diff --git a/slides/images/kde.pdf b/slides/images/kde.pdf index 620e616..0968458 100644 Binary files a/slides/images/kde.pdf and b/slides/images/kde.pdf differ diff --git a/slides/sections/0.md b/slides/sections/0.md index 0185e0e..9170f24 100644 --- a/slides/sections/0.md +++ b/slides/sections/0.md @@ -45,6 +45,11 @@ header-includes: | \hspace{30pt} \Longrightarrow \hspace{30pt} } + % "thus" in text + \DeclareMathOperator{\hence}{% + \quad \longrightarrow \quad + } + % "and" in formulas \DeclareMathOperator{\et}{% \hspace{30pt} \wedge \hspace{30pt} diff --git a/slides/sections/1.md b/slides/sections/1.md index 7fc1a74..09783e8 100644 --- a/slides/sections/1.md +++ b/slides/sections/1.md @@ -84,10 +84,10 @@ utilized in the approximation of the former. . . . -- Kolmogorov - Smirnov: +- Kolmogorov - Smirnov test: - compatibility between expected and observed CDF . . . - Trapani test: - - compatibility between expected and observed momenta + - compatibility between expected and observed moments diff --git a/slides/sections/2.md b/slides/sections/2.md index 780bdfa..5e491f6 100644 --- a/slides/sections/2.md +++ b/slides/sections/2.md @@ -35,7 +35,7 @@ $$ ## Landau mode -- Maximum $\quad \Longrightarrow \quad \partial_x L(\mu) = 0$ +- Maximum $\hence \partial_x L(\mu) = 0$ . . . diff --git a/slides/sections/4.md b/slides/sections/4.md index b7db3ca..1304f4b 100644 --- a/slides/sections/4.md +++ b/slides/sections/4.md @@ -8,7 +8,7 @@ how to estimate their median, mode and FWHM? . . . -- Binning data $\quad \longrightarrow \quad$ result depending on bin-width +- Binning data $\hence$ result depending on bin-width . . . diff --git a/slides/sections/6.md b/slides/sections/6.md index 78929ae..0851093 100644 --- a/slides/sections/6.md +++ b/slides/sections/6.md @@ -27,7 +27,7 @@ Median: ::: {.column width=50%} $$ - \thus \text{Compatible!} + \hence \text{Compatible!} $$ ::: :::: @@ -46,7 +46,7 @@ Mode: ::: {.column width=50%} $$ - \thus \text{Compatible!} + \hence \text{Compatible!} $$ ::: :::: @@ -65,7 +65,7 @@ FWHM: ::: {.column width=50%} $$ - \thus \text{Compatible!} + \hence \text{Compatible!} $$ ::: :::: @@ -89,7 +89,7 @@ $$ reverse sampling -- sampling $y$ uniformly in [0, 1] $\quad \longrightarrow \quad x = Q_M(y)$ +- sampling $y$ uniformly in [0, 1] $\hence x = Q_M(y)$ ## Compatibility results: @@ -104,7 +104,7 @@ Median: ::: {.column width=50%} $$ - \thus \text{Not compatible!} + \hence \text{Not compatible!} $$ ::: :::: @@ -123,7 +123,7 @@ Mode: ::: {.column width=50%} $$ - \thus \text{Compatible!} + \hence \text{Compatible!} $$ ::: :::: @@ -142,7 +142,7 @@ FWHM: ::: {.column width=50%} $$ - \thus \text{Compatible!} + \hence \text{Compatible!} $$ ::: :::: diff --git a/slides/sections/7.md b/slides/sections/7.md index 8b8f533..98a11fc 100644 --- a/slides/sections/7.md +++ b/slides/sections/7.md @@ -62,7 +62,7 @@ Landau sample: ::: {.column width=50%} $$ - \thus \text{Compatible!} + \hence \text{Compatible!} $$ ::: :::: @@ -81,7 +81,7 @@ Moyal sample: ::: {.column width=50%} $$ - \thus \text{Not compatible!} + \hence \text{Not compatible!} $$ ::: :::: diff --git a/slides/sections/8.md b/slides/sections/8.md index 8733fb0..b89500f 100644 --- a/slides/sections/8.md +++ b/slides/sections/8.md @@ -1,12 +1,12 @@ # Trapani test -## Finite/infinite momenta +## Infinite moments For a Landau PDF: \begin{align*} E_L[x] &\longrightarrow + \infty \\ - V_L[x] &\longrightarrow + \infty + V_L[x] \text{undefined} \end{align*} . . . @@ -18,9 +18,9 @@ For a Moyal PDF: \end{align*} -## Finite/infinite momenta +## Infinite moments -- Check whether a momentum is finite or infinite +- Check whether a moment is finite or infinite \begin{align*} \text{infinite} &\thus Landau \\ \text{finite} &\thus Moyal @@ -39,7 +39,7 @@ For a Moyal PDF: - Random variable $\left\{ x_i \right\}$ sampled from a distribution $f$ - Sample moments according to $f$ moments - $H_0$: $\mu_k \longrightarrow + \infty$ - - Statistic with chi-squared distribution + - Statistic with 1 dof chi-squared distribution ::: @@ -129,9 +129,9 @@ $$ If $a_j$ uniformly distributed and $N \rightarrow + \infty$, for the CLT: $$ - \sum_j \zeta_j (u) \quad \text{follows} \quad + \sum_j \zeta_j (u) \hence G \left( \frac{r}{2}, \frac{r}{4} \right) - \thus \vartheta (u) \quad \text{follows} \quad + \thus \vartheta (u) \hence G \left( 0, 1 \right) $$ @@ -139,7 +139,7 @@ $$ - Test statistic: $$ - \chi^2 = \int_{\underbar{u}}^{\bar{u}} du \vartheta^2 (u) + \Theta = \int_{\underbar{u}}^{\bar{u}} du \, \vartheta^2 (u) \psi(u) $$ @@ -147,8 +147,9 @@ $$ According to L. Trapani (10.1016/j.jeconom.2015.08.006): -- $r = o(N)$ +- $r = o(N) \hence r = N^{0.75}$ - $\underbar{u} = 1 \quad \wedge \quad \bar{u} = 1$ +- $\psi(u) = \chi_{[\underbar{u}, \bar{u}]}$ . . . @@ -160,45 +161,90 @@ $$ $$ +## Trapani test + +If $\mu_k \ne + \infty \hence \left\{ a_j \right\}$ are not uniformly distributed +\vspace{20pt} +Rewriting: +$$ + \vartheta (u) = \frac{2}{\sqrt{r}} + \left[ \sum_{j} \zeta_j (u) - \frac{r}{2} \right] + = \frac{2}{\sqrt{r}} + \sum_{j} \left[ \zeta_j (u) - \frac{1}{2} \right] +$$ + +\vspace{20pt} + +Residues become very large $\hence$ $p$-values decreases. + + # Samples results ## Samples results -$N = 50000$ sampled points - . . . Landau sample: :::: {.columns} -::: {.column width=50%} - - $D = 0.004$ - - $p = 0.379$ +::: {.column width=33%} + $$ + \mu_1 + \begin{cases} + \Theta = 0.255 \\ + p = 0.614 + \end{cases} + $$ ::: -::: {.column width=50%} +::: {.column width=33% .c} $$ - \thus \text{Compatible!} + \mu_2 + \begin{cases} + \Theta = 0.432 \\ + p = 0.511 + \end{cases} + $$ +::: + +::: {.column width=33% .c} + $$ + \hence \text{Infinite!} $$ ::: :::: -\vspace{10pt} - . . . +\vspace{20pt} + Moyal sample: :::: {.columns} -::: {.column width=50%} - - $D = 0.153$ - - $p = 0.000$ +::: {.column width=33%} + $$ + \mu_1 + \begin{cases} + \Theta^2 = 106 \\ + p = 0.000 + \end{cases} + $$ ::: -::: {.column width=50%} +::: {.column width=33%} $$ - \thus \text{Not compatible!} + \mu_2 + \begin{cases} + \Theta^2 = 162 \\ + p = 0.000 + \end{cases} + $$ +::: + +::: {.column width=33% .c} + $$ + \hence \text{Finite!} $$ ::: ::::