frac{1}{\sqrt{2 \pi e}} \thus M(x_{\pm}) = \frac{1}{\sqrt{8 \pi e}} +$$ + +. . . + +which leads to: +$$ + x_{\pm} + e^{-x_{\pm}} = 1 + 2 \ln(2) \thus + \begin{cases} + x_+ = 1 + 2 \ln(2) + W_0 \left( - \frac{1}{4 e} \right) \\ + x_- = 1 + 2 \ln(2) + W_{-1} \left( - \frac{1}{4 e} \right) + \end{cases} +$$ + +## Moyal FWHM + +$$ + x_+ - x_- = W_0 \left( - \frac{1}{4 e} \right) + - W_{-1} \left( - \frac{1}{4 e} \right) + = 3.590806098... + = a +$$ + +\begin{align*} + M(z) + &\thus \text{FWHM}_M = a \\ + M_{\mu \sigma}(x) + &\thus \text{FWHM}_M = \sigma \cdot a \\ +\end{align*} diff --git a/slides/sections/4.md b/slides/sections/4.md new file mode 100644 index 0000000..ae9ffed --- /dev/null +++ b/slides/sections/4.md @@ -0,0 +1,21 @@ +# Data sample + +## Data sample + +The $M(x)$ most similar to $L(x)$ is found by imposing: + +- equal mode +$$ + \mu_M = \mu_L \approx −0.22278298... +$$ + +- equal width +$$ + \text{FWHM}_M = \text{FWHM}_L = \sigma \cdot a +$$ + +. . . + +$$ + \implies \sigma_M \approx 1.1191486 +$$